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 Symj⊗det) 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