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.



Congruence of normalized eigenforms at two primes
I assume you allow $f\_3$ to have arbitrary Fourier coefficients (i.e., $\mathfrak{p}\_1$ and $\mathfrak{p}\_2$ are prime ideals in a suitable number field that contains $K$ and the coefficient field of $f\_3$).


In the case of a single pair $(f\_1,\mathfrak{p}\_1)=(f\_2,\mathfrak{p}\_2)$, this is the famous level lowering/raising problem. Assuming the irreducibility of the residual representation, this has been solved by Ribet/Diamond-Taylor. See the following papers :


Ribet, K. A.
On modular representations of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ arising from modular forms.
Invent. Math. 100 (1990), no. 2, 431–476. 


Diamond, Fred ; Taylor, Richard. 
Nonoptimal levels of mod $l$ modular representations.
Invent. Math. 115 (1994), no. 3, 435–462.


You may also look at Ribet's Report on mod $l$ representations of ${\rm Gal}(\overline{\bf Q}/{\bf Q})$ (Proc. Sympos. Pure Math., 55, Part 2) where he explains how to adjust the weight, level and character of the form $f\_3$. On these matters the works of Carayol and Edixhoven are also quite relevant.


In the case of distinct pairs $(f\_1,\mathfrak{p}\_1)$ and $(f\_2,\mathfrak{p}\_2)$, combining the results mentioned above already gives you necessary conditions on $k\_3$ and $N\_3$ to be satisfied for such a form $f\_3$ to exist.


Finally, these notes of Mazur and Stein deal with a particular - yet quite interesting - case :


<https://wstein.org/Tables/serremodpq/>


