LMFDB - Group of Dirichlet characters of modulus 2704

Show commands: PariGP / SageMath

sage: H = DirichletGroup(2704)

pari: g = idealstar(,2704,2)

Character group

sage: G.order() pari: g.no
Order	=	1248
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{4}\times C_{156}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{2704}(2367,\cdot)$, $\chi_{2704}(677,\cdot)$, $\chi_{2704}(1185,\cdot)$

First 32 of 1248 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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$15$	$17$	$19$	$21$	$23$
$\chi_{2704}(1,\cdot)$	2704.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{2704}(3,\cdot)$	2704.cw	156	yes	$-1$	$1$	$e\left(\frac{49}{156}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{49}{78}\right)$	$e\left(\frac{19}{156}\right)$	$e\left(\frac{17}{78}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{2704}(5,\cdot)$	2704.ca	52	yes	$-1$	$1$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{11}{26}\right)$	$-1$	$e\left(\frac{15}{26}\right)$	$1$
$\chi_{2704}(7,\cdot)$	2704.da	156	no	$1$	$1$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{139}{156}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{101}{156}\right)$	$e\left(\frac{113}{156}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(9,\cdot)$	2704.cl	78	no	$1$	$1$	$e\left(\frac{49}{78}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{10}{39}\right)$	$e\left(\frac{19}{78}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(11,\cdot)$	2704.cy	156	yes	$1$	$1$	$e\left(\frac{19}{156}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{101}{156}\right)$	$e\left(\frac{19}{78}\right)$	$e\left(\frac{59}{78}\right)$	$e\left(\frac{49}{156}\right)$	$e\left(\frac{31}{78}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(15,\cdot)$	2704.cq	156	no	$1$	$1$	$e\left(\frac{17}{78}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{113}{156}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{49}{156}\right)$	$e\left(\frac{139}{156}\right)$	$e\left(\frac{37}{78}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{2704}(17,\cdot)$	2704.co	78	no	$1$	$1$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{11}{26}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{31}{78}\right)$	$e\left(\frac{37}{78}\right)$	$e\left(\frac{25}{39}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(19,\cdot)$	2704.bf	12	no	$1$	$1$	$e\left(\frac{5}{12}\right)$	$-1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$
$\chi_{2704}(21,\cdot)$	2704.cf	52	yes	$-1$	$1$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{10}{13}\right)$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{5}{26}\right)$	$1$	$e\left(\frac{4}{13}\right)$	$1$
$\chi_{2704}(23,\cdot)$	2704.x	6	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(25,\cdot)$	2704.bs	26	no	$1$	$1$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{8}{13}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{11}{13}\right)$	$1$	$e\left(\frac{2}{13}\right)$	$1$
$\chi_{2704}(27,\cdot)$	2704.cc	52	yes	$-1$	$1$	$e\left(\frac{49}{52}\right)$	$e\left(\frac{37}{52}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{17}{26}\right)$	$e\left(\frac{2}{13}\right)$	$i$	$e\left(\frac{5}{52}\right)$	$1$
$\chi_{2704}(29,\cdot)$	2704.cv	156	yes	$1$	$1$	$e\left(\frac{7}{156}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{73}{78}\right)$	$e\left(\frac{7}{78}\right)$	$e\left(\frac{25}{156}\right)$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{51}{52}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(31,\cdot)$	2704.ci	52	no	$1$	$1$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{5}{13}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{17}{26}\right)$	$i$	$e\left(\frac{5}{52}\right)$	$1$
$\chi_{2704}(33,\cdot)$	2704.db	156	no	$-1$	$1$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{109}{156}\right)$	$e\left(\frac{34}{39}\right)$	$e\left(\frac{137}{156}\right)$	$e\left(\frac{83}{156}\right)$	$e\left(\frac{35}{78}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{2704}(35,\cdot)$	2704.cw	156	yes	$-1$	$1$	$e\left(\frac{149}{156}\right)$	$e\left(\frac{23}{52}\right)$	$e\left(\frac{22}{39}\right)$	$e\left(\frac{71}{78}\right)$	$e\left(\frac{131}{156}\right)$	$e\left(\frac{31}{78}\right)$	$e\left(\frac{22}{39}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{27}{52}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(37,\cdot)$	2704.cz	156	yes	$-1$	$1$	$e\left(\frac{121}{156}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{11}{156}\right)$	$e\left(\frac{43}{78}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{115}{156}\right)$	$e\left(\frac{25}{78}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{13}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{2704}(41,\cdot)$	2704.cr	156	no	$-1$	$1$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{47}{156}\right)$	$e\left(\frac{5}{39}\right)$	$e\left(\frac{97}{156}\right)$	$e\left(\frac{73}{156}\right)$	$e\left(\frac{43}{78}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(43,\cdot)$	2704.cu	156	yes	$-1$	$1$	$e\left(\frac{35}{156}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{53}{78}\right)$	$e\left(\frac{35}{78}\right)$	$e\left(\frac{47}{156}\right)$	$e\left(\frac{20}{39}\right)$	$e\left(\frac{7}{39}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(45,\cdot)$	2704.cz	156	yes	$-1$	$1$	$e\left(\frac{83}{156}\right)$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{121}{156}\right)$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{17}{39}\right)$	$e\left(\frac{17}{156}\right)$	$e\left(\frac{41}{78}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{2704}(47,\cdot)$	2704.ci	52	no	$1$	$1$	$e\left(\frac{15}{26}\right)$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{37}{52}\right)$	$e\left(\frac{2}{13}\right)$	$e\left(\frac{5}{52}\right)$	$e\left(\frac{11}{52}\right)$	$e\left(\frac{25}{26}\right)$	$-i$	$e\left(\frac{15}{52}\right)$	$1$
$\chi_{2704}(49,\cdot)$	2704.co	78	no	$1$	$1$	$e\left(\frac{4}{39}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{61}{78}\right)$	$e\left(\frac{8}{39}\right)$	$e\left(\frac{23}{78}\right)$	$e\left(\frac{35}{78}\right)$	$e\left(\frac{11}{39}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{2704}(51,\cdot)$	2704.ce	52	yes	$-1$	$1$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{17}{52}\right)$	$e\left(\frac{5}{26}\right)$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{27}{52}\right)$	$e\left(\frac{9}{13}\right)$	$e\left(\frac{9}{13}\right)$	$i$	$e\left(\frac{29}{52}\right)$	$1$
$\chi_{2704}(53,\cdot)$	2704.cd	52	yes	$1$	$1$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{9}{52}\right)$	$e\left(\frac{21}{26}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{4}{13}\right)$	$e\left(\frac{4}{13}\right)$	$-i$	$e\left(\frac{49}{52}\right)$	$-1$
$\chi_{2704}(55,\cdot)$	2704.cp	78	no	$-1$	$1$	$e\left(\frac{1}{39}\right)$	$e\left(\frac{25}{26}\right)$	$e\left(\frac{25}{78}\right)$	$e\left(\frac{2}{39}\right)$	$e\left(\frac{37}{39}\right)$	$e\left(\frac{77}{78}\right)$	$e\left(\frac{32}{39}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{9}{26}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(57,\cdot)$	2704.ch	52	no	$-1$	$1$	$e\left(\frac{19}{26}\right)$	$e\left(\frac{21}{52}\right)$	$e\left(\frac{33}{52}\right)$	$e\left(\frac{6}{13}\right)$	$e\left(\frac{15}{52}\right)$	$e\left(\frac{7}{52}\right)$	$e\left(\frac{23}{26}\right)$	$i$	$e\left(\frac{19}{52}\right)$	$-1$
$\chi_{2704}(59,\cdot)$	2704.cy	156	yes	$1$	$1$	$e\left(\frac{11}{156}\right)$	$e\left(\frac{7}{26}\right)$	$e\left(\frac{1}{156}\right)$	$e\left(\frac{11}{78}\right)$	$e\left(\frac{67}{78}\right)$	$e\left(\frac{53}{156}\right)$	$e\left(\frac{59}{78}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{13}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{2704}(61,\cdot)$	2704.cv	156	yes	$1$	$1$	$e\left(\frac{83}{156}\right)$	$e\left(\frac{43}{52}\right)$	$e\left(\frac{41}{78}\right)$	$e\left(\frac{5}{78}\right)$	$e\left(\frac{29}{156}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{1}{39}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{3}{52}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{2704}(63,\cdot)$	2704.cq	156	no	$1$	$1$	$e\left(\frac{53}{78}\right)$	$e\left(\frac{25}{52}\right)$	$e\left(\frac{155}{156}\right)$	$e\left(\frac{14}{39}\right)$	$e\left(\frac{139}{156}\right)$	$e\left(\frac{25}{156}\right)$	$e\left(\frac{19}{78}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{35}{52}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{2704}(67,\cdot)$	2704.cy	156	yes	$1$	$1$	$e\left(\frac{25}{156}\right)$	$e\left(\frac{23}{26}\right)$	$e\left(\frac{59}{156}\right)$	$e\left(\frac{25}{78}\right)$	$e\left(\frac{53}{78}\right)$	$e\left(\frac{7}{156}\right)$	$e\left(\frac{49}{78}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{13}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{2704}(69,\cdot)$	2704.cx	156	yes	$1$	$1$	$e\left(\frac{101}{156}\right)$	$e\left(\frac{47}{52}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{23}{78}\right)$	$e\left(\frac{149}{156}\right)$	$e\left(\frac{43}{78}\right)$	$e\left(\frac{28}{39}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{19}{52}\right)$	$e\left(\frac{1}{6}\right)$

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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	$2704$
Structure	\(C_{2}\times C_{4}\times C_{156}\)
Order	$1248$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(15\)	\(17\)	\(19\)	\(21\)	\(23\)
\(\chi_{2704}(1,\cdot)\)	2704.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{2704}(3,\cdot)\)	2704.cw	156	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{156}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{49}{78}\right)\)	\(e\left(\frac{19}{156}\right)\)	\(e\left(\frac{17}{78}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{2704}(5,\cdot)\)	2704.ca	52	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(-1\)	\(e\left(\frac{15}{26}\right)\)	\(1\)
\(\chi_{2704}(7,\cdot)\)	2704.da	156	no	\(1\)	\(1\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{139}{156}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{101}{156}\right)\)	\(e\left(\frac{113}{156}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(9,\cdot)\)	2704.cl	78	no	\(1\)	\(1\)	\(e\left(\frac{49}{78}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{10}{39}\right)\)	\(e\left(\frac{19}{78}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(11,\cdot)\)	2704.cy	156	yes	\(1\)	\(1\)	\(e\left(\frac{19}{156}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{101}{156}\right)\)	\(e\left(\frac{19}{78}\right)\)	\(e\left(\frac{59}{78}\right)\)	\(e\left(\frac{49}{156}\right)\)	\(e\left(\frac{31}{78}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(15,\cdot)\)	2704.cq	156	no	\(1\)	\(1\)	\(e\left(\frac{17}{78}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{113}{156}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{49}{156}\right)\)	\(e\left(\frac{139}{156}\right)\)	\(e\left(\frac{37}{78}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{2704}(17,\cdot)\)	2704.co	78	no	\(1\)	\(1\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{11}{26}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{31}{78}\right)\)	\(e\left(\frac{37}{78}\right)\)	\(e\left(\frac{25}{39}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(19,\cdot)\)	2704.bf	12	no	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{2704}(21,\cdot)\)	2704.cf	52	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{10}{13}\right)\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(1\)	\(e\left(\frac{4}{13}\right)\)	\(1\)
\(\chi_{2704}(23,\cdot)\)	2704.x	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(25,\cdot)\)	2704.bs	26	no	\(1\)	\(1\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{8}{13}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(1\)	\(e\left(\frac{2}{13}\right)\)	\(1\)
\(\chi_{2704}(27,\cdot)\)	2704.cc	52	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{52}\right)\)	\(e\left(\frac{37}{52}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(i\)	\(e\left(\frac{5}{52}\right)\)	\(1\)
\(\chi_{2704}(29,\cdot)\)	2704.cv	156	yes	\(1\)	\(1\)	\(e\left(\frac{7}{156}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{73}{78}\right)\)	\(e\left(\frac{7}{78}\right)\)	\(e\left(\frac{25}{156}\right)\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{51}{52}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(31,\cdot)\)	2704.ci	52	no	\(1\)	\(1\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{5}{13}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{17}{26}\right)\)	\(i\)	\(e\left(\frac{5}{52}\right)\)	\(1\)
\(\chi_{2704}(33,\cdot)\)	2704.db	156	no	\(-1\)	\(1\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{109}{156}\right)\)	\(e\left(\frac{34}{39}\right)\)	\(e\left(\frac{137}{156}\right)\)	\(e\left(\frac{83}{156}\right)\)	\(e\left(\frac{35}{78}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{2704}(35,\cdot)\)	2704.cw	156	yes	\(-1\)	\(1\)	\(e\left(\frac{149}{156}\right)\)	\(e\left(\frac{23}{52}\right)\)	\(e\left(\frac{22}{39}\right)\)	\(e\left(\frac{71}{78}\right)\)	\(e\left(\frac{131}{156}\right)\)	\(e\left(\frac{31}{78}\right)\)	\(e\left(\frac{22}{39}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{27}{52}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(37,\cdot)\)	2704.cz	156	yes	\(-1\)	\(1\)	\(e\left(\frac{121}{156}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{11}{156}\right)\)	\(e\left(\frac{43}{78}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{115}{156}\right)\)	\(e\left(\frac{25}{78}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{13}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{2704}(41,\cdot)\)	2704.cr	156	no	\(-1\)	\(1\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{47}{156}\right)\)	\(e\left(\frac{5}{39}\right)\)	\(e\left(\frac{97}{156}\right)\)	\(e\left(\frac{73}{156}\right)\)	\(e\left(\frac{43}{78}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(43,\cdot)\)	2704.cu	156	yes	\(-1\)	\(1\)	\(e\left(\frac{35}{156}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{53}{78}\right)\)	\(e\left(\frac{35}{78}\right)\)	\(e\left(\frac{47}{156}\right)\)	\(e\left(\frac{20}{39}\right)\)	\(e\left(\frac{7}{39}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(45,\cdot)\)	2704.cz	156	yes	\(-1\)	\(1\)	\(e\left(\frac{83}{156}\right)\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{121}{156}\right)\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{17}{39}\right)\)	\(e\left(\frac{17}{156}\right)\)	\(e\left(\frac{41}{78}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{2704}(47,\cdot)\)	2704.ci	52	no	\(1\)	\(1\)	\(e\left(\frac{15}{26}\right)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{37}{52}\right)\)	\(e\left(\frac{2}{13}\right)\)	\(e\left(\frac{5}{52}\right)\)	\(e\left(\frac{11}{52}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(-i\)	\(e\left(\frac{15}{52}\right)\)	\(1\)
\(\chi_{2704}(49,\cdot)\)	2704.co	78	no	\(1\)	\(1\)	\(e\left(\frac{4}{39}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{61}{78}\right)\)	\(e\left(\frac{8}{39}\right)\)	\(e\left(\frac{23}{78}\right)\)	\(e\left(\frac{35}{78}\right)\)	\(e\left(\frac{11}{39}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{2704}(51,\cdot)\)	2704.ce	52	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{17}{52}\right)\)	\(e\left(\frac{5}{26}\right)\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{27}{52}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(e\left(\frac{9}{13}\right)\)	\(i\)	\(e\left(\frac{29}{52}\right)\)	\(1\)
\(\chi_{2704}(53,\cdot)\)	2704.cd	52	yes	\(1\)	\(1\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{9}{52}\right)\)	\(e\left(\frac{21}{26}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(e\left(\frac{4}{13}\right)\)	\(-i\)	\(e\left(\frac{49}{52}\right)\)	\(-1\)
\(\chi_{2704}(55,\cdot)\)	2704.cp	78	no	\(-1\)	\(1\)	\(e\left(\frac{1}{39}\right)\)	\(e\left(\frac{25}{26}\right)\)	\(e\left(\frac{25}{78}\right)\)	\(e\left(\frac{2}{39}\right)\)	\(e\left(\frac{37}{39}\right)\)	\(e\left(\frac{77}{78}\right)\)	\(e\left(\frac{32}{39}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{26}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(57,\cdot)\)	2704.ch	52	no	\(-1\)	\(1\)	\(e\left(\frac{19}{26}\right)\)	\(e\left(\frac{21}{52}\right)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{6}{13}\right)\)	\(e\left(\frac{15}{52}\right)\)	\(e\left(\frac{7}{52}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(i\)	\(e\left(\frac{19}{52}\right)\)	\(-1\)
\(\chi_{2704}(59,\cdot)\)	2704.cy	156	yes	\(1\)	\(1\)	\(e\left(\frac{11}{156}\right)\)	\(e\left(\frac{7}{26}\right)\)	\(e\left(\frac{1}{156}\right)\)	\(e\left(\frac{11}{78}\right)\)	\(e\left(\frac{67}{78}\right)\)	\(e\left(\frac{53}{156}\right)\)	\(e\left(\frac{59}{78}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{13}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{2704}(61,\cdot)\)	2704.cv	156	yes	\(1\)	\(1\)	\(e\left(\frac{83}{156}\right)\)	\(e\left(\frac{43}{52}\right)\)	\(e\left(\frac{41}{78}\right)\)	\(e\left(\frac{5}{78}\right)\)	\(e\left(\frac{29}{156}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{1}{39}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{3}{52}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{2704}(63,\cdot)\)	2704.cq	156	no	\(1\)	\(1\)	\(e\left(\frac{53}{78}\right)\)	\(e\left(\frac{25}{52}\right)\)	\(e\left(\frac{155}{156}\right)\)	\(e\left(\frac{14}{39}\right)\)	\(e\left(\frac{139}{156}\right)\)	\(e\left(\frac{25}{156}\right)\)	\(e\left(\frac{19}{78}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{35}{52}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{2704}(67,\cdot)\)	2704.cy	156	yes	\(1\)	\(1\)	\(e\left(\frac{25}{156}\right)\)	\(e\left(\frac{23}{26}\right)\)	\(e\left(\frac{59}{156}\right)\)	\(e\left(\frac{25}{78}\right)\)	\(e\left(\frac{53}{78}\right)\)	\(e\left(\frac{7}{156}\right)\)	\(e\left(\frac{49}{78}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{13}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{2704}(69,\cdot)\)	2704.cx	156	yes	\(1\)	\(1\)	\(e\left(\frac{101}{156}\right)\)	\(e\left(\frac{47}{52}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{23}{78}\right)\)	\(e\left(\frac{149}{156}\right)\)	\(e\left(\frac{43}{78}\right)\)	\(e\left(\frac{28}{39}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{19}{52}\right)\)	\(e\left(\frac{1}{6}\right)\)

Introduction

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.