By William Stein

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**Example text**

The spaces Mk (Γ1 (N ))(ε) and Mk (Γ1 (N ))(ε ) are canonically isomorphic if ε and ε are conjugate. 10 (Galois Orbit). Given a Dirichlet character ε ∈ D(N, R), this algorithm computes the orbit of ε under the action of G = Gal(F /F ), where F is the prime subfield of Frac(R), so F = Fp or Q. 1. [Order of ζ] Let n be the order of the chosen root ζ ∈ R. 2. [Nontrivial Automorphisms] If char(R) = 0, let A = {a : 2 ≤ a < n and (a, n) = 1}. If char(R) = p > 0, compute the multiplicative order r of p modulo n, and let A = {pm : 1 ≤ m < r}.

2. [Nontrivial Automorphisms] If char(R) = 0, let A = {a : 2 ≤ a < n and (a, n) = 1}. If char(R) = p > 0, compute the multiplicative order r of p modulo n, and let A = {pm : 1 ≤ m < r}. 3. [Compute Orbit] Compute and output the set of unique elements εa for each a ∈ A (there could be repeats, so we output unique elements only). Proof. We prove that the nontrivial automorphisms of ζ in characteristic p are as in Step 2. It is well-known that every automorphism in characteristic p s on ζ ∈ Fp is of the form x → xp , for some s.

That vector has a 1 at the non-pivot column, 0’s at all other non-pivot columns, and for each pivot column, the negative of the entry of A at the non-pivot column in the row with that pivot element. 2. Intersection of Subspaces: Suppose W1 and W2 are subspace of a finite-dimensional vector space V . Let A1 and A2 be matrices whose columns form a basis for W1 and W2 , respectively. Let A = [A1 |A2 ] be the augmented matrix formed from A1 and A2 . Let K be the kernel of the linear transformation defined by A.

### Algorithms for Computing with Modular Forms by William Stein

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