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.

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.