Let $N$ be a natural number and let $\Gamma_1(N)$ be the congruence subgroup of $SL_2(\mathbb{Z})$. Let $M(N)$ denote the space of all integer weight holomorphic modular forms for $\Gamma_1(N)$ whose Fourier coefficients lie in $\mathbb{Z}$. Let $p$ be a prime. Then we have the natural map (reducing the coefficients modulo $p$),

$$ \varphi : M(N)\to (\mathbb{Z}/p\mathbb{Z}) [[q]], $$

and let $M_p(N)$ denote the image of $\varphi$.

I am interested in understanding what is known about $\varphi$. Whether it is surjective? Whether it is injective? In particular, will the lift (under $\varphi$) of a $\mod p$ cusp form be always a cusp form?

If not, what is the kernel? I did some googling and wasn't able to find a complete source on this subject.

Any help is appreciated.

Thank you