The classical modular form database currently contains information for all newforms $f\in S_k^\mathrm{new}(N,\chi)$ of level $N\ge 1$, weight $k\ge 1$, character $\chi\colon (\Z/N\Z)^\times\to \C$, for which any of the following hold:
 $Nk^2\le 4000$;
 $\chi$ trivial and $Nk^2\le 40000$;
 $N \le 24$ and $Nk^2 \le 40000$;
 $N \le 10$ and $Nk^2 \le 100000$;
 $N \le 100$ and $k \le 12$;
 $k > 1$ and $\dim S_k^{\rm new}(N,\chi) \le 100$ and $Nk^2 \le 40000$.
For each newform with qexpansion $f=\sum a_nq^n$ the database contains the integers $\mathrm{tr}(a_n)$ for $1\le n \le 1000$, and when the dimension of $f$ is at most $20$, the algebraic integers $a_n$ for $1\le n \le 1000$ expressed in terms of an explicit basis for the coefficient ring $\Q(f)$. For $1000 < N \le 4000$ this data is available for all $n\le 2000$, and for $4000 < N\le 10000$, for all $n\le 3000$ (these values exceed both the Sturm bound and $30\sqrt{N}$ in every case).
For each embedding $\rho\colon \Q(f)\to \C$ the complex numbers $\rho(a_n)$ are available as floating point numbers with a precision of at least 52 bits (separately, for both real and imaginary parts); this information is available for all newforms, regardless of their dimension, even when algebraic $a_n$ are not available (for the same ranges of $n$ as above).
Dimension tables are available for all newspaces $S_k^\mathrm{new}(N,\chi)$ with $Nk^2 \le 4000$, and also for those with $k>1$ and $Nk^2\le 40000$, and those with $N\le 10$ and $Nk^2\le 100000$. For newspaces in these ranges with $N\le 4000$ we have also computed the first 1000 coefficients of the trace form of the newspace.
Not every invariant of every newform has been computed (this is computationally infeasible). Below is completeness information for some specific invariants:

The following information is available for every newform: level, weight, character, dimension, analytic conductor, all self twists, SatoTate group, trace form, complex embedding data, an upper bound on the analytic rank, and an upper bound on the set of inner twists.

The following information is available for every newform of dimension at most 20, as well as every weight one newform: coefficient field, exact algebraic coefficient data, generators for the coefficient ring, and a lower bound on the index of the coefficient ring in the ring of integers in the coefficient field.

The following information is available for every weight one newform: projective image and projective field.