LMFDB - Group of Dirichlet characters of modulus 135

Show commands: PariGP / SageMath

sage: H = DirichletGroup(135)

pari: g = idealstar(,135,2)

Character group

sage: G.order() pari: g.no
Order	=	72
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{36}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{135}(56,\cdot)$, $\chi_{135}(82,\cdot)$

First 32 of 72 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	$-1$	$1$	$2$	$4$	$7$	$8$	$11$	$13$	$14$	$16$	$17$	$19$
$\chi_{135}(1,\cdot)$	135.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{135}(2,\cdot)$	135.q	36	yes	$1$	$1$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(4,\cdot)$	135.p	18	yes	$1$	$1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{135}(7,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(8,\cdot)$	135.m	12	no	$1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-i$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$i$	$-1$
$\chi_{135}(11,\cdot)$	135.o	18	no	$-1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{135}(13,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{135}(14,\cdot)$	135.n	18	yes	$-1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{135}(16,\cdot)$	135.k	9	no	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{135}(17,\cdot)$	135.m	12	no	$1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$-1$
$\chi_{135}(19,\cdot)$	135.j	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$
$\chi_{135}(22,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{135}(23,\cdot)$	135.q	36	yes	$1$	$1$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{135}(26,\cdot)$	135.c	2	no	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
$\chi_{135}(28,\cdot)$	135.g	4	no	$-1$	$1$	$-i$	$-1$	$-i$	$i$	$1$	$i$	$-1$	$1$	$-i$	$-1$
$\chi_{135}(29,\cdot)$	135.n	18	yes	$-1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{135}(31,\cdot)$	135.k	9	no	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{135}(32,\cdot)$	135.q	36	yes	$1$	$1$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{135}(34,\cdot)$	135.p	18	yes	$1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{135}(37,\cdot)$	135.l	12	no	$-1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{12}\right)$	$-i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$i$	$-1$
$\chi_{135}(38,\cdot)$	135.q	36	yes	$1$	$1$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(41,\cdot)$	135.o	18	no	$-1$	$1$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{135}(43,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(44,\cdot)$	135.h	6	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$
$\chi_{135}(46,\cdot)$	135.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$
$\chi_{135}(47,\cdot)$	135.q	36	yes	$1$	$1$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(49,\cdot)$	135.p	18	yes	$1$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{135}(52,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{135}(53,\cdot)$	135.f	4	no	$1$	$1$	$i$	$-1$	$-i$	$-i$	$-1$	$i$	$1$	$1$	$i$	$-1$
$\chi_{135}(56,\cdot)$	135.o	18	no	$-1$	$1$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{135}(58,\cdot)$	135.r	36	yes	$-1$	$1$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{135}(59,\cdot)$	135.n	18	yes	$-1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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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 $\F_{q}$

Modulus	$135$
Structure	\(C_{2}\times C_{36}\)
Order	$72$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(7\)	\(8\)	\(11\)	\(13\)	\(14\)	\(16\)	\(17\)	\(19\)
\(\chi_{135}(1,\cdot)\)	135.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{135}(2,\cdot)\)	135.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(4,\cdot)\)	135.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{135}(7,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(8,\cdot)\)	135.m	12	no	\(1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(-1\)
\(\chi_{135}(11,\cdot)\)	135.o	18	no	\(-1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{135}(13,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{135}(14,\cdot)\)	135.n	18	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{135}(16,\cdot)\)	135.k	9	no	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{135}(17,\cdot)\)	135.m	12	no	\(1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(-1\)
\(\chi_{135}(19,\cdot)\)	135.j	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)
\(\chi_{135}(22,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{135}(23,\cdot)\)	135.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{135}(26,\cdot)\)	135.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{135}(28,\cdot)\)	135.g	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(-1\)
\(\chi_{135}(29,\cdot)\)	135.n	18	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{135}(31,\cdot)\)	135.k	9	no	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{135}(32,\cdot)\)	135.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{135}(34,\cdot)\)	135.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{135}(37,\cdot)\)	135.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(-1\)
\(\chi_{135}(38,\cdot)\)	135.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(41,\cdot)\)	135.o	18	no	\(-1\)	\(1\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{135}(43,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(44,\cdot)\)	135.h	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)
\(\chi_{135}(46,\cdot)\)	135.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)
\(\chi_{135}(47,\cdot)\)	135.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(49,\cdot)\)	135.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{135}(52,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{135}(53,\cdot)\)	135.f	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(i\)	\(1\)	\(1\)	\(i\)	\(-1\)
\(\chi_{135}(56,\cdot)\)	135.o	18	no	\(-1\)	\(1\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{135}(58,\cdot)\)	135.r	36	yes	\(-1\)	\(1\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{135}(59,\cdot)\)	135.n	18	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

Introduction

L-functions

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Character group

First 32 of 72 characters

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