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Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just wonder why we care about them.

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This is a layman answer. Hecke operators commute and are self-adjoint, hence the modular forms which are eigenvectors wrt. all Hecke operators form a basis of the space of all modular forms (and the same for cusp forms). If $f$ is such an eigenvector then the L-series corresponding to $f$ has multiplicative coefficients, i.e. it can be represented by an Euler product (over all primes). So AFAIK, Hecke operators are important for connections of modular forms with number theory.

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As a very specific example, we can use Hecke operators and the theory of modular forms to quickly get Jacobi's Four Squares Formula.

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