Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}_1$ and $\mathfrak{p}_2$ are distinct prime ideals in $K$. Under what conditions can one find $f_3\in S_{k_3}(\Gamma_0(N_3))$ which is a normalized cuspidal eigenform such that $f_3\equiv f_i\mod \mathfrak{p}_i$ for $i=1,2$.

This is a natural question and is most likely classical. To start, let's not have any assumptions on $N_3$ and $k_3$. Could someone point me to some references, that would be really helpful, I understand what I'm asking may be really well understood, in which case I'd like to know of what is known about all the pairs $(k_3,N_3)$ for which a form $f_3$ satisfying the simultaneous congruence exists.