Multiplicity one bound for cohomological automorphic representations with a fixed level

Let $F$ be a totally real field, and $\mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $π_1=\otimesπ_{1,v}$ and $π_2=\otimesπ_{2,v}$ be distinct cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{F})$ with levels less than or equal to $N$. Let $\mathcal{N}(π_1,π_2)$ be the minimum of the absolute norm of $v \nmid \infty$ such that $π_{1,v} \not \simeq π_{2,v}$ and that $π_{1,v}$ and $π_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $π_1$ and $π_2$, $\mathcal{N}(π_1,π_2) \leq C_N.$ This improves known bounds $\mathcal{N}(π_1,π_2)=O(Q^A) \;\;\; (\text{some } A \text{ depending only on } n),$ where $Q$ is the maximum of the analytic conductors of $π_1$ and $π_2$. This result applies to newforms on $Γ_1(N)$. In particular, assume that $f_1$ and $f_2$ are Hecke eigenforms of weight $k_1$ and $k_2$ on $\mathrm{SL}_2(\mathbb{Z})$, respectively. We prove that if for all $p \in \{2,7\}$, $λ_{f_1}(p)/\sqrt{p}^{(k_1-1)} = λ_{f_2}(p)/\sqrt{p}^{(k_2-1)},$ then $f_1=cf_2$ for some constant $c$. Here, for each prime $p$, $λ_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.

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