LMFDB - Group of Dirichlet characters of modulus 89

Show commands: PariGP / SageMath

sage: H = DirichletGroup(89)

pari: g = idealstar(,89,2)

Character group

sage: G.order() pari: g.no
Order	=	88
sage: H.invariants() pari: g.cyc
Structure	=	$C_{88}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{89}(3,\cdot)$

First 32 of 88 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{89}(1,\cdot)$	89.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{89}(2,\cdot)$	89.e	11	yes	$1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{3}{11}\right)$
$\chi_{89}(3,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{88}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{17}{88}\right)$	$e\left(\frac{81}{88}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{21}{22}\right)$
$\chi_{89}(4,\cdot)$	89.e	11	yes	$1$	$1$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{6}{11}\right)$
$\chi_{89}(5,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{9}{11}\right)$
$\chi_{89}(6,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{17}{88}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{25}{88}\right)$	$e\left(\frac{57}{88}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{5}{22}\right)$
$\chi_{89}(7,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{81}{88}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{57}{88}\right)$	$e\left(\frac{49}{88}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{7}{22}\right)$
$\chi_{89}(8,\cdot)$	89.e	11	yes	$1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{9}{11}\right)$
$\chi_{89}(9,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{10}{11}\right)$
$\chi_{89}(10,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{1}{11}\right)$
$\chi_{89}(11,\cdot)$	89.f	22	yes	$1$	$1$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{2}{11}\right)$
$\chi_{89}(12,\cdot)$	89.d	8	yes	$-1$	$1$	$1$	$e\left(\frac{3}{8}\right)$	$1$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$-i$	$i$	$-1$
$\chi_{89}(13,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{23}{88}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{39}{88}\right)$	$e\left(\frac{15}{88}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{21}{22}\right)$
$\chi_{89}(14,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{9}{88}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{65}{88}\right)$	$e\left(\frac{25}{88}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{89}(15,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{71}{88}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{63}{88}\right)$	$e\left(\frac{31}{88}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{17}{22}\right)$
$\chi_{89}(16,\cdot)$	89.e	11	yes	$1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{1}{11}\right)$
$\chi_{89}(17,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{8}{11}\right)$
$\chi_{89}(18,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{2}{11}\right)$
$\chi_{89}(19,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{67}{88}\right)$	$e\left(\frac{19}{88}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{9}{22}\right)$
$\chi_{89}(20,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{31}{44}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{4}{11}\right)$
$\chi_{89}(21,\cdot)$	89.g	44	yes	$1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{3}{11}\right)$
$\chi_{89}(22,\cdot)$	89.f	22	yes	$1$	$1$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{5}{11}\right)$
$\chi_{89}(23,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{57}{88}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{1}{88}\right)$	$e\left(\frac{41}{88}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{31}{44}\right)$	$e\left(\frac{9}{22}\right)$
$\chi_{89}(24,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{49}{88}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{41}{88}\right)$	$e\left(\frac{9}{88}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{5}{44}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{17}{22}\right)$
$\chi_{89}(25,\cdot)$	89.f	22	yes	$1$	$1$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{7}{11}\right)$
$\chi_{89}(26,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{39}{88}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{47}{88}\right)$	$e\left(\frac{79}{88}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{5}{44}\right)$	$e\left(\frac{5}{22}\right)$
$\chi_{89}(27,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{3}{88}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{51}{88}\right)$	$e\left(\frac{67}{88}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{89}(28,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{25}{88}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{73}{88}\right)$	$e\left(\frac{1}{88}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{89}(29,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{59}{88}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{27}{88}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{7}{22}\right)$
$\chi_{89}(30,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{87}{88}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{71}{88}\right)$	$e\left(\frac{7}{88}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{1}{22}\right)$
$\chi_{89}(31,\cdot)$	89.h	88	yes	$-1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{31}{88}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{87}{88}\right)$	$e\left(\frac{47}{88}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{31}{44}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{89}(32,\cdot)$	89.e	11	yes	$1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{4}{11}\right)$

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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	$89$
Structure	\(C_{88}\)
Order	$88$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{89}(1,\cdot)\)	89.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{89}(2,\cdot)\)	89.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)
\(\chi_{89}(3,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{88}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{17}{88}\right)\)	\(e\left(\frac{81}{88}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)
\(\chi_{89}(4,\cdot)\)	89.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)
\(\chi_{89}(5,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)
\(\chi_{89}(6,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{17}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{25}{88}\right)\)	\(e\left(\frac{57}{88}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{5}{22}\right)\)
\(\chi_{89}(7,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{81}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{57}{88}\right)\)	\(e\left(\frac{49}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)
\(\chi_{89}(8,\cdot)\)	89.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)
\(\chi_{89}(9,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)
\(\chi_{89}(10,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{1}{11}\right)\)
\(\chi_{89}(11,\cdot)\)	89.f	22	yes	\(1\)	\(1\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)
\(\chi_{89}(12,\cdot)\)	89.d	8	yes	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(-i\)	\(i\)	\(-1\)
\(\chi_{89}(13,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{23}{88}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{39}{88}\right)\)	\(e\left(\frac{15}{88}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)
\(\chi_{89}(14,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{9}{88}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{65}{88}\right)\)	\(e\left(\frac{25}{88}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{89}(15,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{71}{88}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{63}{88}\right)\)	\(e\left(\frac{31}{88}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)
\(\chi_{89}(16,\cdot)\)	89.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)
\(\chi_{89}(17,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)
\(\chi_{89}(18,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)
\(\chi_{89}(19,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{67}{88}\right)\)	\(e\left(\frac{19}{88}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{9}{22}\right)\)
\(\chi_{89}(20,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{31}{44}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{4}{11}\right)\)
\(\chi_{89}(21,\cdot)\)	89.g	44	yes	\(1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)
\(\chi_{89}(22,\cdot)\)	89.f	22	yes	\(1\)	\(1\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)
\(\chi_{89}(23,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{57}{88}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{1}{88}\right)\)	\(e\left(\frac{41}{88}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{31}{44}\right)\)	\(e\left(\frac{9}{22}\right)\)
\(\chi_{89}(24,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{49}{88}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{41}{88}\right)\)	\(e\left(\frac{9}{88}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{5}{44}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)
\(\chi_{89}(25,\cdot)\)	89.f	22	yes	\(1\)	\(1\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)
\(\chi_{89}(26,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{39}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{47}{88}\right)\)	\(e\left(\frac{79}{88}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{5}{44}\right)\)	\(e\left(\frac{5}{22}\right)\)
\(\chi_{89}(27,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{3}{88}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{51}{88}\right)\)	\(e\left(\frac{67}{88}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(28,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{25}{88}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{73}{88}\right)\)	\(e\left(\frac{1}{88}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(29,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{59}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{27}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)
\(\chi_{89}(30,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{87}{88}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{71}{88}\right)\)	\(e\left(\frac{7}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{1}{22}\right)\)
\(\chi_{89}(31,\cdot)\)	89.h	88	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{31}{88}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{87}{88}\right)\)	\(e\left(\frac{47}{88}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{31}{44}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{89}(32,\cdot)\)	89.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)

Introduction

L-functions

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Character group

First 32 of 88 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.