# Computing double coset operators in a computer algebra system

I want to do double coset operators computations on modular forms of half integer weight and with character such as the trace operators that map modular forms of congruence subgroups $$\Gamma_0(N)$$ to modular forms of $$\Gamma_0(M)$$ where $$M$$ divides $$N$$. I know that there are commands in SAGE and PARI/GP that can compute action of certain Hecke operators (using modular symbols and trace formulas respectively).

However, can I compute action of arbitrary double coset operators using such computer algebra systems (in particular the trace operator mentioned above)?

• There are formulas that will express the trace operator (at least for $\Gamma_{0}(N)$) in terms of the action of Atkin-Lehner involutions and the $U(d)$ operator from which one might be to get what you want. However, as far as I understand it, there's no direct way to use SAGE or PARI/GP to do modular symbols computations involving half-integer weight modular forms. (I think Sage will use "tricks" to get the Fourier expansions of half-integer weight modular forms from integer weight ones.) Jan 16, 2019 at 2:11
• @JeremyRouse Thanks a lot for your useful comment. What formulas are you referring to that express trace operators in terms of A-L involutions? Please could you give me a reference. Thanks!
– user35360
Jan 16, 2019 at 16:48
• In the half-integer weight setting, you can find such a formula (for the trace from $\Gamma_{0}(4N)$ to $\Gamma_{0}(4N/p)$) on pages 66 and 67 of W. Kohnen's 1982 Crelle paper titled "Newforms of half-integral weight". Jan 16, 2019 at 22:15

From the MAGMA documentation:

Since V2.8, Magma has included packages for modular forms and modular symbols. These were originally developed by William Stein, and are continually being developed further and improved by the Magma group. The modular forms package is, to a large extent, built on top of the modular symbols package. However, it also contains several independent features, notably Eisenstein series, half-integral weight forms and weight 1 forms.

Construction of spaces of modular forms of weight k ≥ 1/2 on Γ0(N) or Γ1(N) (or with specified character)

Decomposition into Eisenstein, cuspidal, and new subspaces

Computation of dimensions (by formulae)

Computation of bases of these spaces, expressing basis elements as q-expansions with desired number of terms

Arithmetic operations for modular forms

Magma is not free but it has an online calculator and detailed documentation at

http://magma.maths.usyd.edu.au/calc/