LMFDB - Group of Dirichlet characters of modulus 513

sage:H = DirichletGroup(513)

pari:g = idealstar(,513,2)

Character group

sage:G.order() pari:g.no
Order	=	324
sage:H.invariants() pari:g.cyc
Structure	=	$C_{18}\times C_{18}$
sage:H.gens() pari:g.gen
Generators	=	$\chi_{513}(191,\cdot)$, $\chi_{513}(325,\cdot)$

First 32 of 324 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{513}(1,\cdot)$	513.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{513}(2,\cdot)$	513.cg	18	yes	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{513}(4,\cdot)$	513.w	9	yes	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{513}(5,\cdot)$	513.cm	18	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{7}{9}\right)$	$-1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{513}(7,\cdot)$	513.bc	9	yes	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{513}(8,\cdot)$	513.k	6	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(10,\cdot)$	513.ce	18	no	$-1$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{513}(11,\cdot)$	513.by	18	yes	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{513}(13,\cdot)$	513.bm	18	yes	$-1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{17}{18}\right)$	$-1$	$e\left(\frac{8}{9}\right)$
$\chi_{513}(14,\cdot)$	513.bi	18	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{4}{9}\right)$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{17}{18}\right)$	$-1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(16,\cdot)$	513.w	9	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{513}(17,\cdot)$	513.bq	18	no	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{4}{9}\right)$	$-1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{513}(20,\cdot)$	513.bz	18	no	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{513}(22,\cdot)$	513.bh	18	yes	$-1$	$1$	$-1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{9}\right)$	$-1$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{18}\right)$	$1$
$\chi_{513}(23,\cdot)$	513.cj	18	yes	$-1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{513}(25,\cdot)$	513.u	9	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{9}\right)$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(26,\cdot)$	513.r	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(28,\cdot)$	513.y	9	no	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{513}(29,\cdot)$	513.cp	18	yes	$1$	$1$	$1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{5}{9}\right)$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{9}\right)$	$1$
$\chi_{513}(31,\cdot)$	513.cb	18	yes	$-1$	$1$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{513}(32,\cdot)$	513.cg	18	yes	$1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{513}(34,\cdot)$	513.bh	18	yes	$-1$	$1$	$-1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{8}{9}\right)$	$-1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{18}\right)$	$1$
$\chi_{513}(35,\cdot)$	513.cf	18	no	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{18}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{513}(37,\cdot)$	513.o	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(40,\cdot)$	513.bh	18	yes	$-1$	$1$	$-1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{4}{9}\right)$	$-1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{17}{18}\right)$	$1$
$\chi_{513}(41,\cdot)$	513.bj	18	yes	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{4}{9}\right)$	$1$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{513}(43,\cdot)$	513.x	9	yes	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{9}\right)$	$1$	$e\left(\frac{4}{9}\right)$
$\chi_{513}(44,\cdot)$	513.cf	18	no	$-1$	$1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{11}{18}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{513}(46,\cdot)$	513.s	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(47,\cdot)$	513.cm	18	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{8}{9}\right)$	$-1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{513}(49,\cdot)$	513.bc	9	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{513}(50,\cdot)$	513.bv	18	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Modulus	$513$
Structure	\(C_{18}\times C_{18}\)
Order	$324$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{513}(1,\cdot)\)	513.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{513}(2,\cdot)\)	513.cg	18	yes	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{513}(4,\cdot)\)	513.w	9	yes	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{513}(5,\cdot)\)	513.cm	18	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(-1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{513}(7,\cdot)\)	513.bc	9	yes	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{513}(8,\cdot)\)	513.k	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(10,\cdot)\)	513.ce	18	no	\(-1\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{513}(11,\cdot)\)	513.by	18	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{513}(13,\cdot)\)	513.bm	18	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{513}(14,\cdot)\)	513.bi	18	yes	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(16,\cdot)\)	513.w	9	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{513}(17,\cdot)\)	513.bq	18	no	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(-1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{513}(20,\cdot)\)	513.bz	18	no	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{513}(22,\cdot)\)	513.bh	18	yes	\(-1\)	\(1\)	\(-1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(-1\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(1\)
\(\chi_{513}(23,\cdot)\)	513.cj	18	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{513}(25,\cdot)\)	513.u	9	yes	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(26,\cdot)\)	513.r	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(28,\cdot)\)	513.y	9	no	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{513}(29,\cdot)\)	513.cp	18	yes	\(1\)	\(1\)	\(1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(1\)
\(\chi_{513}(31,\cdot)\)	513.cb	18	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{513}(32,\cdot)\)	513.cg	18	yes	\(1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{513}(34,\cdot)\)	513.bh	18	yes	\(-1\)	\(1\)	\(-1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(1\)
\(\chi_{513}(35,\cdot)\)	513.cf	18	no	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{513}(37,\cdot)\)	513.o	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(40,\cdot)\)	513.bh	18	yes	\(-1\)	\(1\)	\(-1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(-1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(1\)
\(\chi_{513}(41,\cdot)\)	513.bj	18	yes	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(1\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{513}(43,\cdot)\)	513.x	9	yes	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(1\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{513}(44,\cdot)\)	513.cf	18	no	\(-1\)	\(1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{513}(46,\cdot)\)	513.s	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(47,\cdot)\)	513.cm	18	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{513}(49,\cdot)\)	513.bc	9	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{513}(50,\cdot)\)	513.bv	18	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.