Using spherical Harmonics, I can modular forms on $\Gamma_0(4)$ as follows. Let $u(\vec{m}) \in S^2$ be a spherical harmonic:
$$ \theta(z; u) = \sum_{m \in \mathbb{Z}^4} u(m) e(|m|z) = \sum_{(a,b,c,d) \in \mathbb{Z}^4} u(a,b,c,d) \, e^{2\pi i (a^2 + b^2 + c^2 + d^2)z}$$
and the weight of this modular form is $2 + \ell$ with $\ell = \deg u$. Let's try polynomials degree $6$. We can get three different modular forms for:
- $u(m) = a^6 + b^6 + c^6 + d^6$
- $u(m) = a^4b^2 + (\text{permutations}) $
- $u(m) = a^2 b^2 c^2 + a^2 b^2 d^2 + b^2c^2 d^2 + a^2 b^2 d^2$
Using invariant theory of finite groups, I believe those are the only non-zero possibilities for $\ell = \deg u = 6$. Since we have these polynomials should be invariant under $(a,b,c,d) \mapsto (\pm a, \pm b , \pm c, \pm d)$ as well as permutations of $\{ a,b,c,d\}$.
Additionally, I expect all these examples $\theta(z;u)$ to be cusp forms, since $u(\vec{0}) = 0$ in all three cases. When I consult LMFDB, we have the space of cusp forms contains only one newform. And I don't know enough about modular forms to interpret that. LMFDB offers an isomorphism between two spaces of cusp forms: $$ S_8^\text{old}(4) \simeq S_8^{new}(\Gamma_0(2))^{\oplus 2} $$ In fact LMFDB says there are no holomorphic newforms over at level $4$ ($\Gamma_0(4)$) of weight $k=8$.
For reference:
Let $S_k(N, \chi)$ be the vector space of holomorphic cusp forms of weight $k$ on the subgroup $\Gamma_1(N)$ with characer $\chi$ under $\Gamma_0(N)$.
The subspace spanned by all cusp forms of the form $f(d\tau)$ with $f > \in S_k(M, \phi) $ with $N = Md$ for some $d > 1$ and $\psi$ is a Dirichlet character modulo $M$ which induces $\chi$, is called the oldspace.
The orthogonal complement with respect to the Petersson inner product is called the newspace.
A newform is a function $f$ in the newspace, which is additionally an eigenfunction of all Hecke operators and all Atkin-Lehmer involutions, \dots
This is an enormous amount of jargon. My questions is if these three functions $\theta(z;u)$ are not linearly independent as functions in $S_8(\Gamma_0(4))$ what is the relationship between them? This would be related to the Lagrange sum of four squares theorem that $n = a^2 + b^2 + c^2 + d^2$ always has a non-zero solution in integers.