Except where explicitly noted below, all modular form data has been computed using rigorous algorithms that do not depend on any unproved assumptions or conjectures:

Self twist have been rigorously proved by checking exact algebraic coefficients up to the Sturm bound for the maximum possible level of the twist.

Inner twists have not been computed rigorously in every case, but all exceptions are explicitly noted in the Proved column of the table displaying inner twists. Even when one or more of the inner twists is marked as unproved, it is known that every inner twist is included in the list; in particular, if no inner twists are listed, then there are no nontrivial inner twists.

The analytic ranks have been rigorously verified (by computing winding elements on spaces of modular symbols and using parity for self dual newforms). In cases where the analytic rank has not been verified this is noted by an asterix on the newforms home page. In all cases the listed analytic rank is known to be an upper bound on the true analytic rank.

For weight 1 newforms, the classification of the projective image as $D_n$, $A_4$, $S_4$, $A_5$ has been rigorously verified either by explicitly computing the projective field (as described on the completeness page), or by a computation in the class group of the imaginary quadratic order whose ring class field is known to contain the projective field.
In addition to using mathematically rigorous algorithms whenever possible, we have performed a variety of consistency checks intended to catch any errors in the software packages used to compute modular forms data, or any errors that might have been introduced during postprocessing. The following checks have been performed:

All newforms of weight $k > 1$ and level $N$ satisfying $Nk^2 \le 2000$ have been independently computed using [Magma] and [Pari/GP] . By comparing the results of these computations we have verified that the decompositions of each newspace $S_k^{\rm new}(N,\chi)$ into Galois orbits agree (with matching coefficient fields), that the first 1000 coefficients of the trace forms for each Galois orbit agree, and for newforms of dimension $d\le 20$, that there is an automorphism of the coefficient field that relates the sequences of algebraic eigenvalues $(a_1,\ldots,a_{1000})$ computed by Pari and Magma.

For all newforms of weight $k>1$ and level $N$ satisfying $Nk^2 \le 4000$ we have verified that the trace forms computed by Magma (using modular symbols) agree with the trace forms obtained from complex analytic data computed using the explicit trace formula. This also verifies the dimensions of the coefficient fields.

For newforms of weight $k=1$ and level $N\le 1000$ we have matched the data computed using Pari/GP with the tables computed by Buzzard and Lauder [arXiv:1605.05346] .

For dihedral newforms of weight $k=1$ and level $N\le 4000$ we have matched trace forms with data computed using the explicit trace formula in Pari/GP with data computed independently in both Pari/GP and Magma using class field theoretic methods.