{ "Title": "$q$-expansion of the modular discriminant $\\Delta$", "Definition": "This list contains the coefficients of the modular discrimiant $\\Delta$ CITE{Wiki} in its $q$-expansion, $\\Delta = \\sum_{n=0}^{\\infty} \\tau(n) q^n$. The coefficients are the values of the Ramanujan tau-function.", "Parameters": { "n": { "type": "Z", "constraints": "$n \\geq 0$" } }, "Comments": { "comment-normalization": "Note that different authors use different normalization factors for $\\Delta$ CITE{Wiki}, CITE{WikiTau}. Here, we use the normalization that makes $\\Delta$ a normalized cusp form (of weight 12 and level 1)." }, "Formulas": {}, "Programs": { "program-sage": { "language": "Sage", "code": "numbers = {n: delta_qexp(101)[n] for n in [0..100]}" } }, "References": {}, "Links": { "Wiki": { "title": "Wikipedia: Modular discriminant", "url": "https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Modular_discriminant" }, "WikiTau": { "title": "Wikipedia: Ramanujan tau function", "url": "https://en.wikipedia.org/wiki/Ramanujan_tau_function" } }, "Similar tables": "", "Keywords": "", "Tags": [ "Taylor series", "modular function", "elliptic curves", "number theory" ], "Data properties": { "type": "Z", "complete": "no", "rigour": "exact", "rigour details": "Exact integer values, without numerical rounding." }, "Display properties": { "number-header": "$\\tau(n)$" }, "Numbers": { "0": "0", "1": "1", "2": "-24", "3": "252", "4": "-1472", "5": "4830", "6": "-6048", "7": "-16744", "8": "84480", 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