The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined

Question

There is a basis question which puzzles me for a while. The question is the following:

Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial nebentypus, with the $n$-th Fourier coefficient being denoted by $a_f(n)$. The how about the explicit estimate for the second moment of the coefficients $a_f(n)$ involving the level aspect? I am especially concerned about the error term in $$\sum_{n\sim X} |a_f(n)|^2=c_f X+\text{Error term}.$$

As far as I know, Farrell Brumley et al. ( see https://arxiv.org/pdf/1804.06402.pdf) and Keiju Sono (see https://arxiv.org/pdf/2110.01783.pdf) show that $$\sum_{n\sim X} |a_f(n)|^2=c_f X+O(p^{\frac{3}{4}} X^{\frac{3}{4} }),$$ which means that the triial bound holds whenever $p<X^{1/4}$.

So, if any expert here know some more relevant results on this topic. Please show some guides or the corresponding references, many thanks.

Great thanks in advance.

Answer 1

If $f$ is a $\mathrm{GL}(2)$ newform over $\mathbb{Q}$ with prime level $p$ and $\zeta^{(p)}(s)$ denotes the Riemann zeta function $\zeta(s)$ with the Euler factor at $p$ removed, then

$$L(s,f\times \bar{f}) = \zeta^{(p)}(2s)\sum_{n=1}^{\infty} \frac{|\lambda_{f}(n)|^2}{n^s}.$$

Moreover, we have the factorization

$$L(s,f\times\bar{f}) = \zeta(s) L(s,\mathrm{Ad}~f) = \zeta(s) L(s,\mathrm{Sym}^2 f\otimes\overline{\chi}_{f}),$$

where $\chi_{f}$ is the nebentypus character. The conductor of this $L$-function is $p^2$. We assume that the spectral component $K$ (weight, Laplace eigenvalue, etc.) is absolutely bounded. By applying Corollary 1.4 in this paper (Equation 1.11, in particular), we find that if $0<\epsilon<1/4$, then

$$\sum_{n\leq x}|\lambda_{f}(n)|^2 = \frac{L(1,\mathrm{Ad}~f)}{\zeta^{(p)}(2)}x+O((p x)^{1/2+\epsilon}) = \frac{6L(1,\mathrm{Ad}~f)}{\pi^2}\Big(1-\frac{1}{p^2}\Big)^{-1}x+O((p x)^{1/2+\epsilon}).$$

The implied constant depends on $K$ and $\epsilon$. Since $L(1,\mathrm{Ad}~f)\gg 1/\log p$ (Goldfeld-Hoffstein-Lieman), the asymptotic is nontrivial when $p < x^{1-4\epsilon}$.