In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\Gamma_1(N)$ of weight $k>2$ and $a,b\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. 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 $SL_2(\mathhbb{Z})$ 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, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix. / Mathematics

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