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  "body": "\n## Summary\n\nThe Petersson inner product is an [[inner product|0NB0AWGJBW]] on the space $S_k$ of [[cusp forms|A50Y012RBP]] of weight $k$.\nIt integrates $f \\overline{g}\\, y^k$ against the hyperbolic area over a fundamental domain of the [[modular group|VEKV1EM77B]].\nThe [[Hecke operators|S4SV3QZ8B1]] are self-adjoint for it, so $S_k$ has a basis of common eigenforms.\nThe [[Eisenstein series|EYREVY31BN]] are orthogonal to the cusp forms in a suitable sense.\n\n## Explanations\n\n### L1 human\n\nTo speak of angles and lengths in a space of functions, one needs an inner product.\nFor cusp forms the natural one multiplies $f$ by the conjugate of $g$, adds the weight factor $y^k$, and integrates over one copy of the quotient of the upper half-plane.\nThe weight factor makes the integrand invariant, so the choice of copy does not matter.\nWith this inner product, the Hecke operators behave like symmetric matrices, and so they can be diagonalized together.\n",
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