LMFDB - Group of Dirichlet characters of modulus 53

Show commands: PariGP / SageMath

sage: H = DirichletGroup(53)

pari: g = idealstar(,53,2)

Character group

sage: G.order() pari: g.no
Order	=	52
sage: H.invariants() pari: g.cyc
Structure	=	$C_{52}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{53}(2,\cdot)$

First 32 of 52 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{53}(1,\cdot)$	53.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{53}(2,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{3}{26}\right)$
$\chi_{53}(3,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{29}{52}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{51}{52}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{25}{26}\right)$
$\chi_{53}(4,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{3}{13}\right)$
$\chi_{53}(5,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{37}{52}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{11}{26}\right)$
$\chi_{53}(6,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{1}{13}\right)$
$\chi_{53}(7,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{8}{13}\right)$
$\chi_{53}(8,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{51}{52}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{37}{52}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{9}{26}\right)$
$\chi_{53}(9,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{12}{13}\right)$
$\chi_{53}(10,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{7}{13}\right)$
$\chi_{53}(11,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{9}{13}\right)$
$\chi_{53}(12,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{5}{26}\right)$
$\chi_{53}(13,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{10}{13}\right)$
$\chi_{53}(14,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{29}{52}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{45}{52}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{19}{26}\right)$
$\chi_{53}(15,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{5}{13}\right)$
$\chi_{53}(16,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{6}{13}\right)$
$\chi_{53}(17,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{2}{13}\right)$
$\chi_{53}(18,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{3}{26}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{1}{26}\right)$
$\chi_{53}(19,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{37}{52}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{7}{26}\right)$
$\chi_{53}(20,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{17}{26}\right)$
$\chi_{53}(21,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{31}{52}\right)$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{1}{52}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{41}{52}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{15}{26}\right)$
$\chi_{53}(22,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{21}{26}\right)$
$\chi_{53}(23,\cdot)$	53.c	4	yes	$-1$	$1$	$-i$	$-i$	$-1$	$i$	$-1$	$-1$	$i$	$-1$	$1$	$-1$
$\chi_{53}(24,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{4}{13}\right)$
$\chi_{53}(25,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{11}{13}\right)$
$\chi_{53}(26,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{31}{52}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{23}{26}\right)$
$\chi_{53}(27,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{51}{52}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{23}{26}\right)$
$\chi_{53}(28,\cdot)$	53.d	13	yes	$1$	$1$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{3}{13}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{11}{13}\right)$
$\chi_{53}(29,\cdot)$	53.e	26	yes	$1$	$1$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{1}{26}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{12}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{4}{13}\right)$
$\chi_{53}(30,\cdot)$	53.c	4	yes	$-1$	$1$	$i$	$i$	$-1$	$-i$	$-1$	$-1$	$-i$	$-1$	$1$	$-1$
$\chi_{53}(31,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{41}{52}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{21}{26}\right)$
$\chi_{53}(32,\cdot)$	53.f	52	yes	$-1$	$1$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{27}{52}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{15}{26}\right)$

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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	$53$
Structure	\(C_{52}\)
Order	$52$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{53}(1,\cdot)\)	53.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{53}(2,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{3}{26}\right)\)
\(\chi_{53}(3,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{29}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{51}{52}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{25}{26}\right)\)
\(\chi_{53}(4,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)
\(\chi_{53}(5,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{37}{52}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{11}{26}\right)\)
\(\chi_{53}(6,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)
\(\chi_{53}(7,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)
\(\chi_{53}(8,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{51}{52}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{37}{52}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{9}{26}\right)\)
\(\chi_{53}(9,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)
\(\chi_{53}(10,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)
\(\chi_{53}(11,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)
\(\chi_{53}(12,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{5}{26}\right)\)
\(\chi_{53}(13,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)
\(\chi_{53}(14,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{29}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{45}{52}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{19}{26}\right)\)
\(\chi_{53}(15,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)
\(\chi_{53}(16,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)
\(\chi_{53}(17,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)
\(\chi_{53}(18,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{3}{26}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{1}{26}\right)\)
\(\chi_{53}(19,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{52}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{7}{26}\right)\)
\(\chi_{53}(20,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{17}{26}\right)\)
\(\chi_{53}(21,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{31}{52}\right)\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{1}{52}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{41}{52}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{15}{26}\right)\)
\(\chi_{53}(22,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{21}{26}\right)\)
\(\chi_{53}(23,\cdot)\)	53.c	4	yes	\(-1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{53}(24,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(25,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(26,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{31}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{23}{26}\right)\)
\(\chi_{53}(27,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{51}{52}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{23}{26}\right)\)
\(\chi_{53}(28,\cdot)\)	53.d	13	yes	\(1\)	\(1\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{3}{13}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(29,\cdot)\)	53.e	26	yes	\(1\)	\(1\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{1}{26}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{12}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(30,\cdot)\)	53.c	4	yes	\(-1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{53}(31,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{41}{52}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{21}{26}\right)\)
\(\chi_{53}(32,\cdot)\)	53.f	52	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{27}{52}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{15}{26}\right)\)

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First 32 of 52 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.