LMFDB - Group of Dirichlet characters of modulus 4000

Show commands: PariGP / SageMath

sage: H = DirichletGroup(4000)

pari: g = idealstar(,4000,2)

Character group

sage: G.order() pari: g.no
Order	=	1600
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{4}\times C_{200}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{4000}(2751,\cdot)$, $\chi_{4000}(2501,\cdot)$, $\chi_{4000}(1377,\cdot)$

First 32 of 1600 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{4000}(1,\cdot)$	4000.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{4000}(3,\cdot)$	4000.cy	200	yes	$1$	$1$	$e\left(\frac{23}{200}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{23}{100}\right)$	$e\left(\frac{139}{200}\right)$	$e\left(\frac{71}{200}\right)$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{77}{200}\right)$	$e\left(\frac{63}{200}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{69}{200}\right)$
$\chi_{4000}(7,\cdot)$	4000.bl	20	no	$1$	$1$	$e\left(\frac{1}{5}\right)$	$i$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{4000}(9,\cdot)$	4000.cn	100	no	$1$	$1$	$e\left(\frac{23}{100}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{39}{100}\right)$	$e\left(\frac{71}{100}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{77}{100}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{69}{100}\right)$
$\chi_{4000}(11,\cdot)$	4000.db	200	yes	$-1$	$1$	$e\left(\frac{139}{200}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{39}{100}\right)$	$e\left(\frac{77}{200}\right)$	$e\left(\frac{3}{200}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{111}{200}\right)$	$e\left(\frac{9}{200}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{17}{200}\right)$
$\chi_{4000}(13,\cdot)$	4000.de	200	yes	$-1$	$1$	$e\left(\frac{71}{200}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{71}{100}\right)$	$e\left(\frac{3}{200}\right)$	$e\left(\frac{67}{200}\right)$	$e\left(\frac{97}{100}\right)$	$e\left(\frac{29}{200}\right)$	$e\left(\frac{51}{200}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{13}{200}\right)$
$\chi_{4000}(17,\cdot)$	4000.ct	100	no	$-1$	$1$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{97}{100}\right)$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{33}{50}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{83}{100}\right)$
$\chi_{4000}(19,\cdot)$	4000.dc	200	yes	$-1$	$1$	$e\left(\frac{77}{200}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{77}{100}\right)$	$e\left(\frac{111}{200}\right)$	$e\left(\frac{29}{200}\right)$	$e\left(\frac{16}{25}\right)$	$e\left(\frac{173}{200}\right)$	$e\left(\frac{187}{200}\right)$	$e\left(\frac{33}{100}\right)$	$e\left(\frac{31}{200}\right)$
$\chi_{4000}(21,\cdot)$	4000.dd	200	yes	$1$	$1$	$e\left(\frac{63}{200}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{9}{200}\right)$	$e\left(\frac{51}{200}\right)$	$e\left(\frac{33}{50}\right)$	$e\left(\frac{187}{200}\right)$	$e\left(\frac{153}{200}\right)$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{189}{200}\right)$
$\chi_{4000}(23,\cdot)$	4000.cv	100	no	$1$	$1$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{33}{100}\right)$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{19}{25}\right)$
$\chi_{4000}(27,\cdot)$	4000.cy	200	yes	$1$	$1$	$e\left(\frac{69}{200}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{69}{100}\right)$	$e\left(\frac{17}{200}\right)$	$e\left(\frac{13}{200}\right)$	$e\left(\frac{83}{100}\right)$	$e\left(\frac{31}{200}\right)$	$e\left(\frac{189}{200}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{7}{200}\right)$
$\chi_{4000}(29,\cdot)$	4000.da	200	yes	$1$	$1$	$e\left(\frac{93}{200}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{93}{100}\right)$	$e\left(\frac{199}{200}\right)$	$e\left(\frac{161}{200}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{157}{200}\right)$	$e\left(\frac{183}{200}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{79}{200}\right)$
$\chi_{4000}(31,\cdot)$	4000.cl	50	no	$-1$	$1$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{49}{50}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{29}{50}\right)$
$\chi_{4000}(33,\cdot)$	4000.cq	100	no	$-1$	$1$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{31}{50}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{37}{100}\right)$	$e\left(\frac{59}{100}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{9}{25}\right)$	$e\left(\frac{73}{100}\right)$	$e\left(\frac{43}{100}\right)$
$\chi_{4000}(37,\cdot)$	4000.de	200	yes	$-1$	$1$	$e\left(\frac{81}{200}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{133}{200}\right)$	$e\left(\frac{37}{200}\right)$	$e\left(\frac{67}{100}\right)$	$e\left(\frac{19}{200}\right)$	$e\left(\frac{61}{200}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{43}{200}\right)$
$\chi_{4000}(39,\cdot)$	4000.cx	100	no	$-1$	$1$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{71}{100}\right)$	$e\left(\frac{69}{100}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{53}{100}\right)$	$e\left(\frac{57}{100}\right)$	$e\left(\frac{13}{50}\right)$	$e\left(\frac{41}{100}\right)$
$\chi_{4000}(41,\cdot)$	4000.cw	100	no	$1$	$1$	$e\left(\frac{33}{100}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{33}{50}\right)$	$e\left(\frac{19}{100}\right)$	$e\left(\frac{41}{100}\right)$	$e\left(\frac{3}{25}\right)$	$e\left(\frac{17}{100}\right)$	$e\left(\frac{23}{100}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{99}{100}\right)$
$\chi_{4000}(43,\cdot)$	4000.by	40	no	$1$	$1$	$e\left(\frac{1}{40}\right)$	$-1$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{40}\right)$
$\chi_{4000}(47,\cdot)$	4000.cr	100	no	$1$	$1$	$e\left(\frac{79}{100}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{18}{25}\right)$	$e\left(\frac{33}{100}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{57}{100}\right)$	$e\left(\frac{37}{100}\right)$
$\chi_{4000}(49,\cdot)$	4000.be	10	no	$1$	$1$	$e\left(\frac{2}{5}\right)$	$-1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{4000}(51,\cdot)$	4000.cc	40	no	$-1$	$1$	$e\left(\frac{29}{40}\right)$	$i$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{7}{40}\right)$
$\chi_{4000}(53,\cdot)$	4000.cz	200	yes	$-1$	$1$	$e\left(\frac{113}{200}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{13}{100}\right)$	$e\left(\frac{9}{200}\right)$	$e\left(\frac{101}{200}\right)$	$e\left(\frac{41}{100}\right)$	$e\left(\frac{87}{200}\right)$	$e\left(\frac{153}{200}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{139}{200}\right)$
$\chi_{4000}(57,\cdot)$	4000.i	4	no	$-1$	$1$	$-1$	$-i$	$1$	$i$	$-1$	$i$	$i$	$i$	$i$	$-1$
$\chi_{4000}(59,\cdot)$	4000.dc	200	yes	$-1$	$1$	$e\left(\frac{51}{200}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{51}{100}\right)$	$e\left(\frac{193}{200}\right)$	$e\left(\frac{27}{200}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{99}{200}\right)$	$e\left(\frac{181}{200}\right)$	$e\left(\frac{79}{100}\right)$	$e\left(\frac{153}{200}\right)$
$\chi_{4000}(61,\cdot)$	4000.dd	200	yes	$1$	$1$	$e\left(\frac{9}{200}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{9}{100}\right)$	$e\left(\frac{87}{200}\right)$	$e\left(\frac{93}{200}\right)$	$e\left(\frac{19}{50}\right)$	$e\left(\frac{141}{200}\right)$	$e\left(\frac{79}{200}\right)$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{27}{200}\right)$
$\chi_{4000}(63,\cdot)$	4000.cs	100	no	$1$	$1$	$e\left(\frac{43}{100}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{61}{100}\right)$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{2}{25}\right)$	$e\left(\frac{19}{100}\right)$	$e\left(\frac{29}{100}\right)$
$\chi_{4000}(67,\cdot)$	4000.df	200	yes	$1$	$1$	$e\left(\frac{107}{200}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{7}{100}\right)$	$e\left(\frac{51}{200}\right)$	$e\left(\frac{139}{200}\right)$	$e\left(\frac{99}{100}\right)$	$e\left(\frac{93}{200}\right)$	$e\left(\frac{167}{200}\right)$	$e\left(\frac{39}{50}\right)$	$e\left(\frac{121}{200}\right)$
$\chi_{4000}(69,\cdot)$	4000.da	200	yes	$1$	$1$	$e\left(\frac{7}{200}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{7}{100}\right)$	$e\left(\frac{101}{200}\right)$	$e\left(\frac{139}{200}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{143}{200}\right)$	$e\left(\frac{117}{200}\right)$	$e\left(\frac{53}{100}\right)$	$e\left(\frac{21}{200}\right)$
$\chi_{4000}(71,\cdot)$	4000.cm	100	no	$-1$	$1$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{83}{100}\right)$	$e\left(\frac{14}{25}\right)$	$e\left(\frac{21}{100}\right)$	$e\left(\frac{49}{100}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{87}{100}\right)$
$\chi_{4000}(73,\cdot)$	4000.cp	100	no	$-1$	$1$	$e\left(\frac{31}{50}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{91}{100}\right)$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{43}{100}\right)$	$e\left(\frac{63}{100}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{71}{100}\right)$	$e\left(\frac{43}{50}\right)$
$\chi_{4000}(77,\cdot)$	4000.cz	200	yes	$-1$	$1$	$e\left(\frac{179}{200}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{79}{100}\right)$	$e\left(\frac{147}{200}\right)$	$e\left(\frac{183}{200}\right)$	$e\left(\frac{3}{100}\right)$	$e\left(\frac{21}{200}\right)$	$e\left(\frac{99}{200}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{137}{200}\right)$
$\chi_{4000}(79,\cdot)$	4000.cg	50	no	$-1$	$1$	$e\left(\frac{37}{50}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{19}{25}\right)$	$e\left(\frac{47}{50}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{11}{50}\right)$

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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	$4000$
Structure	\(C_{2}\times C_{4}\times C_{200}\)
Order	$1600$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{4000}(1,\cdot)\)	4000.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{4000}(3,\cdot)\)	4000.cy	200	yes	\(1\)	\(1\)	\(e\left(\frac{23}{200}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{23}{100}\right)\)	\(e\left(\frac{139}{200}\right)\)	\(e\left(\frac{71}{200}\right)\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{77}{200}\right)\)	\(e\left(\frac{63}{200}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{69}{200}\right)\)
\(\chi_{4000}(7,\cdot)\)	4000.bl	20	no	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(i\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{4000}(9,\cdot)\)	4000.cn	100	no	\(1\)	\(1\)	\(e\left(\frac{23}{100}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{39}{100}\right)\)	\(e\left(\frac{71}{100}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{77}{100}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{69}{100}\right)\)
\(\chi_{4000}(11,\cdot)\)	4000.db	200	yes	\(-1\)	\(1\)	\(e\left(\frac{139}{200}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{39}{100}\right)\)	\(e\left(\frac{77}{200}\right)\)	\(e\left(\frac{3}{200}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{111}{200}\right)\)	\(e\left(\frac{9}{200}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{17}{200}\right)\)
\(\chi_{4000}(13,\cdot)\)	4000.de	200	yes	\(-1\)	\(1\)	\(e\left(\frac{71}{200}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{71}{100}\right)\)	\(e\left(\frac{3}{200}\right)\)	\(e\left(\frac{67}{200}\right)\)	\(e\left(\frac{97}{100}\right)\)	\(e\left(\frac{29}{200}\right)\)	\(e\left(\frac{51}{200}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{13}{200}\right)\)
\(\chi_{4000}(17,\cdot)\)	4000.ct	100	no	\(-1\)	\(1\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{97}{100}\right)\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{33}{50}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{83}{100}\right)\)
\(\chi_{4000}(19,\cdot)\)	4000.dc	200	yes	\(-1\)	\(1\)	\(e\left(\frac{77}{200}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{77}{100}\right)\)	\(e\left(\frac{111}{200}\right)\)	\(e\left(\frac{29}{200}\right)\)	\(e\left(\frac{16}{25}\right)\)	\(e\left(\frac{173}{200}\right)\)	\(e\left(\frac{187}{200}\right)\)	\(e\left(\frac{33}{100}\right)\)	\(e\left(\frac{31}{200}\right)\)
\(\chi_{4000}(21,\cdot)\)	4000.dd	200	yes	\(1\)	\(1\)	\(e\left(\frac{63}{200}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{9}{200}\right)\)	\(e\left(\frac{51}{200}\right)\)	\(e\left(\frac{33}{50}\right)\)	\(e\left(\frac{187}{200}\right)\)	\(e\left(\frac{153}{200}\right)\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{189}{200}\right)\)
\(\chi_{4000}(23,\cdot)\)	4000.cv	100	no	\(1\)	\(1\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{33}{100}\right)\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{19}{25}\right)\)
\(\chi_{4000}(27,\cdot)\)	4000.cy	200	yes	\(1\)	\(1\)	\(e\left(\frac{69}{200}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{69}{100}\right)\)	\(e\left(\frac{17}{200}\right)\)	\(e\left(\frac{13}{200}\right)\)	\(e\left(\frac{83}{100}\right)\)	\(e\left(\frac{31}{200}\right)\)	\(e\left(\frac{189}{200}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{7}{200}\right)\)
\(\chi_{4000}(29,\cdot)\)	4000.da	200	yes	\(1\)	\(1\)	\(e\left(\frac{93}{200}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{93}{100}\right)\)	\(e\left(\frac{199}{200}\right)\)	\(e\left(\frac{161}{200}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{157}{200}\right)\)	\(e\left(\frac{183}{200}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{79}{200}\right)\)
\(\chi_{4000}(31,\cdot)\)	4000.cl	50	no	\(-1\)	\(1\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{49}{50}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{29}{50}\right)\)
\(\chi_{4000}(33,\cdot)\)	4000.cq	100	no	\(-1\)	\(1\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{31}{50}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{37}{100}\right)\)	\(e\left(\frac{59}{100}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{9}{25}\right)\)	\(e\left(\frac{73}{100}\right)\)	\(e\left(\frac{43}{100}\right)\)
\(\chi_{4000}(37,\cdot)\)	4000.de	200	yes	\(-1\)	\(1\)	\(e\left(\frac{81}{200}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{133}{200}\right)\)	\(e\left(\frac{37}{200}\right)\)	\(e\left(\frac{67}{100}\right)\)	\(e\left(\frac{19}{200}\right)\)	\(e\left(\frac{61}{200}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{43}{200}\right)\)
\(\chi_{4000}(39,\cdot)\)	4000.cx	100	no	\(-1\)	\(1\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{71}{100}\right)\)	\(e\left(\frac{69}{100}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{53}{100}\right)\)	\(e\left(\frac{57}{100}\right)\)	\(e\left(\frac{13}{50}\right)\)	\(e\left(\frac{41}{100}\right)\)
\(\chi_{4000}(41,\cdot)\)	4000.cw	100	no	\(1\)	\(1\)	\(e\left(\frac{33}{100}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{33}{50}\right)\)	\(e\left(\frac{19}{100}\right)\)	\(e\left(\frac{41}{100}\right)\)	\(e\left(\frac{3}{25}\right)\)	\(e\left(\frac{17}{100}\right)\)	\(e\left(\frac{23}{100}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{99}{100}\right)\)
\(\chi_{4000}(43,\cdot)\)	4000.by	40	no	\(1\)	\(1\)	\(e\left(\frac{1}{40}\right)\)	\(-1\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{40}\right)\)
\(\chi_{4000}(47,\cdot)\)	4000.cr	100	no	\(1\)	\(1\)	\(e\left(\frac{79}{100}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{18}{25}\right)\)	\(e\left(\frac{33}{100}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{57}{100}\right)\)	\(e\left(\frac{37}{100}\right)\)
\(\chi_{4000}(49,\cdot)\)	4000.be	10	no	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{4000}(51,\cdot)\)	4000.cc	40	no	\(-1\)	\(1\)	\(e\left(\frac{29}{40}\right)\)	\(i\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{7}{40}\right)\)
\(\chi_{4000}(53,\cdot)\)	4000.cz	200	yes	\(-1\)	\(1\)	\(e\left(\frac{113}{200}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{13}{100}\right)\)	\(e\left(\frac{9}{200}\right)\)	\(e\left(\frac{101}{200}\right)\)	\(e\left(\frac{41}{100}\right)\)	\(e\left(\frac{87}{200}\right)\)	\(e\left(\frac{153}{200}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{139}{200}\right)\)
\(\chi_{4000}(57,\cdot)\)	4000.i	4	no	\(-1\)	\(1\)	\(-1\)	\(-i\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(i\)	\(i\)	\(-1\)
\(\chi_{4000}(59,\cdot)\)	4000.dc	200	yes	\(-1\)	\(1\)	\(e\left(\frac{51}{200}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{51}{100}\right)\)	\(e\left(\frac{193}{200}\right)\)	\(e\left(\frac{27}{200}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{99}{200}\right)\)	\(e\left(\frac{181}{200}\right)\)	\(e\left(\frac{79}{100}\right)\)	\(e\left(\frac{153}{200}\right)\)
\(\chi_{4000}(61,\cdot)\)	4000.dd	200	yes	\(1\)	\(1\)	\(e\left(\frac{9}{200}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{9}{100}\right)\)	\(e\left(\frac{87}{200}\right)\)	\(e\left(\frac{93}{200}\right)\)	\(e\left(\frac{19}{50}\right)\)	\(e\left(\frac{141}{200}\right)\)	\(e\left(\frac{79}{200}\right)\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{27}{200}\right)\)
\(\chi_{4000}(63,\cdot)\)	4000.cs	100	no	\(1\)	\(1\)	\(e\left(\frac{43}{100}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{61}{100}\right)\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{2}{25}\right)\)	\(e\left(\frac{19}{100}\right)\)	\(e\left(\frac{29}{100}\right)\)
\(\chi_{4000}(67,\cdot)\)	4000.df	200	yes	\(1\)	\(1\)	\(e\left(\frac{107}{200}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{7}{100}\right)\)	\(e\left(\frac{51}{200}\right)\)	\(e\left(\frac{139}{200}\right)\)	\(e\left(\frac{99}{100}\right)\)	\(e\left(\frac{93}{200}\right)\)	\(e\left(\frac{167}{200}\right)\)	\(e\left(\frac{39}{50}\right)\)	\(e\left(\frac{121}{200}\right)\)
\(\chi_{4000}(69,\cdot)\)	4000.da	200	yes	\(1\)	\(1\)	\(e\left(\frac{7}{200}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{7}{100}\right)\)	\(e\left(\frac{101}{200}\right)\)	\(e\left(\frac{139}{200}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{143}{200}\right)\)	\(e\left(\frac{117}{200}\right)\)	\(e\left(\frac{53}{100}\right)\)	\(e\left(\frac{21}{200}\right)\)
\(\chi_{4000}(71,\cdot)\)	4000.cm	100	no	\(-1\)	\(1\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{83}{100}\right)\)	\(e\left(\frac{14}{25}\right)\)	\(e\left(\frac{21}{100}\right)\)	\(e\left(\frac{49}{100}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{87}{100}\right)\)
\(\chi_{4000}(73,\cdot)\)	4000.cp	100	no	\(-1\)	\(1\)	\(e\left(\frac{31}{50}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{91}{100}\right)\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{43}{100}\right)\)	\(e\left(\frac{63}{100}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{71}{100}\right)\)	\(e\left(\frac{43}{50}\right)\)
\(\chi_{4000}(77,\cdot)\)	4000.cz	200	yes	\(-1\)	\(1\)	\(e\left(\frac{179}{200}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{79}{100}\right)\)	\(e\left(\frac{147}{200}\right)\)	\(e\left(\frac{183}{200}\right)\)	\(e\left(\frac{3}{100}\right)\)	\(e\left(\frac{21}{200}\right)\)	\(e\left(\frac{99}{200}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{137}{200}\right)\)
\(\chi_{4000}(79,\cdot)\)	4000.cg	50	no	\(-1\)	\(1\)	\(e\left(\frac{37}{50}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{19}{25}\right)\)	\(e\left(\frac{47}{50}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{11}{50}\right)\)

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.