Exact formulas for dimensions of Atkin-Lehner eigenspaces follow from trace formulas of Yamauchi and Skoruppa-Zagier. Skoruppa-Zagier corrected some clerical errors in Yamauchi's paper. See:
Nils-Peter Skoruppa and Don Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), no. 1, 113–146.
In the case of squarefree level, I worked things out explicitly in this paper:
Kimball Martin, Refined dimensions of cusp forms, and equidistribution and bias of signs, J. Number Theory 188 (2018), 1–17.
The main focus is dimensions of new parts of Atkin-Lehner eigenspaces, but dimensions which include oldspaces can be computed similarly. Either you can take the newspace dimensions, and add in the old form contribution, or just use the trace formulas on the full spaces $S_k(N)$ to get a formula mimicking what I did for the newspaces.
Code to compute dimensions (and a link to my paper) in the squarefree level case is available here:
https://math.ou.edu/~kmartin/data/
Using these formulas should be faster than the direct calculations in David Loeffler's answer. You can also code up the non-squarefree level case using Skoruppa-Zagier's trace formula without too much trouble.
Also, in case you're not familiar with it, the LMFDB modular forms page has tabulated a lot of data for newforms, including Atkin-Lehner signs. You can use this check dimensions in many cases.