Is it true that for any even $k$ and $N,$ there always exist a Hecke eigenform with integer Fourier coefficient of weight $k$ and level $N$ ?
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$\begingroup$ Please use a highlevel tag like "nt.numbertheory". I added this tag now. $\endgroup$– GH from MOJul 3, 2020 at 23:43

$\begingroup$ lmfdb.org $\endgroup$– PastenJul 4, 2020 at 0:15

1$\begingroup$ Are you asking about holomorphic modular forms? Cusp forms? In either case, no as $M_2(1) = 0$. If you exclude this example and allow Eisenstien series, then yes. $\endgroup$– KimballJul 4, 2020 at 1:33

2$\begingroup$ There are lots of Eisenstein series with integer coefficients, as several people have pointed out. CM forms (coming from Hecke characters over im. quad. fields) also often have small coefficient fields. On the other hand, I believe it's expected that if you fix an $N$ and look at cuspidal, nonCM eigenforms of level $N$, trivial character and weight $k$, then the degrees of the coefficient fields go to $\infty$ with $k$. The LMFDB database that @Pasten links to is a great way of getting a feel for this  you can search by degree of coefficient field. $\endgroup$– David LoefflerJul 4, 2020 at 8:14
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No, eg. $k$ odd and $N=1$. If $k=4m+6n$ then can't you just restrict an Eisenstein series for $N=1$ to a smaller group?