LMFDB - Group of Dirichlet characters of modulus 229

Show commands: PariGP / SageMath

sage: H = DirichletGroup(229)

pari: g = idealstar(,229,2)

Character group

sage: G.order() pari: g.no
Order	=	228
sage: H.invariants() pari: g.cyc
Structure	=	$C_{228}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{229}(6,\cdot)$

First 32 of 228 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{229}(1,\cdot)$	229.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{229}(2,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{71}{76}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{33}{38}\right)$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{7}{76}\right)$	$e\left(\frac{65}{76}\right)$	$e\left(\frac{61}{76}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{73}{76}\right)$	$e\left(\frac{35}{38}\right)$
$\chi_{229}(3,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{43}{57}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{23}{57}\right)$	$e\left(\frac{52}{57}\right)$	$e\left(\frac{35}{57}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{29}{57}\right)$	$e\left(\frac{32}{57}\right)$	$e\left(\frac{15}{19}\right)$
$\chi_{229}(4,\cdot)$	229.h	38	yes	$1$	$1$	$e\left(\frac{33}{38}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{7}{38}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{23}{38}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{35}{38}\right)$	$e\left(\frac{16}{19}\right)$
$\chi_{229}(5,\cdot)$	229.k	114	yes	$1$	$1$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{23}{57}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{7}{57}\right)$	$e\left(\frac{49}{114}\right)$	$e\left(\frac{113}{114}\right)$	$e\left(\frac{3}{38}\right)$	$e\left(\frac{46}{57}\right)$	$e\left(\frac{17}{114}\right)$	$e\left(\frac{12}{19}\right)$
$\chi_{229}(6,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{7}{76}\right)$	$e\left(\frac{52}{57}\right)$	$e\left(\frac{7}{38}\right)$	$e\left(\frac{49}{114}\right)$	$e\left(\frac{1}{228}\right)$	$e\left(\frac{107}{228}\right)$	$e\left(\frac{21}{76}\right)$	$e\left(\frac{47}{57}\right)$	$e\left(\frac{119}{228}\right)$	$e\left(\frac{27}{38}\right)$
$\chi_{229}(7,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{65}{76}\right)$	$e\left(\frac{35}{57}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{113}{114}\right)$	$e\left(\frac{107}{228}\right)$	$e\left(\frac{49}{228}\right)$	$e\left(\frac{43}{76}\right)$	$e\left(\frac{13}{57}\right)$	$e\left(\frac{193}{228}\right)$	$e\left(\frac{1}{38}\right)$
$\chi_{229}(8,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{61}{76}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{23}{38}\right)$	$e\left(\frac{3}{38}\right)$	$e\left(\frac{21}{76}\right)$	$e\left(\frac{43}{76}\right)$	$e\left(\frac{31}{76}\right)$	$e\left(\frac{18}{19}\right)$	$e\left(\frac{67}{76}\right)$	$e\left(\frac{29}{38}\right)$
$\chi_{229}(9,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{29}{57}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{46}{57}\right)$	$e\left(\frac{47}{57}\right)$	$e\left(\frac{13}{57}\right)$	$e\left(\frac{18}{19}\right)$	$e\left(\frac{1}{57}\right)$	$e\left(\frac{7}{57}\right)$	$e\left(\frac{11}{19}\right)$
$\chi_{229}(10,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{73}{76}\right)$	$e\left(\frac{32}{57}\right)$	$e\left(\frac{35}{38}\right)$	$e\left(\frac{17}{114}\right)$	$e\left(\frac{119}{228}\right)$	$e\left(\frac{193}{228}\right)$	$e\left(\frac{67}{76}\right)$	$e\left(\frac{7}{57}\right)$	$e\left(\frac{25}{228}\right)$	$e\left(\frac{21}{38}\right)$
$\chi_{229}(11,\cdot)$	229.h	38	yes	$1$	$1$	$e\left(\frac{35}{38}\right)$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{29}{38}\right)$	$e\left(\frac{11}{19}\right)$	$e\left(\frac{21}{38}\right)$	$e\left(\frac{2}{19}\right)$
$\chi_{229}(12,\cdot)$	229.k	114	yes	$1$	$1$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{4}{57}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{26}{57}\right)$	$e\left(\frac{11}{114}\right)$	$e\left(\frac{37}{114}\right)$	$e\left(\frac{3}{38}\right)$	$e\left(\frac{8}{57}\right)$	$e\left(\frac{55}{114}\right)$	$e\left(\frac{12}{19}\right)$
$\chi_{229}(13,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{59}{76}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{21}{38}\right)$	$e\left(\frac{11}{38}\right)$	$e\left(\frac{39}{76}\right)$	$e\left(\frac{69}{76}\right)$	$e\left(\frac{25}{76}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{5}{76}\right)$	$e\left(\frac{5}{38}\right)$
$\chi_{229}(14,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{44}{57}\right)$	$e\left(\frac{11}{19}\right)$	$e\left(\frac{1}{57}\right)$	$e\left(\frac{32}{57}\right)$	$e\left(\frac{4}{57}\right)$	$e\left(\frac{7}{19}\right)$	$e\left(\frac{31}{57}\right)$	$e\left(\frac{46}{57}\right)$	$e\left(\frac{18}{19}\right)$
$\chi_{229}(15,\cdot)$	229.h	38	yes	$1$	$1$	$e\left(\frac{7}{38}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{7}{19}\right)$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{13}{38}\right)$	$e\left(\frac{23}{38}\right)$	$e\left(\frac{21}{38}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{8}{19}\right)$
$\chi_{229}(16,\cdot)$	229.g	19	yes	$1$	$1$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{2}{19}\right)$	$e\left(\frac{7}{19}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{4}{19}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{13}{19}\right)$
$\chi_{229}(17,\cdot)$	229.g	19	yes	$1$	$1$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{11}{19}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{12}{19}\right)$
$\chi_{229}(18,\cdot)$	229.f	12	yes	$-1$	$1$	$i$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{12}\right)$	$-i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{12}\right)$	$-1$
$\chi_{229}(19,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{53}{57}\right)$	$e\left(\frac{18}{19}\right)$	$e\left(\frac{31}{57}\right)$	$e\left(\frac{23}{57}\right)$	$e\left(\frac{10}{57}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{49}{57}\right)$	$e\left(\frac{1}{57}\right)$	$e\left(\frac{7}{19}\right)$
$\chi_{229}(20,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{17}{19}\right)$	$e\left(\frac{41}{57}\right)$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{10}{57}\right)$	$e\left(\frac{35}{57}\right)$	$e\left(\frac{40}{57}\right)$	$e\left(\frac{13}{19}\right)$	$e\left(\frac{25}{57}\right)$	$e\left(\frac{4}{57}\right)$	$e\left(\frac{9}{19}\right)$
$\chi_{229}(21,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{1}{76}\right)$	$e\left(\frac{7}{19}\right)$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{15}{38}\right)$	$e\left(\frac{29}{76}\right)$	$e\left(\frac{63}{76}\right)$	$e\left(\frac{3}{76}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{31}{76}\right)$	$e\left(\frac{31}{38}\right)$
$\chi_{229}(22,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{65}{76}\right)$	$e\left(\frac{18}{19}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{25}{38}\right)$	$e\left(\frac{61}{76}\right)$	$e\left(\frac{67}{76}\right)$	$e\left(\frac{43}{76}\right)$	$e\left(\frac{17}{19}\right)$	$e\left(\frac{39}{76}\right)$	$e\left(\frac{1}{38}\right)$
$\chi_{229}(23,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{3}{76}\right)$	$e\left(\frac{25}{57}\right)$	$e\left(\frac{3}{38}\right)$	$e\left(\frac{97}{114}\right)$	$e\left(\frac{109}{228}\right)$	$e\left(\frac{35}{228}\right)$	$e\left(\frac{9}{76}\right)$	$e\left(\frac{50}{57}\right)$	$e\left(\frac{203}{228}\right)$	$e\left(\frac{17}{38}\right)$
$\chi_{229}(24,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{73}{76}\right)$	$e\left(\frac{13}{57}\right)$	$e\left(\frac{35}{38}\right)$	$e\left(\frac{55}{114}\right)$	$e\left(\frac{43}{228}\right)$	$e\left(\frac{41}{228}\right)$	$e\left(\frac{67}{76}\right)$	$e\left(\frac{26}{57}\right)$	$e\left(\frac{101}{228}\right)$	$e\left(\frac{21}{38}\right)$
$\chi_{229}(25,\cdot)$	229.i	57	yes	$1$	$1$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{46}{57}\right)$	$e\left(\frac{2}{19}\right)$	$e\left(\frac{14}{57}\right)$	$e\left(\frac{49}{57}\right)$	$e\left(\frac{56}{57}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{35}{57}\right)$	$e\left(\frac{17}{57}\right)$	$e\left(\frac{5}{19}\right)$
$\chi_{229}(26,\cdot)$	229.h	38	yes	$1$	$1$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{17}{19}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{23}{38}\right)$	$e\left(\frac{29}{38}\right)$	$e\left(\frac{5}{38}\right)$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{1}{38}\right)$	$e\left(\frac{1}{19}\right)$
$\chi_{229}(27,\cdot)$	229.g	19	yes	$1$	$1$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{18}{19}\right)$	$e\left(\frac{4}{19}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{13}{19}\right)$	$e\left(\frac{7}{19}\right)$
$\chi_{229}(28,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{55}{76}\right)$	$e\left(\frac{53}{57}\right)$	$e\left(\frac{17}{38}\right)$	$e\left(\frac{5}{114}\right)$	$e\left(\frac{149}{228}\right)$	$e\left(\frac{211}{228}\right)$	$e\left(\frac{13}{76}\right)$	$e\left(\frac{49}{57}\right)$	$e\left(\frac{175}{228}\right)$	$e\left(\frac{33}{38}\right)$
$\chi_{229}(29,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{27}{76}\right)$	$e\left(\frac{16}{57}\right)$	$e\left(\frac{27}{38}\right)$	$e\left(\frac{37}{114}\right)$	$e\left(\frac{145}{228}\right)$	$e\left(\frac{11}{228}\right)$	$e\left(\frac{5}{76}\right)$	$e\left(\frac{32}{57}\right)$	$e\left(\frac{155}{228}\right)$	$e\left(\frac{1}{38}\right)$
$\chi_{229}(30,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{9}{76}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{9}{38}\right)$	$e\left(\frac{21}{38}\right)$	$e\left(\frac{33}{76}\right)$	$e\left(\frac{35}{76}\right)$	$e\left(\frac{27}{76}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{51}{76}\right)$	$e\left(\frac{13}{38}\right)$
$\chi_{229}(31,\cdot)$	229.l	228	yes	$-1$	$1$	$e\left(\frac{51}{76}\right)$	$e\left(\frac{26}{57}\right)$	$e\left(\frac{13}{38}\right)$	$e\left(\frac{53}{114}\right)$	$e\left(\frac{29}{228}\right)$	$e\left(\frac{139}{228}\right)$	$e\left(\frac{1}{76}\right)$	$e\left(\frac{52}{57}\right)$	$e\left(\frac{31}{228}\right)$	$e\left(\frac{23}{38}\right)$
$\chi_{229}(32,\cdot)$	229.j	76	yes	$-1$	$1$	$e\left(\frac{51}{76}\right)$	$e\left(\frac{15}{19}\right)$	$e\left(\frac{13}{38}\right)$	$e\left(\frac{5}{38}\right)$	$e\left(\frac{35}{76}\right)$	$e\left(\frac{21}{76}\right)$	$e\left(\frac{1}{76}\right)$	$e\left(\frac{11}{19}\right)$	$e\left(\frac{61}{76}\right)$	$e\left(\frac{23}{38}\right)$

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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	$229$
Structure	\(C_{228}\)
Order	$228$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{229}(1,\cdot)\)	229.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{229}(2,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{71}{76}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{33}{38}\right)\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{7}{76}\right)\)	\(e\left(\frac{65}{76}\right)\)	\(e\left(\frac{61}{76}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{73}{76}\right)\)	\(e\left(\frac{35}{38}\right)\)
\(\chi_{229}(3,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{43}{57}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{23}{57}\right)\)	\(e\left(\frac{52}{57}\right)\)	\(e\left(\frac{35}{57}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{29}{57}\right)\)	\(e\left(\frac{32}{57}\right)\)	\(e\left(\frac{15}{19}\right)\)
\(\chi_{229}(4,\cdot)\)	229.h	38	yes	\(1\)	\(1\)	\(e\left(\frac{33}{38}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{7}{38}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{23}{38}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{35}{38}\right)\)	\(e\left(\frac{16}{19}\right)\)
\(\chi_{229}(5,\cdot)\)	229.k	114	yes	\(1\)	\(1\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{23}{57}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{7}{57}\right)\)	\(e\left(\frac{49}{114}\right)\)	\(e\left(\frac{113}{114}\right)\)	\(e\left(\frac{3}{38}\right)\)	\(e\left(\frac{46}{57}\right)\)	\(e\left(\frac{17}{114}\right)\)	\(e\left(\frac{12}{19}\right)\)
\(\chi_{229}(6,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{76}\right)\)	\(e\left(\frac{52}{57}\right)\)	\(e\left(\frac{7}{38}\right)\)	\(e\left(\frac{49}{114}\right)\)	\(e\left(\frac{1}{228}\right)\)	\(e\left(\frac{107}{228}\right)\)	\(e\left(\frac{21}{76}\right)\)	\(e\left(\frac{47}{57}\right)\)	\(e\left(\frac{119}{228}\right)\)	\(e\left(\frac{27}{38}\right)\)
\(\chi_{229}(7,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{65}{76}\right)\)	\(e\left(\frac{35}{57}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{113}{114}\right)\)	\(e\left(\frac{107}{228}\right)\)	\(e\left(\frac{49}{228}\right)\)	\(e\left(\frac{43}{76}\right)\)	\(e\left(\frac{13}{57}\right)\)	\(e\left(\frac{193}{228}\right)\)	\(e\left(\frac{1}{38}\right)\)
\(\chi_{229}(8,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{61}{76}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{23}{38}\right)\)	\(e\left(\frac{3}{38}\right)\)	\(e\left(\frac{21}{76}\right)\)	\(e\left(\frac{43}{76}\right)\)	\(e\left(\frac{31}{76}\right)\)	\(e\left(\frac{18}{19}\right)\)	\(e\left(\frac{67}{76}\right)\)	\(e\left(\frac{29}{38}\right)\)
\(\chi_{229}(9,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{29}{57}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{46}{57}\right)\)	\(e\left(\frac{47}{57}\right)\)	\(e\left(\frac{13}{57}\right)\)	\(e\left(\frac{18}{19}\right)\)	\(e\left(\frac{1}{57}\right)\)	\(e\left(\frac{7}{57}\right)\)	\(e\left(\frac{11}{19}\right)\)
\(\chi_{229}(10,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{73}{76}\right)\)	\(e\left(\frac{32}{57}\right)\)	\(e\left(\frac{35}{38}\right)\)	\(e\left(\frac{17}{114}\right)\)	\(e\left(\frac{119}{228}\right)\)	\(e\left(\frac{193}{228}\right)\)	\(e\left(\frac{67}{76}\right)\)	\(e\left(\frac{7}{57}\right)\)	\(e\left(\frac{25}{228}\right)\)	\(e\left(\frac{21}{38}\right)\)
\(\chi_{229}(11,\cdot)\)	229.h	38	yes	\(1\)	\(1\)	\(e\left(\frac{35}{38}\right)\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{29}{38}\right)\)	\(e\left(\frac{11}{19}\right)\)	\(e\left(\frac{21}{38}\right)\)	\(e\left(\frac{2}{19}\right)\)
\(\chi_{229}(12,\cdot)\)	229.k	114	yes	\(1\)	\(1\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{4}{57}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{26}{57}\right)\)	\(e\left(\frac{11}{114}\right)\)	\(e\left(\frac{37}{114}\right)\)	\(e\left(\frac{3}{38}\right)\)	\(e\left(\frac{8}{57}\right)\)	\(e\left(\frac{55}{114}\right)\)	\(e\left(\frac{12}{19}\right)\)
\(\chi_{229}(13,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{59}{76}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{21}{38}\right)\)	\(e\left(\frac{11}{38}\right)\)	\(e\left(\frac{39}{76}\right)\)	\(e\left(\frac{69}{76}\right)\)	\(e\left(\frac{25}{76}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{5}{76}\right)\)	\(e\left(\frac{5}{38}\right)\)
\(\chi_{229}(14,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{44}{57}\right)\)	\(e\left(\frac{11}{19}\right)\)	\(e\left(\frac{1}{57}\right)\)	\(e\left(\frac{32}{57}\right)\)	\(e\left(\frac{4}{57}\right)\)	\(e\left(\frac{7}{19}\right)\)	\(e\left(\frac{31}{57}\right)\)	\(e\left(\frac{46}{57}\right)\)	\(e\left(\frac{18}{19}\right)\)
\(\chi_{229}(15,\cdot)\)	229.h	38	yes	\(1\)	\(1\)	\(e\left(\frac{7}{38}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{7}{19}\right)\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{13}{38}\right)\)	\(e\left(\frac{23}{38}\right)\)	\(e\left(\frac{21}{38}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{8}{19}\right)\)
\(\chi_{229}(16,\cdot)\)	229.g	19	yes	\(1\)	\(1\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{2}{19}\right)\)	\(e\left(\frac{7}{19}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{4}{19}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{13}{19}\right)\)
\(\chi_{229}(17,\cdot)\)	229.g	19	yes	\(1\)	\(1\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{11}{19}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)
\(\chi_{229}(18,\cdot)\)	229.f	12	yes	\(-1\)	\(1\)	\(i\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-1\)
\(\chi_{229}(19,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{53}{57}\right)\)	\(e\left(\frac{18}{19}\right)\)	\(e\left(\frac{31}{57}\right)\)	\(e\left(\frac{23}{57}\right)\)	\(e\left(\frac{10}{57}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{49}{57}\right)\)	\(e\left(\frac{1}{57}\right)\)	\(e\left(\frac{7}{19}\right)\)
\(\chi_{229}(20,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{17}{19}\right)\)	\(e\left(\frac{41}{57}\right)\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{10}{57}\right)\)	\(e\left(\frac{35}{57}\right)\)	\(e\left(\frac{40}{57}\right)\)	\(e\left(\frac{13}{19}\right)\)	\(e\left(\frac{25}{57}\right)\)	\(e\left(\frac{4}{57}\right)\)	\(e\left(\frac{9}{19}\right)\)
\(\chi_{229}(21,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{76}\right)\)	\(e\left(\frac{7}{19}\right)\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{15}{38}\right)\)	\(e\left(\frac{29}{76}\right)\)	\(e\left(\frac{63}{76}\right)\)	\(e\left(\frac{3}{76}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{31}{76}\right)\)	\(e\left(\frac{31}{38}\right)\)
\(\chi_{229}(22,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{65}{76}\right)\)	\(e\left(\frac{18}{19}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{25}{38}\right)\)	\(e\left(\frac{61}{76}\right)\)	\(e\left(\frac{67}{76}\right)\)	\(e\left(\frac{43}{76}\right)\)	\(e\left(\frac{17}{19}\right)\)	\(e\left(\frac{39}{76}\right)\)	\(e\left(\frac{1}{38}\right)\)
\(\chi_{229}(23,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{76}\right)\)	\(e\left(\frac{25}{57}\right)\)	\(e\left(\frac{3}{38}\right)\)	\(e\left(\frac{97}{114}\right)\)	\(e\left(\frac{109}{228}\right)\)	\(e\left(\frac{35}{228}\right)\)	\(e\left(\frac{9}{76}\right)\)	\(e\left(\frac{50}{57}\right)\)	\(e\left(\frac{203}{228}\right)\)	\(e\left(\frac{17}{38}\right)\)
\(\chi_{229}(24,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{73}{76}\right)\)	\(e\left(\frac{13}{57}\right)\)	\(e\left(\frac{35}{38}\right)\)	\(e\left(\frac{55}{114}\right)\)	\(e\left(\frac{43}{228}\right)\)	\(e\left(\frac{41}{228}\right)\)	\(e\left(\frac{67}{76}\right)\)	\(e\left(\frac{26}{57}\right)\)	\(e\left(\frac{101}{228}\right)\)	\(e\left(\frac{21}{38}\right)\)
\(\chi_{229}(25,\cdot)\)	229.i	57	yes	\(1\)	\(1\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{46}{57}\right)\)	\(e\left(\frac{2}{19}\right)\)	\(e\left(\frac{14}{57}\right)\)	\(e\left(\frac{49}{57}\right)\)	\(e\left(\frac{56}{57}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{35}{57}\right)\)	\(e\left(\frac{17}{57}\right)\)	\(e\left(\frac{5}{19}\right)\)
\(\chi_{229}(26,\cdot)\)	229.h	38	yes	\(1\)	\(1\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{17}{19}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{23}{38}\right)\)	\(e\left(\frac{29}{38}\right)\)	\(e\left(\frac{5}{38}\right)\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{1}{38}\right)\)	\(e\left(\frac{1}{19}\right)\)
\(\chi_{229}(27,\cdot)\)	229.g	19	yes	\(1\)	\(1\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{18}{19}\right)\)	\(e\left(\frac{4}{19}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{13}{19}\right)\)	\(e\left(\frac{7}{19}\right)\)
\(\chi_{229}(28,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{55}{76}\right)\)	\(e\left(\frac{53}{57}\right)\)	\(e\left(\frac{17}{38}\right)\)	\(e\left(\frac{5}{114}\right)\)	\(e\left(\frac{149}{228}\right)\)	\(e\left(\frac{211}{228}\right)\)	\(e\left(\frac{13}{76}\right)\)	\(e\left(\frac{49}{57}\right)\)	\(e\left(\frac{175}{228}\right)\)	\(e\left(\frac{33}{38}\right)\)
\(\chi_{229}(29,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{27}{76}\right)\)	\(e\left(\frac{16}{57}\right)\)	\(e\left(\frac{27}{38}\right)\)	\(e\left(\frac{37}{114}\right)\)	\(e\left(\frac{145}{228}\right)\)	\(e\left(\frac{11}{228}\right)\)	\(e\left(\frac{5}{76}\right)\)	\(e\left(\frac{32}{57}\right)\)	\(e\left(\frac{155}{228}\right)\)	\(e\left(\frac{1}{38}\right)\)
\(\chi_{229}(30,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{76}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{9}{38}\right)\)	\(e\left(\frac{21}{38}\right)\)	\(e\left(\frac{33}{76}\right)\)	\(e\left(\frac{35}{76}\right)\)	\(e\left(\frac{27}{76}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{51}{76}\right)\)	\(e\left(\frac{13}{38}\right)\)
\(\chi_{229}(31,\cdot)\)	229.l	228	yes	\(-1\)	\(1\)	\(e\left(\frac{51}{76}\right)\)	\(e\left(\frac{26}{57}\right)\)	\(e\left(\frac{13}{38}\right)\)	\(e\left(\frac{53}{114}\right)\)	\(e\left(\frac{29}{228}\right)\)	\(e\left(\frac{139}{228}\right)\)	\(e\left(\frac{1}{76}\right)\)	\(e\left(\frac{52}{57}\right)\)	\(e\left(\frac{31}{228}\right)\)	\(e\left(\frac{23}{38}\right)\)
\(\chi_{229}(32,\cdot)\)	229.j	76	yes	\(-1\)	\(1\)	\(e\left(\frac{51}{76}\right)\)	\(e\left(\frac{15}{19}\right)\)	\(e\left(\frac{13}{38}\right)\)	\(e\left(\frac{5}{38}\right)\)	\(e\left(\frac{35}{76}\right)\)	\(e\left(\frac{21}{76}\right)\)	\(e\left(\frac{1}{76}\right)\)	\(e\left(\frac{11}{19}\right)\)	\(e\left(\frac{61}{76}\right)\)	\(e\left(\frac{23}{38}\right)\)

Introduction

L-functions

Modular forms

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Character group

First 32 of 228 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.