SUMMARY ORDER
Isaac Dayan appeals from his conviction by a jury and sentence before Judge Baer for one count of conspiracy to commit bank fraud, one count of bank fraud in connection with a check-kiting scheme, and one count of bank fraud for fraudulently obtaining a loan. The facts and procedural history are recounted in our opinion addressing appellant’s claim of a defective indictment and of error in the restitution order, filed this day (February 5, 2007).
With regard to his conviction, Dayan argues that: (i) it was an impermissible constructive amendment to the indictment to allow the jury to convict on findings that the loan application was false in any respect but sales records; (ii) Count Two did not state an offense, and the government’s evidence on Count Two was insufficient as a matter of law; (iii) the jury should not have been given a conscious avoidance instruction; (iv) the government should not have been allowed to present evidence of Dayan’s transfers of money from his business accounts to his brokerage accounts; (v) the jury instructions on bank fraud were erroneous; and (vi) the jury should not have been told that it could convict Dayan of bank fraud against an Antwerp bank that was not proven to be a financial institution within the meaning of the bank fraud statute. We conclude that these arguments lack merit and affirm.
a) Constructive Amendment of an Indictment
Count Four of the indictment charged Dayan with bank fraud for submitting a loan application to Fleet National Bank that “falsely and fraudulently reported sales of approximately $8,660,505 for the year 1999” for his business Fortune Diamond Importers. Dayan argues that he was convicted on Count Four because the government relied at trial on other false statements in his application, such as the statement that he owed no back taxes.
Judge Baer instructed the jury that it could convict on finding any false statement in the loan application that was material to the scheme. When the jury asked during deliberations whether it could convict on Count Four based on false statements in the loan application other than the sales figures, the judge sent the jury a copy of the instructions summarizing the bank fraud charges and stating the elements of bank fraud.
To demonstrate that an indictment was constructively amended, the defendant must show “that either the proof at trial or the trial court’s jury instructions so altered an essential element of the charge that, upon review, it is uncertain whether the defendant was convicted of conduct that was the subject of the grand jury’s indictment.” United States v. Salmonese, 352 [87]*87F.3d 608, 620 (2d Cir.2003) (internal quotation marks omitted). “The Fifth Amendment’s indictment clause prohibits ‘constructive amendments’ of an indictment when the government’s evidence and the jury instructions ‘modify essential elements of the offense charged to the point that there is a substantial likelihood that the defendant may have been convicted of an offense other than the one charged by the grand jury.’ ” LanFranco v. Murray, 313 F.3d 112, 118 n. 1 (2d Cir.2002). An “essential element” of the offense is modified only when the defendant was not “given notice of the core of criminality to be proven at trial.” See id. (internal quotation marks omitted). The evidence at trial must “broaden[] the basis of conviction beyond that charged in the indictment.” United States v. Patino, 962 F.2d 263, 265 (2d Cir.1992).
By contrast, a variance occurs where “the charging terms of the indictment are left unaltered, but the evidence offered at trial proves facts materially different from those alleged in the indictment.” United States v. Thomas, 274 F.3d 655, 670 (2d Cir.2001) (internal quotation marks omitted). A variance is not prejudicial, and not grounds for reversal, “where the allegation and proof substantially correspond, where the variance is not of a character that could have misled the defendant at the trial, and where the variance is not such as to deprive the accused of his right to be protected against another prosecution for the same offense.” United States v. Mucciante, 21 F.3d 1228, 1236 (2d Cir.1994) (internal quotation marks omitted).
The false sales figures were contained within a “to wit” clause, and our holdings show that details contained in a “to wit” clause do not per se bar a trial court from instructing a jury that it may convict even if only related and unmentioned details were proven at trial. See United States v. Rivera, 415 F.3d 284, 287 (2d Cir.2005) (constructive amendment, variance); United States v. Danielson, 199 F.3d 666, 669 (2d Cir.1999) (constructive amendment). Each of these decisions held that deviation from the details described in the “to wit” clauses was not fatal to the conviction.
Nor was there any prejudicial variance. Although the government did elicit an admission from Dayan that he had failed to report his tax liability on the loan application to Fleet, they also devoted substantial time and testimony to their original theory that he falsified his sales figures on the application. There is simply no risk of appellant’s defense having been impaired by a lack of notice of what crimes were charged. Further, the identification of the loan applications in the indictment clearly avoids the risk of Dayan being tried twice for the same offenses.
b) Offense Stated in Count Two
We review a claim that an indictment does not state an offense only to determine whether the indictment “charges a crime with sufficient precision to inform the defendant of the charges he must meet and with enough detail that he may plead double jeopardy in a future prosecution based on the same set of events.” United States v. Walsh, 194 F.3d 37, 44 (2d Cir.1999) (internal quotation marks omitted). An indictment is sufficient if “it, first, contains the elements of the offense charged and fairly informs a defendant of the charge against which he must defend, and, second, enables him to plead an acquittal or conviction in bar of future prosecutions for the same offense.” Hamling v. United States, 418 U.S. 87, 117, 94 S.Ct. 2887, 41 L.Ed.2d 590 (1974).
Dayan argues that the indictment failed to allege that the checks that he received from others and deposited were not covered by funds in the others’ accounts. He [88]*88also argues that it failed to allege the false representations to a financial institution necessary to commit “embellished” check-kiting under 18 U.S.C. § 1344(2), which requires false representations in addition to check-kiting itself. See United States v. Burnett, 10 F.3d 74, 78-79 (2d Cir.1993).
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SUMMARY ORDER
Isaac Dayan appeals from his conviction by a jury and sentence before Judge Baer for one count of conspiracy to commit bank fraud, one count of bank fraud in connection with a check-kiting scheme, and one count of bank fraud for fraudulently obtaining a loan. The facts and procedural history are recounted in our opinion addressing appellant’s claim of a defective indictment and of error in the restitution order, filed this day (February 5, 2007).
With regard to his conviction, Dayan argues that: (i) it was an impermissible constructive amendment to the indictment to allow the jury to convict on findings that the loan application was false in any respect but sales records; (ii) Count Two did not state an offense, and the government’s evidence on Count Two was insufficient as a matter of law; (iii) the jury should not have been given a conscious avoidance instruction; (iv) the government should not have been allowed to present evidence of Dayan’s transfers of money from his business accounts to his brokerage accounts; (v) the jury instructions on bank fraud were erroneous; and (vi) the jury should not have been told that it could convict Dayan of bank fraud against an Antwerp bank that was not proven to be a financial institution within the meaning of the bank fraud statute. We conclude that these arguments lack merit and affirm.
a) Constructive Amendment of an Indictment
Count Four of the indictment charged Dayan with bank fraud for submitting a loan application to Fleet National Bank that “falsely and fraudulently reported sales of approximately $8,660,505 for the year 1999” for his business Fortune Diamond Importers. Dayan argues that he was convicted on Count Four because the government relied at trial on other false statements in his application, such as the statement that he owed no back taxes.
Judge Baer instructed the jury that it could convict on finding any false statement in the loan application that was material to the scheme. When the jury asked during deliberations whether it could convict on Count Four based on false statements in the loan application other than the sales figures, the judge sent the jury a copy of the instructions summarizing the bank fraud charges and stating the elements of bank fraud.
To demonstrate that an indictment was constructively amended, the defendant must show “that either the proof at trial or the trial court’s jury instructions so altered an essential element of the charge that, upon review, it is uncertain whether the defendant was convicted of conduct that was the subject of the grand jury’s indictment.” United States v. Salmonese, 352 [87]*87F.3d 608, 620 (2d Cir.2003) (internal quotation marks omitted). “The Fifth Amendment’s indictment clause prohibits ‘constructive amendments’ of an indictment when the government’s evidence and the jury instructions ‘modify essential elements of the offense charged to the point that there is a substantial likelihood that the defendant may have been convicted of an offense other than the one charged by the grand jury.’ ” LanFranco v. Murray, 313 F.3d 112, 118 n. 1 (2d Cir.2002). An “essential element” of the offense is modified only when the defendant was not “given notice of the core of criminality to be proven at trial.” See id. (internal quotation marks omitted). The evidence at trial must “broaden[] the basis of conviction beyond that charged in the indictment.” United States v. Patino, 962 F.2d 263, 265 (2d Cir.1992).
By contrast, a variance occurs where “the charging terms of the indictment are left unaltered, but the evidence offered at trial proves facts materially different from those alleged in the indictment.” United States v. Thomas, 274 F.3d 655, 670 (2d Cir.2001) (internal quotation marks omitted). A variance is not prejudicial, and not grounds for reversal, “where the allegation and proof substantially correspond, where the variance is not of a character that could have misled the defendant at the trial, and where the variance is not such as to deprive the accused of his right to be protected against another prosecution for the same offense.” United States v. Mucciante, 21 F.3d 1228, 1236 (2d Cir.1994) (internal quotation marks omitted).
The false sales figures were contained within a “to wit” clause, and our holdings show that details contained in a “to wit” clause do not per se bar a trial court from instructing a jury that it may convict even if only related and unmentioned details were proven at trial. See United States v. Rivera, 415 F.3d 284, 287 (2d Cir.2005) (constructive amendment, variance); United States v. Danielson, 199 F.3d 666, 669 (2d Cir.1999) (constructive amendment). Each of these decisions held that deviation from the details described in the “to wit” clauses was not fatal to the conviction.
Nor was there any prejudicial variance. Although the government did elicit an admission from Dayan that he had failed to report his tax liability on the loan application to Fleet, they also devoted substantial time and testimony to their original theory that he falsified his sales figures on the application. There is simply no risk of appellant’s defense having been impaired by a lack of notice of what crimes were charged. Further, the identification of the loan applications in the indictment clearly avoids the risk of Dayan being tried twice for the same offenses.
b) Offense Stated in Count Two
We review a claim that an indictment does not state an offense only to determine whether the indictment “charges a crime with sufficient precision to inform the defendant of the charges he must meet and with enough detail that he may plead double jeopardy in a future prosecution based on the same set of events.” United States v. Walsh, 194 F.3d 37, 44 (2d Cir.1999) (internal quotation marks omitted). An indictment is sufficient if “it, first, contains the elements of the offense charged and fairly informs a defendant of the charge against which he must defend, and, second, enables him to plead an acquittal or conviction in bar of future prosecutions for the same offense.” Hamling v. United States, 418 U.S. 87, 117, 94 S.Ct. 2887, 41 L.Ed.2d 590 (1974).
Dayan argues that the indictment failed to allege that the checks that he received from others and deposited were not covered by funds in the others’ accounts. He [88]*88also argues that it failed to allege the false representations to a financial institution necessary to commit “embellished” check-kiting under 18 U.S.C. § 1344(2), which requires false representations in addition to check-kiting itself. See United States v. Burnett, 10 F.3d 74, 78-79 (2d Cir.1993).
Section 1344 defines bank fraud as “a scheme or artifice—(1) to defraud a financial institution; or (2) to obtain any of the moneys, funds, credits, assets, securities, or other property owned by, or under the custody or control of, a financial institution, by means of false or fraudulent pretenses, representations, or promises.” The present indictment tracks the language of both sections of the statute. It sufficiently describes a scheme to defraud a financial institution under Section 1344(1), because Paragraph Seven, which alleges that the counter-parties did not have adequate funds in their accounts to cover the checks Dayan deposited, is incorporated in the other counts. The false representations required by Section 1344(2) were sufficiently alleged by Paragraph Seven’s charge of falsely representing inflated sales figures to obtain credit for his businesses, again incorporated in the counts charging Dayan with using those sales figures to falsify loan applications. The indictment clearly informed Dayan of the charges against him and was easily sufficient to ensure that he will not be prosecuted twice for the same offense. Hamling, 418 U.S. at 117, 94 S.Ct. 2887.
Appellant also contends that the evidence on Count Two was insufficient because the government did not prove that the co-conspirators lacked sufficient funds to cover their checks to Dayan. In reviewing insufficiency claims, we may “overturn the conviction only if no rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.” Walsh, 194 F.3d at 51 (citation omitted). At trial, Haas testified that he issued checks to Dayan that his account could not cover, and that he had informed Dayan of that. Dayan admitted that he had written checks to the alleged co-conspirators that his accounts could not cover. Further, the scheme fell apart in February and March of 2001 as checks written by other alleged co-conspirators to Dayan began to bounce. This evidence was easily sufficient to allow a reasonable trier of fact to find that the co-conspirators lacked sufficient funds to cover their checks.
c) Admissibility of Evidence
We review rulings on the admissibility of evidence for abuse of discretion. United States v. Salameh, 152 F.3d 88, 110 (2d Cir.1998). The government introduced evidence at the trial that, during the alleged check-kiting scheme, Dayan deposited nearly $3 million from one of the accounts used in the scheme to brokerage accounts at Merrill Lynch. The government argued that this evidence showed Dayan had personally profited from the fraud. The district court admitted the evidence on the grounds that although it was not very probative, it was also not very prejudicial. Although the checks Dayan sent to Merrill Lynch for the brokerage account never bounced, the government offered evidence that he often sent checks to Merrill Lynch with insufficient funds in his account and then deposited checks from his co-conspirators to cover them. We find no abuse of discretion in the admission of this evidence.
d) Jury Instructions
We review claims of error in jury instructions de novo, reversing only if “viewing the charge as a whole, there was a prejudicial error.” United States v. Aina-Marshall, 336 F.3d 167, 170 (2d Cir.2003). Dayan challenges three distinct aspects of [89]*89the jury instructions: the instruction on conscious avoidance, the definition of bank fraud, and the lack of any instruction on the definition of a financial institution.
1) Conscious Avoidance
“A conscious-avoidance charge is appropriate when (a) the element of knowledge is in dispute, and (b) the evidence would permit a rational juror to conclude beyond a reasonable doubt ‘that the defendant was aware of a high probability of the fact in dispute and consciously avoided confirming that fact.’” United States v. Zedner, 401 F.3d 36, 50 (2d Cir.2005) (citations omitted), rev’d on other grounds, — U.S. -, 126 S.Ct. 1976, 164 L.Ed.2d 749 (2006).
We find no error in the district court’s instructions regarding the elements of the conscious avoidance doctrine. The only issue, therefore, is whether the instruction should have been given at all.
There was no error in giving such an instruction. Dayan testified that he did not know his conduct was wrong and that the banks knew of his conduct and were “fine with it.” He also argued that he did not know the purpose of the check-trading was bank fraud. However, the government proved several suspicious circumstances that would have alerted Dayan to the fraud unless he consciously avoided knowledge: the need for Dayan and the counter-parties to make deposits simultaneously; receiving multiple checks at the same time from the same party; differing numbers of checks used to make and repay the “loans”; and consistently exchanging slightly unequal amounts. Because Dayan’s knowledge was at issue and there was evidence that would allow a jury to find conscious avoidance, the jury instruction was proper.
2) Definition of Bank Fraud
Under 18 U.S.C. § 1344, bank fraud is defined as the knowing execution of “a scheme or artifice—(1) to defraud a financial institution; or (2) to obtain any of the moneys, funds, credits, assets, securities, or other property owned by, or under the custody or control of, a financial institution, by means of false or fraudulent pretenses, representations, or promises.” As there was no objection to the portions of the jury charge now challenged on appeal, we review for plain error. Fed. R.Crim.P. 52(b), Johnson v. United States, 520 U.S. 461, 466-67, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997).
Dayan challenges three parts of the jury charge: (i) that the jury could consider whether the banks at issue had given their “informed consent” as a substitute for intent to deceive; (ii) that the jury could convict based on the omission or concealment of a material fact, and (iii) the instruction on materiality. The district court’s charge largely followed the standard jury instructions for bank fraud as set out in 2A O’Malley et al., Federal Jury Practice & Instructions §§ 47.09-47.11 (5th ed.2000). Elaborating further on the elements, the district judge instructed the jury that “[t]o act with ‘intent to defraud’ means to act willfully and with intent to deceive for the purpose of causing some financial loss or risk of financial loss to another—here, to obtain money or property from a bank without the company’s informed consent.” The judge also instructed the jury that it could convict Dayan of bank fraud under Section 1344(2) if it found Dayan deceived the bank through “intentional omission or concealment of facts that make what was written, said, or done deliberately misleading.” Although he instructed the jury that it must find the false representations and pretenses “mate[90]*90rial” to convict Dayan of a Section 1344(2) offense, it is not clear whether the instruction was also intended to apply to the Section 1344(1) charge.
The phrase “informed consent” does not appear in any of the model jury instructions. Rather, the language was taken from the government’s suggested jury instructions. Considering the jury instructions as a whole, United States v. Mitchell, 328 F.3d 77, 82 (2d Cir.2003), we conclude that the jury was accurately advised on the applicable law despite the reference to the banks’ “informed consent,” because the court thereafter clearly instructed the jury that “ ‘[ejssential to a scheme to defraud is fraudulent intent’ and that ‘the proof must demonstrate that the defendant had a conscious, knowing intent to defraud and that the defendant contemplated or intended some harm to the property rights of the victim.’ ”
As for the “intentional omission or concealment” portion of the charge, Dayan’s own counsel had suggested very similar language. When a defendant asks for the instruction that he or she later challenges as erroneous, the “invited error” doctrine bars the defendant from challenging that instruction on appeal. United States v. Young, 745 F.2d 733, 752 (2d Cir.1984).
Dayan proposed a jury instruction that echoed the last phrase of Section 1344(2), stating “[a] pretense, representation or promise is false or fraudulent if it was falsely made with the intent to deceive. Deceitful statements of half truth, the concealment of material facts, and the expression of an opinion not honestly entertained may constitute false or fraudulent pretenses, representations or promises under the bank fraud statute.” (Emphasis added). As the district court’s charge echoed the defense’s request in only slightly different language, the invited error doctrine bars Dayan from challenging this instruction.
Dayan also asserts that the district court failed to instruct the jury that materiality is an element of bank fraud under Section 1344(1), as well as Section 1344(2). Under Neder v. United States, 527 U.S. 1, 25, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999), a falsehood must be material under the federal bank fraud statutes. Although the judge instructed the jury on materiality, it is not clear whether his instruction applied only to the charge under Section 1344(2), or to both charges. Nevertheless, because this portion of the charge could be read to apply to both sections of 1344, there was no plain error.
3) Definition of Financial Institution
Dayan argues that the omission from the jury charge of the jurisdictional element of bank fraud was error because trial testimony included a reference to Antwerpse Diamantbank (“ADB”), a foreign bank.
A foreign bank may be a financial institution if it is “a branch or agency of a foreign bank (as such terms are defined in paragraphs (1) and (3) of section 1(b) of the International Banking Act of 1978).” 18 U.S.C. § 20(9). The definitions of “branch” and “agency” in the International Banking Act of 1978 both require, inter alia, an “office or ... place of business of a foreign bank located in any State of the United States.” 12 U.S.C. § 3101(1), (3).
Dayan argues that it was error not to instruct the jury about the jurisdictional elements for bank fraud involving a foreign bank. However, Dayan’s trial counsel [91]*91waived any such instructions at the charge conference:
THE COURT: What about the federally insured concept, which goes on for three pages? Maybe I’m wrong. Maybe only two. The No. 23, two pages, a page and a sentence. Is there some need, you really believe, you haven’t brought out and anybody would doubt that these are federally insured banks?
MR. WEINSTEIN: It’s very clear, your Honor.
THE COURT: I’m taking it out.
MR. WEINSTEIN: As long as that’s something they’re waiving as well.
THE COURT: As long as you have no problem. If you have a problem about where this took place, you ought to tell me.
MR: WOHL: I have no problem.
If, as Dayan now claims, he intended to waive only the federally insured aspect of the jurisdictional element, he surely would have questioned the omission of the foreign bank aspect after the charge was given.
Moreover, ADB’s role in this trial was entirely peripheral, its only connection being that it lent Haas $3 million during the time of the scheme. ADB’s subsequent losses were due to Haas’ failure to pay back that line of credit, which was arguably a collateral result of the check-kiting scheme, and occurred a year after Dayan stopped exchanging checks with Haas. On these facts, the district court’s failure to charge the jury on the definition of a financial institution was not such a miscarriage of justice, if a miscarriage at all, as to vitiate a waiver of the issue.
We therefore affirm the conviction but vacate and remand the restitution order for reasons stated in the accompanying published opinion.