LMFDB - Group of Dirichlet characters of modulus 86

Show commands: PariGP / SageMath

sage: H = DirichletGroup(86)

pari: g = idealstar(,86,2)

Character group

sage: G.order() pari: g.no
Order	=	42
sage: H.invariants() pari: g.cyc
Structure	=	$C_{42}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{86}(3,\cdot)$

First 32 of 42 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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{86}(1,\cdot)$	86.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{86}(3,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{6}{7}\right)$
$\chi_{86}(5,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{3}{7}\right)$
$\chi_{86}(7,\cdot)$	86.d	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$
$\chi_{86}(9,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{5}{7}\right)$
$\chi_{86}(11,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{6}{7}\right)$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{5}{7}\right)$
$\chi_{86}(13,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{3}{7}\right)$
$\chi_{86}(15,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{2}{7}\right)$
$\chi_{86}(17,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{86}(19,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{2}{7}\right)$
$\chi_{86}(21,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{7}\right)$	$1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$
$\chi_{86}(23,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{5}{7}\right)$
$\chi_{86}(25,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{6}{7}\right)$
$\chi_{86}(27,\cdot)$	86.f	14	no	$-1$	$1$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{11}{14}\right)$	$-1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{86}(29,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(31,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(33,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{86}(35,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{7}\right)$	$1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{7}\right)$
$\chi_{86}(37,\cdot)$	86.d	6	no	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$
$\chi_{86}(39,\cdot)$	86.f	14	no	$-1$	$1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{9}{14}\right)$	$-1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{2}{7}\right)$
$\chi_{86}(41,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{4}{7}\right)$	$1$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(45,\cdot)$	86.f	14	no	$-1$	$1$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{1}{14}\right)$	$-1$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(47,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{1}{7}\right)$	$1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{7}\right)$
$\chi_{86}(49,\cdot)$	86.c	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$
$\chi_{86}(51,\cdot)$	86.f	14	no	$-1$	$1$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{3}{14}\right)$	$-1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{3}{7}\right)$
$\chi_{86}(53,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{86}(55,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(57,\cdot)$	86.g	21	no	$1$	$1$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{86}(59,\cdot)$	86.e	7	no	$1$	$1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{2}{7}\right)$	$1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{86}(61,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{6}{7}\right)$
$\chi_{86}(63,\cdot)$	86.h	42	no	$-1$	$1$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{5}{7}\right)$
$\chi_{86}(65,\cdot)$	86.f	14	no	$-1$	$1$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{13}{14}\right)$	$-1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{6}{7}\right)$

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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	$86$
Structure	\(C_{42}\)
Order	$42$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{86}(1,\cdot)\)	86.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{86}(3,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(5,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(7,\cdot)\)	86.d	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)
\(\chi_{86}(9,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(11,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(13,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(15,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(17,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(19,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(21,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(23,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(25,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(27,\cdot)\)	86.f	14	no	\(-1\)	\(1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(-1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(29,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(31,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(33,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(35,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(37,\cdot)\)	86.d	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)
\(\chi_{86}(39,\cdot)\)	86.f	14	no	\(-1\)	\(1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(41,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(1\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(45,\cdot)\)	86.f	14	no	\(-1\)	\(1\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(-1\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(47,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(49,\cdot)\)	86.c	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)
\(\chi_{86}(51,\cdot)\)	86.f	14	no	\(-1\)	\(1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(-1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(53,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(55,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(57,\cdot)\)	86.g	21	no	\(1\)	\(1\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(59,\cdot)\)	86.e	7	no	\(1\)	\(1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(61,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(63,\cdot)\)	86.h	42	no	\(-1\)	\(1\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(65,\cdot)\)	86.f	14	no	\(-1\)	\(1\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(-1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)

Introduction

L-functions

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Character group

First 32 of 42 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.