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Fungrim entry: e46697

zerosτHE2k ⁣(τ)={γτ:τzeroszFE2k ⁣(z)andγPSL2(Z)}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}
Assumptions:kZ2k \in \mathbb{Z}_{\ge 2}
TeX:
\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}

k \in \mathbb{Z}_{\ge 2}
Definitions:
Fungrim symbol Notation Short description
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
HHH\mathbb{H} Upper complex half-plane
ModularGroupActionγτ\gamma \circ \tau Action of modular group
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
PSL2ZPSL2(Z)\operatorname{PSL}_2(\mathbb{Z}) Modular group (canonical representatives)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("e46697"),
    Formula(Equal(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, HH)), Set(ModularGroupAction(gamma, tau), For(Tuple(gamma, tau)), And(Element(tau, Zeros(EisensteinE(Mul(2, k), z), ForElement(z, ModularGroupFundamentalDomain))), Element(gamma, PSL2Z))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))))

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2019-11-19 15:10:20.037976 UTC