LMFDB - Group of Dirichlet characters of modulus 617

Show commands: PariGP / SageMath

sage: H = DirichletGroup(617)

pari: g = idealstar(,617,2)

Character group

sage: G.order() pari: g.no
Order	=	616
sage: H.invariants() pari: g.cyc
Structure	=	$C_{616}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{617}(3,\cdot)$

First 32 of 616 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{617}(1,\cdot)$	617.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{617}(2,\cdot)$	617.n	154	yes	$1$	$1$	$e\left(\frac{29}{77}\right)$	$e\left(\frac{107}{154}\right)$	$e\left(\frac{58}{77}\right)$	$e\left(\frac{5}{154}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{51}{77}\right)$	$e\left(\frac{10}{77}\right)$	$e\left(\frac{30}{77}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{16}{77}\right)$
$\chi_{617}(3,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{107}{154}\right)$	$e\left(\frac{1}{616}\right)$	$e\left(\frac{30}{77}\right)$	$e\left(\frac{321}{616}\right)$	$e\left(\frac{39}{56}\right)$	$e\left(\frac{71}{308}\right)$	$e\left(\frac{13}{154}\right)$	$e\left(\frac{1}{308}\right)$	$e\left(\frac{19}{88}\right)$	$e\left(\frac{57}{308}\right)$
$\chi_{617}(4,\cdot)$	617.l	77	yes	$1$	$1$	$e\left(\frac{58}{77}\right)$	$e\left(\frac{30}{77}\right)$	$e\left(\frac{39}{77}\right)$	$e\left(\frac{5}{77}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{25}{77}\right)$	$e\left(\frac{20}{77}\right)$	$e\left(\frac{60}{77}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{32}{77}\right)$
$\chi_{617}(5,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{5}{154}\right)$	$e\left(\frac{321}{616}\right)$	$e\left(\frac{5}{77}\right)$	$e\left(\frac{169}{616}\right)$	$e\left(\frac{31}{56}\right)$	$e\left(\frac{307}{308}\right)$	$e\left(\frac{15}{154}\right)$	$e\left(\frac{13}{308}\right)$	$e\left(\frac{27}{88}\right)$	$e\left(\frac{125}{308}\right)$
$\chi_{617}(6,\cdot)$	617.k	56	yes	$-1$	$1$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{39}{56}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{31}{56}\right)$	$e\left(\frac{43}{56}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{11}{28}\right)$
$\chi_{617}(7,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{51}{77}\right)$	$e\left(\frac{71}{308}\right)$	$e\left(\frac{25}{77}\right)$	$e\left(\frac{307}{308}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{113}{154}\right)$	$e\left(\frac{76}{77}\right)$	$e\left(\frac{71}{154}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{43}{154}\right)$
$\chi_{617}(8,\cdot)$	617.n	154	yes	$1$	$1$	$e\left(\frac{10}{77}\right)$	$e\left(\frac{13}{154}\right)$	$e\left(\frac{20}{77}\right)$	$e\left(\frac{15}{154}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{76}{77}\right)$	$e\left(\frac{30}{77}\right)$	$e\left(\frac{13}{77}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{48}{77}\right)$
$\chi_{617}(9,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{30}{77}\right)$	$e\left(\frac{1}{308}\right)$	$e\left(\frac{60}{77}\right)$	$e\left(\frac{13}{308}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{71}{154}\right)$	$e\left(\frac{13}{77}\right)$	$e\left(\frac{1}{154}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{57}{154}\right)$
$\chi_{617}(10,\cdot)$	617.m	88	yes	$-1$	$1$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{19}{88}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{27}{88}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{63}{88}\right)$	$e\left(\frac{27}{44}\right)$
$\chi_{617}(11,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{16}{77}\right)$	$e\left(\frac{57}{308}\right)$	$e\left(\frac{32}{77}\right)$	$e\left(\frac{125}{308}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{43}{154}\right)$	$e\left(\frac{48}{77}\right)$	$e\left(\frac{57}{154}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{15}{154}\right)$
$\chi_{617}(12,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{69}{154}\right)$	$e\left(\frac{241}{616}\right)$	$e\left(\frac{69}{77}\right)$	$e\left(\frac{361}{616}\right)$	$e\left(\frac{47}{56}\right)$	$e\left(\frac{171}{308}\right)$	$e\left(\frac{53}{154}\right)$	$e\left(\frac{241}{308}\right)$	$e\left(\frac{3}{88}\right)$	$e\left(\frac{185}{308}\right)$
$\chi_{617}(13,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{71}{154}\right)$	$e\left(\frac{277}{616}\right)$	$e\left(\frac{71}{77}\right)$	$e\left(\frac{213}{616}\right)$	$e\left(\frac{51}{56}\right)$	$e\left(\frac{263}{308}\right)$	$e\left(\frac{59}{154}\right)$	$e\left(\frac{277}{308}\right)$	$e\left(\frac{71}{88}\right)$	$e\left(\frac{81}{308}\right)$
$\chi_{617}(14,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{3}{77}\right)$	$e\left(\frac{285}{308}\right)$	$e\left(\frac{6}{77}\right)$	$e\left(\frac{9}{308}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{61}{154}\right)$	$e\left(\frac{9}{77}\right)$	$e\left(\frac{131}{154}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{75}{154}\right)$
$\chi_{617}(15,\cdot)$	617.j	44	yes	$1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{35}{44}\right)$	$i$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{617}(16,\cdot)$	617.l	77	yes	$1$	$1$	$e\left(\frac{39}{77}\right)$	$e\left(\frac{60}{77}\right)$	$e\left(\frac{1}{77}\right)$	$e\left(\frac{10}{77}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{50}{77}\right)$	$e\left(\frac{40}{77}\right)$	$e\left(\frac{43}{77}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{64}{77}\right)$
$\chi_{617}(17,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{137}{154}\right)$	$e\left(\frac{387}{616}\right)$	$e\left(\frac{60}{77}\right)$	$e\left(\frac{411}{616}\right)$	$e\left(\frac{29}{56}\right)$	$e\left(\frac{65}{308}\right)$	$e\left(\frac{103}{154}\right)$	$e\left(\frac{79}{308}\right)$	$e\left(\frac{49}{88}\right)$	$e\left(\frac{191}{308}\right)$
$\chi_{617}(18,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{59}{77}\right)$	$e\left(\frac{215}{308}\right)$	$e\left(\frac{41}{77}\right)$	$e\left(\frac{23}{308}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{19}{154}\right)$	$e\left(\frac{23}{77}\right)$	$e\left(\frac{61}{154}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{89}{154}\right)$
$\chi_{617}(19,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{39}{77}\right)$	$e\left(\frac{9}{308}\right)$	$e\left(\frac{1}{77}\right)$	$e\left(\frac{117}{308}\right)$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{23}{154}\right)$	$e\left(\frac{40}{77}\right)$	$e\left(\frac{9}{154}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{51}{154}\right)$
$\chi_{617}(20,\cdot)$	617.k	56	yes	$-1$	$1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{51}{56}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{19}{56}\right)$	$e\left(\frac{39}{56}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{23}{28}\right)$
$\chi_{617}(21,\cdot)$	617.k	56	yes	$-1$	$1$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{13}{56}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{29}{56}\right)$	$e\left(\frac{33}{56}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{28}\right)$
$\chi_{617}(22,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{45}{77}\right)$	$e\left(\frac{271}{308}\right)$	$e\left(\frac{13}{77}\right)$	$e\left(\frac{135}{308}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{145}{154}\right)$	$e\left(\frac{58}{77}\right)$	$e\left(\frac{117}{154}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{47}{154}\right)$
$\chi_{617}(23,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{103}{154}\right)$	$e\left(\frac{237}{616}\right)$	$e\left(\frac{26}{77}\right)$	$e\left(\frac{309}{616}\right)$	$e\left(\frac{3}{56}\right)$	$e\left(\frac{195}{308}\right)$	$e\left(\frac{1}{154}\right)$	$e\left(\frac{237}{308}\right)$	$e\left(\frac{15}{88}\right)$	$e\left(\frac{265}{308}\right)$
$\chi_{617}(24,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{127}{154}\right)$	$e\left(\frac{53}{616}\right)$	$e\left(\frac{50}{77}\right)$	$e\left(\frac{381}{616}\right)$	$e\left(\frac{51}{56}\right)$	$e\left(\frac{67}{308}\right)$	$e\left(\frac{73}{154}\right)$	$e\left(\frac{53}{308}\right)$	$e\left(\frac{39}{88}\right)$	$e\left(\frac{249}{308}\right)$
$\chi_{617}(25,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{5}{77}\right)$	$e\left(\frac{13}{308}\right)$	$e\left(\frac{10}{77}\right)$	$e\left(\frac{169}{308}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{153}{154}\right)$	$e\left(\frac{15}{77}\right)$	$e\left(\frac{13}{154}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{125}{154}\right)$
$\chi_{617}(26,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{129}{154}\right)$	$e\left(\frac{89}{616}\right)$	$e\left(\frac{52}{77}\right)$	$e\left(\frac{233}{616}\right)$	$e\left(\frac{55}{56}\right)$	$e\left(\frac{159}{308}\right)$	$e\left(\frac{79}{154}\right)$	$e\left(\frac{89}{308}\right)$	$e\left(\frac{19}{88}\right)$	$e\left(\frac{145}{308}\right)$
$\chi_{617}(27,\cdot)$	617.p	616	yes	$-1$	$1$	$e\left(\frac{13}{154}\right)$	$e\left(\frac{3}{616}\right)$	$e\left(\frac{13}{77}\right)$	$e\left(\frac{347}{616}\right)$	$e\left(\frac{5}{56}\right)$	$e\left(\frac{213}{308}\right)$	$e\left(\frac{39}{154}\right)$	$e\left(\frac{3}{308}\right)$	$e\left(\frac{57}{88}\right)$	$e\left(\frac{171}{308}\right)$
$\chi_{617}(28,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{32}{77}\right)$	$e\left(\frac{191}{308}\right)$	$e\left(\frac{64}{77}\right)$	$e\left(\frac{19}{308}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{9}{154}\right)$	$e\left(\frac{19}{77}\right)$	$e\left(\frac{37}{154}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{107}{154}\right)$
$\chi_{617}(29,\cdot)$	617.m	88	yes	$-1$	$1$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{31}{88}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{88}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{31}{44}\right)$	$e\left(\frac{75}{88}\right)$	$e\left(\frac{7}{44}\right)$
$\chi_{617}(30,\cdot)$	617.o	308	yes	$1$	$1$	$e\left(\frac{8}{77}\right)$	$e\left(\frac{67}{308}\right)$	$e\left(\frac{16}{77}\right)$	$e\left(\frac{255}{308}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{137}{154}\right)$	$e\left(\frac{24}{77}\right)$	$e\left(\frac{67}{154}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{123}{154}\right)$
$\chi_{617}(31,\cdot)$	617.f	11	yes	$1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{2}{11}\right)$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{4}{11}\right)$
$\chi_{617}(32,\cdot)$	617.n	154	yes	$1$	$1$	$e\left(\frac{68}{77}\right)$	$e\left(\frac{73}{154}\right)$	$e\left(\frac{59}{77}\right)$	$e\left(\frac{25}{154}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{24}{77}\right)$	$e\left(\frac{50}{77}\right)$	$e\left(\frac{73}{77}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{3}{77}\right)$

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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	$617$
Structure	\(C_{616}\)
Order	$616$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{617}(1,\cdot)\)	617.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{617}(2,\cdot)\)	617.n	154	yes	\(1\)	\(1\)	\(e\left(\frac{29}{77}\right)\)	\(e\left(\frac{107}{154}\right)\)	\(e\left(\frac{58}{77}\right)\)	\(e\left(\frac{5}{154}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{51}{77}\right)\)	\(e\left(\frac{10}{77}\right)\)	\(e\left(\frac{30}{77}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{16}{77}\right)\)
\(\chi_{617}(3,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{107}{154}\right)\)	\(e\left(\frac{1}{616}\right)\)	\(e\left(\frac{30}{77}\right)\)	\(e\left(\frac{321}{616}\right)\)	\(e\left(\frac{39}{56}\right)\)	\(e\left(\frac{71}{308}\right)\)	\(e\left(\frac{13}{154}\right)\)	\(e\left(\frac{1}{308}\right)\)	\(e\left(\frac{19}{88}\right)\)	\(e\left(\frac{57}{308}\right)\)
\(\chi_{617}(4,\cdot)\)	617.l	77	yes	\(1\)	\(1\)	\(e\left(\frac{58}{77}\right)\)	\(e\left(\frac{30}{77}\right)\)	\(e\left(\frac{39}{77}\right)\)	\(e\left(\frac{5}{77}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{25}{77}\right)\)	\(e\left(\frac{20}{77}\right)\)	\(e\left(\frac{60}{77}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{32}{77}\right)\)
\(\chi_{617}(5,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{154}\right)\)	\(e\left(\frac{321}{616}\right)\)	\(e\left(\frac{5}{77}\right)\)	\(e\left(\frac{169}{616}\right)\)	\(e\left(\frac{31}{56}\right)\)	\(e\left(\frac{307}{308}\right)\)	\(e\left(\frac{15}{154}\right)\)	\(e\left(\frac{13}{308}\right)\)	\(e\left(\frac{27}{88}\right)\)	\(e\left(\frac{125}{308}\right)\)
\(\chi_{617}(6,\cdot)\)	617.k	56	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{39}{56}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{31}{56}\right)\)	\(e\left(\frac{43}{56}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{11}{28}\right)\)
\(\chi_{617}(7,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{51}{77}\right)\)	\(e\left(\frac{71}{308}\right)\)	\(e\left(\frac{25}{77}\right)\)	\(e\left(\frac{307}{308}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{113}{154}\right)\)	\(e\left(\frac{76}{77}\right)\)	\(e\left(\frac{71}{154}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{43}{154}\right)\)
\(\chi_{617}(8,\cdot)\)	617.n	154	yes	\(1\)	\(1\)	\(e\left(\frac{10}{77}\right)\)	\(e\left(\frac{13}{154}\right)\)	\(e\left(\frac{20}{77}\right)\)	\(e\left(\frac{15}{154}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{76}{77}\right)\)	\(e\left(\frac{30}{77}\right)\)	\(e\left(\frac{13}{77}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{48}{77}\right)\)
\(\chi_{617}(9,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{30}{77}\right)\)	\(e\left(\frac{1}{308}\right)\)	\(e\left(\frac{60}{77}\right)\)	\(e\left(\frac{13}{308}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{71}{154}\right)\)	\(e\left(\frac{13}{77}\right)\)	\(e\left(\frac{1}{154}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{57}{154}\right)\)
\(\chi_{617}(10,\cdot)\)	617.m	88	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{19}{88}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{27}{88}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{63}{88}\right)\)	\(e\left(\frac{27}{44}\right)\)
\(\chi_{617}(11,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{16}{77}\right)\)	\(e\left(\frac{57}{308}\right)\)	\(e\left(\frac{32}{77}\right)\)	\(e\left(\frac{125}{308}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{43}{154}\right)\)	\(e\left(\frac{48}{77}\right)\)	\(e\left(\frac{57}{154}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{15}{154}\right)\)
\(\chi_{617}(12,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{69}{154}\right)\)	\(e\left(\frac{241}{616}\right)\)	\(e\left(\frac{69}{77}\right)\)	\(e\left(\frac{361}{616}\right)\)	\(e\left(\frac{47}{56}\right)\)	\(e\left(\frac{171}{308}\right)\)	\(e\left(\frac{53}{154}\right)\)	\(e\left(\frac{241}{308}\right)\)	\(e\left(\frac{3}{88}\right)\)	\(e\left(\frac{185}{308}\right)\)
\(\chi_{617}(13,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{71}{154}\right)\)	\(e\left(\frac{277}{616}\right)\)	\(e\left(\frac{71}{77}\right)\)	\(e\left(\frac{213}{616}\right)\)	\(e\left(\frac{51}{56}\right)\)	\(e\left(\frac{263}{308}\right)\)	\(e\left(\frac{59}{154}\right)\)	\(e\left(\frac{277}{308}\right)\)	\(e\left(\frac{71}{88}\right)\)	\(e\left(\frac{81}{308}\right)\)
\(\chi_{617}(14,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{3}{77}\right)\)	\(e\left(\frac{285}{308}\right)\)	\(e\left(\frac{6}{77}\right)\)	\(e\left(\frac{9}{308}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{61}{154}\right)\)	\(e\left(\frac{9}{77}\right)\)	\(e\left(\frac{131}{154}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{75}{154}\right)\)
\(\chi_{617}(15,\cdot)\)	617.j	44	yes	\(1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(i\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{617}(16,\cdot)\)	617.l	77	yes	\(1\)	\(1\)	\(e\left(\frac{39}{77}\right)\)	\(e\left(\frac{60}{77}\right)\)	\(e\left(\frac{1}{77}\right)\)	\(e\left(\frac{10}{77}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{50}{77}\right)\)	\(e\left(\frac{40}{77}\right)\)	\(e\left(\frac{43}{77}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{64}{77}\right)\)
\(\chi_{617}(17,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{137}{154}\right)\)	\(e\left(\frac{387}{616}\right)\)	\(e\left(\frac{60}{77}\right)\)	\(e\left(\frac{411}{616}\right)\)	\(e\left(\frac{29}{56}\right)\)	\(e\left(\frac{65}{308}\right)\)	\(e\left(\frac{103}{154}\right)\)	\(e\left(\frac{79}{308}\right)\)	\(e\left(\frac{49}{88}\right)\)	\(e\left(\frac{191}{308}\right)\)
\(\chi_{617}(18,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{59}{77}\right)\)	\(e\left(\frac{215}{308}\right)\)	\(e\left(\frac{41}{77}\right)\)	\(e\left(\frac{23}{308}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{19}{154}\right)\)	\(e\left(\frac{23}{77}\right)\)	\(e\left(\frac{61}{154}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{89}{154}\right)\)
\(\chi_{617}(19,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{39}{77}\right)\)	\(e\left(\frac{9}{308}\right)\)	\(e\left(\frac{1}{77}\right)\)	\(e\left(\frac{117}{308}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{23}{154}\right)\)	\(e\left(\frac{40}{77}\right)\)	\(e\left(\frac{9}{154}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{51}{154}\right)\)
\(\chi_{617}(20,\cdot)\)	617.k	56	yes	\(-1\)	\(1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{51}{56}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{19}{56}\right)\)	\(e\left(\frac{39}{56}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{23}{28}\right)\)
\(\chi_{617}(21,\cdot)\)	617.k	56	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{13}{56}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{29}{56}\right)\)	\(e\left(\frac{33}{56}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{13}{28}\right)\)
\(\chi_{617}(22,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{45}{77}\right)\)	\(e\left(\frac{271}{308}\right)\)	\(e\left(\frac{13}{77}\right)\)	\(e\left(\frac{135}{308}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{145}{154}\right)\)	\(e\left(\frac{58}{77}\right)\)	\(e\left(\frac{117}{154}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{47}{154}\right)\)
\(\chi_{617}(23,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{103}{154}\right)\)	\(e\left(\frac{237}{616}\right)\)	\(e\left(\frac{26}{77}\right)\)	\(e\left(\frac{309}{616}\right)\)	\(e\left(\frac{3}{56}\right)\)	\(e\left(\frac{195}{308}\right)\)	\(e\left(\frac{1}{154}\right)\)	\(e\left(\frac{237}{308}\right)\)	\(e\left(\frac{15}{88}\right)\)	\(e\left(\frac{265}{308}\right)\)
\(\chi_{617}(24,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{127}{154}\right)\)	\(e\left(\frac{53}{616}\right)\)	\(e\left(\frac{50}{77}\right)\)	\(e\left(\frac{381}{616}\right)\)	\(e\left(\frac{51}{56}\right)\)	\(e\left(\frac{67}{308}\right)\)	\(e\left(\frac{73}{154}\right)\)	\(e\left(\frac{53}{308}\right)\)	\(e\left(\frac{39}{88}\right)\)	\(e\left(\frac{249}{308}\right)\)
\(\chi_{617}(25,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{5}{77}\right)\)	\(e\left(\frac{13}{308}\right)\)	\(e\left(\frac{10}{77}\right)\)	\(e\left(\frac{169}{308}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{153}{154}\right)\)	\(e\left(\frac{15}{77}\right)\)	\(e\left(\frac{13}{154}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{125}{154}\right)\)
\(\chi_{617}(26,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{129}{154}\right)\)	\(e\left(\frac{89}{616}\right)\)	\(e\left(\frac{52}{77}\right)\)	\(e\left(\frac{233}{616}\right)\)	\(e\left(\frac{55}{56}\right)\)	\(e\left(\frac{159}{308}\right)\)	\(e\left(\frac{79}{154}\right)\)	\(e\left(\frac{89}{308}\right)\)	\(e\left(\frac{19}{88}\right)\)	\(e\left(\frac{145}{308}\right)\)
\(\chi_{617}(27,\cdot)\)	617.p	616	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{154}\right)\)	\(e\left(\frac{3}{616}\right)\)	\(e\left(\frac{13}{77}\right)\)	\(e\left(\frac{347}{616}\right)\)	\(e\left(\frac{5}{56}\right)\)	\(e\left(\frac{213}{308}\right)\)	\(e\left(\frac{39}{154}\right)\)	\(e\left(\frac{3}{308}\right)\)	\(e\left(\frac{57}{88}\right)\)	\(e\left(\frac{171}{308}\right)\)
\(\chi_{617}(28,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{32}{77}\right)\)	\(e\left(\frac{191}{308}\right)\)	\(e\left(\frac{64}{77}\right)\)	\(e\left(\frac{19}{308}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{9}{154}\right)\)	\(e\left(\frac{19}{77}\right)\)	\(e\left(\frac{37}{154}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{107}{154}\right)\)
\(\chi_{617}(29,\cdot)\)	617.m	88	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{31}{88}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{88}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{31}{44}\right)\)	\(e\left(\frac{75}{88}\right)\)	\(e\left(\frac{7}{44}\right)\)
\(\chi_{617}(30,\cdot)\)	617.o	308	yes	\(1\)	\(1\)	\(e\left(\frac{8}{77}\right)\)	\(e\left(\frac{67}{308}\right)\)	\(e\left(\frac{16}{77}\right)\)	\(e\left(\frac{255}{308}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{137}{154}\right)\)	\(e\left(\frac{24}{77}\right)\)	\(e\left(\frac{67}{154}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{123}{154}\right)\)
\(\chi_{617}(31,\cdot)\)	617.f	11	yes	\(1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)
\(\chi_{617}(32,\cdot)\)	617.n	154	yes	\(1\)	\(1\)	\(e\left(\frac{68}{77}\right)\)	\(e\left(\frac{73}{154}\right)\)	\(e\left(\frac{59}{77}\right)\)	\(e\left(\frac{25}{154}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{24}{77}\right)\)	\(e\left(\frac{50}{77}\right)\)	\(e\left(\frac{73}{77}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{3}{77}\right)\)

Introduction

L-functions

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Character group

First 32 of 616 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.