LMFDB - Group of Dirichlet characters of modulus 218

Show commands: PariGP / SageMath

sage: H = DirichletGroup(218)

pari: g = idealstar(,218,2)

Character group

sage: G.order() pari: g.no
Order	=	108
sage: H.invariants() pari: g.cyc
Structure	=	$C_{108}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{218}(115,\cdot)$

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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{218}(1,\cdot)$	218.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{218}(3,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{27}\right)$
$\chi_{218}(5,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{20}{27}\right)$
$\chi_{218}(7,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{27}\right)$
$\chi_{218}(9,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{16}{27}\right)$
$\chi_{218}(11,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{85}{108}\right)$	$e\left(\frac{53}{108}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{19}{27}\right)$
$\chi_{218}(13,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{53}{108}\right)$	$e\left(\frac{61}{108}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{2}{27}\right)$
$\chi_{218}(15,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{27}\right)$
$\chi_{218}(17,\cdot)$	218.j	36	no	$-1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{218}(19,\cdot)$	218.j	36	no	$-1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{218}(21,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{10}{27}\right)$
$\chi_{218}(23,\cdot)$	218.j	36	no	$-1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{218}(25,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{13}{27}\right)$
$\chi_{218}(27,\cdot)$	218.f	9	no	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{218}(29,\cdot)$	218.k	54	no	$1$	$1$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{7}{54}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{26}{27}\right)$
$\chi_{218}(31,\cdot)$	218.k	54	no	$1$	$1$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{25}{27}\right)$
$\chi_{218}(33,\cdot)$	218.d	4	no	$-1$	$1$	$1$	$1$	$1$	$1$	$-i$	$-i$	$1$	$i$	$-i$	$1$
$\chi_{218}(35,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{22}{27}\right)$
$\chi_{218}(37,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{91}{108}\right)$	$e\left(\frac{11}{108}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{7}{27}\right)$
$\chi_{218}(39,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{49}{108}\right)$	$e\left(\frac{89}{108}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{10}{27}\right)$
$\chi_{218}(41,\cdot)$	218.g	12	no	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$i$	$e\left(\frac{1}{3}\right)$
$\chi_{218}(43,\cdot)$	218.h	18	no	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{218}(45,\cdot)$	218.c	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{218}(47,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{7}{108}\right)$	$e\left(\frac{59}{108}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{13}{27}\right)$
$\chi_{218}(49,\cdot)$	218.i	27	no	$1$	$1$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{4}{27}\right)$
$\chi_{218}(51,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{47}{108}\right)$	$e\left(\frac{103}{108}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{14}{27}\right)$
$\chi_{218}(53,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{71}{108}\right)$	$e\left(\frac{43}{108}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{20}{27}\right)$
$\chi_{218}(55,\cdot)$	218.j	36	no	$-1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{218}(57,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{65}{108}\right)$	$e\left(\frac{85}{108}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{5}{27}\right)$
$\chi_{218}(59,\cdot)$	218.l	108	no	$-1$	$1$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{13}{108}\right)$	$e\left(\frac{17}{108}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{1}{27}\right)$
$\chi_{218}(61,\cdot)$	218.k	54	no	$1$	$1$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{14}{27}\right)$
$\chi_{218}(63,\cdot)$	218.c	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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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	$218$
Structure	\(C_{108}\)
Order	$108$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{218}(1,\cdot)\)	218.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{218}(3,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{27}\right)\)
\(\chi_{218}(5,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{20}{27}\right)\)
\(\chi_{218}(7,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{27}\right)\)
\(\chi_{218}(9,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{16}{27}\right)\)
\(\chi_{218}(11,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{85}{108}\right)\)	\(e\left(\frac{53}{108}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{19}{27}\right)\)
\(\chi_{218}(13,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{53}{108}\right)\)	\(e\left(\frac{61}{108}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{2}{27}\right)\)
\(\chi_{218}(15,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{27}\right)\)
\(\chi_{218}(17,\cdot)\)	218.j	36	no	\(-1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{218}(19,\cdot)\)	218.j	36	no	\(-1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{218}(21,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{10}{27}\right)\)
\(\chi_{218}(23,\cdot)\)	218.j	36	no	\(-1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{218}(25,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{13}{27}\right)\)
\(\chi_{218}(27,\cdot)\)	218.f	9	no	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{218}(29,\cdot)\)	218.k	54	no	\(1\)	\(1\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{7}{54}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{26}{27}\right)\)
\(\chi_{218}(31,\cdot)\)	218.k	54	no	\(1\)	\(1\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{25}{27}\right)\)
\(\chi_{218}(33,\cdot)\)	218.d	4	no	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-i\)	\(-i\)	\(1\)	\(i\)	\(-i\)	\(1\)
\(\chi_{218}(35,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{22}{27}\right)\)
\(\chi_{218}(37,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{91}{108}\right)\)	\(e\left(\frac{11}{108}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{7}{27}\right)\)
\(\chi_{218}(39,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{49}{108}\right)\)	\(e\left(\frac{89}{108}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{10}{27}\right)\)
\(\chi_{218}(41,\cdot)\)	218.g	12	no	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{218}(43,\cdot)\)	218.h	18	no	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{218}(45,\cdot)\)	218.c	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{218}(47,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{7}{108}\right)\)	\(e\left(\frac{59}{108}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{13}{27}\right)\)
\(\chi_{218}(49,\cdot)\)	218.i	27	no	\(1\)	\(1\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{4}{27}\right)\)
\(\chi_{218}(51,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{47}{108}\right)\)	\(e\left(\frac{103}{108}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{14}{27}\right)\)
\(\chi_{218}(53,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{71}{108}\right)\)	\(e\left(\frac{43}{108}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{20}{27}\right)\)
\(\chi_{218}(55,\cdot)\)	218.j	36	no	\(-1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{218}(57,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{65}{108}\right)\)	\(e\left(\frac{85}{108}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{5}{27}\right)\)
\(\chi_{218}(59,\cdot)\)	218.l	108	no	\(-1\)	\(1\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{13}{108}\right)\)	\(e\left(\frac{17}{108}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{1}{27}\right)\)
\(\chi_{218}(61,\cdot)\)	218.k	54	no	\(1\)	\(1\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{14}{27}\right)\)
\(\chi_{218}(63,\cdot)\)	218.c	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Introduction

L-functions

Modular forms

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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.