LMFDB - Group of Dirichlet characters of modulus 1960

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(1960)

pari: g = idealstar(,1960,2)

Character group

sage: G.order() pari: g.no
Order	=	672
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{2}\times C_{2}\times C_{84}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{1960}(1471,\cdot)$, $\chi_{1960}(981,\cdot)$, $\chi_{1960}(1177,\cdot)$, $\chi_{1960}(1081,\cdot)$

First 32 of 672 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	$-1$	$1$	$3$	$9$	$11$	$13$	$17$	$19$	$23$	$27$	$29$	$31$
$\chi_{1960}(1,\cdot)$	1960.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{1960}(3,\cdot)$	1960.do	84	yes	$-1$	$1$	$e\left(\frac{23}{84}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{55}{84}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(9,\cdot)$	1960.db	42	no	$1$	$1$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{1960}(11,\cdot)$	1960.di	42	no	$-1$	$1$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{1960}(13,\cdot)$	1960.cr	28	yes	$1$	$1$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{11}{28}\right)$	$-1$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{1}{7}\right)$	$-1$
$\chi_{1960}(17,\cdot)$	1960.dp	84	no	$1$	$1$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{1960}(19,\cdot)$	1960.ba	6	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{1960}(23,\cdot)$	1960.dl	84	no	$1$	$1$	$e\left(\frac{55}{84}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{29}{42}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(27,\cdot)$	1960.cp	28	yes	$-1$	$1$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{1}{28}\right)$	$1$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{2}{7}\right)$	$1$
$\chi_{1960}(29,\cdot)$	1960.ca	14	yes	$1$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{14}\right)$	$-1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{14}\right)$	$1$
$\chi_{1960}(31,\cdot)$	1960.bc	6	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(33,\cdot)$	1960.dp	84	no	$1$	$1$	$e\left(\frac{19}{84}\right)$	$e\left(\frac{19}{42}\right)$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{13}{84}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{29}{84}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(37,\cdot)$	1960.dk	84	yes	$-1$	$1$	$e\left(\frac{1}{84}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{25}{84}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{1960}(39,\cdot)$	1960.dg	42	no	$-1$	$1$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{1960}(41,\cdot)$	1960.cg	14	no	$-1$	$1$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{13}{14}\right)$	$-1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{3}{7}\right)$	$-1$
$\chi_{1960}(43,\cdot)$	1960.cn	28	yes	$1$	$1$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{9}{28}\right)$	$-1$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{4}{7}\right)$	$-1$
$\chi_{1960}(47,\cdot)$	1960.dn	84	no	$-1$	$1$	$e\left(\frac{31}{84}\right)$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{19}{84}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{65}{84}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{1960}(51,\cdot)$	1960.di	42	no	$-1$	$1$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{10}{21}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(53,\cdot)$	1960.dk	84	yes	$-1$	$1$	$e\left(\frac{83}{84}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{1}{42}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{25}{84}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(57,\cdot)$	1960.co	28	no	$-1$	$1$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{19}{28}\right)$	$-1$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{13}{14}\right)$	$1$
$\chi_{1960}(59,\cdot)$	1960.dj	42	yes	$1$	$1$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(61,\cdot)$	1960.dd	42	no	$-1$	$1$	$e\left(\frac{16}{21}\right)$	$e\left(\frac{11}{21}\right)$	$e\left(\frac{41}{42}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{23}{42}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(67,\cdot)$	1960.bs	12	no	$1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$1$	$e\left(\frac{1}{6}\right)$
$\chi_{1960}(69,\cdot)$	1960.cl	14	yes	$-1$	$1$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{5}{7}\right)$	$1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{3}{14}\right)$	$-1$
$\chi_{1960}(71,\cdot)$	1960.ci	14	no	$-1$	$1$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{2}{7}\right)$	$-1$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{2}{7}\right)$	$-1$
$\chi_{1960}(73,\cdot)$	1960.dp	84	no	$1$	$1$	$e\left(\frac{11}{84}\right)$	$e\left(\frac{11}{42}\right)$	$e\left(\frac{5}{21}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{65}{84}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{61}{84}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{1960}(79,\cdot)$	1960.bd	6	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(81,\cdot)$	1960.cm	21	no	$1$	$1$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{17}{21}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{8}{21}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{21}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(83,\cdot)$	1960.cp	28	yes	$-1$	$1$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{3}{28}\right)$	$1$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{6}{7}\right)$	$1$
$\chi_{1960}(87,\cdot)$	1960.dn	84	no	$-1$	$1$	$e\left(\frac{59}{84}\right)$	$e\left(\frac{17}{42}\right)$	$e\left(\frac{25}{42}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{47}{84}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{37}{84}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{1960}(89,\cdot)$	1960.dh	42	no	$-1$	$1$	$e\left(\frac{1}{21}\right)$	$e\left(\frac{2}{21}\right)$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{4}{21}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{13}{42}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{1960}(93,\cdot)$	1960.dk	84	yes	$-1$	$1$	$e\left(\frac{79}{84}\right)$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{43}{84}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{41}{84}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{3}\right)$

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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	$1960$
Structure	\(C_{2}\times C_{2}\times C_{2}\times C_{84}\)
Order	$672$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(27\)	\(29\)	\(31\)
\(\chi_{1960}(1,\cdot)\)	1960.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{1960}(3,\cdot)\)	1960.do	84	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{84}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{55}{84}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(9,\cdot)\)	1960.db	42	no	\(1\)	\(1\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{1960}(11,\cdot)\)	1960.di	42	no	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{1960}(13,\cdot)\)	1960.cr	28	yes	\(1\)	\(1\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(-1\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(-1\)
\(\chi_{1960}(17,\cdot)\)	1960.dp	84	no	\(1\)	\(1\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{1960}(19,\cdot)\)	1960.ba	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{1960}(23,\cdot)\)	1960.dl	84	no	\(1\)	\(1\)	\(e\left(\frac{55}{84}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(27,\cdot)\)	1960.cp	28	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(1\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(1\)
\(\chi_{1960}(29,\cdot)\)	1960.ca	14	yes	\(1\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(1\)
\(\chi_{1960}(31,\cdot)\)	1960.bc	6	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(33,\cdot)\)	1960.dp	84	no	\(1\)	\(1\)	\(e\left(\frac{19}{84}\right)\)	\(e\left(\frac{19}{42}\right)\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{13}{84}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{29}{84}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(37,\cdot)\)	1960.dk	84	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{84}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{25}{84}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{1960}(39,\cdot)\)	1960.dg	42	no	\(-1\)	\(1\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{1960}(41,\cdot)\)	1960.cg	14	no	\(-1\)	\(1\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(-1\)
\(\chi_{1960}(43,\cdot)\)	1960.cn	28	yes	\(1\)	\(1\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(-1\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(-1\)
\(\chi_{1960}(47,\cdot)\)	1960.dn	84	no	\(-1\)	\(1\)	\(e\left(\frac{31}{84}\right)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{19}{84}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{65}{84}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{1960}(51,\cdot)\)	1960.di	42	no	\(-1\)	\(1\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{10}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(53,\cdot)\)	1960.dk	84	yes	\(-1\)	\(1\)	\(e\left(\frac{83}{84}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{25}{84}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(57,\cdot)\)	1960.co	28	no	\(-1\)	\(1\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(-1\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(1\)
\(\chi_{1960}(59,\cdot)\)	1960.dj	42	yes	\(1\)	\(1\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(61,\cdot)\)	1960.dd	42	no	\(-1\)	\(1\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{11}{21}\right)\)	\(e\left(\frac{41}{42}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{23}{42}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{20}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(67,\cdot)\)	1960.bs	12	no	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{1960}(69,\cdot)\)	1960.cl	14	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(-1\)
\(\chi_{1960}(71,\cdot)\)	1960.ci	14	no	\(-1\)	\(1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(-1\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(-1\)
\(\chi_{1960}(73,\cdot)\)	1960.dp	84	no	\(1\)	\(1\)	\(e\left(\frac{11}{84}\right)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{65}{84}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{61}{84}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{1960}(79,\cdot)\)	1960.bd	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(81,\cdot)\)	1960.cm	21	no	\(1\)	\(1\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(83,\cdot)\)	1960.cp	28	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(1\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(1\)
\(\chi_{1960}(87,\cdot)\)	1960.dn	84	no	\(-1\)	\(1\)	\(e\left(\frac{59}{84}\right)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{47}{84}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{37}{84}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{1960}(89,\cdot)\)	1960.dh	42	no	\(-1\)	\(1\)	\(e\left(\frac{1}{21}\right)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{1960}(93,\cdot)\)	1960.dk	84	yes	\(-1\)	\(1\)	\(e\left(\frac{79}{84}\right)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{43}{84}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{41}{84}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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.