LMFDB - Group of Dirichlet characters of modulus 500

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(500)

pari: g = idealstar(,500,2)

Character group

sage: G.order() pari: g.no
Order	=	200
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{100}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{500}(251,\cdot)$, $\chi_{500}(377,\cdot)$

First 32 of 200 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{500}(1,\cdot)$	500.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{500}(3,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{99}{100}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{73}{100}\right)$	$e\left(\frac{11}{100}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{67}{100}\right)$	$e\left(\frac{97}{100}\right)$
$\chi_{500}(7,\cdot)$	500.l	20	no	$1$	$1$	$e\left(\frac{9}{20}\right)$	$-i$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{500}(9,\cdot)$	500.o	50	no	$1$	$1$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{47}{50}\right)$
$\chi_{500}(11,\cdot)$	500.p	50	yes	$-1$	$1$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{13}{50}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{9}{50}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{23}{50}\right)$
$\chi_{500}(13,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{73}{100}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{21}{100}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{1}{50}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{9}{100}\right)$	$e\left(\frac{19}{100}\right)$
$\chi_{500}(17,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{11}{100}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{33}{100}\right)$
$\chi_{500}(19,\cdot)$	500.n	50	yes	$-1$	$1$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{9}{50}\right)$	$e\left(\frac{1}{50}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{14}{25}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{7}{25}\right)$
$\chi_{500}(21,\cdot)$	500.m	25	no	$1$	$1$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{14}{25}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{8}{25}\right)$
$\chi_{500}(23,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{67}{100}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{9}{100}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{11}{100}\right)$	$e\left(\frac{1}{100}\right)$
$\chi_{500}(27,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{97}{100}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{19}{100}\right)$	$e\left(\frac{33}{100}\right)$	$e\left(\frac{7}{25}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{1}{100}\right)$	$e\left(\frac{91}{100}\right)$
$\chi_{500}(29,\cdot)$	500.o	50	no	$1$	$1$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{17}{25}\right)$	$e\left(\frac{3}{25}\right)$	$e\left(\frac{9}{50}\right)$	$e\left(\frac{13}{50}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{1}{50}\right)$
$\chi_{500}(31,\cdot)$	500.p	50	yes	$-1$	$1$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{29}{50}\right)$
$\chi_{500}(33,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{31}{50}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{37}{100}\right)$	$e\left(\frac{59}{100}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{9}{25}\right)$	$e\left(\frac{73}{100}\right)$	$e\left(\frac{43}{100}\right)$
$\chi_{500}(37,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{3}{100}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{31}{100}\right)$	$e\left(\frac{17}{100}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{17}{25}\right)$	$e\left(\frac{99}{100}\right)$	$e\left(\frac{9}{100}\right)$
$\chi_{500}(39,\cdot)$	500.n	50	yes	$-1$	$1$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{39}{50}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{4}{25}\right)$
$\chi_{500}(41,\cdot)$	500.m	25	no	$1$	$1$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{3}{25}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{6}{25}\right)$
$\chi_{500}(43,\cdot)$	500.l	20	no	$1$	$1$	$e\left(\frac{3}{20}\right)$	$i$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{500}(47,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{83}{100}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{57}{100}\right)$	$e\left(\frac{87}{100}\right)$
$\chi_{500}(49,\cdot)$	500.i	10	no	$1$	$1$	$e\left(\frac{9}{10}\right)$	$-1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{500}(51,\cdot)$	500.j	10	no	$-1$	$1$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{500}(53,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{69}{100}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{13}{100}\right)$	$e\left(\frac{91}{100}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{77}{100}\right)$	$e\left(\frac{7}{100}\right)$
$\chi_{500}(57,\cdot)$	500.f	4	no	$-1$	$1$	$-i$	$i$	$-1$	$1$	$-i$	$i$	$-1$	$1$	$-i$	$i$
$\chi_{500}(59,\cdot)$	500.n	50	yes	$-1$	$1$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{13}{50}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{31}{50}\right)$	$e\left(\frac{7}{25}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{16}{25}\right)$
$\chi_{500}(61,\cdot)$	500.m	25	no	$1$	$1$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{14}{25}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{9}{25}\right)$	$e\left(\frac{19}{25}\right)$
$\chi_{500}(63,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{43}{100}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{19}{100}\right)$	$e\left(\frac{29}{100}\right)$
$\chi_{500}(67,\cdot)$	500.r	100	yes	$1$	$1$	$e\left(\frac{41}{100}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{7}{100}\right)$	$e\left(\frac{49}{100}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{53}{100}\right)$	$e\left(\frac{23}{100}\right)$
$\chi_{500}(69,\cdot)$	500.o	50	no	$1$	$1$	$e\left(\frac{33}{50}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{39}{50}\right)$	$e\left(\frac{49}{50}\right)$
$\chi_{500}(71,\cdot)$	500.p	50	yes	$-1$	$1$	$e\left(\frac{27}{50}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{14}{25}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{31}{50}\right)$
$\chi_{500}(73,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{37}{100}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{49}{100}\right)$	$e\left(\frac{43}{100}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{21}{100}\right)$	$e\left(\frac{11}{100}\right)$
$\chi_{500}(77,\cdot)$	500.q	100	no	$-1$	$1$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{27}{50}\right)$	$e\left(\frac{9}{25}\right)$	$e\left(\frac{79}{100}\right)$	$e\left(\frac{53}{100}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{3}{25}\right)$	$e\left(\frac{91}{100}\right)$	$e\left(\frac{81}{100}\right)$
$\chi_{500}(79,\cdot)$	500.n	50	yes	$-1$	$1$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{41}{50}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{13}{50}\right)$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{18}{25}\right)$

Click here to search among the remaining 168 characters.

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	$500$
Structure	\(C_{2}\times C_{100}\)
Order	$200$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{500}(1,\cdot)\)	500.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{500}(3,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{99}{100}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{73}{100}\right)\)	\(e\left(\frac{11}{100}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{67}{100}\right)\)	\(e\left(\frac{97}{100}\right)\)
\(\chi_{500}(7,\cdot)\)	500.l	20	no	\(1\)	\(1\)	\(e\left(\frac{9}{20}\right)\)	\(-i\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{500}(9,\cdot)\)	500.o	50	no	\(1\)	\(1\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{47}{50}\right)\)
\(\chi_{500}(11,\cdot)\)	500.p	50	yes	\(-1\)	\(1\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{13}{50}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{9}{50}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{23}{50}\right)\)
\(\chi_{500}(13,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{73}{100}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{21}{100}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{1}{50}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{9}{100}\right)\)	\(e\left(\frac{19}{100}\right)\)
\(\chi_{500}(17,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{11}{100}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{33}{100}\right)\)
\(\chi_{500}(19,\cdot)\)	500.n	50	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{9}{50}\right)\)	\(e\left(\frac{1}{50}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{14}{25}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{7}{25}\right)\)
\(\chi_{500}(21,\cdot)\)	500.m	25	no	\(1\)	\(1\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{14}{25}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{8}{25}\right)\)
\(\chi_{500}(23,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{67}{100}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{9}{100}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{11}{100}\right)\)	\(e\left(\frac{1}{100}\right)\)
\(\chi_{500}(27,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{97}{100}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{19}{100}\right)\)	\(e\left(\frac{33}{100}\right)\)	\(e\left(\frac{7}{25}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{1}{100}\right)\)	\(e\left(\frac{91}{100}\right)\)
\(\chi_{500}(29,\cdot)\)	500.o	50	no	\(1\)	\(1\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{17}{25}\right)\)	\(e\left(\frac{3}{25}\right)\)	\(e\left(\frac{9}{50}\right)\)	\(e\left(\frac{13}{50}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{1}{50}\right)\)
\(\chi_{500}(31,\cdot)\)	500.p	50	yes	\(-1\)	\(1\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{29}{50}\right)\)
\(\chi_{500}(33,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{31}{50}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{37}{100}\right)\)	\(e\left(\frac{59}{100}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{9}{25}\right)\)	\(e\left(\frac{73}{100}\right)\)	\(e\left(\frac{43}{100}\right)\)
\(\chi_{500}(37,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{3}{100}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{31}{100}\right)\)	\(e\left(\frac{17}{100}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{17}{25}\right)\)	\(e\left(\frac{99}{100}\right)\)	\(e\left(\frac{9}{100}\right)\)
\(\chi_{500}(39,\cdot)\)	500.n	50	yes	\(-1\)	\(1\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{39}{50}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{4}{25}\right)\)
\(\chi_{500}(41,\cdot)\)	500.m	25	no	\(1\)	\(1\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{3}{25}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{6}{25}\right)\)
\(\chi_{500}(43,\cdot)\)	500.l	20	no	\(1\)	\(1\)	\(e\left(\frac{3}{20}\right)\)	\(i\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{500}(47,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{83}{100}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{57}{100}\right)\)	\(e\left(\frac{87}{100}\right)\)
\(\chi_{500}(49,\cdot)\)	500.i	10	no	\(1\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{500}(51,\cdot)\)	500.j	10	no	\(-1\)	\(1\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{500}(53,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{69}{100}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{13}{100}\right)\)	\(e\left(\frac{91}{100}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{77}{100}\right)\)	\(e\left(\frac{7}{100}\right)\)
\(\chi_{500}(57,\cdot)\)	500.f	4	no	\(-1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(i\)
\(\chi_{500}(59,\cdot)\)	500.n	50	yes	\(-1\)	\(1\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{13}{50}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{31}{50}\right)\)	\(e\left(\frac{7}{25}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{16}{25}\right)\)
\(\chi_{500}(61,\cdot)\)	500.m	25	no	\(1\)	\(1\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{14}{25}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{9}{25}\right)\)	\(e\left(\frac{19}{25}\right)\)
\(\chi_{500}(63,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{43}{100}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{19}{100}\right)\)	\(e\left(\frac{29}{100}\right)\)
\(\chi_{500}(67,\cdot)\)	500.r	100	yes	\(1\)	\(1\)	\(e\left(\frac{41}{100}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{7}{100}\right)\)	\(e\left(\frac{49}{100}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{53}{100}\right)\)	\(e\left(\frac{23}{100}\right)\)
\(\chi_{500}(69,\cdot)\)	500.o	50	no	\(1\)	\(1\)	\(e\left(\frac{33}{50}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{39}{50}\right)\)	\(e\left(\frac{49}{50}\right)\)
\(\chi_{500}(71,\cdot)\)	500.p	50	yes	\(-1\)	\(1\)	\(e\left(\frac{27}{50}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{14}{25}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{31}{50}\right)\)
\(\chi_{500}(73,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{37}{100}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{49}{100}\right)\)	\(e\left(\frac{43}{100}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{21}{100}\right)\)	\(e\left(\frac{11}{100}\right)\)
\(\chi_{500}(77,\cdot)\)	500.q	100	no	\(-1\)	\(1\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{27}{50}\right)\)	\(e\left(\frac{9}{25}\right)\)	\(e\left(\frac{79}{100}\right)\)	\(e\left(\frac{53}{100}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{3}{25}\right)\)	\(e\left(\frac{91}{100}\right)\)	\(e\left(\frac{81}{100}\right)\)
\(\chi_{500}(79,\cdot)\)	500.n	50	yes	\(-1\)	\(1\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{41}{50}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{13}{50}\right)\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{18}{25}\right)\)

Introduction

L-functions

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Character group

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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.