LMFDB - Group of Dirichlet characters of modulus 621

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(621)

gp:g = idealstar(,621,2)

magma:G = FullDirichletGroup(621);

Character group

Order	=	396	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{2}\times C_{198}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{621}(461,\cdot)$, $\chi_{621}(28,\cdot)$	comment:Generators sage:G.gens() gp:g.gen magma:Generators(G);

First 32 of 396 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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{621}(1,\cdot)$	621.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{621}(2,\cdot)$	621.w	198	yes	$-1$	$1$	$e\left(\frac{47}{198}\right)$	$e\left(\frac{47}{99}\right)$	$e\left(\frac{73}{198}\right)$	$e\left(\frac{61}{99}\right)$	$e\left(\frac{47}{66}\right)$	$e\left(\frac{20}{33}\right)$	$e\left(\frac{107}{198}\right)$	$e\left(\frac{71}{99}\right)$	$e\left(\frac{169}{198}\right)$	$e\left(\frac{94}{99}\right)$
$\chi_{621}(4,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{47}{99}\right)$	$e\left(\frac{94}{99}\right)$	$e\left(\frac{73}{99}\right)$	$e\left(\frac{23}{99}\right)$	$e\left(\frac{14}{33}\right)$	$e\left(\frac{7}{33}\right)$	$e\left(\frac{8}{99}\right)$	$e\left(\frac{43}{99}\right)$	$e\left(\frac{70}{99}\right)$	$e\left(\frac{89}{99}\right)$
$\chi_{621}(5,\cdot)$	621.v	198	yes	$1$	$1$	$e\left(\frac{73}{198}\right)$	$e\left(\frac{73}{99}\right)$	$e\left(\frac{43}{99}\right)$	$e\left(\frac{61}{198}\right)$	$e\left(\frac{7}{66}\right)$	$e\left(\frac{53}{66}\right)$	$e\left(\frac{2}{99}\right)$	$e\left(\frac{85}{99}\right)$	$e\left(\frac{67}{99}\right)$	$e\left(\frac{47}{99}\right)$
$\chi_{621}(7,\cdot)$	621.x	198	yes	$-1$	$1$	$e\left(\frac{61}{99}\right)$	$e\left(\frac{23}{99}\right)$	$e\left(\frac{61}{198}\right)$	$e\left(\frac{125}{198}\right)$	$e\left(\frac{28}{33}\right)$	$e\left(\frac{61}{66}\right)$	$e\left(\frac{65}{198}\right)$	$e\left(\frac{20}{99}\right)$	$e\left(\frac{49}{198}\right)$	$e\left(\frac{46}{99}\right)$
$\chi_{621}(8,\cdot)$	621.r	66	no	$-1$	$1$	$e\left(\frac{47}{66}\right)$	$e\left(\frac{14}{33}\right)$	$e\left(\frac{7}{66}\right)$	$e\left(\frac{28}{33}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{41}{66}\right)$	$e\left(\frac{5}{33}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{28}{33}\right)$
$\chi_{621}(10,\cdot)$	621.t	66	no	$-1$	$1$	$e\left(\frac{20}{33}\right)$	$e\left(\frac{7}{33}\right)$	$e\left(\frac{53}{66}\right)$	$e\left(\frac{61}{66}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{19}{33}\right)$	$e\left(\frac{35}{66}\right)$	$e\left(\frac{14}{33}\right)$
$\chi_{621}(11,\cdot)$	621.v	198	yes	$1$	$1$	$e\left(\frac{107}{198}\right)$	$e\left(\frac{8}{99}\right)$	$e\left(\frac{2}{99}\right)$	$e\left(\frac{65}{198}\right)$	$e\left(\frac{41}{66}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{7}{99}\right)$	$e\left(\frac{50}{99}\right)$	$e\left(\frac{86}{99}\right)$	$e\left(\frac{16}{99}\right)$
$\chi_{621}(13,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{71}{99}\right)$	$e\left(\frac{43}{99}\right)$	$e\left(\frac{85}{99}\right)$	$e\left(\frac{20}{99}\right)$	$e\left(\frac{5}{33}\right)$	$e\left(\frac{19}{33}\right)$	$e\left(\frac{50}{99}\right)$	$e\left(\frac{46}{99}\right)$	$e\left(\frac{91}{99}\right)$	$e\left(\frac{86}{99}\right)$
$\chi_{621}(14,\cdot)$	621.v	198	yes	$1$	$1$	$e\left(\frac{169}{198}\right)$	$e\left(\frac{70}{99}\right)$	$e\left(\frac{67}{99}\right)$	$e\left(\frac{49}{198}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{35}{66}\right)$	$e\left(\frac{86}{99}\right)$	$e\left(\frac{91}{99}\right)$	$e\left(\frac{10}{99}\right)$	$e\left(\frac{41}{99}\right)$
$\chi_{621}(16,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{94}{99}\right)$	$e\left(\frac{89}{99}\right)$	$e\left(\frac{47}{99}\right)$	$e\left(\frac{46}{99}\right)$	$e\left(\frac{28}{33}\right)$	$e\left(\frac{14}{33}\right)$	$e\left(\frac{16}{99}\right)$	$e\left(\frac{86}{99}\right)$	$e\left(\frac{41}{99}\right)$	$e\left(\frac{79}{99}\right)$
$\chi_{621}(17,\cdot)$	621.s	66	no	$1$	$1$	$e\left(\frac{31}{66}\right)$	$e\left(\frac{31}{33}\right)$	$e\left(\frac{16}{33}\right)$	$e\left(\frac{25}{66}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{23}{33}\right)$	$e\left(\frac{4}{33}\right)$	$e\left(\frac{28}{33}\right)$	$e\left(\frac{29}{33}\right)$
$\chi_{621}(19,\cdot)$	621.t	66	no	$-1$	$1$	$e\left(\frac{1}{33}\right)$	$e\left(\frac{2}{33}\right)$	$e\left(\frac{1}{66}\right)$	$e\left(\frac{41}{66}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{53}{66}\right)$	$e\left(\frac{29}{33}\right)$	$e\left(\frac{43}{66}\right)$	$e\left(\frac{4}{33}\right)$
$\chi_{621}(20,\cdot)$	621.v	198	yes	$1$	$1$	$e\left(\frac{167}{198}\right)$	$e\left(\frac{68}{99}\right)$	$e\left(\frac{17}{99}\right)$	$e\left(\frac{107}{198}\right)$	$e\left(\frac{35}{66}\right)$	$e\left(\frac{1}{66}\right)$	$e\left(\frac{10}{99}\right)$	$e\left(\frac{29}{99}\right)$	$e\left(\frac{38}{99}\right)$	$e\left(\frac{37}{99}\right)$
$\chi_{621}(22,\cdot)$	621.k	18	yes	$-1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{621}(25,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{73}{99}\right)$	$e\left(\frac{47}{99}\right)$	$e\left(\frac{86}{99}\right)$	$e\left(\frac{61}{99}\right)$	$e\left(\frac{7}{33}\right)$	$e\left(\frac{20}{33}\right)$	$e\left(\frac{4}{99}\right)$	$e\left(\frac{71}{99}\right)$	$e\left(\frac{35}{99}\right)$	$e\left(\frac{94}{99}\right)$
$\chi_{621}(26,\cdot)$	621.p	22	no	$-1$	$1$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{9}{11}\right)$
$\chi_{621}(28,\cdot)$	621.n	22	no	$-1$	$1$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{4}{11}\right)$
$\chi_{621}(29,\cdot)$	621.w	198	yes	$-1$	$1$	$e\left(\frac{137}{198}\right)$	$e\left(\frac{38}{99}\right)$	$e\left(\frac{19}{198}\right)$	$e\left(\frac{43}{99}\right)$	$e\left(\frac{5}{66}\right)$	$e\left(\frac{26}{33}\right)$	$e\left(\frac{17}{198}\right)$	$e\left(\frac{89}{99}\right)$	$e\left(\frac{25}{198}\right)$	$e\left(\frac{76}{99}\right)$
$\chi_{621}(31,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{65}{99}\right)$	$e\left(\frac{31}{99}\right)$	$e\left(\frac{82}{99}\right)$	$e\left(\frac{95}{99}\right)$	$e\left(\frac{32}{33}\right)$	$e\left(\frac{16}{33}\right)$	$e\left(\frac{89}{99}\right)$	$e\left(\frac{70}{99}\right)$	$e\left(\frac{61}{99}\right)$	$e\left(\frac{62}{99}\right)$
$\chi_{621}(32,\cdot)$	621.w	198	yes	$-1$	$1$	$e\left(\frac{37}{198}\right)$	$e\left(\frac{37}{99}\right)$	$e\left(\frac{167}{198}\right)$	$e\left(\frac{8}{99}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{1}{33}\right)$	$e\left(\frac{139}{198}\right)$	$e\left(\frac{58}{99}\right)$	$e\left(\frac{53}{198}\right)$	$e\left(\frac{74}{99}\right)$
$\chi_{621}(34,\cdot)$	621.x	198	yes	$-1$	$1$	$e\left(\frac{70}{99}\right)$	$e\left(\frac{41}{99}\right)$	$e\left(\frac{169}{198}\right)$	$e\left(\frac{197}{198}\right)$	$e\left(\frac{4}{33}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{47}{198}\right)$	$e\left(\frac{83}{99}\right)$	$e\left(\frac{139}{198}\right)$	$e\left(\frac{82}{99}\right)$
$\chi_{621}(35,\cdot)$	621.r	66	no	$-1$	$1$	$e\left(\frac{65}{66}\right)$	$e\left(\frac{32}{33}\right)$	$e\left(\frac{49}{66}\right)$	$e\left(\frac{31}{33}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{23}{66}\right)$	$e\left(\frac{2}{33}\right)$	$e\left(\frac{61}{66}\right)$	$e\left(\frac{31}{33}\right)$
$\chi_{621}(37,\cdot)$	621.t	66	no	$-1$	$1$	$e\left(\frac{8}{33}\right)$	$e\left(\frac{16}{33}\right)$	$e\left(\frac{41}{66}\right)$	$e\left(\frac{31}{66}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{61}{66}\right)$	$e\left(\frac{1}{33}\right)$	$e\left(\frac{47}{66}\right)$	$e\left(\frac{32}{33}\right)$
$\chi_{621}(38,\cdot)$	621.v	198	yes	$1$	$1$	$e\left(\frac{53}{198}\right)$	$e\left(\frac{53}{99}\right)$	$e\left(\frac{38}{99}\right)$	$e\left(\frac{47}{198}\right)$	$e\left(\frac{53}{66}\right)$	$e\left(\frac{43}{66}\right)$	$e\left(\frac{34}{99}\right)$	$e\left(\frac{59}{99}\right)$	$e\left(\frac{50}{99}\right)$	$e\left(\frac{7}{99}\right)$
$\chi_{621}(40,\cdot)$	621.x	198	yes	$-1$	$1$	$e\left(\frac{8}{99}\right)$	$e\left(\frac{16}{99}\right)$	$e\left(\frac{107}{198}\right)$	$e\left(\frac{31}{198}\right)$	$e\left(\frac{8}{33}\right)$	$e\left(\frac{41}{66}\right)$	$e\left(\frac{127}{198}\right)$	$e\left(\frac{1}{99}\right)$	$e\left(\frac{47}{198}\right)$	$e\left(\frac{32}{99}\right)$
$\chi_{621}(41,\cdot)$	621.w	198	yes	$-1$	$1$	$e\left(\frac{7}{198}\right)$	$e\left(\frac{7}{99}\right)$	$e\left(\frac{53}{198}\right)$	$e\left(\frac{47}{99}\right)$	$e\left(\frac{7}{66}\right)$	$e\left(\frac{10}{33}\right)$	$e\left(\frac{37}{198}\right)$	$e\left(\frac{19}{99}\right)$	$e\left(\frac{101}{198}\right)$	$e\left(\frac{14}{99}\right)$
$\chi_{621}(43,\cdot)$	621.x	198	yes	$-1$	$1$	$e\left(\frac{67}{99}\right)$	$e\left(\frac{35}{99}\right)$	$e\left(\frac{67}{198}\right)$	$e\left(\frac{173}{198}\right)$	$e\left(\frac{1}{33}\right)$	$e\left(\frac{1}{66}\right)$	$e\left(\frac{185}{198}\right)$	$e\left(\frac{95}{99}\right)$	$e\left(\frac{109}{198}\right)$	$e\left(\frac{70}{99}\right)$
$\chi_{621}(44,\cdot)$	621.s	66	no	$1$	$1$	$e\left(\frac{1}{66}\right)$	$e\left(\frac{1}{33}\right)$	$e\left(\frac{25}{33}\right)$	$e\left(\frac{37}{66}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{5}{33}\right)$	$e\left(\frac{31}{33}\right)$	$e\left(\frac{19}{33}\right)$	$e\left(\frac{2}{33}\right)$
$\chi_{621}(47,\cdot)$	621.l	18	no	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{621}(49,\cdot)$	621.u	99	yes	$1$	$1$	$e\left(\frac{23}{99}\right)$	$e\left(\frac{46}{99}\right)$	$e\left(\frac{61}{99}\right)$	$e\left(\frac{26}{99}\right)$	$e\left(\frac{23}{33}\right)$	$e\left(\frac{28}{33}\right)$	$e\left(\frac{65}{99}\right)$	$e\left(\frac{40}{99}\right)$	$e\left(\frac{49}{99}\right)$	$e\left(\frac{92}{99}\right)$
$\chi_{621}(50,\cdot)$	621.w	198	yes	$-1$	$1$	$e\left(\frac{193}{198}\right)$	$e\left(\frac{94}{99}\right)$	$e\left(\frac{47}{198}\right)$	$e\left(\frac{23}{99}\right)$	$e\left(\frac{61}{66}\right)$	$e\left(\frac{7}{33}\right)$	$e\left(\frac{115}{198}\right)$	$e\left(\frac{43}{99}\right)$	$e\left(\frac{41}{198}\right)$	$e\left(\frac{89}{99}\right)$

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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 $\F_{q}$
Belyi maps

Modulus	$621$
Structure	\(C_{2}\times C_{198}\)
Order	$396$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{621}(1,\cdot)\)	621.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{621}(2,\cdot)\)	621.w	198	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{198}\right)\)	\(e\left(\frac{47}{99}\right)\)	\(e\left(\frac{73}{198}\right)\)	\(e\left(\frac{61}{99}\right)\)	\(e\left(\frac{47}{66}\right)\)	\(e\left(\frac{20}{33}\right)\)	\(e\left(\frac{107}{198}\right)\)	\(e\left(\frac{71}{99}\right)\)	\(e\left(\frac{169}{198}\right)\)	\(e\left(\frac{94}{99}\right)\)
\(\chi_{621}(4,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{47}{99}\right)\)	\(e\left(\frac{94}{99}\right)\)	\(e\left(\frac{73}{99}\right)\)	\(e\left(\frac{23}{99}\right)\)	\(e\left(\frac{14}{33}\right)\)	\(e\left(\frac{7}{33}\right)\)	\(e\left(\frac{8}{99}\right)\)	\(e\left(\frac{43}{99}\right)\)	\(e\left(\frac{70}{99}\right)\)	\(e\left(\frac{89}{99}\right)\)
\(\chi_{621}(5,\cdot)\)	621.v	198	yes	\(1\)	\(1\)	\(e\left(\frac{73}{198}\right)\)	\(e\left(\frac{73}{99}\right)\)	\(e\left(\frac{43}{99}\right)\)	\(e\left(\frac{61}{198}\right)\)	\(e\left(\frac{7}{66}\right)\)	\(e\left(\frac{53}{66}\right)\)	\(e\left(\frac{2}{99}\right)\)	\(e\left(\frac{85}{99}\right)\)	\(e\left(\frac{67}{99}\right)\)	\(e\left(\frac{47}{99}\right)\)
\(\chi_{621}(7,\cdot)\)	621.x	198	yes	\(-1\)	\(1\)	\(e\left(\frac{61}{99}\right)\)	\(e\left(\frac{23}{99}\right)\)	\(e\left(\frac{61}{198}\right)\)	\(e\left(\frac{125}{198}\right)\)	\(e\left(\frac{28}{33}\right)\)	\(e\left(\frac{61}{66}\right)\)	\(e\left(\frac{65}{198}\right)\)	\(e\left(\frac{20}{99}\right)\)	\(e\left(\frac{49}{198}\right)\)	\(e\left(\frac{46}{99}\right)\)
\(\chi_{621}(8,\cdot)\)	621.r	66	no	\(-1\)	\(1\)	\(e\left(\frac{47}{66}\right)\)	\(e\left(\frac{14}{33}\right)\)	\(e\left(\frac{7}{66}\right)\)	\(e\left(\frac{28}{33}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{41}{66}\right)\)	\(e\left(\frac{5}{33}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{28}{33}\right)\)
\(\chi_{621}(10,\cdot)\)	621.t	66	no	\(-1\)	\(1\)	\(e\left(\frac{20}{33}\right)\)	\(e\left(\frac{7}{33}\right)\)	\(e\left(\frac{53}{66}\right)\)	\(e\left(\frac{61}{66}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{19}{33}\right)\)	\(e\left(\frac{35}{66}\right)\)	\(e\left(\frac{14}{33}\right)\)
\(\chi_{621}(11,\cdot)\)	621.v	198	yes	\(1\)	\(1\)	\(e\left(\frac{107}{198}\right)\)	\(e\left(\frac{8}{99}\right)\)	\(e\left(\frac{2}{99}\right)\)	\(e\left(\frac{65}{198}\right)\)	\(e\left(\frac{41}{66}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{7}{99}\right)\)	\(e\left(\frac{50}{99}\right)\)	\(e\left(\frac{86}{99}\right)\)	\(e\left(\frac{16}{99}\right)\)
\(\chi_{621}(13,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{71}{99}\right)\)	\(e\left(\frac{43}{99}\right)\)	\(e\left(\frac{85}{99}\right)\)	\(e\left(\frac{20}{99}\right)\)	\(e\left(\frac{5}{33}\right)\)	\(e\left(\frac{19}{33}\right)\)	\(e\left(\frac{50}{99}\right)\)	\(e\left(\frac{46}{99}\right)\)	\(e\left(\frac{91}{99}\right)\)	\(e\left(\frac{86}{99}\right)\)
\(\chi_{621}(14,\cdot)\)	621.v	198	yes	\(1\)	\(1\)	\(e\left(\frac{169}{198}\right)\)	\(e\left(\frac{70}{99}\right)\)	\(e\left(\frac{67}{99}\right)\)	\(e\left(\frac{49}{198}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{35}{66}\right)\)	\(e\left(\frac{86}{99}\right)\)	\(e\left(\frac{91}{99}\right)\)	\(e\left(\frac{10}{99}\right)\)	\(e\left(\frac{41}{99}\right)\)
\(\chi_{621}(16,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{94}{99}\right)\)	\(e\left(\frac{89}{99}\right)\)	\(e\left(\frac{47}{99}\right)\)	\(e\left(\frac{46}{99}\right)\)	\(e\left(\frac{28}{33}\right)\)	\(e\left(\frac{14}{33}\right)\)	\(e\left(\frac{16}{99}\right)\)	\(e\left(\frac{86}{99}\right)\)	\(e\left(\frac{41}{99}\right)\)	\(e\left(\frac{79}{99}\right)\)
\(\chi_{621}(17,\cdot)\)	621.s	66	no	\(1\)	\(1\)	\(e\left(\frac{31}{66}\right)\)	\(e\left(\frac{31}{33}\right)\)	\(e\left(\frac{16}{33}\right)\)	\(e\left(\frac{25}{66}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{23}{33}\right)\)	\(e\left(\frac{4}{33}\right)\)	\(e\left(\frac{28}{33}\right)\)	\(e\left(\frac{29}{33}\right)\)
\(\chi_{621}(19,\cdot)\)	621.t	66	no	\(-1\)	\(1\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{2}{33}\right)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{41}{66}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{53}{66}\right)\)	\(e\left(\frac{29}{33}\right)\)	\(e\left(\frac{43}{66}\right)\)	\(e\left(\frac{4}{33}\right)\)
\(\chi_{621}(20,\cdot)\)	621.v	198	yes	\(1\)	\(1\)	\(e\left(\frac{167}{198}\right)\)	\(e\left(\frac{68}{99}\right)\)	\(e\left(\frac{17}{99}\right)\)	\(e\left(\frac{107}{198}\right)\)	\(e\left(\frac{35}{66}\right)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{10}{99}\right)\)	\(e\left(\frac{29}{99}\right)\)	\(e\left(\frac{38}{99}\right)\)	\(e\left(\frac{37}{99}\right)\)
\(\chi_{621}(22,\cdot)\)	621.k	18	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{621}(25,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{73}{99}\right)\)	\(e\left(\frac{47}{99}\right)\)	\(e\left(\frac{86}{99}\right)\)	\(e\left(\frac{61}{99}\right)\)	\(e\left(\frac{7}{33}\right)\)	\(e\left(\frac{20}{33}\right)\)	\(e\left(\frac{4}{99}\right)\)	\(e\left(\frac{71}{99}\right)\)	\(e\left(\frac{35}{99}\right)\)	\(e\left(\frac{94}{99}\right)\)
\(\chi_{621}(26,\cdot)\)	621.p	22	no	\(-1\)	\(1\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)
\(\chi_{621}(28,\cdot)\)	621.n	22	no	\(-1\)	\(1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{4}{11}\right)\)
\(\chi_{621}(29,\cdot)\)	621.w	198	yes	\(-1\)	\(1\)	\(e\left(\frac{137}{198}\right)\)	\(e\left(\frac{38}{99}\right)\)	\(e\left(\frac{19}{198}\right)\)	\(e\left(\frac{43}{99}\right)\)	\(e\left(\frac{5}{66}\right)\)	\(e\left(\frac{26}{33}\right)\)	\(e\left(\frac{17}{198}\right)\)	\(e\left(\frac{89}{99}\right)\)	\(e\left(\frac{25}{198}\right)\)	\(e\left(\frac{76}{99}\right)\)
\(\chi_{621}(31,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{65}{99}\right)\)	\(e\left(\frac{31}{99}\right)\)	\(e\left(\frac{82}{99}\right)\)	\(e\left(\frac{95}{99}\right)\)	\(e\left(\frac{32}{33}\right)\)	\(e\left(\frac{16}{33}\right)\)	\(e\left(\frac{89}{99}\right)\)	\(e\left(\frac{70}{99}\right)\)	\(e\left(\frac{61}{99}\right)\)	\(e\left(\frac{62}{99}\right)\)
\(\chi_{621}(32,\cdot)\)	621.w	198	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{198}\right)\)	\(e\left(\frac{37}{99}\right)\)	\(e\left(\frac{167}{198}\right)\)	\(e\left(\frac{8}{99}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{139}{198}\right)\)	\(e\left(\frac{58}{99}\right)\)	\(e\left(\frac{53}{198}\right)\)	\(e\left(\frac{74}{99}\right)\)
\(\chi_{621}(34,\cdot)\)	621.x	198	yes	\(-1\)	\(1\)	\(e\left(\frac{70}{99}\right)\)	\(e\left(\frac{41}{99}\right)\)	\(e\left(\frac{169}{198}\right)\)	\(e\left(\frac{197}{198}\right)\)	\(e\left(\frac{4}{33}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{47}{198}\right)\)	\(e\left(\frac{83}{99}\right)\)	\(e\left(\frac{139}{198}\right)\)	\(e\left(\frac{82}{99}\right)\)
\(\chi_{621}(35,\cdot)\)	621.r	66	no	\(-1\)	\(1\)	\(e\left(\frac{65}{66}\right)\)	\(e\left(\frac{32}{33}\right)\)	\(e\left(\frac{49}{66}\right)\)	\(e\left(\frac{31}{33}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{23}{66}\right)\)	\(e\left(\frac{2}{33}\right)\)	\(e\left(\frac{61}{66}\right)\)	\(e\left(\frac{31}{33}\right)\)
\(\chi_{621}(37,\cdot)\)	621.t	66	no	\(-1\)	\(1\)	\(e\left(\frac{8}{33}\right)\)	\(e\left(\frac{16}{33}\right)\)	\(e\left(\frac{41}{66}\right)\)	\(e\left(\frac{31}{66}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{61}{66}\right)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{47}{66}\right)\)	\(e\left(\frac{32}{33}\right)\)
\(\chi_{621}(38,\cdot)\)	621.v	198	yes	\(1\)	\(1\)	\(e\left(\frac{53}{198}\right)\)	\(e\left(\frac{53}{99}\right)\)	\(e\left(\frac{38}{99}\right)\)	\(e\left(\frac{47}{198}\right)\)	\(e\left(\frac{53}{66}\right)\)	\(e\left(\frac{43}{66}\right)\)	\(e\left(\frac{34}{99}\right)\)	\(e\left(\frac{59}{99}\right)\)	\(e\left(\frac{50}{99}\right)\)	\(e\left(\frac{7}{99}\right)\)
\(\chi_{621}(40,\cdot)\)	621.x	198	yes	\(-1\)	\(1\)	\(e\left(\frac{8}{99}\right)\)	\(e\left(\frac{16}{99}\right)\)	\(e\left(\frac{107}{198}\right)\)	\(e\left(\frac{31}{198}\right)\)	\(e\left(\frac{8}{33}\right)\)	\(e\left(\frac{41}{66}\right)\)	\(e\left(\frac{127}{198}\right)\)	\(e\left(\frac{1}{99}\right)\)	\(e\left(\frac{47}{198}\right)\)	\(e\left(\frac{32}{99}\right)\)
\(\chi_{621}(41,\cdot)\)	621.w	198	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{198}\right)\)	\(e\left(\frac{7}{99}\right)\)	\(e\left(\frac{53}{198}\right)\)	\(e\left(\frac{47}{99}\right)\)	\(e\left(\frac{7}{66}\right)\)	\(e\left(\frac{10}{33}\right)\)	\(e\left(\frac{37}{198}\right)\)	\(e\left(\frac{19}{99}\right)\)	\(e\left(\frac{101}{198}\right)\)	\(e\left(\frac{14}{99}\right)\)
\(\chi_{621}(43,\cdot)\)	621.x	198	yes	\(-1\)	\(1\)	\(e\left(\frac{67}{99}\right)\)	\(e\left(\frac{35}{99}\right)\)	\(e\left(\frac{67}{198}\right)\)	\(e\left(\frac{173}{198}\right)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{185}{198}\right)\)	\(e\left(\frac{95}{99}\right)\)	\(e\left(\frac{109}{198}\right)\)	\(e\left(\frac{70}{99}\right)\)
\(\chi_{621}(44,\cdot)\)	621.s	66	no	\(1\)	\(1\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{25}{33}\right)\)	\(e\left(\frac{37}{66}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{5}{33}\right)\)	\(e\left(\frac{31}{33}\right)\)	\(e\left(\frac{19}{33}\right)\)	\(e\left(\frac{2}{33}\right)\)
\(\chi_{621}(47,\cdot)\)	621.l	18	no	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{621}(49,\cdot)\)	621.u	99	yes	\(1\)	\(1\)	\(e\left(\frac{23}{99}\right)\)	\(e\left(\frac{46}{99}\right)\)	\(e\left(\frac{61}{99}\right)\)	\(e\left(\frac{26}{99}\right)\)	\(e\left(\frac{23}{33}\right)\)	\(e\left(\frac{28}{33}\right)\)	\(e\left(\frac{65}{99}\right)\)	\(e\left(\frac{40}{99}\right)\)	\(e\left(\frac{49}{99}\right)\)	\(e\left(\frac{92}{99}\right)\)
\(\chi_{621}(50,\cdot)\)	621.w	198	yes	\(-1\)	\(1\)	\(e\left(\frac{193}{198}\right)\)	\(e\left(\frac{94}{99}\right)\)	\(e\left(\frac{47}{198}\right)\)	\(e\left(\frac{23}{99}\right)\)	\(e\left(\frac{61}{66}\right)\)	\(e\left(\frac{7}{33}\right)\)	\(e\left(\frac{115}{198}\right)\)	\(e\left(\frac{43}{99}\right)\)	\(e\left(\frac{41}{198}\right)\)	\(e\left(\frac{89}{99}\right)\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.