This page gives Fourier coefficients of Siegel modular cusp forms of degree 2 and level 1 with character. For the character see Section 12 of [10]. 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 (with character): \[\begin{split} (j,k) \in \{ &[0, 5], [6, 3], [6, 5], [8, 4], [12, 2], [0, 9], [8, 5], [12, 3], [2, 9], [6, 7], [10, 5], [12, 4], [14, 3], [0, 11], [4, 9],\\ &[6, 8], [8, 7], [14, 4], [16, 3], [18, 2], [2, 11], [8, 8], [10, 7], [16, 4], [20, 2], [0, 13], [4, 11], [6, 10], [10, 8], [14, 6],\\ &[20, 3], [2, 13], [26, 2], [4, 14], [28, 2], [2, 16], [4, 16], [34, 2], [2, 20], [0, 30], [0, 34], [0, 36], [0, 38] \} \end{split}\]

In all these cases the dimension of the space of cusp forms is 1. For the cases where $k=2$ where no dimension formula is known we refer to the paper [10].

If $(\tau_{11}, \tau_{12}, \tau_{22})$ are the coordinates on the Siegel upper half
space of degree 2 we set
\[
q_1=\exp(\pi i \tau_{11}),\quad q_2=\exp(\pi i \tau_{22})\quad\text{and}\quad u=\exp(\pi i \tau_{12}).
\]
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 $k$.