Nobody has ever written down a “black‐box’’ Arthur–Selberg–trace‐formula routine that spits out Fourier–expansions of the (2,2)–forms the way one does for weight 2 on SL₂ by modular symbols. In fact the consensus in the community is

1. For the full Siegel modular group Sp₄(ℤ) the only nonzero cuspidal scalar forms occur in weight ≥10 (Igusa’s χ₁₀, χ₁₂, …) so there is really nothing to compute at weight 2 on Sp₄(ℤ).

2. The only natural place one finds genuine weight 2 Siegel forms is on smaller groups (e.g. the paramodular group K(N)); such forms are exactly the ones predicted by the Brumer–Kramer paramodular conjecture for abelian surfaces.

3. No one has managed to drive the full trace‐formula machine in weight 2, level variable GSp₄ and actually output Fourier coefficients. The obstacles (continuous spectrum, endoscopy, intertwining operators, massive combinatorics of orbital integrals, etc.) make it a forbidding programming project.

What has been done instead is

• Jacobi‐expansion methods (Skoruppa, Ibukiyama, Poor–Yuen). By expressing a Siegel form as a Fourier–Jacobi expansion one reduces to computing vector‐valued modular forms / Jacobi forms. Skoruppa carried this out in level 1 for high scalar weights (≥5); Ibukiyama extended many of these ideas. But these methods simply never reach the “critical’’ scalar–weight 2 on Sp₄(ℤ) (there are no such forms at full level), and on K(N) they become much more intricate.

• Paramodular computations (Poor–Yuen, many collaborators). In a sequence of papers and talks Poor and Yuen have described and implemented a Fourier–Jacobi‐expansion algorithm for the paramodular groups K(N) in weight 2, and have determined newforms (and their Hecke eigenvalues) for a range of prime levels N (up to a few hundred in practice). That code is not part of any public computer‐algebra system (it lives in the authors’ private repositories), but the data for levels up to about 300 have appeared in print or on their websites in connection with the paramodular conjecture.

• Point‐counting and cohomology (Faber–van der Geer–Bergström). One can compute traces of Hecke operators Tₚ on the cohomology of local systems over the moduli space A₂ of principally polarized abelian surfaces (equivalently, on Siegel modular forms of all weights on Sp₄(ℤ)) by counting genus 2 curves over 𝔽ₚ. This approach does give you Tr Tₚ on S_{k₁,k₂}(Sp₄(ℤ)) for many (k₁,k₂) (including (2,2)), and tables of these traces for p≤37 and k₁+k₂ up to 20 are in the literature. However it does not produce a basis of Fourier expansions, only traces.

In short:

 – No one has ever implemented the trace‐formula in weight (2,2), level Γ₀(N) or K(N), and tabulated actual q‐expansions.
 – The only explicit q‐expansions in weight 2 (i.e. Siegel weight (2,2)) live on the paramodular groups, and come from Jacobi‐expansion code of Poor–Yuen (not publicly released).
 – If you want mere traces of Tₚ on S_{2,2} you can get them at level 1 (Sp₄(ℤ)) up to p=37 from the point‐counting work of Faber–van der Geer–Bergström, but again there is no public “plug-and-play’’ implementation of the trace formula itself.