LMFDB - Group of Dirichlet characters of modulus 324

Show commands: PariGP / SageMath

sage: H = DirichletGroup(324)

pari: g = idealstar(,324,2)

Character group

sage: G.order() pari: g.no
Order	=	108
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{54}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{324}(163,\cdot)$, $\chi_{324}(245,\cdot)$

First 32 of 108 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	$-1$	$1$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$25$	$29$	$31$
$\chi_{324}(1,\cdot)$	324.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{324}(5,\cdot)$	324.o	54	no	$-1$	$1$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{14}{27}\right)$
$\chi_{324}(7,\cdot)$	324.n	54	yes	$-1$	$1$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{19}{54}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{23}{54}\right)$
$\chi_{324}(11,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{19}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{17}{54}\right)$
$\chi_{324}(13,\cdot)$	324.m	27	no	$1$	$1$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{26}{27}\right)$
$\chi_{324}(17,\cdot)$	324.k	18	no	$-1$	$1$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{324}(19,\cdot)$	324.j	18	no	$-1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{18}\right)$
$\chi_{324}(23,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{31}{54}\right)$
$\chi_{324}(25,\cdot)$	324.m	27	no	$1$	$1$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{1}{27}\right)$
$\chi_{324}(29,\cdot)$	324.o	54	no	$-1$	$1$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{19}{54}\right)$	$e\left(\frac{19}{27}\right)$
$\chi_{324}(31,\cdot)$	324.n	54	yes	$-1$	$1$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{49}{54}\right)$
$\chi_{324}(35,\cdot)$	324.l	18	no	$1$	$1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{17}{18}\right)$
$\chi_{324}(37,\cdot)$	324.i	9	no	$1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{324}(41,\cdot)$	324.o	54	no	$-1$	$1$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{17}{27}\right)$
$\chi_{324}(43,\cdot)$	324.n	54	yes	$-1$	$1$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{1}{54}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{35}{54}\right)$
$\chi_{324}(47,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{5}{54}\right)$
$\chi_{324}(49,\cdot)$	324.m	27	no	$1$	$1$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{23}{27}\right)$
$\chi_{324}(53,\cdot)$	324.g	6	no	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{324}(55,\cdot)$	324.f	6	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{324}(59,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{37}{54}\right)$
$\chi_{324}(61,\cdot)$	324.m	27	no	$1$	$1$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{7}{27}\right)$
$\chi_{324}(65,\cdot)$	324.o	54	no	$-1$	$1$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{13}{27}\right)$
$\chi_{324}(67,\cdot)$	324.n	54	yes	$-1$	$1$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{43}{54}\right)$
$\chi_{324}(71,\cdot)$	324.l	18	no	$1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{7}{18}\right)$
$\chi_{324}(73,\cdot)$	324.i	9	no	$1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{324}(77,\cdot)$	324.o	54	no	$-1$	$1$	$e\left(\frac{19}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{47}{54}\right)$	$e\left(\frac{20}{27}\right)$
$\chi_{324}(79,\cdot)$	324.n	54	yes	$-1$	$1$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{47}{54}\right)$
$\chi_{324}(83,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{43}{54}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{47}{54}\right)$
$\chi_{324}(85,\cdot)$	324.m	27	no	$1$	$1$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{20}{27}\right)$
$\chi_{324}(89,\cdot)$	324.k	18	no	$-1$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{324}(91,\cdot)$	324.j	18	no	$-1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{18}\right)$
$\chi_{324}(95,\cdot)$	324.p	54	yes	$1$	$1$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{29}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{43}{54}\right)$

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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	$324$
Structure	\(C_{2}\times C_{54}\)
Order	$108$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(5\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(25\)	\(29\)	\(31\)
\(\chi_{324}(1,\cdot)\)	324.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{324}(5,\cdot)\)	324.o	54	no	\(-1\)	\(1\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)
\(\chi_{324}(7,\cdot)\)	324.n	54	yes	\(-1\)	\(1\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{19}{54}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{23}{54}\right)\)
\(\chi_{324}(11,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{19}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{17}{54}\right)\)
\(\chi_{324}(13,\cdot)\)	324.m	27	no	\(1\)	\(1\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)
\(\chi_{324}(17,\cdot)\)	324.k	18	no	\(-1\)	\(1\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{324}(19,\cdot)\)	324.j	18	no	\(-1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)
\(\chi_{324}(23,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{31}{54}\right)\)
\(\chi_{324}(25,\cdot)\)	324.m	27	no	\(1\)	\(1\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)
\(\chi_{324}(29,\cdot)\)	324.o	54	no	\(-1\)	\(1\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{19}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)
\(\chi_{324}(31,\cdot)\)	324.n	54	yes	\(-1\)	\(1\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{49}{54}\right)\)
\(\chi_{324}(35,\cdot)\)	324.l	18	no	\(1\)	\(1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)
\(\chi_{324}(37,\cdot)\)	324.i	9	no	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{324}(41,\cdot)\)	324.o	54	no	\(-1\)	\(1\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)
\(\chi_{324}(43,\cdot)\)	324.n	54	yes	\(-1\)	\(1\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{1}{54}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{35}{54}\right)\)
\(\chi_{324}(47,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{5}{54}\right)\)
\(\chi_{324}(49,\cdot)\)	324.m	27	no	\(1\)	\(1\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)
\(\chi_{324}(53,\cdot)\)	324.g	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{324}(55,\cdot)\)	324.f	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{324}(59,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{37}{54}\right)\)
\(\chi_{324}(61,\cdot)\)	324.m	27	no	\(1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)
\(\chi_{324}(65,\cdot)\)	324.o	54	no	\(-1\)	\(1\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)
\(\chi_{324}(67,\cdot)\)	324.n	54	yes	\(-1\)	\(1\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{43}{54}\right)\)
\(\chi_{324}(71,\cdot)\)	324.l	18	no	\(1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{7}{18}\right)\)
\(\chi_{324}(73,\cdot)\)	324.i	9	no	\(1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{324}(77,\cdot)\)	324.o	54	no	\(-1\)	\(1\)	\(e\left(\frac{19}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{47}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)
\(\chi_{324}(79,\cdot)\)	324.n	54	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{47}{54}\right)\)
\(\chi_{324}(83,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{43}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{47}{54}\right)\)
\(\chi_{324}(85,\cdot)\)	324.m	27	no	\(1\)	\(1\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)
\(\chi_{324}(89,\cdot)\)	324.k	18	no	\(-1\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{324}(91,\cdot)\)	324.j	18	no	\(-1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)
\(\chi_{324}(95,\cdot)\)	324.p	54	yes	\(1\)	\(1\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{29}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{43}{54}\right)\)

Introduction

L-functions

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Character group

First 32 of 108 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.