LMFDB - Group of Dirichlet characters of modulus 1312

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1312)

pari: g = idealstar(,1312,2)

Character group

sage: G.order() pari: g.no
Order	=	640
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{8}\times C_{40}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{1312}(575,\cdot)$, $\chi_{1312}(165,\cdot)$, $\chi_{1312}(129,\cdot)$

First 32 of 640 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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{1312}(1,\cdot)$	1312.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{1312}(3,\cdot)$	1312.bc	8	yes	$1$	$1$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-1$	$-1$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-1$	$e\left(\frac{1}{8}\right)$
$\chi_{1312}(5,\cdot)$	1312.de	40	yes	$1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{7}{10}\right)$	$i$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{13}{40}\right)$
$\chi_{1312}(7,\cdot)$	1312.dh	40	no	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{40}\right)$	$-i$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{1312}(9,\cdot)$	1312.i	4	no	$1$	$1$	$-1$	$i$	$-i$	$1$	$1$	$-1$	$-i$	$-i$	$1$	$i$
$\chi_{1312}(11,\cdot)$	1312.cn	40	yes	$1$	$1$	$-1$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{27}{40}\right)$	$1$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{7}{40}\right)$
$\chi_{1312}(13,\cdot)$	1312.cv	40	yes	$-1$	$1$	$i$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{39}{40}\right)$	$-1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{40}\right)$
$\chi_{1312}(15,\cdot)$	1312.cz	40	no	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{23}{40}\right)$	$-i$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1312}(17,\cdot)$	1312.db	40	no	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{7}{40}\right)$	$-i$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{1}{20}\right)$
$\chi_{1312}(19,\cdot)$	1312.cn	40	yes	$1$	$1$	$-1$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{1}{40}\right)$	$1$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{21}{40}\right)$
$\chi_{1312}(21,\cdot)$	1312.de	40	yes	$1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{9}{10}\right)$	$i$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{1}{40}\right)$
$\chi_{1312}(23,\cdot)$	1312.ch	20	no	$-1$	$1$	$i$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{1312}(25,\cdot)$	1312.cf	20	no	$1$	$1$	$i$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{2}{5}\right)$	$-1$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1312}(27,\cdot)$	1312.bc	8	yes	$1$	$1$	$-i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-1$	$-1$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-1$	$e\left(\frac{3}{8}\right)$
$\chi_{1312}(29,\cdot)$	1312.ct	40	yes	$-1$	$1$	$-i$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{23}{40}\right)$	$-1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{13}{40}\right)$
$\chi_{1312}(31,\cdot)$	1312.bx	10	no	$-1$	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{1312}(33,\cdot)$	1312.ci	20	no	$1$	$1$	$-i$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{11}{20}\right)$	$-1$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{1312}(35,\cdot)$	1312.cn	40	yes	$1$	$1$	$-1$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{29}{40}\right)$	$1$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{9}{40}\right)$
$\chi_{1312}(37,\cdot)$	1312.dd	40	yes	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{9}{20}\right)$	$-i$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{33}{40}\right)$
$\chi_{1312}(39,\cdot)$	1312.ca	20	no	$-1$	$1$	$-1$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{17}{20}\right)$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{1312}(43,\cdot)$	1312.df	40	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{1}{10}\right)$	$i$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{9}{40}\right)$
$\chi_{1312}(45,\cdot)$	1312.cw	40	yes	$1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{9}{20}\right)$	$i$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{23}{40}\right)$
$\chi_{1312}(47,\cdot)$	1312.cz	40	no	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{19}{40}\right)$	$-i$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1312}(49,\cdot)$	1312.cc	20	no	$1$	$1$	$-i$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{20}\right)$	$-1$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{1312}(51,\cdot)$	1312.cx	40	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{1}{20}\right)$	$i$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{7}{40}\right)$
$\chi_{1312}(53,\cdot)$	1312.dj	40	yes	$-1$	$1$	$1$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{23}{40}\right)$	$1$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{23}{40}\right)$
$\chi_{1312}(55,\cdot)$	1312.x	8	no	$1$	$1$	$e\left(\frac{1}{8}\right)$	$-1$	$e\left(\frac{3}{8}\right)$	$i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-1$
$\chi_{1312}(57,\cdot)$	1312.cg	20	no	$1$	$1$	$-i$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{9}{10}\right)$	$-1$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1312}(59,\cdot)$	1312.cx	40	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{7}{20}\right)$	$-i$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{9}{40}\right)$
$\chi_{1312}(61,\cdot)$	1312.de	40	yes	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{9}{10}\right)$	$-i$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{31}{40}\right)$
$\chi_{1312}(63,\cdot)$	1312.cy	40	no	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{31}{40}\right)$	$-i$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{3}{20}\right)$
$\chi_{1312}(65,\cdot)$	1312.da	40	no	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{27}{40}\right)$	$-i$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{11}{20}\right)$

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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	$1312$
Structure	\(C_{2}\times C_{8}\times C_{40}\)
Order	$640$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{1312}(1,\cdot)\)	1312.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{1312}(3,\cdot)\)	1312.bc	8	yes	\(1\)	\(1\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(-1\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{1312}(5,\cdot)\)	1312.de	40	yes	\(1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(i\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)
\(\chi_{1312}(7,\cdot)\)	1312.dh	40	no	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(-i\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{1312}(9,\cdot)\)	1312.i	4	no	\(1\)	\(1\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(1\)	\(-1\)	\(-i\)	\(-i\)	\(1\)	\(i\)
\(\chi_{1312}(11,\cdot)\)	1312.cn	40	yes	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(1\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{7}{40}\right)\)
\(\chi_{1312}(13,\cdot)\)	1312.cv	40	yes	\(-1\)	\(1\)	\(i\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{40}\right)\)
\(\chi_{1312}(15,\cdot)\)	1312.cz	40	no	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(-i\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1312}(17,\cdot)\)	1312.db	40	no	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(-i\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{1}{20}\right)\)
\(\chi_{1312}(19,\cdot)\)	1312.cn	40	yes	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(1\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{21}{40}\right)\)
\(\chi_{1312}(21,\cdot)\)	1312.de	40	yes	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(i\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{1}{40}\right)\)
\(\chi_{1312}(23,\cdot)\)	1312.ch	20	no	\(-1\)	\(1\)	\(i\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{1312}(25,\cdot)\)	1312.cf	20	no	\(1\)	\(1\)	\(i\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1312}(27,\cdot)\)	1312.bc	8	yes	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(-1\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{1312}(29,\cdot)\)	1312.ct	40	yes	\(-1\)	\(1\)	\(-i\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(-1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{13}{40}\right)\)
\(\chi_{1312}(31,\cdot)\)	1312.bx	10	no	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{1312}(33,\cdot)\)	1312.ci	20	no	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(-1\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{1312}(35,\cdot)\)	1312.cn	40	yes	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{9}{40}\right)\)
\(\chi_{1312}(37,\cdot)\)	1312.dd	40	yes	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(-i\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{33}{40}\right)\)
\(\chi_{1312}(39,\cdot)\)	1312.ca	20	no	\(-1\)	\(1\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{1312}(43,\cdot)\)	1312.df	40	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(i\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)
\(\chi_{1312}(45,\cdot)\)	1312.cw	40	yes	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(i\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)
\(\chi_{1312}(47,\cdot)\)	1312.cz	40	no	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(-i\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1312}(49,\cdot)\)	1312.cc	20	no	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(-1\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{1312}(51,\cdot)\)	1312.cx	40	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(i\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{7}{40}\right)\)
\(\chi_{1312}(53,\cdot)\)	1312.dj	40	yes	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(1\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{23}{40}\right)\)
\(\chi_{1312}(55,\cdot)\)	1312.x	8	no	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)
\(\chi_{1312}(57,\cdot)\)	1312.cg	20	no	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1312}(59,\cdot)\)	1312.cx	40	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(-i\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)
\(\chi_{1312}(61,\cdot)\)	1312.de	40	yes	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(-i\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{31}{40}\right)\)
\(\chi_{1312}(63,\cdot)\)	1312.cy	40	no	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(-i\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{3}{20}\right)\)
\(\chi_{1312}(65,\cdot)\)	1312.da	40	no	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(-i\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{11}{20}\right)\)

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.