N=11, M=23:

We have

$$\begin{eqnarray*}g &=& q + aq^2 + (-2a - 1)q^3 + (-a - 1)q^4 + 2aq^5 + (a - 2)q^... ...2a + 2)q^{10} + (-2a - 4)q^{11} + (a +3)q^{12} + 3q^{13}+\cdots, \end{eqnarray*}$$

with a²+a-1=0; and

$$\begin{eqnarray*}(1 + M - a_M(f))(1 + M + a_M(f)) &=& 5^2\cdot 23,\\ \mbox{\rm... ...m}((1 + N - a_N(g))(1 + N + a_N(g))) &=& 2^4\cdot5\cdot11\cdot19 \end{eqnarray*}$$

We must take p=23; for q take each of 2,5,19. The relevant congruences are $f\equiv h_4\pmod{23}$, and $g\equiv h_3, h_4\pmod{2}$, $g\equiv h_2\pmod{5}$, $g\equiv h_1\pmod{19}$. We do not have maximal success; only $\rho_{f,23}\times\rho_{g,2}$ arises from $S_2(\Gamma_0(11\cdot 23))$.