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An adelic reference for $GL(2, \mathbb{Z})$ is also "Traces of Hecke operators" by Knightly and Li. You choose matrix coefficent at the infinite place and the characteristic function $GL_2(\mathbb{Z}_p)$ times the center, and plug it into the Arthur trace formula. They do that almost for weight $k \geq 3$, but they work with matrix coefficient. This gives the dimension formula. This is likely to be more fruitful for generalization to higher rank or deeper level, where not always a classical treatment is available.