What are modular forms or cusps forms, resp. ?
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We start with defining their common domains $\mathbb{H}\_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $Sp(g,\mathbb{Z})$ is the subgroup of $SL(2g,\mathbb{Z})$ such that all elements satisfy $M=J\_g^t M J\_g$ with $J\_g$ being the canonical almost complex structure.


$Sp(g,\mathbb{Z})$ acts on $\mathbb{H}\_g$ by $M(Z)=(AZ+B)(CZ+D)^{-1}$ where A,B,C and D are the block matrix entries of M.


Let $\rho : GL(g,\mathbb{C}) \to GL(V)$ be a rational representation on a finite dimensional $\mathbb{C}$-vector space then the associated modular forms are the holomorphic functions $f : \mathbb{H}\_g \to V$ satisfying $f(M(Z))=\rho(CZ+D)f(Z)$ for all $M \in Sp(g,\mathbb{Z})$.


Cusps forms can be easily characterized as the elements of Siegel's $\Phi$ operator's kernel.


Modular forms in genus 2
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If g equals 2 then the observed representations are the ones of $GL(2,\mathbb{C})$. We know from representation theory that all irreducible representations are isomorphic to a rep of the type $det^k \otimes Sym^{2j}(\rho\_{standard})$. 


We denote by $\rho\_{standard}$ the standard representation $X \mapsto X$. $Sym^{2j}(\rho\_{standard})$ is the associated symmetric product $GL(2,\mathbb{C}) \to Sym^{2j}(\mathbb{C}^2)$. $det$ is just the 1 dimensional determinant representation $GL(2,\mathbb{C}) \to \mathbb{C}$.


For $k\geq 4$ Tsushima has given a dimension formula for the vector space of cusps forms in 



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> Ryuji Tsushima. An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z). Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 59:139–142, 1983.
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Satoh and Ibukiyama gave (but partly didn't publish AFAIK) generators for the modules of vector valued modular forms to the representations $det^k \otimes Sym^{2j}(\rho\_{standard})$ 
with running k and fixed j in ${1,2,3}$. 


EDIT: Ibukiyama sent me the following article



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> Tomoyoshi Ibukiyama. Vector Valued Siegel Modular Forms of Symmetric Tensor Weight of Small Degrees. Commentarii Mathematici Universitatis Sancti Pauli, 61, No. 1:51–75, 2012.
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And odd weighgt for $Sym^{6}$, i.e. $det^{2k+1} \otimes Sym^{6}(\rho\_{standard})$, was treated here



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> Christiaan van Dorp. Generators for a module of vector-valued Siegel modular forms of degree 2, <http://arxiv.org/abs/1301.2910> .
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While editing let me just add Satoh's article to get a (rather) conclusive list



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> Takakazu Satoh. On certain vector valued Siegel modular forms of degree two. Mathematische Annalen, 274:365–387, 1986.
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end of edit.


The actual question
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So the next question for me was are there cusps forms to $det^3 \otimes Sym^{2j}(\rho\_{standard})$ and can they ( at least a single one) be given explicitly, in particular for j=4 ?


cheers 
Tom


p.s. please excuse all mistakes I made but it was the first time for me publishing on such a plattform.



Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)
Though I haven't seen it, I've heard that Tomoya Kiyuna, a student at Kyushu
University, has done this for the case $j=4$ as part of his master's thesis. In particular, he finds eighteen explicit generators of the module of vector-valued Siegel modular forms of the symmetric tensor 8. 


