I'm studying modular forms, but I can't understand the definition of diamond operator.
Why can I define for all $\alpha$ with $\delta \equiv d$? I can't understand the reason why two different matrix $\alpha$ and $\alpha'$ with $\delta \equiv \delta'\equiv d$ give the same operator.
If $\alpha, \alpha'$ are two matrices in $\Gamma_0(N)$ with bottom right entry congruent to $d \pmod N$, then they are equal up to left multiplication by an element of $\Gamma_1(N)$. Indeed, this is just saying that the map which sends $\alpha$ to its bottom right entry, is a group homomorphism $$\Gamma_0(N) \to (\mathbb Z / N \mathbb Z)^\times$$ with kernel $\Gamma_1(N)$. So if $\alpha = \gamma \alpha'$ with $\gamma \in \Gamma_1(N)$, then $$f[\alpha']_k = f[\gamma \alpha]_k = (f[\gamma]_k)[\alpha]_k = f[\alpha]_k$$ because $f \in \mathcal M_k(\Gamma_1(N))$.