Unformatted text preview: 101 show that each matrix A from A N is contained in the left-hand-side. This follows from the well known fact that each integral matrix can be transformed by integral row and column transformations to the unique matrix of the form ` n n ´ , where n | n . The last assertion can be checked by using elementary number theory. When N = p is prime, we obviously have # A p = p + 1. Now, if N is not prime we have # A N = ψ ( N ) = N Y p | N (1 + p- 1 ) = μ ,N . This can be proved by using the multiplicative property of the function ψ ( n ) and the formula ψ ( N ) = X d | N d ( d, N d ) φ (( d, N d )) , where φ is the Euler function. Lemma 10.3 . Let f ( τ ) be a modular function with respect to Γ(1) which is holomor- phic on H and admits the Fourier expansion f = P ∞ n =- r c n q n . Then f is a polyno- mial in j ( τ ) with coefficients in the subring of C generated by the Fourier coefficients c ,...,c- r ....
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- Fall '09