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][1] has tabulated a lot of data for newforms, including Atkin-Lehner signs.  You can use this check dimensions in many cases.


  [1]: http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/