LMFDB - Group of Dirichlet characters of modulus 81

Show commands: PariGP / SageMath

sage: H = DirichletGroup(81)

pari: g = idealstar(,81,2)

Character group

sage: G.order() pari: g.no
Order	=	54
sage: H.invariants() pari: g.cyc
Structure	=	$C_{54}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{81}(2,\cdot)$

First 32 of 54 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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{81}(1,\cdot)$	81.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{81}(2,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{1}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{2}{27}\right)$
$\chi_{81}(4,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{4}{27}\right)$
$\chi_{81}(5,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{19}{27}\right)$
$\chi_{81}(7,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{5}{27}\right)$
$\chi_{81}(8,\cdot)$	81.f	18	no	$-1$	$1$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{81}(10,\cdot)$	81.e	9	no	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{81}(11,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{54}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{26}{27}\right)$
$\chi_{81}(13,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{16}{27}\right)$
$\chi_{81}(14,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{19}{54}\right)$	$e\left(\frac{7}{27}\right)$
$\chi_{81}(16,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{8}{27}\right)$
$\chi_{81}(17,\cdot)$	81.f	18	no	$-1$	$1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{81}(19,\cdot)$	81.e	9	no	$1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{81}(20,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{54}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{47}{54}\right)$	$e\left(\frac{23}{27}\right)$
$\chi_{81}(22,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{1}{27}\right)$
$\chi_{81}(23,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{22}{27}\right)$
$\chi_{81}(25,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{11}{27}\right)$
$\chi_{81}(26,\cdot)$	81.d	6	no	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{81}(28,\cdot)$	81.c	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{81}(29,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{20}{27}\right)$
$\chi_{81}(31,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{13}{27}\right)$
$\chi_{81}(32,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{7}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{10}{27}\right)$
$\chi_{81}(34,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{14}{27}\right)$
$\chi_{81}(35,\cdot)$	81.f	18	no	$-1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{81}(37,\cdot)$	81.e	9	no	$1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{81}(38,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{47}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{17}{27}\right)$
$\chi_{81}(40,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{25}{27}\right)$
$\chi_{81}(41,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{25}{27}\right)$
$\chi_{81}(43,\cdot)$	81.g	27	yes	$1$	$1$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{17}{27}\right)$
$\chi_{81}(44,\cdot)$	81.f	18	no	$-1$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{81}(46,\cdot)$	81.e	9	no	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{81}(47,\cdot)$	81.h	54	yes	$-1$	$1$	$e\left(\frac{7}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{14}{27}\right)$

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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	$81$
Structure	\(C_{54}\)
Order	$54$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{81}(1,\cdot)\)	81.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{81}(2,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{2}{27}\right)\)
\(\chi_{81}(4,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)
\(\chi_{81}(5,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)
\(\chi_{81}(7,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)
\(\chi_{81}(8,\cdot)\)	81.f	18	no	\(-1\)	\(1\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{81}(10,\cdot)\)	81.e	9	no	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{81}(11,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{54}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{26}{27}\right)\)
\(\chi_{81}(13,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)
\(\chi_{81}(14,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{19}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)
\(\chi_{81}(16,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)
\(\chi_{81}(17,\cdot)\)	81.f	18	no	\(-1\)	\(1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{81}(19,\cdot)\)	81.e	9	no	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{81}(20,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{47}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)
\(\chi_{81}(22,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)
\(\chi_{81}(23,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{22}{27}\right)\)
\(\chi_{81}(25,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)
\(\chi_{81}(26,\cdot)\)	81.d	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{81}(28,\cdot)\)	81.c	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{81}(29,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)
\(\chi_{81}(31,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)
\(\chi_{81}(32,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{7}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{10}{27}\right)\)
\(\chi_{81}(34,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)
\(\chi_{81}(35,\cdot)\)	81.f	18	no	\(-1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{81}(37,\cdot)\)	81.e	9	no	\(1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{81}(38,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{47}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)
\(\chi_{81}(40,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)
\(\chi_{81}(41,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{25}{27}\right)\)
\(\chi_{81}(43,\cdot)\)	81.g	27	yes	\(1\)	\(1\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)
\(\chi_{81}(44,\cdot)\)	81.f	18	no	\(-1\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{81}(46,\cdot)\)	81.e	9	no	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{81}(47,\cdot)\)	81.h	54	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)

Introduction

L-functions

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Character group

First 32 of 54 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.