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  "body": "\n## Statement\n\nLet $N \\ge 1$ and $k \\ge 2$ be integers.\nLet $f = \\sum_{n \\ge 1} a_n q^n$ be a [[cusp form|A50Y012RBP]] of weight $k$ for the [[congruence subgroup|1H6R856P3C]] $\\Gamma_0(N)$, with $q = e^{2 \\pi i \\tau}$.\nAssume that $a_1 = 1$ and that $f$ is an eigenvector of the [[Hecke operator|S4SV3QZ8B1]] $T_p$ for every [[prime|01826KMG2H]] $p$ not dividing $N$.\nThen for every prime $p$ not dividing $N$,\n$$\\abs{a_p} \\le 2 p^{(k-1)/2}.$$\nIn particular, the [[Ramanujan tau function|6DNZAGCSQS]] satisfies $\\abs{\\tau(p)} \\le 2 p^{11/2}$ for every prime $p$.\n\n## Remarks\n\nRamanujan conjectured the bound for $\\tau(p)$ in 1916, and Petersson extended the conjecture to other modular forms.\nDeligne proved it in 1974 as a consequence of his proof of the Weil conjectures.\nEarlier work of Eichler, Shimura and Deligne had identified $a_p$ with the trace of Frobenius on an étale cohomology group, whose eigenvalues have absolute value $p^{(k-1)/2}$ by the Weil conjectures.\nExample: $\\tau(2) = -24$ and $2 \\cdot 2^{11/2} = 2^{13/2} \\approx 90.5$; $\\tau(3) = 252$ and $2 \\cdot 3^{11/2} \\approx 841.8$.\nThe normalization $a_1 = 1$ and the eigenform hypothesis are needed: the cusp form $100\\Delta$ has $a_2 = -2400$, which breaks the bound.\nThe analogous statement for Maass forms is still open.\n\n## Explanations\n\n### L1 human\n\nThe coefficients of a Hecke eigenform look random, but they cannot be too large.\nAt a prime $p$, the coefficient is at most about $2 p^{(k-1)/2}$, roughly the square root of the trivial size $p^{k-1}$.\nFor Ramanujan's function this says $\\abs{\\tau(p)} \\le 2 p^{11/2}$.\nIt took almost sixty years, and a revolution in algebraic geometry, to prove it.\n\n### L2 human\n\nFor a normalized eigenform, the Euler factor at $p \\nmid N$ is $1 - a_p p^{-s} + p^{k-1-2s}$.\nWriting it as $(1 - \\alpha_p p^{-s})(1 - \\beta_p p^{-s})$, the bound is equivalent to $\\abs{\\alpha_p} = \\abs{\\beta_p} = p^{(k-1)/2}$.\nSo the theorem is an analogue of the Riemann hypothesis for the local factors of the L-function of $f$.\nTogether with multiplicativity, it gives $\\abs{a_n} \\le d(n)\\, n^{(k-1)/2}$ for $n$ prime to $N$, where $d(n)$ is the number of divisors of $n$.\nThe [[Sato–Tate conjecture|TRGBJYPBXH]] describes how the normalized values $a_p / (2 p^{(k-1)/2}) \\in [-1, 1]$ are distributed.\n",
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