N=11, M=19:

A basis for $S_2(\Gamma_0(19))$ is

$$g=q - 2q^3 - 2q^4 + 3q^5 - q^7 + q^9 + 3q^{11} + 4q^{12} - 4q^{13}+\cdots.$$

We find that

$$\begin{eqnarray*}(1 + M - a_M(f))(1 + M + a_M(f)) &=& 2^4\cdot 5^2,\\ (1 + N - a_N(g))(1 + N + a_N(g)) &=& 2^5\cdot 5. \end{eqnarray*}$$

The only pair of primes for which $\rho$ is irreducible is p=2, q=5. The four classes of newforms at level $11\cdot{}19$ are $h_1,\ldots,h_4$. The relevant congruences are $f\equiv h_1,h_2,h_3,h_4\pmod{2}$and $g\equiv h_4\pmod{5}$. Therefore $\rho_{f,2}\times\rho_{g,5}$ arises from the eigenform h₄; the coefficient a₂ of h₄satisfies the irreducible polynomial x⁷ + x⁶ - 14x⁵ - 10x⁴ + 59x³ + 27x² - 66x - 30. Again, we have maximal success.