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 $pi_1=otimespi_{1,v}$ and $pi_2=otimespi_{2,v}$ be distinct cohomological cuspidal automorphic representations of $mathrm{GL}_n(mathbb{A}_{F})$ with levels less than or equal to $N$. Let $mathcal{N}(pi_1,pi_2)$ be the minimum of the absolute norm of $v mid infty$ such that $pi_{1,v} ot simeq pi_{2,v}$ and that $pi_{1,v}$ and $pi_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $pi_1$ and $pi_2$, $$mathcal{N}(pi_1,pi_2) leq C_N.$$ This improves known bounds $$ mathcal{N}(pi_1,pi_2)=O(Q^A) ;;; (text{some } A text{ depending only on } n), $$ where $Q$ is the maximum of the analytic conductors of $pi_1$ and $pi_2$. This result applies to newforms on $Gamma_1(N)$. 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 $p in {2,7}$, $$lambda_{f_1}(p)/sqrt{p}^{(k_1-1)} = lambda_{f_2}(p)/sqrt{p}^{(k_2-1)},$$ then $f_1=cf_2$ for some constant $c$. Here, for each prime $p$, $lambda_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.