There is an amazing sequence T_{1}, T_{2}, T_{3}, … of commuting linear maps on modular symbols. The corresponding systems of eigenvalues

{a_{1}, a_{2}, a_{3}, … }

are the coefficients a_{n} of the q-expansions of modular forms.

When n = p is prime to N, we have

T_{p}(\{α, β\}) = [(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) + _{r mod p}^{∑} (\begin{matrix}1 & r \\ 0 & p \end{matrix})] \{α, β\}.

Example: N = 11

T_{2}({0, 1/5}) = {0, 2/5} + {0, 1/10} + {1/2, 3/5}
= −2{0, 1/5}
= a_{2}{0, 1/5}

Consequently, the modular form of level 11 is

f = q + a_{2}q^{2} + a_{3}q^{3} + ⋯,

where T_{ℓ}({0, 1/5}) = a_{ℓ}{0, 1/5}. There is a deep connection with elliptic curves (due to Shimura):

a_{ℓ} = ℓ + 1 − #E(F_{ℓ}),

where E is y^{2} + y = x^{3} − x^{2} − 10x − 20.