This page gives Fourier coefficients of Siegel modular cusp forms of degree 2 and level 1. One can get Fourier coefficients of forms of weight $(j,k)$ (that is, corresponding to $\Sym^j \otimes \det^k$) for the following cases (ordered by cohomological weight $j+2k-3$).

Level 1 (no character): \[\begin{split} (j,k) \in \{ &[0, 10], [6, 8], [0, 12], [4, 10], [8, 8], [12, 6], [6, 10], [8, 9], [12, 7], [0, 14], [4, 12], [6, 11], [10, 9], [14, 7],\\ &[16, 6], [18, 5], [2, 14], [8, 11], [16, 7], [18, 6], [20, 5], [6, 13], [10, 11], [24, 4], [4, 15], [28, 4], [4, 17], [30, 4],\\ & [34, 4], [36, 3], [2, 21], [2, 23], [42, 3], [44, 3], [2, 25], [46, 3], [50, 3], [0, 35], [0, 39], [0, 41], [0, 43] \} \end{split}\]

In all these cases the dimension of the space of cusp forms is 1.

If $(\tau_{11}, \tau_{12}, \tau_{22})$ are the coordinates on the Siegel upper half
space of degree 2 we set
\[
q_1=\exp(2 \pi i \tau_{11}),\quad q_2=\exp(2 \pi i \tau_{22})\quad\text{and}\quad u=\exp(2 \pi i \tau_{12}).
\]
The Fourier expansion is a sum
\[
\sum_n a(n) q_1^{n_1} q_2^{n_2} u^{n_{12}}
\]
with $n=(n_1,n_2,n_{12})$ ranging over the integral triples such that
$n_1 x^2+n_{12} xy+n_2 y^2$ is positive semi-definite. Here $a(n)$ is a vector of
length $j+1$ (in $\Sym^j(\mathbb{C}^2)$).

(If $x,y$ are coordinates on $\mathbb{C}^2$ then $x^n,x^ny,\ldots ,y^n$ are coordinates on $\Sym^{n}(\mathbb{C}^2)$.)

We collect terms and write the Fourier expansion as \[ \sum_{n_1> 0,n_2 > 0} A(n_1,n_2) q_1^{n_1} q_2^{n_2} \] with $A(n_1,n_2)$ a vector of length $j+1$ with entries that are Laurent polynomials in $u$.

This page can provide you for a given weight $(j,k)$ with

- the coefficient $A(n_1,n_2)$ for $(n_1,n_2)$ in a given range depending on $(j,k)$,
- the beginning of the Fourier development in the form of a list $[\ldots ,[[n_1,n_2],A(n_1,n_2)], \ldots]$,
- the possibility to download the Fourier series as far as available here, in comma-separated format.

There are {{ Coefficients.length }} non-zero coefficients available with . The first terms are Look up a coefficient:

There is no data available for these values of $j$ and $k$.