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 if you want dimensions including old space, that's a preliminary step. 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.