LMFDB - Group of Dirichlet characters of modulus 289

Show commands: PariGP / SageMath

sage: H = DirichletGroup(289)

pari: g = idealstar(,289,2)

Character group

sage: G.order() pari: g.no
Order	=	272
sage: H.invariants() pari: g.cyc
Structure	=	$C_{272}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{289}(3,\cdot)$

First 32 of 272 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{289}(1,\cdot)$	289.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{289}(2,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{131}{136}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{9}{136}\right)$
$\chi_{289}(3,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{1}{272}\right)$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{191}{272}\right)$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{13}{136}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{147}{272}\right)$	$e\left(\frac{23}{272}\right)$
$\chi_{289}(4,\cdot)$	289.h	68	yes	$1$	$1$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{15}{17}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{9}{68}\right)$
$\chi_{289}(5,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{131}{136}\right)$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{217}{272}\right)$	$e\left(\frac{219}{272}\right)$	$e\left(\frac{135}{272}\right)$	$e\left(\frac{121}{136}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{207}{272}\right)$	$e\left(\frac{99}{272}\right)$
$\chi_{289}(6,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{191}{272}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{219}{272}\right)$	$e\left(\frac{33}{272}\right)$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{35}{136}\right)$	$e\left(\frac{55}{136}\right)$	$e\left(\frac{61}{272}\right)$	$e\left(\frac{41}{272}\right)$
$\chi_{289}(7,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{135}{272}\right)$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{89}{272}\right)$	$e\left(\frac{111}{136}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{209}{272}\right)$	$e\left(\frac{29}{272}\right)$
$\chi_{289}(8,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{13}{136}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{121}{136}\right)$	$e\left(\frac{35}{136}\right)$	$e\left(\frac{111}{136}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{27}{136}\right)$
$\chi_{289}(9,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{55}{136}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{23}{136}\right)$
$\chi_{289}(10,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{147}{272}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{207}{272}\right)$	$e\left(\frac{61}{272}\right)$	$e\left(\frac{209}{272}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{121}{272}\right)$	$e\left(\frac{117}{272}\right)$
$\chi_{289}(11,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{23}{272}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{99}{272}\right)$	$e\left(\frac{41}{272}\right)$	$e\left(\frac{29}{272}\right)$	$e\left(\frac{27}{136}\right)$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{117}{272}\right)$	$e\left(\frac{257}{272}\right)$
$\chi_{289}(12,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{109}{272}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{209}{272}\right)$	$e\left(\frac{147}{272}\right)$	$e\left(\frac{31}{272}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{109}{136}\right)$	$e\left(\frac{247}{272}\right)$	$e\left(\frac{59}{272}\right)$
$\chi_{289}(13,\cdot)$	289.h	68	yes	$1$	$1$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{14}{17}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{39}{68}\right)$
$\chi_{289}(14,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{73}{272}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{125}{272}\right)$	$e\left(\frac{71}{272}\right)$	$e\left(\frac{163}{272}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{73}{136}\right)$	$e\left(\frac{123}{272}\right)$	$e\left(\frac{47}{272}\right)$
$\chi_{289}(15,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{45}{68}\right)$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{69}{136}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{41}{136}\right)$	$e\left(\frac{61}{136}\right)$
$\chi_{289}(16,\cdot)$	289.g	34	yes	$1$	$1$	$e\left(\frac{15}{17}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{11}{17}\right)$	$e\left(\frac{10}{17}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{9}{34}\right)$
$\chi_{289}(18,\cdot)$	289.f	17	yes	$1$	$1$	$e\left(\frac{2}{17}\right)$	$e\left(\frac{12}{17}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{11}{17}\right)$	$e\left(\frac{14}{17}\right)$	$e\left(\frac{7}{17}\right)$	$e\left(\frac{6}{17}\right)$	$e\left(\frac{7}{17}\right)$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{4}{17}\right)$
$\chi_{289}(19,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{25}{136}\right)$
$\chi_{289}(20,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{55}{136}\right)$	$e\left(\frac{65}{272}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{197}{272}\right)$	$e\left(\frac{175}{272}\right)$	$e\left(\frac{11}{272}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{65}{136}\right)$	$e\left(\frac{35}{272}\right)$	$e\left(\frac{135}{272}\right)$
$\chi_{289}(21,\cdot)$	289.h	68	yes	$1$	$1$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{16}{17}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{13}{68}\right)$
$\chi_{289}(22,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{213}{272}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{89}{272}\right)$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{103}{272}\right)$	$e\left(\frac{49}{136}\right)$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{31}{272}\right)$	$e\left(\frac{3}{272}\right)$
$\chi_{289}(23,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{105}{136}\right)$	$e\left(\frac{223}{272}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{203}{272}\right)$	$e\left(\frac{161}{272}\right)$	$e\left(\frac{21}{272}\right)$	$e\left(\frac{43}{136}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{141}{272}\right)$	$e\left(\frac{233}{272}\right)$
$\chi_{289}(24,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{27}{272}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{199}{272}\right)$	$e\left(\frac{261}{272}\right)$	$e\left(\frac{105}{272}\right)$	$e\left(\frac{79}{136}\right)$	$e\left(\frac{27}{136}\right)$	$e\left(\frac{161}{272}\right)$	$e\left(\frac{77}{272}\right)$
$\chi_{289}(25,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{71}{136}\right)$	$e\left(\frac{99}{136}\right)$
$\chi_{289}(26,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{131}{136}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{87}{136}\right)$
$\chi_{289}(27,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{13}{136}\right)$	$e\left(\frac{3}{272}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{143}{272}\right)$	$e\left(\frac{29}{272}\right)$	$e\left(\frac{193}{272}\right)$	$e\left(\frac{39}{136}\right)$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{169}{272}\right)$	$e\left(\frac{69}{272}\right)$
$\chi_{289}(28,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{97}{136}\right)$	$e\left(\frac{263}{272}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{115}{272}\right)$	$e\left(\frac{185}{272}\right)$	$e\left(\frac{237}{272}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{127}{136}\right)$	$e\left(\frac{37}{272}\right)$	$e\left(\frac{65}{272}\right)$
$\chi_{289}(29,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{43}{136}\right)$	$e\left(\frac{125}{272}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{65}{272}\right)$	$e\left(\frac{211}{272}\right)$	$e\left(\frac{63}{272}\right)$	$e\left(\frac{129}{136}\right)$	$e\left(\frac{125}{136}\right)$	$e\left(\frac{151}{272}\right)$	$e\left(\frac{155}{272}\right)$
$\chi_{289}(30,\cdot)$	289.h	68	yes	$1$	$1$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{35}{68}\right)$
$\chi_{289}(31,\cdot)$	289.j	272	yes	$-1$	$1$	$e\left(\frac{39}{136}\right)$	$e\left(\frac{9}{272}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{157}{272}\right)$	$e\left(\frac{87}{272}\right)$	$e\left(\frac{35}{272}\right)$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{235}{272}\right)$	$e\left(\frac{207}{272}\right)$
$\chi_{289}(32,\cdot)$	289.i	136	yes	$1$	$1$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{67}{136}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{111}{136}\right)$	$e\left(\frac{13}{136}\right)$	$e\left(\frac{49}{136}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{45}{136}\right)$
$\chi_{289}(33,\cdot)$	289.g	34	yes	$1$	$1$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{9}{17}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{3}{17}\right)$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{1}{34}\right)$

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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	$289$
Structure	\(C_{272}\)
Order	$272$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{289}(1,\cdot)\)	289.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{289}(2,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{131}{136}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)
\(\chi_{289}(3,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{1}{272}\right)\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{191}{272}\right)\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{13}{136}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{147}{272}\right)\)	\(e\left(\frac{23}{272}\right)\)
\(\chi_{289}(4,\cdot)\)	289.h	68	yes	\(1\)	\(1\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{15}{17}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{9}{68}\right)\)
\(\chi_{289}(5,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{131}{136}\right)\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{217}{272}\right)\)	\(e\left(\frac{219}{272}\right)\)	\(e\left(\frac{135}{272}\right)\)	\(e\left(\frac{121}{136}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{207}{272}\right)\)	\(e\left(\frac{99}{272}\right)\)
\(\chi_{289}(6,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{191}{272}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{219}{272}\right)\)	\(e\left(\frac{33}{272}\right)\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{35}{136}\right)\)	\(e\left(\frac{55}{136}\right)\)	\(e\left(\frac{61}{272}\right)\)	\(e\left(\frac{41}{272}\right)\)
\(\chi_{289}(7,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{135}{272}\right)\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{89}{272}\right)\)	\(e\left(\frac{111}{136}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{209}{272}\right)\)	\(e\left(\frac{29}{272}\right)\)
\(\chi_{289}(8,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{13}{136}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{121}{136}\right)\)	\(e\left(\frac{35}{136}\right)\)	\(e\left(\frac{111}{136}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{27}{136}\right)\)
\(\chi_{289}(9,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{55}{136}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{23}{136}\right)\)
\(\chi_{289}(10,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{147}{272}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{207}{272}\right)\)	\(e\left(\frac{61}{272}\right)\)	\(e\left(\frac{209}{272}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{121}{272}\right)\)	\(e\left(\frac{117}{272}\right)\)
\(\chi_{289}(11,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{23}{272}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{99}{272}\right)\)	\(e\left(\frac{41}{272}\right)\)	\(e\left(\frac{29}{272}\right)\)	\(e\left(\frac{27}{136}\right)\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{117}{272}\right)\)	\(e\left(\frac{257}{272}\right)\)
\(\chi_{289}(12,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{109}{272}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{209}{272}\right)\)	\(e\left(\frac{147}{272}\right)\)	\(e\left(\frac{31}{272}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{109}{136}\right)\)	\(e\left(\frac{247}{272}\right)\)	\(e\left(\frac{59}{272}\right)\)
\(\chi_{289}(13,\cdot)\)	289.h	68	yes	\(1\)	\(1\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{14}{17}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{39}{68}\right)\)
\(\chi_{289}(14,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{73}{272}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{125}{272}\right)\)	\(e\left(\frac{71}{272}\right)\)	\(e\left(\frac{163}{272}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{73}{136}\right)\)	\(e\left(\frac{123}{272}\right)\)	\(e\left(\frac{47}{272}\right)\)
\(\chi_{289}(15,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{45}{68}\right)\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{69}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{41}{136}\right)\)	\(e\left(\frac{61}{136}\right)\)
\(\chi_{289}(16,\cdot)\)	289.g	34	yes	\(1\)	\(1\)	\(e\left(\frac{15}{17}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{11}{17}\right)\)	\(e\left(\frac{10}{17}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{9}{34}\right)\)
\(\chi_{289}(18,\cdot)\)	289.f	17	yes	\(1\)	\(1\)	\(e\left(\frac{2}{17}\right)\)	\(e\left(\frac{12}{17}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{11}{17}\right)\)	\(e\left(\frac{14}{17}\right)\)	\(e\left(\frac{7}{17}\right)\)	\(e\left(\frac{6}{17}\right)\)	\(e\left(\frac{7}{17}\right)\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{4}{17}\right)\)
\(\chi_{289}(19,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{25}{136}\right)\)
\(\chi_{289}(20,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{55}{136}\right)\)	\(e\left(\frac{65}{272}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{197}{272}\right)\)	\(e\left(\frac{175}{272}\right)\)	\(e\left(\frac{11}{272}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{65}{136}\right)\)	\(e\left(\frac{35}{272}\right)\)	\(e\left(\frac{135}{272}\right)\)
\(\chi_{289}(21,\cdot)\)	289.h	68	yes	\(1\)	\(1\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{16}{17}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{13}{68}\right)\)
\(\chi_{289}(22,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{213}{272}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{89}{272}\right)\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{103}{272}\right)\)	\(e\left(\frac{49}{136}\right)\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{31}{272}\right)\)	\(e\left(\frac{3}{272}\right)\)
\(\chi_{289}(23,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{105}{136}\right)\)	\(e\left(\frac{223}{272}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{203}{272}\right)\)	\(e\left(\frac{161}{272}\right)\)	\(e\left(\frac{21}{272}\right)\)	\(e\left(\frac{43}{136}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{141}{272}\right)\)	\(e\left(\frac{233}{272}\right)\)
\(\chi_{289}(24,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{27}{272}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{199}{272}\right)\)	\(e\left(\frac{261}{272}\right)\)	\(e\left(\frac{105}{272}\right)\)	\(e\left(\frac{79}{136}\right)\)	\(e\left(\frac{27}{136}\right)\)	\(e\left(\frac{161}{272}\right)\)	\(e\left(\frac{77}{272}\right)\)
\(\chi_{289}(25,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{71}{136}\right)\)	\(e\left(\frac{99}{136}\right)\)
\(\chi_{289}(26,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{131}{136}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{87}{136}\right)\)
\(\chi_{289}(27,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{136}\right)\)	\(e\left(\frac{3}{272}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{143}{272}\right)\)	\(e\left(\frac{29}{272}\right)\)	\(e\left(\frac{193}{272}\right)\)	\(e\left(\frac{39}{136}\right)\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{169}{272}\right)\)	\(e\left(\frac{69}{272}\right)\)
\(\chi_{289}(28,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{97}{136}\right)\)	\(e\left(\frac{263}{272}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{115}{272}\right)\)	\(e\left(\frac{185}{272}\right)\)	\(e\left(\frac{237}{272}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{127}{136}\right)\)	\(e\left(\frac{37}{272}\right)\)	\(e\left(\frac{65}{272}\right)\)
\(\chi_{289}(29,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{43}{136}\right)\)	\(e\left(\frac{125}{272}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{65}{272}\right)\)	\(e\left(\frac{211}{272}\right)\)	\(e\left(\frac{63}{272}\right)\)	\(e\left(\frac{129}{136}\right)\)	\(e\left(\frac{125}{136}\right)\)	\(e\left(\frac{151}{272}\right)\)	\(e\left(\frac{155}{272}\right)\)
\(\chi_{289}(30,\cdot)\)	289.h	68	yes	\(1\)	\(1\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{35}{68}\right)\)
\(\chi_{289}(31,\cdot)\)	289.j	272	yes	\(-1\)	\(1\)	\(e\left(\frac{39}{136}\right)\)	\(e\left(\frac{9}{272}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{157}{272}\right)\)	\(e\left(\frac{87}{272}\right)\)	\(e\left(\frac{35}{272}\right)\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{235}{272}\right)\)	\(e\left(\frac{207}{272}\right)\)
\(\chi_{289}(32,\cdot)\)	289.i	136	yes	\(1\)	\(1\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{67}{136}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{111}{136}\right)\)	\(e\left(\frac{13}{136}\right)\)	\(e\left(\frac{49}{136}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{45}{136}\right)\)
\(\chi_{289}(33,\cdot)\)	289.g	34	yes	\(1\)	\(1\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{9}{17}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{3}{17}\right)\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{1}{34}\right)\)

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L-functions

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