New PDF release: Algorithms for Computing with Modular Forms

By William Stein

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Atkin and Lehner also prove that one can completely determine Sk (Γ0 (M ))new just from the information of how the Hecke operators act on it (their “multiplicity one” theory). D. thesis under A. Ogg (see [Li75]). If N | N | N , then the maps αd from Mk (Γ0 (N )) to Mk (Γ0 (N )) factor through Mk (Γ0 (N )). Thus in the definition of Mk (Γ0 (N ))old and Mk (Γ0 (N ))new , it would suffice to consider only proper divisors N of N such that N/N is prime. 4 below). 2. 3. The dimension of the new subspace is dim Sk (Γ0 (N ))new = M |N µ(N/M ) · dim Sk (Γ0 (M )), where the sum is over the positive divisors of N , and for an integer R,   if p3 | R for some p 0 µ(R) = −2 otherwise,   p||R where the product is over primes that exactly divide n.

Then x = gi + bN1 = gi + (bN1 /N ) · N and (x, N ) = 1, which completes the proof. 9 (Extension of Character). Given a Dirichlet character ε ∈ D(N, R) and a multiple N of N , this algorithm finds a characters ε ∈ D(N , R), such that ε (a) = ε(a), for all a ∈ Z with (a, N ) = 1. 1. [Minimal Generators] Compute the minimal generators gi for (Z/N Z)∗ . 2. [Evaluate] Compute ε(gi ) for each i. Since (gi , N ) = 1, we also have (gi , N ) = 1. 3. [Output Character] Output the character defined by [ε(g0 ), .

What we denote by a//b above. Finally, gcd is not included with Python. gcd function. 656–657] as the solution to exercise 51 on page 379. See in particular the paragraph right in the middle of page 657, which describes the algorithm. Knuth says this rational reconstruction algorithm is due to Wang, Kornerup, and Gregory from around 1983. 656–657]. 1) and similarly for U . When computing the usual extended gcd, at the end v2 = gcd(a, m) and v0 , v1 give a representation of the v2 as a Z-linear combination of m and a.

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Algorithms for Computing with Modular Forms by William Stein


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