Modular forms are arithmetic objects

Question

What does arithmetic object exactly means? In an article, I found the following statement:

modular forms are arithmetic objects.

What this should means?

Bests.

Answer 1

This has several interpretations, as the discussion on MO linked in the comments shows.

One interpretation is that Hecke eigenforms are the basic objects in class field theory for $GL_2$ over $\mathbb Q$: consider e.g. the Taylor--Wiles modularity theorem for elliptic curves over $\mathbb Q$, which states that the $L$-function of any such curve coincides with the $L$-function of a Hecke eigenform.

More generally, spaces of modular forms are spanned by modular forms whose $q$-expansions are integers, or algebraic integers, so modular forms give rise to very particular sequences (the sequences of $q$-expansion coefficients) of integers or algebraic integers.

For Hecke eigenforms, the interpretation of these sequences goes back to the relation with elliptic curves, or, more generally, Galois representations, mentioned above. For theta series, these sequences count representations of numbers by integral quadratic forms.

Even more vaguely, as the preceding discussion attempts to indicate, there is a huge amount of number theory connected to, or more precisely, arising from, modular forms, especially from their $q$-expansions.