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The value of the character on g1 is χ(g1 ) = (e2πi/N1 )a1 . Thus the integers ai are integers modulo Ni , and this representation is basically the same as the one we described in the previous paragraph (and which the author does not like). 1 This exercise is about the structure of the units of Z/pn Z. (a) If p is odd and n is a positive integer, prove that (Z/pn Z)∗ is cyclic. (b) If n ≥ 3 prove that (Z/2n Z)∗ is a direct sum of the cylclic subgroups −1 and 5 , of orders 2 and 2n−2 , respectively.

The algorithm for computing the sum over A3 (N ) is similar, but we omit it. The following table contains the dimension of Sk (N, ε) for some sample values of N and k. , the product of the generators of D(N, C∗ )). 2. Cohen and Oesterle also give dimension formulas for spaces of half-integral weight modular forms, which we do not give in this chapter. Also [CO77] does not contain any proofs that their claimed formulas are correct, but instead say only that “Les formules qui les donnent sont connues de beaucoup de gens et il existe plusieurs m´ethodes permettant de les obtenir (th´eor`eme 54CHAPTER 4.

Fix a prime divisor p | N and let r = vp (N ). Then  r r −1  if 2 · vp (c) ≤ r and 2 | r, p 2 + p 2 r−1 λ(p, N, vp (c)) = 2 · p 2 if 2 · vp (c) ≤ r and 2 r,   r−vp (c) 2·p if 2 · vp (c) > r The formula can be used to compute dim Mk (N, ε), dim Sk (N, ε), and dim Ek (N, ε) for any N , ε, k = 1, by using that dim Sk (N, ε) = 0 dim Mk (N, ε) = 0 dim M0 (N, ε) = 1 if k ≤ 0 if k < 0 if k = 0 One thing that is not straightforward when implementing an algorithm to compute the above dimension formulas is how to efficiently compute the sets A4 (N ) and A6 (N ).

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