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  "body": "\n## Definition\n\nLet $\\mathbb{H} = \\set{z = x + iy : y \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]].\nLet $\\Gamma$ be a [[congruence subgroup|1H6R856P3C]] of $\\mathrm{SL}_2(\\Z)$, acting on $\\mathbb{H}$ by $\\gamma z = \\frac{az + b}{cz + d}$ for $\\gamma = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$.\nLet $\\Delta = -y^2 \\left( \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} \\right)$ be the hyperbolic [[Laplace operator|F4NWZNTRMD]].\nA *Maass wave form* for $\\Gamma$ is a [[smooth function|0V33BNKZ9S]] $f \\colon \\mathbb{H} \\to \\C$ such that:\n\n1. $f(\\gamma z) = f(z)$ for all $\\gamma \\in \\Gamma$ and $z \\in \\mathbb{H}$.\n2. There is $\\lambda \\in \\C$ with $\\Delta f = \\lambda f$.\n3. $f$ has moderate growth at every cusp: for every $\\sigma \\in \\mathrm{SL}_2(\\Z)$ there is $N \u003e 0$ with $f(\\sigma z) = O(y^N)$ as $y \\to \\infty$, uniformly in $x$.\n\nThe number $\\lambda$ is the [[eigenvalue|0BQAA2BK3Q]] of $f$.\nA *Maass cusp form* is a Maass form whose constant term $\\int_0^1 f(\\sigma(x + iy)) \\, dx$ vanishes at every cusp, that is, for every $\\sigma \\in \\mathrm{SL}_2(\\Z)$.\n\n## Remarks\n\nExample: for $\\Gamma = \\mathrm{SL}_2(\\Z)$ and $\\mathrm{Re}(s) \u003e 1$, the real-analytic Eisenstein series\n$$E(z, s) = \\frac{1}{2} \\sum_{\\substack{(c, d) \\in \\Z^2 \\\\ \\gcd(c, d) = 1}} \\frac{y^s}{\\abs{cz + d}^{2s}}$$\nis a Maass form with eigenvalue $s(1 - s)$.\nEach term is $(\\mathrm{Im}\\, \\gamma z)^s$ for a matrix $\\gamma$ with bottom row $(c, d)$, so the sum is $\\Gamma$-invariant, and $\\Delta y^s = -y^2 s(s-1) y^{s-2} = s(1-s) y^s$ term by term.\nIt is not a cusp form, since its constant term is $y^s + \\varphi(s) y^{1-s}$ for some function $\\varphi$.\n\nNon-example: $f(z) = y^s$ satisfies clauses 2 and 3 and is invariant under $z \\mapsto z + 1$, but it is not invariant under $z \\mapsto -1/z$, so clause 1 fails.\n\nA cusp form decays rapidly at the cusps and is square-integrable on $\\Gamma \\backslash \\mathbb{H}$; the eigenvalue of a nonzero Maass cusp form is then real and positive.\nOne often writes $\\lambda = \\frac{1}{4} + r^2$ or $\\lambda = s(1 - s)$.\nSome authors also allow a weight or a character in clause 1.\n",
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