Coefficient in the Fourier expansion of the cusp form

Question

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here.

How one can compute the coefficient $a(n)$ when $n$ is rather large ? for example, what is the coefficient $a(2015)$ ?

Answer 1

There are a couple of computer software packages which are very good at this sort of thing, such as Sage.

masiao@fermat:~$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath Version 6.7, Release Date: 2015-05-17                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage: F = Newforms(4, 6)[0]
sage: F[2015]
-96066432

(That took about 40 seconds to run, by the way. There are surely much faster approaches for this form, making use of the product expansion; but Sage is using general algorithms applicable to any modular form.)

The book "Modular Forms: A Computational Approach", by the founder of the Sage project, William Stein, gives a beautiful and down-to-earth account of how computations like this one are done. It's available for free as an e-book.