Let r_{o}, …, r_{p} be coset representatives for Γ_{0}(p) in SL_{2}(Z). So

SL_{2}(Z) = Γ_{0}(p)r_{o} ∪ Γ_{0}(p)r_{1} ∪ ⋯ ∪ Γ_{0}(p)r_{p}.

E.g.,

(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 2 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 3 & 1 \end{matrix}) , …, (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Let V be the p + 1-dimensional vector space with basis

x_{0}, x_{1}, …, x_{p}.

Theorem (Manin). There is an isomorphism

V/R = ModSym (Γ_{0}(p)),

where x_{i} maps to r_{i}{0, ∞} = {r_{i}(0), r_{i}(∞)}, and the relations R are described explicitly below.

The subspace R of relations is the subspace generated by

x_{i} + x_{i}S
x_{i} + x_{i}T + x_{i}T^{2},

where

S = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) and T = (\begin{matrix}1 & −1 \\ 1 & 0 \end{matrix}) ,

and i = 0, …, p.

Generating modular symbols: {0, ∞}, {0, 1}, {0, 1/2}, …, {0, 1/10}, {∞, 0} Doing the linear algebra, we find that {0, ∞}, {0, 1/5}, {0, 1/7} are a basis. And, e.g.,

{0, 1/2} = −{0, 1/5}
{0, 1/3} = −{0, 1/7}
{0, 1/4} =  {0, 1/5} − {0, 1/7}