Cusp forms with different orders at infinity

Question

Let’s assume we have cusp forms $f_1,...,f_n\in S_k(\Gamma_1(N))$ which they are zero at infinity with different orders. Then, how can we show that they are linearly independent?

Answer 1

You can write the cusp forms as $$f_i(q)=q^{c_i}g_i(q)$$ where the $c_i$ are distinct, and $g_i(0)\ne0$.

Then given a linear dependency $$\sum_i a_i f_i(q)=0$$ consider the term $a_if_i(q)$ with minimum $c_i$ amongst the terms with $a_i\ne0$. Divide by $q^{c_i}$ and set $q=0$ to get a contradiction.