Quadratic forms of modular forms

In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of the products of their common off-diagonal coefficients. This phenomenon could be interpreted as a mixed $L^4$-norm problem. We also define the $\ell^p$-norm of a holomorphic cusp form via its expansion with respect to an orthonormal Hecke basis. We then establish a conditional upper bound for the $\ell^p$-norm, and deduce that the coefficients of quadratic forms of holomorphic cusp forms in the Hecke basis are not uniformly small, being dominated by small-amplitude components. This behavior is consistent with the expected distribution of orthogonal families of $L$-functions.