LMFDB - Group of Dirichlet characters of modulus 169

Show commands: PariGP / SageMath

sage: H = DirichletGroup(169)

pari: g = idealstar(,169,2)

Character group

sage: G.order() pari: g.no
Order	=	156
sage: H.invariants() pari: g.cyc
Structure	=	$C_{156}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{169}(2,\cdot)$

First 32 of 156 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{169}(1,\cdot)$	169.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{169}(2,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{1}{156}\right)$	$e\left(\frac{31}{39}\right)$	$e\left(\frac{1}{78}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{125}{156}\right)$	$e\left(\frac{107}{156}\right)$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{103}{156}\right)$
$\chi_{169}(3,\cdot)$	169.i	39	yes	$1$	$1$	$e\left(\frac{31}{39}\right)$	$e\left(\frac{22}{39}\right)$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{5}{39}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{34}{39}\right)$
$\chi_{169}(4,\cdot)$	169.k	78	yes	$1$	$1$	$e\left(\frac{1}{78}\right)$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{1}{39}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{47}{78}\right)$	$e\left(\frac{29}{78}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{7}{39}\right)$	$e\left(\frac{5}{39}\right)$	$e\left(\frac{25}{78}\right)$
$\chi_{169}(5,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{27}{52}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{49}{52}\right)$
$\chi_{169}(6,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{125}{156}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{47}{78}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{25}{156}\right)$	$e\left(\frac{115}{156}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{1}{78}\right)$	$e\left(\frac{83}{156}\right)$
$\chi_{169}(7,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{107}{156}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{29}{78}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{115}{156}\right)$	$e\left(\frac{61}{156}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{67}{78}\right)$	$e\left(\frac{101}{156}\right)$
$\chi_{169}(8,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{51}{52}\right)$
$\chi_{169}(9,\cdot)$	169.i	39	yes	$1$	$1$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{5}{39}\right)$	$e\left(\frac{7}{39}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{10}{39}\right)$	$e\left(\frac{35}{39}\right)$	$e\left(\frac{29}{39}\right)$
$\chi_{169}(10,\cdot)$	169.k	78	yes	$1$	$1$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{5}{39}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{1}{78}\right)$	$e\left(\frac{67}{78}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{35}{39}\right)$	$e\left(\frac{25}{39}\right)$	$e\left(\frac{47}{78}\right)$
$\chi_{169}(11,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{103}{156}\right)$	$e\left(\frac{34}{39}\right)$	$e\left(\frac{25}{78}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{83}{156}\right)$	$e\left(\frac{101}{156}\right)$	$e\left(\frac{51}{52}\right)$	$e\left(\frac{29}{39}\right)$	$e\left(\frac{47}{78}\right)$	$e\left(\frac{1}{156}\right)$
$\chi_{169}(12,\cdot)$	169.h	26	yes	$1$	$1$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{5}{26}\right)$
$\chi_{169}(14,\cdot)$	169.g	13	yes	$1$	$1$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{4}{13}\right)$
$\chi_{169}(15,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{133}{156}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{55}{78}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{89}{156}\right)$	$e\left(\frac{35}{156}\right)$	$e\left(\frac{29}{52}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{41}{78}\right)$	$e\left(\frac{127}{156}\right)$
$\chi_{169}(16,\cdot)$	169.i	39	yes	$1$	$1$	$e\left(\frac{1}{39}\right)$	$e\left(\frac{7}{39}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{8}{39}\right)$	$e\left(\frac{29}{39}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{10}{39}\right)$	$e\left(\frac{25}{39}\right)$
$\chi_{169}(17,\cdot)$	169.k	78	yes	$1$	$1$	$e\left(\frac{73}{78}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{34}{39}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{77}{78}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{31}{78}\right)$
$\chi_{169}(18,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{31}{52}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{27}{52}\right)$	$e\left(\frac{41}{52}\right)$	$e\left(\frac{41}{52}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{21}{52}\right)$
$\chi_{169}(19,\cdot)$	169.f	12	no	$-1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$
$\chi_{169}(20,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{11}{156}\right)$	$e\left(\frac{29}{39}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{127}{156}\right)$	$e\left(\frac{85}{156}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{19}{39}\right)$	$e\left(\frac{55}{78}\right)$	$e\left(\frac{41}{156}\right)$
$\chi_{169}(21,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{27}{52}\right)$
$\chi_{169}(22,\cdot)$	169.c	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{169}(23,\cdot)$	169.e	6	no	$1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{169}(24,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{127}{156}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{49}{78}\right)$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{119}{156}\right)$	$e\left(\frac{17}{156}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{35}{39}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{133}{156}\right)$
$\chi_{169}(25,\cdot)$	169.h	26	yes	$1$	$1$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{23}{26}\right)$
$\chi_{169}(27,\cdot)$	169.g	13	yes	$1$	$1$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{8}{13}\right)$
$\chi_{169}(28,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{109}{156}\right)$	$e\left(\frac{25}{39}\right)$	$e\left(\frac{31}{78}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{53}{156}\right)$	$e\left(\frac{119}{156}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{11}{39}\right)$	$e\left(\frac{77}{78}\right)$	$e\left(\frac{151}{156}\right)$
$\chi_{169}(29,\cdot)$	169.i	39	yes	$1$	$1$	$e\left(\frac{10}{39}\right)$	$e\left(\frac{31}{39}\right)$	$e\left(\frac{20}{39}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{22}{39}\right)$	$e\left(\frac{16}{39}\right)$
$\chi_{169}(30,\cdot)$	169.k	78	yes	$1$	$1$	$e\left(\frac{67}{78}\right)$	$e\left(\frac{20}{39}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{29}{78}\right)$	$e\left(\frac{71}{78}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{1}{39}\right)$	$e\left(\frac{23}{39}\right)$	$e\left(\frac{37}{78}\right)$
$\chi_{169}(31,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{45}{52}\right)$
$\chi_{169}(32,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{5}{156}\right)$	$e\left(\frac{38}{39}\right)$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{1}{156}\right)$	$e\left(\frac{67}{156}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{25}{78}\right)$	$e\left(\frac{47}{156}\right)$
$\chi_{169}(33,\cdot)$	169.l	156	yes	$-1$	$1$	$e\left(\frac{71}{156}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{71}{78}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{139}{156}\right)$	$e\left(\frac{109}{156}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{34}{39}\right)$	$e\left(\frac{43}{78}\right)$	$e\left(\frac{137}{156}\right)$
$\chi_{169}(34,\cdot)$	169.j	52	yes	$-1$	$1$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{41}{52}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{3}{52}\right)$

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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	$169$
Structure	\(C_{156}\)
Order	$156$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{169}(1,\cdot)\)	169.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{169}(2,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{156}\right)\)	\(e\left(\frac{31}{39}\right)\)	\(e\left(\frac{1}{78}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{125}{156}\right)\)	\(e\left(\frac{107}{156}\right)\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{103}{156}\right)\)
\(\chi_{169}(3,\cdot)\)	169.i	39	yes	\(1\)	\(1\)	\(e\left(\frac{31}{39}\right)\)	\(e\left(\frac{22}{39}\right)\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{5}{39}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{34}{39}\right)\)
\(\chi_{169}(4,\cdot)\)	169.k	78	yes	\(1\)	\(1\)	\(e\left(\frac{1}{78}\right)\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{1}{39}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{47}{78}\right)\)	\(e\left(\frac{29}{78}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{7}{39}\right)\)	\(e\left(\frac{5}{39}\right)\)	\(e\left(\frac{25}{78}\right)\)
\(\chi_{169}(5,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{27}{52}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{49}{52}\right)\)
\(\chi_{169}(6,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{125}{156}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{47}{78}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{25}{156}\right)\)	\(e\left(\frac{115}{156}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{1}{78}\right)\)	\(e\left(\frac{83}{156}\right)\)
\(\chi_{169}(7,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{107}{156}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{29}{78}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{115}{156}\right)\)	\(e\left(\frac{61}{156}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{67}{78}\right)\)	\(e\left(\frac{101}{156}\right)\)
\(\chi_{169}(8,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{51}{52}\right)\)
\(\chi_{169}(9,\cdot)\)	169.i	39	yes	\(1\)	\(1\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{5}{39}\right)\)	\(e\left(\frac{7}{39}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{10}{39}\right)\)	\(e\left(\frac{35}{39}\right)\)	\(e\left(\frac{29}{39}\right)\)
\(\chi_{169}(10,\cdot)\)	169.k	78	yes	\(1\)	\(1\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{5}{39}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{1}{78}\right)\)	\(e\left(\frac{67}{78}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{35}{39}\right)\)	\(e\left(\frac{25}{39}\right)\)	\(e\left(\frac{47}{78}\right)\)
\(\chi_{169}(11,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{103}{156}\right)\)	\(e\left(\frac{34}{39}\right)\)	\(e\left(\frac{25}{78}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{83}{156}\right)\)	\(e\left(\frac{101}{156}\right)\)	\(e\left(\frac{51}{52}\right)\)	\(e\left(\frac{29}{39}\right)\)	\(e\left(\frac{47}{78}\right)\)	\(e\left(\frac{1}{156}\right)\)
\(\chi_{169}(12,\cdot)\)	169.h	26	yes	\(1\)	\(1\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{5}{26}\right)\)
\(\chi_{169}(14,\cdot)\)	169.g	13	yes	\(1\)	\(1\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)
\(\chi_{169}(15,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{133}{156}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{55}{78}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{89}{156}\right)\)	\(e\left(\frac{35}{156}\right)\)	\(e\left(\frac{29}{52}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{41}{78}\right)\)	\(e\left(\frac{127}{156}\right)\)
\(\chi_{169}(16,\cdot)\)	169.i	39	yes	\(1\)	\(1\)	\(e\left(\frac{1}{39}\right)\)	\(e\left(\frac{7}{39}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{8}{39}\right)\)	\(e\left(\frac{29}{39}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{10}{39}\right)\)	\(e\left(\frac{25}{39}\right)\)
\(\chi_{169}(17,\cdot)\)	169.k	78	yes	\(1\)	\(1\)	\(e\left(\frac{73}{78}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{34}{39}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{77}{78}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{31}{78}\right)\)
\(\chi_{169}(18,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{31}{52}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{27}{52}\right)\)	\(e\left(\frac{41}{52}\right)\)	\(e\left(\frac{41}{52}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{21}{52}\right)\)
\(\chi_{169}(19,\cdot)\)	169.f	12	no	\(-1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)
\(\chi_{169}(20,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{11}{156}\right)\)	\(e\left(\frac{29}{39}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{127}{156}\right)\)	\(e\left(\frac{85}{156}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{19}{39}\right)\)	\(e\left(\frac{55}{78}\right)\)	\(e\left(\frac{41}{156}\right)\)
\(\chi_{169}(21,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{27}{52}\right)\)
\(\chi_{169}(22,\cdot)\)	169.c	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{169}(23,\cdot)\)	169.e	6	no	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{169}(24,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{127}{156}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{49}{78}\right)\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{119}{156}\right)\)	\(e\left(\frac{17}{156}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{35}{39}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{133}{156}\right)\)
\(\chi_{169}(25,\cdot)\)	169.h	26	yes	\(1\)	\(1\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{23}{26}\right)\)
\(\chi_{169}(27,\cdot)\)	169.g	13	yes	\(1\)	\(1\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)
\(\chi_{169}(28,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{109}{156}\right)\)	\(e\left(\frac{25}{39}\right)\)	\(e\left(\frac{31}{78}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{53}{156}\right)\)	\(e\left(\frac{119}{156}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{11}{39}\right)\)	\(e\left(\frac{77}{78}\right)\)	\(e\left(\frac{151}{156}\right)\)
\(\chi_{169}(29,\cdot)\)	169.i	39	yes	\(1\)	\(1\)	\(e\left(\frac{10}{39}\right)\)	\(e\left(\frac{31}{39}\right)\)	\(e\left(\frac{20}{39}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{22}{39}\right)\)	\(e\left(\frac{16}{39}\right)\)
\(\chi_{169}(30,\cdot)\)	169.k	78	yes	\(1\)	\(1\)	\(e\left(\frac{67}{78}\right)\)	\(e\left(\frac{20}{39}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{29}{78}\right)\)	\(e\left(\frac{71}{78}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{1}{39}\right)\)	\(e\left(\frac{23}{39}\right)\)	\(e\left(\frac{37}{78}\right)\)
\(\chi_{169}(31,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{45}{52}\right)\)
\(\chi_{169}(32,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{156}\right)\)	\(e\left(\frac{38}{39}\right)\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{1}{156}\right)\)	\(e\left(\frac{67}{156}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{25}{78}\right)\)	\(e\left(\frac{47}{156}\right)\)
\(\chi_{169}(33,\cdot)\)	169.l	156	yes	\(-1\)	\(1\)	\(e\left(\frac{71}{156}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{71}{78}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{139}{156}\right)\)	\(e\left(\frac{109}{156}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{34}{39}\right)\)	\(e\left(\frac{43}{78}\right)\)	\(e\left(\frac{137}{156}\right)\)
\(\chi_{169}(34,\cdot)\)	169.j	52	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{41}{52}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{3}{52}\right)\)

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First 32 of 156 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.