Let $\ell$ be a prime number, let $k$ be a finite field of characteristic $\ell$, and let

\begin{align*} \bar{\rho}_{f,\lambda}: G_{\mathbb{Q}} &\to GL_2(k),\\ \bar{\rho}_{g,\lambda}: G_{\mathbb{Q}} &\to GL_2(k) \end{align*}

be the residual Galois representations attached to normalized cuspidal newforms $f$ and $g$ of levels $N_f$ and $N_g$, weights $\kappa_f$ and $\kappa_g$, and nebentypus characters $\varepsilon_f$ and $\varepsilon_g$, respectively. Suppose that there is a finite set $S$ of rational primes containing $\ell$ and every prime divisor of $N_fN_g$ such that both residual representations are unramified at every prime $p \notin S$ and, for every prime $p \notin S$,

\begin{align*} a_p(f) &\equiv a_p(g) \quad \text{in } k,\\ \varepsilon_f(p)p^{\kappa_f-1} &\equiv \varepsilon_g(p)p^{\kappa_g-1} \quad \text{in } k. \end{align*}

Then the semisimplifications of the two residual representations are isomorphic:

\begin{align*} \bar{\rho}_{f,\lambda}^{\mathrm{ss}} \cong \bar{\rho}_{g,\lambda}^{\mathrm{ss}} \end{align*}

as $k$-linear representations of $G_{\mathbb{Q}}$.