LMFDB - Group of Dirichlet characters of modulus 419

Show commands: PariGP / SageMath

sage: H = DirichletGroup(419)

pari: g = idealstar(,419,2)

Character group

sage: G.order() pari: g.no
Order	=	418
sage: H.invariants() pari: g.cyc
Structure	=	$C_{418}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{419}(2,\cdot)$

First 32 of 418 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{419}(1,\cdot)$	419.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{419}(2,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{1}{418}\right)$	$e\left(\frac{50}{209}\right)$	$e\left(\frac{1}{209}\right)$	$e\left(\frac{136}{209}\right)$	$e\left(\frac{101}{418}\right)$	$e\left(\frac{2}{19}\right)$	$e\left(\frac{3}{418}\right)$	$e\left(\frac{100}{209}\right)$	$e\left(\frac{273}{418}\right)$	$e\left(\frac{287}{418}\right)$
$\chi_{419}(3,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{50}{209}\right)$	$e\left(\frac{193}{209}\right)$	$e\left(\frac{100}{209}\right)$	$e\left(\frac{15}{209}\right)$	$e\left(\frac{34}{209}\right)$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{150}{209}\right)$	$e\left(\frac{177}{209}\right)$	$e\left(\frac{65}{209}\right)$	$e\left(\frac{138}{209}\right)$
$\chi_{419}(4,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{1}{209}\right)$	$e\left(\frac{100}{209}\right)$	$e\left(\frac{2}{209}\right)$	$e\left(\frac{63}{209}\right)$	$e\left(\frac{101}{209}\right)$	$e\left(\frac{4}{19}\right)$	$e\left(\frac{3}{209}\right)$	$e\left(\frac{200}{209}\right)$	$e\left(\frac{64}{209}\right)$	$e\left(\frac{78}{209}\right)$
$\chi_{419}(5,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{136}{209}\right)$	$e\left(\frac{15}{209}\right)$	$e\left(\frac{63}{209}\right)$	$e\left(\frac{208}{209}\right)$	$e\left(\frac{151}{209}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{199}{209}\right)$	$e\left(\frac{30}{209}\right)$	$e\left(\frac{135}{209}\right)$	$e\left(\frac{158}{209}\right)$
$\chi_{419}(6,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{101}{418}\right)$	$e\left(\frac{34}{209}\right)$	$e\left(\frac{101}{209}\right)$	$e\left(\frac{151}{209}\right)$	$e\left(\frac{169}{418}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{303}{418}\right)$	$e\left(\frac{68}{209}\right)$	$e\left(\frac{403}{418}\right)$	$e\left(\frac{145}{418}\right)$
$\chi_{419}(7,\cdot)$	419.d	19	yes	$1$	$1$	$e\left(\frac{2}{19}\right)$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{4}{19}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{12}{19}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{4}{19}\right)$
$\chi_{419}(8,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{3}{418}\right)$	$e\left(\frac{150}{209}\right)$	$e\left(\frac{3}{209}\right)$	$e\left(\frac{199}{209}\right)$	$e\left(\frac{303}{418}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{9}{418}\right)$	$e\left(\frac{91}{209}\right)$	$e\left(\frac{401}{418}\right)$	$e\left(\frac{25}{418}\right)$
$\chi_{419}(9,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{100}{209}\right)$	$e\left(\frac{177}{209}\right)$	$e\left(\frac{200}{209}\right)$	$e\left(\frac{30}{209}\right)$	$e\left(\frac{68}{209}\right)$	$e\left(\frac{1}{19}\right)$	$e\left(\frac{91}{209}\right)$	$e\left(\frac{145}{209}\right)$	$e\left(\frac{130}{209}\right)$	$e\left(\frac{67}{209}\right)$
$\chi_{419}(10,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{273}{418}\right)$	$e\left(\frac{65}{209}\right)$	$e\left(\frac{64}{209}\right)$	$e\left(\frac{135}{209}\right)$	$e\left(\frac{403}{418}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{401}{418}\right)$	$e\left(\frac{130}{209}\right)$	$e\left(\frac{125}{418}\right)$	$e\left(\frac{185}{418}\right)$
$\chi_{419}(11,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{287}{418}\right)$	$e\left(\frac{138}{209}\right)$	$e\left(\frac{78}{209}\right)$	$e\left(\frac{158}{209}\right)$	$e\left(\frac{145}{418}\right)$	$e\left(\frac{4}{19}\right)$	$e\left(\frac{25}{418}\right)$	$e\left(\frac{67}{209}\right)$	$e\left(\frac{185}{418}\right)$	$e\left(\frac{23}{418}\right)$
$\chi_{419}(12,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{51}{209}\right)$	$e\left(\frac{84}{209}\right)$	$e\left(\frac{102}{209}\right)$	$e\left(\frac{78}{209}\right)$	$e\left(\frac{135}{209}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{153}{209}\right)$	$e\left(\frac{168}{209}\right)$	$e\left(\frac{129}{209}\right)$	$e\left(\frac{7}{209}\right)$
$\chi_{419}(13,\cdot)$	419.c	11	yes	$1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{11}\right)$	$1$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{8}{11}\right)$
$\chi_{419}(14,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{45}{418}\right)$	$e\left(\frac{160}{209}\right)$	$e\left(\frac{45}{209}\right)$	$e\left(\frac{59}{209}\right)$	$e\left(\frac{365}{418}\right)$	$e\left(\frac{14}{19}\right)$	$e\left(\frac{135}{418}\right)$	$e\left(\frac{111}{209}\right)$	$e\left(\frac{163}{418}\right)$	$e\left(\frac{375}{418}\right)$
$\chi_{419}(15,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{186}{209}\right)$	$e\left(\frac{208}{209}\right)$	$e\left(\frac{163}{209}\right)$	$e\left(\frac{14}{209}\right)$	$e\left(\frac{185}{209}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{140}{209}\right)$	$e\left(\frac{207}{209}\right)$	$e\left(\frac{200}{209}\right)$	$e\left(\frac{87}{209}\right)$
$\chi_{419}(16,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{2}{209}\right)$	$e\left(\frac{200}{209}\right)$	$e\left(\frac{4}{209}\right)$	$e\left(\frac{126}{209}\right)$	$e\left(\frac{202}{209}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{6}{209}\right)$	$e\left(\frac{191}{209}\right)$	$e\left(\frac{128}{209}\right)$	$e\left(\frac{156}{209}\right)$
$\chi_{419}(17,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{393}{418}\right)$	$e\left(\frac{4}{209}\right)$	$e\left(\frac{184}{209}\right)$	$e\left(\frac{153}{209}\right)$	$e\left(\frac{401}{418}\right)$	$e\left(\frac{7}{19}\right)$	$e\left(\frac{343}{418}\right)$	$e\left(\frac{8}{209}\right)$	$e\left(\frac{281}{418}\right)$	$e\left(\frac{349}{418}\right)$
$\chi_{419}(18,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{201}{418}\right)$	$e\left(\frac{18}{209}\right)$	$e\left(\frac{201}{209}\right)$	$e\left(\frac{166}{209}\right)$	$e\left(\frac{237}{418}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{185}{418}\right)$	$e\left(\frac{36}{209}\right)$	$e\left(\frac{115}{418}\right)$	$e\left(\frac{3}{418}\right)$
$\chi_{419}(19,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{339}{418}\right)$	$e\left(\frac{21}{209}\right)$	$e\left(\frac{130}{209}\right)$	$e\left(\frac{124}{209}\right)$	$e\left(\frac{381}{418}\right)$	$e\left(\frac{13}{19}\right)$	$e\left(\frac{181}{418}\right)$	$e\left(\frac{42}{209}\right)$	$e\left(\frac{169}{418}\right)$	$e\left(\frac{317}{418}\right)$
$\chi_{419}(20,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{137}{209}\right)$	$e\left(\frac{115}{209}\right)$	$e\left(\frac{65}{209}\right)$	$e\left(\frac{62}{209}\right)$	$e\left(\frac{43}{209}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{202}{209}\right)$	$e\left(\frac{21}{209}\right)$	$e\left(\frac{199}{209}\right)$	$e\left(\frac{27}{209}\right)$
$\chi_{419}(21,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{72}{209}\right)$	$e\left(\frac{94}{209}\right)$	$e\left(\frac{144}{209}\right)$	$e\left(\frac{147}{209}\right)$	$e\left(\frac{166}{209}\right)$	$e\left(\frac{3}{19}\right)$	$e\left(\frac{7}{209}\right)$	$e\left(\frac{188}{209}\right)$	$e\left(\frac{10}{209}\right)$	$e\left(\frac{182}{209}\right)$
$\chi_{419}(22,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{144}{209}\right)$	$e\left(\frac{188}{209}\right)$	$e\left(\frac{79}{209}\right)$	$e\left(\frac{85}{209}\right)$	$e\left(\frac{123}{209}\right)$	$e\left(\frac{6}{19}\right)$	$e\left(\frac{14}{209}\right)$	$e\left(\frac{167}{209}\right)$	$e\left(\frac{20}{209}\right)$	$e\left(\frac{155}{209}\right)$
$\chi_{419}(23,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{140}{209}\right)$	$e\left(\frac{206}{209}\right)$	$e\left(\frac{71}{209}\right)$	$e\left(\frac{42}{209}\right)$	$e\left(\frac{137}{209}\right)$	$e\left(\frac{9}{19}\right)$	$e\left(\frac{2}{209}\right)$	$e\left(\frac{203}{209}\right)$	$e\left(\frac{182}{209}\right)$	$e\left(\frac{52}{209}\right)$
$\chi_{419}(24,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{103}{418}\right)$	$e\left(\frac{134}{209}\right)$	$e\left(\frac{103}{209}\right)$	$e\left(\frac{5}{209}\right)$	$e\left(\frac{371}{418}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{309}{418}\right)$	$e\left(\frac{59}{209}\right)$	$e\left(\frac{113}{418}\right)$	$e\left(\frac{301}{418}\right)$
$\chi_{419}(25,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{63}{209}\right)$	$e\left(\frac{30}{209}\right)$	$e\left(\frac{126}{209}\right)$	$e\left(\frac{207}{209}\right)$	$e\left(\frac{93}{209}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{189}{209}\right)$	$e\left(\frac{60}{209}\right)$	$e\left(\frac{61}{209}\right)$	$e\left(\frac{107}{209}\right)$
$\chi_{419}(26,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{305}{418}\right)$	$e\left(\frac{202}{209}\right)$	$e\left(\frac{96}{209}\right)$	$e\left(\frac{98}{209}\right)$	$e\left(\frac{291}{418}\right)$	$e\left(\frac{2}{19}\right)$	$e\left(\frac{79}{418}\right)$	$e\left(\frac{195}{209}\right)$	$e\left(\frac{83}{418}\right)$	$e\left(\frac{173}{418}\right)$
$\chi_{419}(27,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{150}{209}\right)$	$e\left(\frac{161}{209}\right)$	$e\left(\frac{91}{209}\right)$	$e\left(\frac{45}{209}\right)$	$e\left(\frac{102}{209}\right)$	$e\left(\frac{11}{19}\right)$	$e\left(\frac{32}{209}\right)$	$e\left(\frac{113}{209}\right)$	$e\left(\frac{195}{209}\right)$	$e\left(\frac{205}{209}\right)$
$\chi_{419}(28,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{23}{209}\right)$	$e\left(\frac{1}{209}\right)$	$e\left(\frac{46}{209}\right)$	$e\left(\frac{195}{209}\right)$	$e\left(\frac{24}{209}\right)$	$e\left(\frac{16}{19}\right)$	$e\left(\frac{69}{209}\right)$	$e\left(\frac{2}{209}\right)$	$e\left(\frac{9}{209}\right)$	$e\left(\frac{122}{209}\right)$
$\chi_{419}(29,\cdot)$	419.g	209	yes	$1$	$1$	$e\left(\frac{25}{209}\right)$	$e\left(\frac{201}{209}\right)$	$e\left(\frac{50}{209}\right)$	$e\left(\frac{112}{209}\right)$	$e\left(\frac{17}{209}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{75}{209}\right)$	$e\left(\frac{193}{209}\right)$	$e\left(\frac{137}{209}\right)$	$e\left(\frac{69}{209}\right)$
$\chi_{419}(30,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{373}{418}\right)$	$e\left(\frac{49}{209}\right)$	$e\left(\frac{164}{209}\right)$	$e\left(\frac{150}{209}\right)$	$e\left(\frac{53}{418}\right)$	$e\left(\frac{5}{19}\right)$	$e\left(\frac{283}{418}\right)$	$e\left(\frac{98}{209}\right)$	$e\left(\frac{255}{418}\right)$	$e\left(\frac{43}{418}\right)$
$\chi_{419}(31,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{327}{418}\right)$	$e\left(\frac{48}{209}\right)$	$e\left(\frac{118}{209}\right)$	$e\left(\frac{164}{209}\right)$	$e\left(\frac{5}{418}\right)$	$e\left(\frac{8}{19}\right)$	$e\left(\frac{145}{418}\right)$	$e\left(\frac{96}{209}\right)$	$e\left(\frac{237}{418}\right)$	$e\left(\frac{217}{418}\right)$
$\chi_{419}(32,\cdot)$	419.h	418	yes	$-1$	$1$	$e\left(\frac{5}{418}\right)$	$e\left(\frac{41}{209}\right)$	$e\left(\frac{5}{209}\right)$	$e\left(\frac{53}{209}\right)$	$e\left(\frac{87}{418}\right)$	$e\left(\frac{10}{19}\right)$	$e\left(\frac{15}{418}\right)$	$e\left(\frac{82}{209}\right)$	$e\left(\frac{111}{418}\right)$	$e\left(\frac{181}{418}\right)$

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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	$419$
Structure	\(C_{418}\)
Order	$418$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{419}(1,\cdot)\)	419.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{419}(2,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{418}\right)\)	\(e\left(\frac{50}{209}\right)\)	\(e\left(\frac{1}{209}\right)\)	\(e\left(\frac{136}{209}\right)\)	\(e\left(\frac{101}{418}\right)\)	\(e\left(\frac{2}{19}\right)\)	\(e\left(\frac{3}{418}\right)\)	\(e\left(\frac{100}{209}\right)\)	\(e\left(\frac{273}{418}\right)\)	\(e\left(\frac{287}{418}\right)\)
\(\chi_{419}(3,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{50}{209}\right)\)	\(e\left(\frac{193}{209}\right)\)	\(e\left(\frac{100}{209}\right)\)	\(e\left(\frac{15}{209}\right)\)	\(e\left(\frac{34}{209}\right)\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{150}{209}\right)\)	\(e\left(\frac{177}{209}\right)\)	\(e\left(\frac{65}{209}\right)\)	\(e\left(\frac{138}{209}\right)\)
\(\chi_{419}(4,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{1}{209}\right)\)	\(e\left(\frac{100}{209}\right)\)	\(e\left(\frac{2}{209}\right)\)	\(e\left(\frac{63}{209}\right)\)	\(e\left(\frac{101}{209}\right)\)	\(e\left(\frac{4}{19}\right)\)	\(e\left(\frac{3}{209}\right)\)	\(e\left(\frac{200}{209}\right)\)	\(e\left(\frac{64}{209}\right)\)	\(e\left(\frac{78}{209}\right)\)
\(\chi_{419}(5,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{136}{209}\right)\)	\(e\left(\frac{15}{209}\right)\)	\(e\left(\frac{63}{209}\right)\)	\(e\left(\frac{208}{209}\right)\)	\(e\left(\frac{151}{209}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{199}{209}\right)\)	\(e\left(\frac{30}{209}\right)\)	\(e\left(\frac{135}{209}\right)\)	\(e\left(\frac{158}{209}\right)\)
\(\chi_{419}(6,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{101}{418}\right)\)	\(e\left(\frac{34}{209}\right)\)	\(e\left(\frac{101}{209}\right)\)	\(e\left(\frac{151}{209}\right)\)	\(e\left(\frac{169}{418}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{303}{418}\right)\)	\(e\left(\frac{68}{209}\right)\)	\(e\left(\frac{403}{418}\right)\)	\(e\left(\frac{145}{418}\right)\)
\(\chi_{419}(7,\cdot)\)	419.d	19	yes	\(1\)	\(1\)	\(e\left(\frac{2}{19}\right)\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{4}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{12}{19}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{4}{19}\right)\)
\(\chi_{419}(8,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{418}\right)\)	\(e\left(\frac{150}{209}\right)\)	\(e\left(\frac{3}{209}\right)\)	\(e\left(\frac{199}{209}\right)\)	\(e\left(\frac{303}{418}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{9}{418}\right)\)	\(e\left(\frac{91}{209}\right)\)	\(e\left(\frac{401}{418}\right)\)	\(e\left(\frac{25}{418}\right)\)
\(\chi_{419}(9,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{100}{209}\right)\)	\(e\left(\frac{177}{209}\right)\)	\(e\left(\frac{200}{209}\right)\)	\(e\left(\frac{30}{209}\right)\)	\(e\left(\frac{68}{209}\right)\)	\(e\left(\frac{1}{19}\right)\)	\(e\left(\frac{91}{209}\right)\)	\(e\left(\frac{145}{209}\right)\)	\(e\left(\frac{130}{209}\right)\)	\(e\left(\frac{67}{209}\right)\)
\(\chi_{419}(10,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{273}{418}\right)\)	\(e\left(\frac{65}{209}\right)\)	\(e\left(\frac{64}{209}\right)\)	\(e\left(\frac{135}{209}\right)\)	\(e\left(\frac{403}{418}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{401}{418}\right)\)	\(e\left(\frac{130}{209}\right)\)	\(e\left(\frac{125}{418}\right)\)	\(e\left(\frac{185}{418}\right)\)
\(\chi_{419}(11,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{287}{418}\right)\)	\(e\left(\frac{138}{209}\right)\)	\(e\left(\frac{78}{209}\right)\)	\(e\left(\frac{158}{209}\right)\)	\(e\left(\frac{145}{418}\right)\)	\(e\left(\frac{4}{19}\right)\)	\(e\left(\frac{25}{418}\right)\)	\(e\left(\frac{67}{209}\right)\)	\(e\left(\frac{185}{418}\right)\)	\(e\left(\frac{23}{418}\right)\)
\(\chi_{419}(12,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{51}{209}\right)\)	\(e\left(\frac{84}{209}\right)\)	\(e\left(\frac{102}{209}\right)\)	\(e\left(\frac{78}{209}\right)\)	\(e\left(\frac{135}{209}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{153}{209}\right)\)	\(e\left(\frac{168}{209}\right)\)	\(e\left(\frac{129}{209}\right)\)	\(e\left(\frac{7}{209}\right)\)
\(\chi_{419}(13,\cdot)\)	419.c	11	yes	\(1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(1\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)
\(\chi_{419}(14,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{45}{418}\right)\)	\(e\left(\frac{160}{209}\right)\)	\(e\left(\frac{45}{209}\right)\)	\(e\left(\frac{59}{209}\right)\)	\(e\left(\frac{365}{418}\right)\)	\(e\left(\frac{14}{19}\right)\)	\(e\left(\frac{135}{418}\right)\)	\(e\left(\frac{111}{209}\right)\)	\(e\left(\frac{163}{418}\right)\)	\(e\left(\frac{375}{418}\right)\)
\(\chi_{419}(15,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{186}{209}\right)\)	\(e\left(\frac{208}{209}\right)\)	\(e\left(\frac{163}{209}\right)\)	\(e\left(\frac{14}{209}\right)\)	\(e\left(\frac{185}{209}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{140}{209}\right)\)	\(e\left(\frac{207}{209}\right)\)	\(e\left(\frac{200}{209}\right)\)	\(e\left(\frac{87}{209}\right)\)
\(\chi_{419}(16,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{2}{209}\right)\)	\(e\left(\frac{200}{209}\right)\)	\(e\left(\frac{4}{209}\right)\)	\(e\left(\frac{126}{209}\right)\)	\(e\left(\frac{202}{209}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{6}{209}\right)\)	\(e\left(\frac{191}{209}\right)\)	\(e\left(\frac{128}{209}\right)\)	\(e\left(\frac{156}{209}\right)\)
\(\chi_{419}(17,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{393}{418}\right)\)	\(e\left(\frac{4}{209}\right)\)	\(e\left(\frac{184}{209}\right)\)	\(e\left(\frac{153}{209}\right)\)	\(e\left(\frac{401}{418}\right)\)	\(e\left(\frac{7}{19}\right)\)	\(e\left(\frac{343}{418}\right)\)	\(e\left(\frac{8}{209}\right)\)	\(e\left(\frac{281}{418}\right)\)	\(e\left(\frac{349}{418}\right)\)
\(\chi_{419}(18,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{201}{418}\right)\)	\(e\left(\frac{18}{209}\right)\)	\(e\left(\frac{201}{209}\right)\)	\(e\left(\frac{166}{209}\right)\)	\(e\left(\frac{237}{418}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{185}{418}\right)\)	\(e\left(\frac{36}{209}\right)\)	\(e\left(\frac{115}{418}\right)\)	\(e\left(\frac{3}{418}\right)\)
\(\chi_{419}(19,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{339}{418}\right)\)	\(e\left(\frac{21}{209}\right)\)	\(e\left(\frac{130}{209}\right)\)	\(e\left(\frac{124}{209}\right)\)	\(e\left(\frac{381}{418}\right)\)	\(e\left(\frac{13}{19}\right)\)	\(e\left(\frac{181}{418}\right)\)	\(e\left(\frac{42}{209}\right)\)	\(e\left(\frac{169}{418}\right)\)	\(e\left(\frac{317}{418}\right)\)
\(\chi_{419}(20,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{137}{209}\right)\)	\(e\left(\frac{115}{209}\right)\)	\(e\left(\frac{65}{209}\right)\)	\(e\left(\frac{62}{209}\right)\)	\(e\left(\frac{43}{209}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{202}{209}\right)\)	\(e\left(\frac{21}{209}\right)\)	\(e\left(\frac{199}{209}\right)\)	\(e\left(\frac{27}{209}\right)\)
\(\chi_{419}(21,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{72}{209}\right)\)	\(e\left(\frac{94}{209}\right)\)	\(e\left(\frac{144}{209}\right)\)	\(e\left(\frac{147}{209}\right)\)	\(e\left(\frac{166}{209}\right)\)	\(e\left(\frac{3}{19}\right)\)	\(e\left(\frac{7}{209}\right)\)	\(e\left(\frac{188}{209}\right)\)	\(e\left(\frac{10}{209}\right)\)	\(e\left(\frac{182}{209}\right)\)
\(\chi_{419}(22,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{144}{209}\right)\)	\(e\left(\frac{188}{209}\right)\)	\(e\left(\frac{79}{209}\right)\)	\(e\left(\frac{85}{209}\right)\)	\(e\left(\frac{123}{209}\right)\)	\(e\left(\frac{6}{19}\right)\)	\(e\left(\frac{14}{209}\right)\)	\(e\left(\frac{167}{209}\right)\)	\(e\left(\frac{20}{209}\right)\)	\(e\left(\frac{155}{209}\right)\)
\(\chi_{419}(23,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{140}{209}\right)\)	\(e\left(\frac{206}{209}\right)\)	\(e\left(\frac{71}{209}\right)\)	\(e\left(\frac{42}{209}\right)\)	\(e\left(\frac{137}{209}\right)\)	\(e\left(\frac{9}{19}\right)\)	\(e\left(\frac{2}{209}\right)\)	\(e\left(\frac{203}{209}\right)\)	\(e\left(\frac{182}{209}\right)\)	\(e\left(\frac{52}{209}\right)\)
\(\chi_{419}(24,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{103}{418}\right)\)	\(e\left(\frac{134}{209}\right)\)	\(e\left(\frac{103}{209}\right)\)	\(e\left(\frac{5}{209}\right)\)	\(e\left(\frac{371}{418}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{309}{418}\right)\)	\(e\left(\frac{59}{209}\right)\)	\(e\left(\frac{113}{418}\right)\)	\(e\left(\frac{301}{418}\right)\)
\(\chi_{419}(25,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{63}{209}\right)\)	\(e\left(\frac{30}{209}\right)\)	\(e\left(\frac{126}{209}\right)\)	\(e\left(\frac{207}{209}\right)\)	\(e\left(\frac{93}{209}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{189}{209}\right)\)	\(e\left(\frac{60}{209}\right)\)	\(e\left(\frac{61}{209}\right)\)	\(e\left(\frac{107}{209}\right)\)
\(\chi_{419}(26,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{305}{418}\right)\)	\(e\left(\frac{202}{209}\right)\)	\(e\left(\frac{96}{209}\right)\)	\(e\left(\frac{98}{209}\right)\)	\(e\left(\frac{291}{418}\right)\)	\(e\left(\frac{2}{19}\right)\)	\(e\left(\frac{79}{418}\right)\)	\(e\left(\frac{195}{209}\right)\)	\(e\left(\frac{83}{418}\right)\)	\(e\left(\frac{173}{418}\right)\)
\(\chi_{419}(27,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{150}{209}\right)\)	\(e\left(\frac{161}{209}\right)\)	\(e\left(\frac{91}{209}\right)\)	\(e\left(\frac{45}{209}\right)\)	\(e\left(\frac{102}{209}\right)\)	\(e\left(\frac{11}{19}\right)\)	\(e\left(\frac{32}{209}\right)\)	\(e\left(\frac{113}{209}\right)\)	\(e\left(\frac{195}{209}\right)\)	\(e\left(\frac{205}{209}\right)\)
\(\chi_{419}(28,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{23}{209}\right)\)	\(e\left(\frac{1}{209}\right)\)	\(e\left(\frac{46}{209}\right)\)	\(e\left(\frac{195}{209}\right)\)	\(e\left(\frac{24}{209}\right)\)	\(e\left(\frac{16}{19}\right)\)	\(e\left(\frac{69}{209}\right)\)	\(e\left(\frac{2}{209}\right)\)	\(e\left(\frac{9}{209}\right)\)	\(e\left(\frac{122}{209}\right)\)
\(\chi_{419}(29,\cdot)\)	419.g	209	yes	\(1\)	\(1\)	\(e\left(\frac{25}{209}\right)\)	\(e\left(\frac{201}{209}\right)\)	\(e\left(\frac{50}{209}\right)\)	\(e\left(\frac{112}{209}\right)\)	\(e\left(\frac{17}{209}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{75}{209}\right)\)	\(e\left(\frac{193}{209}\right)\)	\(e\left(\frac{137}{209}\right)\)	\(e\left(\frac{69}{209}\right)\)
\(\chi_{419}(30,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{373}{418}\right)\)	\(e\left(\frac{49}{209}\right)\)	\(e\left(\frac{164}{209}\right)\)	\(e\left(\frac{150}{209}\right)\)	\(e\left(\frac{53}{418}\right)\)	\(e\left(\frac{5}{19}\right)\)	\(e\left(\frac{283}{418}\right)\)	\(e\left(\frac{98}{209}\right)\)	\(e\left(\frac{255}{418}\right)\)	\(e\left(\frac{43}{418}\right)\)
\(\chi_{419}(31,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{327}{418}\right)\)	\(e\left(\frac{48}{209}\right)\)	\(e\left(\frac{118}{209}\right)\)	\(e\left(\frac{164}{209}\right)\)	\(e\left(\frac{5}{418}\right)\)	\(e\left(\frac{8}{19}\right)\)	\(e\left(\frac{145}{418}\right)\)	\(e\left(\frac{96}{209}\right)\)	\(e\left(\frac{237}{418}\right)\)	\(e\left(\frac{217}{418}\right)\)
\(\chi_{419}(32,\cdot)\)	419.h	418	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{418}\right)\)	\(e\left(\frac{41}{209}\right)\)	\(e\left(\frac{5}{209}\right)\)	\(e\left(\frac{53}{209}\right)\)	\(e\left(\frac{87}{418}\right)\)	\(e\left(\frac{10}{19}\right)\)	\(e\left(\frac{15}{418}\right)\)	\(e\left(\frac{82}{209}\right)\)	\(e\left(\frac{111}{418}\right)\)	\(e\left(\frac{181}{418}\right)\)

Introduction

L-functions

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First 32 of 418 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.