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$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The Voronoi formula on $\GL(3)$ with ramification", Zhou has formulated that, for any integers $a,c$ satisfying $(a,c)=1$ with $p|c$, one roughly has the corresponding relation: $$\sum _{n\sim X}{A_f(n,1)e\left(\frac{an}{c}\right)} \omega \left ({n} \right) \leftrightsquigarrow c \sum_{\substack{m_2\neq 0}} \sum_{m_1|c} \frac{A_f(m_1,m_2)}{m_1|m_2|} S(a, m_2;c/m_1) \mho \left ( \frac{m^2_1|m_2|X}{c^3} ;\omega\right) .$$

Now, consider $F$ be an automorphic form on $\GL_4$ of level $p$. My question is: in the case of $p|c$, whether or not we have a similar formula like: $$ (1) \quad \sum_{n\sim X} A_{F}(n,1,1) e \left( \frac{an}{c} \right) \omega \left ({n} \right) \leftrightsquigarrow c\sum_{d_1|c}\sum_{d_2|\frac{c}{d_1}}\sum_{m\neq 0} \frac{\overline{A_F(m, d_2,d_1)}}{|m|d_2d_1} {KL}_2(\overline{a}, m,c;(1,1) , (d_1,d_2) )\, \mho^\prime \left ( \frac{mXd^2_2 d^3_1}{c^4} ;\omega\right),\; ? $$ where the hyper-Kloosterman sum is defined as $${KL}_2(n, m,c;(1,1), (d_1,d_2))= \sideset{_{}^{}} {^{\ast}_{}}\sum\limits_{x_1 \bmod \frac{c}{d_1} } \,\, \sideset{_{}^{}}{^{\ast}_{}}\sum\limits_{x_2 \bmod \frac{c}{d_1d_2} } e{\left (\frac{d_1 x_1 n}{c}+\frac{d_2\overline{x_1x_2}}{\frac{c}{d_1}}+\frac{{x_2}m}{ \frac{c}{d_1d_2}}\right )} .$$ Note that when the level is trivial, the formula above for $\GL_4$ is already known; see, for example, V. Chandee and X. Li's paper (The second moment of $\GL(4) \times \GL(2)$ $L$-functions at special points). Besides, in the special case where $(p,c)=1$, one has an analog for $\GL_4$ Maass form, which has a similar structure like Eqn. (2) in Zhou's above mentioned paper; see, for example, Andrew Corbett's paper. I still can not be sure what looks like for the Voronoi formula on $\GL(4)$ when $p|c$.

If any expert here knows something on this question, please show a guide or certain references.

Many many thanks in advance.

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    $\begingroup$ Corbett's paper deals with this particular situation (and much much more). Of course, one has to rewrite his adelic statement back into a classical statement. $\endgroup$
    – Peter Humphries
    Commented Nov 26 at 14:53
  • $\begingroup$ @PeterHumphries Thanks, Peter, I will check this again. It seems that Corbett was more concerned about the co-prime situation, as it mentioned "...Of particular interest are those primes at which the level and modulus are jointly ramified..." in the Abstract. $\endgroup$
    – hofnumber
    Commented Nov 26 at 23:21
  • $\begingroup$ I think there is a translation error on your behalf: if the level and the modulus are jointly ramified, then they are not coprime. $\endgroup$
    – Peter Humphries
    Commented Nov 27 at 0:18
  • $\begingroup$ @PeterHumphries Many thanks for pointing out this blur. $\endgroup$
    – hofnumber
    Commented Nov 27 at 0:28
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    $\begingroup$ The assumption that f has prime level p greatly restricts the possibilities for the local component of the corresponding automorphic representation at p. This makes it relatively easy to work out how Corbett's formula looks explicitly in this case. $\endgroup$
    – Windom Earle
    Commented Nov 27 at 14:51

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