LMFDB - Group of Dirichlet characters of modulus 173

Show commands: PariGP / SageMath

sage: H = DirichletGroup(173)

pari: g = idealstar(,173,2)

Character group

sage: G.order() pari: g.no
Order	=	172
sage: H.invariants() pari: g.cyc
Structure	=	$C_{172}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{173}(2,\cdot)$

First 32 of 172 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{173}(1,\cdot)$	173.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{173}(2,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{1}{172}\right)$	$e\left(\frac{27}{172}\right)$	$e\left(\frac{1}{86}\right)$	$e\left(\frac{39}{172}\right)$	$e\left(\frac{7}{43}\right)$	$e\left(\frac{95}{172}\right)$	$e\left(\frac{3}{172}\right)$	$e\left(\frac{27}{86}\right)$	$e\left(\frac{10}{43}\right)$	$e\left(\frac{23}{172}\right)$
$\chi_{173}(3,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{27}{172}\right)$	$e\left(\frac{41}{172}\right)$	$e\left(\frac{27}{86}\right)$	$e\left(\frac{21}{172}\right)$	$e\left(\frac{17}{43}\right)$	$e\left(\frac{157}{172}\right)$	$e\left(\frac{81}{172}\right)$	$e\left(\frac{41}{86}\right)$	$e\left(\frac{12}{43}\right)$	$e\left(\frac{105}{172}\right)$
$\chi_{173}(4,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{1}{86}\right)$	$e\left(\frac{27}{86}\right)$	$e\left(\frac{1}{43}\right)$	$e\left(\frac{39}{86}\right)$	$e\left(\frac{14}{43}\right)$	$e\left(\frac{9}{86}\right)$	$e\left(\frac{3}{86}\right)$	$e\left(\frac{27}{43}\right)$	$e\left(\frac{20}{43}\right)$	$e\left(\frac{23}{86}\right)$
$\chi_{173}(5,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{39}{172}\right)$	$e\left(\frac{21}{172}\right)$	$e\left(\frac{39}{86}\right)$	$e\left(\frac{145}{172}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{93}{172}\right)$	$e\left(\frac{117}{172}\right)$	$e\left(\frac{21}{86}\right)$	$e\left(\frac{3}{43}\right)$	$e\left(\frac{37}{172}\right)$
$\chi_{173}(6,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{7}{43}\right)$	$e\left(\frac{17}{43}\right)$	$e\left(\frac{14}{43}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{24}{43}\right)$	$e\left(\frac{20}{43}\right)$	$e\left(\frac{21}{43}\right)$	$e\left(\frac{34}{43}\right)$	$e\left(\frac{22}{43}\right)$	$e\left(\frac{32}{43}\right)$
$\chi_{173}(7,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{95}{172}\right)$	$e\left(\frac{157}{172}\right)$	$e\left(\frac{9}{86}\right)$	$e\left(\frac{93}{172}\right)$	$e\left(\frac{20}{43}\right)$	$e\left(\frac{81}{172}\right)$	$e\left(\frac{113}{172}\right)$	$e\left(\frac{71}{86}\right)$	$e\left(\frac{4}{43}\right)$	$e\left(\frac{121}{172}\right)$
$\chi_{173}(8,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{3}{172}\right)$	$e\left(\frac{81}{172}\right)$	$e\left(\frac{3}{86}\right)$	$e\left(\frac{117}{172}\right)$	$e\left(\frac{21}{43}\right)$	$e\left(\frac{113}{172}\right)$	$e\left(\frac{9}{172}\right)$	$e\left(\frac{81}{86}\right)$	$e\left(\frac{30}{43}\right)$	$e\left(\frac{69}{172}\right)$
$\chi_{173}(9,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{27}{86}\right)$	$e\left(\frac{41}{86}\right)$	$e\left(\frac{27}{43}\right)$	$e\left(\frac{21}{86}\right)$	$e\left(\frac{34}{43}\right)$	$e\left(\frac{71}{86}\right)$	$e\left(\frac{81}{86}\right)$	$e\left(\frac{41}{43}\right)$	$e\left(\frac{24}{43}\right)$	$e\left(\frac{19}{86}\right)$
$\chi_{173}(10,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{10}{43}\right)$	$e\left(\frac{12}{43}\right)$	$e\left(\frac{20}{43}\right)$	$e\left(\frac{3}{43}\right)$	$e\left(\frac{22}{43}\right)$	$e\left(\frac{4}{43}\right)$	$e\left(\frac{30}{43}\right)$	$e\left(\frac{24}{43}\right)$	$e\left(\frac{13}{43}\right)$	$e\left(\frac{15}{43}\right)$
$\chi_{173}(11,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{23}{172}\right)$	$e\left(\frac{105}{172}\right)$	$e\left(\frac{23}{86}\right)$	$e\left(\frac{37}{172}\right)$	$e\left(\frac{32}{43}\right)$	$e\left(\frac{121}{172}\right)$	$e\left(\frac{69}{172}\right)$	$e\left(\frac{19}{86}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{13}{172}\right)$
$\chi_{173}(12,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{29}{172}\right)$	$e\left(\frac{95}{172}\right)$	$e\left(\frac{29}{86}\right)$	$e\left(\frac{99}{172}\right)$	$e\left(\frac{31}{43}\right)$	$e\left(\frac{3}{172}\right)$	$e\left(\frac{87}{172}\right)$	$e\left(\frac{9}{86}\right)$	$e\left(\frac{32}{43}\right)$	$e\left(\frac{151}{172}\right)$
$\chi_{173}(13,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{65}{86}\right)$	$e\left(\frac{35}{86}\right)$	$e\left(\frac{22}{43}\right)$	$e\left(\frac{41}{86}\right)$	$e\left(\frac{7}{43}\right)$	$e\left(\frac{69}{86}\right)$	$e\left(\frac{23}{86}\right)$	$e\left(\frac{35}{43}\right)$	$e\left(\frac{10}{43}\right)$	$e\left(\frac{33}{86}\right)$
$\chi_{173}(14,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{24}{43}\right)$	$e\left(\frac{3}{43}\right)$	$e\left(\frac{5}{43}\right)$	$e\left(\frac{33}{43}\right)$	$e\left(\frac{27}{43}\right)$	$e\left(\frac{1}{43}\right)$	$e\left(\frac{29}{43}\right)$	$e\left(\frac{6}{43}\right)$	$e\left(\frac{14}{43}\right)$	$e\left(\frac{36}{43}\right)$
$\chi_{173}(15,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{33}{86}\right)$	$e\left(\frac{31}{86}\right)$	$e\left(\frac{33}{43}\right)$	$e\left(\frac{83}{86}\right)$	$e\left(\frac{32}{43}\right)$	$e\left(\frac{39}{86}\right)$	$e\left(\frac{13}{86}\right)$	$e\left(\frac{31}{43}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{71}{86}\right)$
$\chi_{173}(16,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{1}{43}\right)$	$e\left(\frac{27}{43}\right)$	$e\left(\frac{2}{43}\right)$	$e\left(\frac{39}{43}\right)$	$e\left(\frac{28}{43}\right)$	$e\left(\frac{9}{43}\right)$	$e\left(\frac{3}{43}\right)$	$e\left(\frac{11}{43}\right)$	$e\left(\frac{40}{43}\right)$	$e\left(\frac{23}{43}\right)$
$\chi_{173}(17,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{73}{172}\right)$	$e\left(\frac{79}{172}\right)$	$e\left(\frac{73}{86}\right)$	$e\left(\frac{95}{172}\right)$	$e\left(\frac{38}{43}\right)$	$e\left(\frac{55}{172}\right)$	$e\left(\frac{47}{172}\right)$	$e\left(\frac{79}{86}\right)$	$e\left(\frac{42}{43}\right)$	$e\left(\frac{131}{172}\right)$
$\chi_{173}(18,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{55}{172}\right)$	$e\left(\frac{109}{172}\right)$	$e\left(\frac{55}{86}\right)$	$e\left(\frac{81}{172}\right)$	$e\left(\frac{41}{43}\right)$	$e\left(\frac{65}{172}\right)$	$e\left(\frac{165}{172}\right)$	$e\left(\frac{23}{86}\right)$	$e\left(\frac{34}{43}\right)$	$e\left(\frac{61}{172}\right)$
$\chi_{173}(19,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{33}{172}\right)$	$e\left(\frac{31}{172}\right)$	$e\left(\frac{33}{86}\right)$	$e\left(\frac{83}{172}\right)$	$e\left(\frac{16}{43}\right)$	$e\left(\frac{39}{172}\right)$	$e\left(\frac{99}{172}\right)$	$e\left(\frac{31}{86}\right)$	$e\left(\frac{29}{43}\right)$	$e\left(\frac{71}{172}\right)$
$\chi_{173}(20,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{41}{172}\right)$	$e\left(\frac{75}{172}\right)$	$e\left(\frac{41}{86}\right)$	$e\left(\frac{51}{172}\right)$	$e\left(\frac{29}{43}\right)$	$e\left(\frac{111}{172}\right)$	$e\left(\frac{123}{172}\right)$	$e\left(\frac{75}{86}\right)$	$e\left(\frac{23}{43}\right)$	$e\left(\frac{83}{172}\right)$
$\chi_{173}(21,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{61}{86}\right)$	$e\left(\frac{13}{86}\right)$	$e\left(\frac{18}{43}\right)$	$e\left(\frac{57}{86}\right)$	$e\left(\frac{37}{43}\right)$	$e\left(\frac{33}{86}\right)$	$e\left(\frac{11}{86}\right)$	$e\left(\frac{13}{43}\right)$	$e\left(\frac{16}{43}\right)$	$e\left(\frac{27}{86}\right)$
$\chi_{173}(22,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{6}{43}\right)$	$e\left(\frac{33}{43}\right)$	$e\left(\frac{12}{43}\right)$	$e\left(\frac{19}{43}\right)$	$e\left(\frac{39}{43}\right)$	$e\left(\frac{11}{43}\right)$	$e\left(\frac{18}{43}\right)$	$e\left(\frac{23}{43}\right)$	$e\left(\frac{25}{43}\right)$	$e\left(\frac{9}{43}\right)$
$\chi_{173}(23,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{5}{43}\right)$	$e\left(\frac{6}{43}\right)$	$e\left(\frac{10}{43}\right)$	$e\left(\frac{23}{43}\right)$	$e\left(\frac{11}{43}\right)$	$e\left(\frac{2}{43}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{12}{43}\right)$	$e\left(\frac{28}{43}\right)$	$e\left(\frac{29}{43}\right)$
$\chi_{173}(24,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{15}{86}\right)$	$e\left(\frac{61}{86}\right)$	$e\left(\frac{15}{43}\right)$	$e\left(\frac{69}{86}\right)$	$e\left(\frac{38}{43}\right)$	$e\left(\frac{49}{86}\right)$	$e\left(\frac{45}{86}\right)$	$e\left(\frac{18}{43}\right)$	$e\left(\frac{42}{43}\right)$	$e\left(\frac{1}{86}\right)$
$\chi_{173}(25,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{39}{86}\right)$	$e\left(\frac{21}{86}\right)$	$e\left(\frac{39}{43}\right)$	$e\left(\frac{59}{86}\right)$	$e\left(\frac{30}{43}\right)$	$e\left(\frac{7}{86}\right)$	$e\left(\frac{31}{86}\right)$	$e\left(\frac{21}{43}\right)$	$e\left(\frac{6}{43}\right)$	$e\left(\frac{37}{86}\right)$
$\chi_{173}(26,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{131}{172}\right)$	$e\left(\frac{97}{172}\right)$	$e\left(\frac{45}{86}\right)$	$e\left(\frac{121}{172}\right)$	$e\left(\frac{14}{43}\right)$	$e\left(\frac{61}{172}\right)$	$e\left(\frac{49}{172}\right)$	$e\left(\frac{11}{86}\right)$	$e\left(\frac{20}{43}\right)$	$e\left(\frac{89}{172}\right)$
$\chi_{173}(27,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{81}{172}\right)$	$e\left(\frac{123}{172}\right)$	$e\left(\frac{81}{86}\right)$	$e\left(\frac{63}{172}\right)$	$e\left(\frac{8}{43}\right)$	$e\left(\frac{127}{172}\right)$	$e\left(\frac{71}{172}\right)$	$e\left(\frac{37}{86}\right)$	$e\left(\frac{36}{43}\right)$	$e\left(\frac{143}{172}\right)$
$\chi_{173}(28,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{97}{172}\right)$	$e\left(\frac{39}{172}\right)$	$e\left(\frac{11}{86}\right)$	$e\left(\frac{171}{172}\right)$	$e\left(\frac{34}{43}\right)$	$e\left(\frac{99}{172}\right)$	$e\left(\frac{119}{172}\right)$	$e\left(\frac{39}{86}\right)$	$e\left(\frac{24}{43}\right)$	$e\left(\frac{167}{172}\right)$
$\chi_{173}(29,\cdot)$	173.d	43	yes	$1$	$1$	$e\left(\frac{36}{43}\right)$	$e\left(\frac{26}{43}\right)$	$e\left(\frac{29}{43}\right)$	$e\left(\frac{28}{43}\right)$	$e\left(\frac{19}{43}\right)$	$e\left(\frac{23}{43}\right)$	$e\left(\frac{22}{43}\right)$	$e\left(\frac{9}{43}\right)$	$e\left(\frac{21}{43}\right)$	$e\left(\frac{11}{43}\right)$
$\chi_{173}(30,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{67}{172}\right)$	$e\left(\frac{89}{172}\right)$	$e\left(\frac{67}{86}\right)$	$e\left(\frac{33}{172}\right)$	$e\left(\frac{39}{43}\right)$	$e\left(\frac{1}{172}\right)$	$e\left(\frac{29}{172}\right)$	$e\left(\frac{3}{86}\right)$	$e\left(\frac{25}{43}\right)$	$e\left(\frac{165}{172}\right)$
$\chi_{173}(31,\cdot)$	173.e	86	yes	$1$	$1$	$e\left(\frac{51}{86}\right)$	$e\left(\frac{1}{86}\right)$	$e\left(\frac{8}{43}\right)$	$e\left(\frac{11}{86}\right)$	$e\left(\frac{26}{43}\right)$	$e\left(\frac{29}{86}\right)$	$e\left(\frac{67}{86}\right)$	$e\left(\frac{1}{43}\right)$	$e\left(\frac{31}{43}\right)$	$e\left(\frac{55}{86}\right)$
$\chi_{173}(32,\cdot)$	173.f	172	yes	$-1$	$1$	$e\left(\frac{5}{172}\right)$	$e\left(\frac{135}{172}\right)$	$e\left(\frac{5}{86}\right)$	$e\left(\frac{23}{172}\right)$	$e\left(\frac{35}{43}\right)$	$e\left(\frac{131}{172}\right)$	$e\left(\frac{15}{172}\right)$	$e\left(\frac{49}{86}\right)$	$e\left(\frac{7}{43}\right)$	$e\left(\frac{115}{172}\right)$

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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	$173$
Structure	\(C_{172}\)
Order	$172$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{173}(1,\cdot)\)	173.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{173}(2,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{172}\right)\)	\(e\left(\frac{27}{172}\right)\)	\(e\left(\frac{1}{86}\right)\)	\(e\left(\frac{39}{172}\right)\)	\(e\left(\frac{7}{43}\right)\)	\(e\left(\frac{95}{172}\right)\)	\(e\left(\frac{3}{172}\right)\)	\(e\left(\frac{27}{86}\right)\)	\(e\left(\frac{10}{43}\right)\)	\(e\left(\frac{23}{172}\right)\)
\(\chi_{173}(3,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{27}{172}\right)\)	\(e\left(\frac{41}{172}\right)\)	\(e\left(\frac{27}{86}\right)\)	\(e\left(\frac{21}{172}\right)\)	\(e\left(\frac{17}{43}\right)\)	\(e\left(\frac{157}{172}\right)\)	\(e\left(\frac{81}{172}\right)\)	\(e\left(\frac{41}{86}\right)\)	\(e\left(\frac{12}{43}\right)\)	\(e\left(\frac{105}{172}\right)\)
\(\chi_{173}(4,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{1}{86}\right)\)	\(e\left(\frac{27}{86}\right)\)	\(e\left(\frac{1}{43}\right)\)	\(e\left(\frac{39}{86}\right)\)	\(e\left(\frac{14}{43}\right)\)	\(e\left(\frac{9}{86}\right)\)	\(e\left(\frac{3}{86}\right)\)	\(e\left(\frac{27}{43}\right)\)	\(e\left(\frac{20}{43}\right)\)	\(e\left(\frac{23}{86}\right)\)
\(\chi_{173}(5,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{39}{172}\right)\)	\(e\left(\frac{21}{172}\right)\)	\(e\left(\frac{39}{86}\right)\)	\(e\left(\frac{145}{172}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{93}{172}\right)\)	\(e\left(\frac{117}{172}\right)\)	\(e\left(\frac{21}{86}\right)\)	\(e\left(\frac{3}{43}\right)\)	\(e\left(\frac{37}{172}\right)\)
\(\chi_{173}(6,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{7}{43}\right)\)	\(e\left(\frac{17}{43}\right)\)	\(e\left(\frac{14}{43}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{24}{43}\right)\)	\(e\left(\frac{20}{43}\right)\)	\(e\left(\frac{21}{43}\right)\)	\(e\left(\frac{34}{43}\right)\)	\(e\left(\frac{22}{43}\right)\)	\(e\left(\frac{32}{43}\right)\)
\(\chi_{173}(7,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{95}{172}\right)\)	\(e\left(\frac{157}{172}\right)\)	\(e\left(\frac{9}{86}\right)\)	\(e\left(\frac{93}{172}\right)\)	\(e\left(\frac{20}{43}\right)\)	\(e\left(\frac{81}{172}\right)\)	\(e\left(\frac{113}{172}\right)\)	\(e\left(\frac{71}{86}\right)\)	\(e\left(\frac{4}{43}\right)\)	\(e\left(\frac{121}{172}\right)\)
\(\chi_{173}(8,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{172}\right)\)	\(e\left(\frac{81}{172}\right)\)	\(e\left(\frac{3}{86}\right)\)	\(e\left(\frac{117}{172}\right)\)	\(e\left(\frac{21}{43}\right)\)	\(e\left(\frac{113}{172}\right)\)	\(e\left(\frac{9}{172}\right)\)	\(e\left(\frac{81}{86}\right)\)	\(e\left(\frac{30}{43}\right)\)	\(e\left(\frac{69}{172}\right)\)
\(\chi_{173}(9,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{27}{86}\right)\)	\(e\left(\frac{41}{86}\right)\)	\(e\left(\frac{27}{43}\right)\)	\(e\left(\frac{21}{86}\right)\)	\(e\left(\frac{34}{43}\right)\)	\(e\left(\frac{71}{86}\right)\)	\(e\left(\frac{81}{86}\right)\)	\(e\left(\frac{41}{43}\right)\)	\(e\left(\frac{24}{43}\right)\)	\(e\left(\frac{19}{86}\right)\)
\(\chi_{173}(10,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{10}{43}\right)\)	\(e\left(\frac{12}{43}\right)\)	\(e\left(\frac{20}{43}\right)\)	\(e\left(\frac{3}{43}\right)\)	\(e\left(\frac{22}{43}\right)\)	\(e\left(\frac{4}{43}\right)\)	\(e\left(\frac{30}{43}\right)\)	\(e\left(\frac{24}{43}\right)\)	\(e\left(\frac{13}{43}\right)\)	\(e\left(\frac{15}{43}\right)\)
\(\chi_{173}(11,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{172}\right)\)	\(e\left(\frac{105}{172}\right)\)	\(e\left(\frac{23}{86}\right)\)	\(e\left(\frac{37}{172}\right)\)	\(e\left(\frac{32}{43}\right)\)	\(e\left(\frac{121}{172}\right)\)	\(e\left(\frac{69}{172}\right)\)	\(e\left(\frac{19}{86}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{13}{172}\right)\)
\(\chi_{173}(12,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{172}\right)\)	\(e\left(\frac{95}{172}\right)\)	\(e\left(\frac{29}{86}\right)\)	\(e\left(\frac{99}{172}\right)\)	\(e\left(\frac{31}{43}\right)\)	\(e\left(\frac{3}{172}\right)\)	\(e\left(\frac{87}{172}\right)\)	\(e\left(\frac{9}{86}\right)\)	\(e\left(\frac{32}{43}\right)\)	\(e\left(\frac{151}{172}\right)\)
\(\chi_{173}(13,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{65}{86}\right)\)	\(e\left(\frac{35}{86}\right)\)	\(e\left(\frac{22}{43}\right)\)	\(e\left(\frac{41}{86}\right)\)	\(e\left(\frac{7}{43}\right)\)	\(e\left(\frac{69}{86}\right)\)	\(e\left(\frac{23}{86}\right)\)	\(e\left(\frac{35}{43}\right)\)	\(e\left(\frac{10}{43}\right)\)	\(e\left(\frac{33}{86}\right)\)
\(\chi_{173}(14,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{24}{43}\right)\)	\(e\left(\frac{3}{43}\right)\)	\(e\left(\frac{5}{43}\right)\)	\(e\left(\frac{33}{43}\right)\)	\(e\left(\frac{27}{43}\right)\)	\(e\left(\frac{1}{43}\right)\)	\(e\left(\frac{29}{43}\right)\)	\(e\left(\frac{6}{43}\right)\)	\(e\left(\frac{14}{43}\right)\)	\(e\left(\frac{36}{43}\right)\)
\(\chi_{173}(15,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{33}{86}\right)\)	\(e\left(\frac{31}{86}\right)\)	\(e\left(\frac{33}{43}\right)\)	\(e\left(\frac{83}{86}\right)\)	\(e\left(\frac{32}{43}\right)\)	\(e\left(\frac{39}{86}\right)\)	\(e\left(\frac{13}{86}\right)\)	\(e\left(\frac{31}{43}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{71}{86}\right)\)
\(\chi_{173}(16,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{1}{43}\right)\)	\(e\left(\frac{27}{43}\right)\)	\(e\left(\frac{2}{43}\right)\)	\(e\left(\frac{39}{43}\right)\)	\(e\left(\frac{28}{43}\right)\)	\(e\left(\frac{9}{43}\right)\)	\(e\left(\frac{3}{43}\right)\)	\(e\left(\frac{11}{43}\right)\)	\(e\left(\frac{40}{43}\right)\)	\(e\left(\frac{23}{43}\right)\)
\(\chi_{173}(17,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{73}{172}\right)\)	\(e\left(\frac{79}{172}\right)\)	\(e\left(\frac{73}{86}\right)\)	\(e\left(\frac{95}{172}\right)\)	\(e\left(\frac{38}{43}\right)\)	\(e\left(\frac{55}{172}\right)\)	\(e\left(\frac{47}{172}\right)\)	\(e\left(\frac{79}{86}\right)\)	\(e\left(\frac{42}{43}\right)\)	\(e\left(\frac{131}{172}\right)\)
\(\chi_{173}(18,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{55}{172}\right)\)	\(e\left(\frac{109}{172}\right)\)	\(e\left(\frac{55}{86}\right)\)	\(e\left(\frac{81}{172}\right)\)	\(e\left(\frac{41}{43}\right)\)	\(e\left(\frac{65}{172}\right)\)	\(e\left(\frac{165}{172}\right)\)	\(e\left(\frac{23}{86}\right)\)	\(e\left(\frac{34}{43}\right)\)	\(e\left(\frac{61}{172}\right)\)
\(\chi_{173}(19,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{33}{172}\right)\)	\(e\left(\frac{31}{172}\right)\)	\(e\left(\frac{33}{86}\right)\)	\(e\left(\frac{83}{172}\right)\)	\(e\left(\frac{16}{43}\right)\)	\(e\left(\frac{39}{172}\right)\)	\(e\left(\frac{99}{172}\right)\)	\(e\left(\frac{31}{86}\right)\)	\(e\left(\frac{29}{43}\right)\)	\(e\left(\frac{71}{172}\right)\)
\(\chi_{173}(20,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{41}{172}\right)\)	\(e\left(\frac{75}{172}\right)\)	\(e\left(\frac{41}{86}\right)\)	\(e\left(\frac{51}{172}\right)\)	\(e\left(\frac{29}{43}\right)\)	\(e\left(\frac{111}{172}\right)\)	\(e\left(\frac{123}{172}\right)\)	\(e\left(\frac{75}{86}\right)\)	\(e\left(\frac{23}{43}\right)\)	\(e\left(\frac{83}{172}\right)\)
\(\chi_{173}(21,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{61}{86}\right)\)	\(e\left(\frac{13}{86}\right)\)	\(e\left(\frac{18}{43}\right)\)	\(e\left(\frac{57}{86}\right)\)	\(e\left(\frac{37}{43}\right)\)	\(e\left(\frac{33}{86}\right)\)	\(e\left(\frac{11}{86}\right)\)	\(e\left(\frac{13}{43}\right)\)	\(e\left(\frac{16}{43}\right)\)	\(e\left(\frac{27}{86}\right)\)
\(\chi_{173}(22,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{6}{43}\right)\)	\(e\left(\frac{33}{43}\right)\)	\(e\left(\frac{12}{43}\right)\)	\(e\left(\frac{19}{43}\right)\)	\(e\left(\frac{39}{43}\right)\)	\(e\left(\frac{11}{43}\right)\)	\(e\left(\frac{18}{43}\right)\)	\(e\left(\frac{23}{43}\right)\)	\(e\left(\frac{25}{43}\right)\)	\(e\left(\frac{9}{43}\right)\)
\(\chi_{173}(23,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{5}{43}\right)\)	\(e\left(\frac{6}{43}\right)\)	\(e\left(\frac{10}{43}\right)\)	\(e\left(\frac{23}{43}\right)\)	\(e\left(\frac{11}{43}\right)\)	\(e\left(\frac{2}{43}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{12}{43}\right)\)	\(e\left(\frac{28}{43}\right)\)	\(e\left(\frac{29}{43}\right)\)
\(\chi_{173}(24,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{15}{86}\right)\)	\(e\left(\frac{61}{86}\right)\)	\(e\left(\frac{15}{43}\right)\)	\(e\left(\frac{69}{86}\right)\)	\(e\left(\frac{38}{43}\right)\)	\(e\left(\frac{49}{86}\right)\)	\(e\left(\frac{45}{86}\right)\)	\(e\left(\frac{18}{43}\right)\)	\(e\left(\frac{42}{43}\right)\)	\(e\left(\frac{1}{86}\right)\)
\(\chi_{173}(25,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{39}{86}\right)\)	\(e\left(\frac{21}{86}\right)\)	\(e\left(\frac{39}{43}\right)\)	\(e\left(\frac{59}{86}\right)\)	\(e\left(\frac{30}{43}\right)\)	\(e\left(\frac{7}{86}\right)\)	\(e\left(\frac{31}{86}\right)\)	\(e\left(\frac{21}{43}\right)\)	\(e\left(\frac{6}{43}\right)\)	\(e\left(\frac{37}{86}\right)\)
\(\chi_{173}(26,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{131}{172}\right)\)	\(e\left(\frac{97}{172}\right)\)	\(e\left(\frac{45}{86}\right)\)	\(e\left(\frac{121}{172}\right)\)	\(e\left(\frac{14}{43}\right)\)	\(e\left(\frac{61}{172}\right)\)	\(e\left(\frac{49}{172}\right)\)	\(e\left(\frac{11}{86}\right)\)	\(e\left(\frac{20}{43}\right)\)	\(e\left(\frac{89}{172}\right)\)
\(\chi_{173}(27,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{81}{172}\right)\)	\(e\left(\frac{123}{172}\right)\)	\(e\left(\frac{81}{86}\right)\)	\(e\left(\frac{63}{172}\right)\)	\(e\left(\frac{8}{43}\right)\)	\(e\left(\frac{127}{172}\right)\)	\(e\left(\frac{71}{172}\right)\)	\(e\left(\frac{37}{86}\right)\)	\(e\left(\frac{36}{43}\right)\)	\(e\left(\frac{143}{172}\right)\)
\(\chi_{173}(28,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{97}{172}\right)\)	\(e\left(\frac{39}{172}\right)\)	\(e\left(\frac{11}{86}\right)\)	\(e\left(\frac{171}{172}\right)\)	\(e\left(\frac{34}{43}\right)\)	\(e\left(\frac{99}{172}\right)\)	\(e\left(\frac{119}{172}\right)\)	\(e\left(\frac{39}{86}\right)\)	\(e\left(\frac{24}{43}\right)\)	\(e\left(\frac{167}{172}\right)\)
\(\chi_{173}(29,\cdot)\)	173.d	43	yes	\(1\)	\(1\)	\(e\left(\frac{36}{43}\right)\)	\(e\left(\frac{26}{43}\right)\)	\(e\left(\frac{29}{43}\right)\)	\(e\left(\frac{28}{43}\right)\)	\(e\left(\frac{19}{43}\right)\)	\(e\left(\frac{23}{43}\right)\)	\(e\left(\frac{22}{43}\right)\)	\(e\left(\frac{9}{43}\right)\)	\(e\left(\frac{21}{43}\right)\)	\(e\left(\frac{11}{43}\right)\)
\(\chi_{173}(30,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{67}{172}\right)\)	\(e\left(\frac{89}{172}\right)\)	\(e\left(\frac{67}{86}\right)\)	\(e\left(\frac{33}{172}\right)\)	\(e\left(\frac{39}{43}\right)\)	\(e\left(\frac{1}{172}\right)\)	\(e\left(\frac{29}{172}\right)\)	\(e\left(\frac{3}{86}\right)\)	\(e\left(\frac{25}{43}\right)\)	\(e\left(\frac{165}{172}\right)\)
\(\chi_{173}(31,\cdot)\)	173.e	86	yes	\(1\)	\(1\)	\(e\left(\frac{51}{86}\right)\)	\(e\left(\frac{1}{86}\right)\)	\(e\left(\frac{8}{43}\right)\)	\(e\left(\frac{11}{86}\right)\)	\(e\left(\frac{26}{43}\right)\)	\(e\left(\frac{29}{86}\right)\)	\(e\left(\frac{67}{86}\right)\)	\(e\left(\frac{1}{43}\right)\)	\(e\left(\frac{31}{43}\right)\)	\(e\left(\frac{55}{86}\right)\)
\(\chi_{173}(32,\cdot)\)	173.f	172	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{172}\right)\)	\(e\left(\frac{135}{172}\right)\)	\(e\left(\frac{5}{86}\right)\)	\(e\left(\frac{23}{172}\right)\)	\(e\left(\frac{35}{43}\right)\)	\(e\left(\frac{131}{172}\right)\)	\(e\left(\frac{15}{172}\right)\)	\(e\left(\frac{49}{86}\right)\)	\(e\left(\frac{7}{43}\right)\)	\(e\left(\frac{115}{172}\right)\)

Introduction

L-functions

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Character group

First 32 of 172 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.