  In this paper, we study solutions to $h=af^2+bfg+g^2$, where $f,g,h$ are
Hecke newforms with respect to $\Gamma_1(N)$ of weight $k>2$ and $a,b\neq 0$.
We show that the number of solutions is finite for all $N$. Assuming Maeda's
conjecture, we prove that the Petersson inner product $\langle f^2,g\rangle$ is
nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for
$SL_2(\mathbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we
obtain that, assuming Maeda's conjecture, identities between cusp eigenforms
for $SL_2(\mathbb{Z})$ of the form $X^2+\sum_{i=1}^n \alpha_iY_i=0$ all are
forced by dimension considerations. We also give a proof using polynomial
identities between eigenforms that the $j$-function is algebraic on zeros of
Eisenstein series of weight $12k$.
