LMFDB - Group of Dirichlet characters of modulus 2500

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(2500)

pari: g = idealstar(,2500,2)

Character group

sage: G.order() pari: g.no
Order	=	1000
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{500}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{2500}(1251,\cdot)$, $\chi_{2500}(1877,\cdot)$

First 32 of 1000 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{2500}(1,\cdot)$	2500.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{2500}(3,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{199}{500}\right)$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{199}{250}\right)$	$e\left(\frac{91}{250}\right)$	$e\left(\frac{373}{500}\right)$	$e\left(\frac{11}{500}\right)$	$e\left(\frac{119}{125}\right)$	$e\left(\frac{86}{125}\right)$	$e\left(\frac{367}{500}\right)$	$e\left(\frac{97}{500}\right)$
$\chi_{2500}(7,\cdot)$	2500.r	100	no	$1$	$1$	$e\left(\frac{29}{100}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{83}{100}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{57}{100}\right)$	$e\left(\frac{87}{100}\right)$
$\chi_{2500}(9,\cdot)$	2500.u	250	no	$1$	$1$	$e\left(\frac{199}{250}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{74}{125}\right)$	$e\left(\frac{91}{125}\right)$	$e\left(\frac{123}{250}\right)$	$e\left(\frac{11}{250}\right)$	$e\left(\frac{113}{125}\right)$	$e\left(\frac{47}{125}\right)$	$e\left(\frac{117}{250}\right)$	$e\left(\frac{97}{250}\right)$
$\chi_{2500}(11,\cdot)$	2500.t	250	yes	$-1$	$1$	$e\left(\frac{91}{250}\right)$	$e\left(\frac{11}{50}\right)$	$e\left(\frac{91}{125}\right)$	$e\left(\frac{163}{250}\right)$	$e\left(\frac{41}{125}\right)$	$e\left(\frac{87}{125}\right)$	$e\left(\frac{109}{250}\right)$	$e\left(\frac{73}{125}\right)$	$e\left(\frac{203}{250}\right)$	$e\left(\frac{23}{250}\right)$
$\chi_{2500}(13,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{373}{500}\right)$	$e\left(\frac{83}{100}\right)$	$e\left(\frac{123}{250}\right)$	$e\left(\frac{41}{125}\right)$	$e\left(\frac{321}{500}\right)$	$e\left(\frac{47}{500}\right)$	$e\left(\frac{51}{250}\right)$	$e\left(\frac{72}{125}\right)$	$e\left(\frac{409}{500}\right)$	$e\left(\frac{119}{500}\right)$
$\chi_{2500}(17,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{11}{500}\right)$	$e\left(\frac{81}{100}\right)$	$e\left(\frac{11}{250}\right)$	$e\left(\frac{87}{125}\right)$	$e\left(\frac{47}{500}\right)$	$e\left(\frac{429}{500}\right)$	$e\left(\frac{157}{250}\right)$	$e\left(\frac{104}{125}\right)$	$e\left(\frac{63}{500}\right)$	$e\left(\frac{33}{500}\right)$
$\chi_{2500}(19,\cdot)$	2500.v	250	yes	$-1$	$1$	$e\left(\frac{119}{125}\right)$	$e\left(\frac{24}{25}\right)$	$e\left(\frac{113}{125}\right)$	$e\left(\frac{109}{250}\right)$	$e\left(\frac{51}{250}\right)$	$e\left(\frac{157}{250}\right)$	$e\left(\frac{237}{250}\right)$	$e\left(\frac{114}{125}\right)$	$e\left(\frac{102}{125}\right)$	$e\left(\frac{107}{125}\right)$
$\chi_{2500}(21,\cdot)$	2500.s	125	no	$1$	$1$	$e\left(\frac{86}{125}\right)$	$e\left(\frac{6}{25}\right)$	$e\left(\frac{47}{125}\right)$	$e\left(\frac{73}{125}\right)$	$e\left(\frac{72}{125}\right)$	$e\left(\frac{104}{125}\right)$	$e\left(\frac{114}{125}\right)$	$e\left(\frac{116}{125}\right)$	$e\left(\frac{38}{125}\right)$	$e\left(\frac{8}{125}\right)$
$\chi_{2500}(23,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{367}{500}\right)$	$e\left(\frac{57}{100}\right)$	$e\left(\frac{117}{250}\right)$	$e\left(\frac{203}{250}\right)$	$e\left(\frac{409}{500}\right)$	$e\left(\frac{63}{500}\right)$	$e\left(\frac{102}{125}\right)$	$e\left(\frac{38}{125}\right)$	$e\left(\frac{11}{500}\right)$	$e\left(\frac{101}{500}\right)$
$\chi_{2500}(27,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{97}{500}\right)$	$e\left(\frac{87}{100}\right)$	$e\left(\frac{97}{250}\right)$	$e\left(\frac{23}{250}\right)$	$e\left(\frac{119}{500}\right)$	$e\left(\frac{33}{500}\right)$	$e\left(\frac{107}{125}\right)$	$e\left(\frac{8}{125}\right)$	$e\left(\frac{101}{500}\right)$	$e\left(\frac{291}{500}\right)$
$\chi_{2500}(29,\cdot)$	2500.u	250	no	$1$	$1$	$e\left(\frac{167}{250}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{42}{125}\right)$	$e\left(\frac{28}{125}\right)$	$e\left(\frac{9}{250}\right)$	$e\left(\frac{13}{250}\right)$	$e\left(\frac{54}{125}\right)$	$e\left(\frac{101}{125}\right)$	$e\left(\frac{161}{250}\right)$	$e\left(\frac{1}{250}\right)$
$\chi_{2500}(31,\cdot)$	2500.t	250	yes	$-1$	$1$	$e\left(\frac{193}{250}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{68}{125}\right)$	$e\left(\frac{49}{250}\right)$	$e\left(\frac{43}{125}\right)$	$e\left(\frac{76}{125}\right)$	$e\left(\frac{157}{250}\right)$	$e\left(\frac{104}{125}\right)$	$e\left(\frac{219}{250}\right)$	$e\left(\frac{79}{250}\right)$
$\chi_{2500}(33,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{381}{500}\right)$	$e\left(\frac{51}{100}\right)$	$e\left(\frac{131}{250}\right)$	$e\left(\frac{2}{125}\right)$	$e\left(\frac{37}{500}\right)$	$e\left(\frac{359}{500}\right)$	$e\left(\frac{97}{250}\right)$	$e\left(\frac{34}{125}\right)$	$e\left(\frac{273}{500}\right)$	$e\left(\frac{143}{500}\right)$
$\chi_{2500}(37,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{203}{500}\right)$	$e\left(\frac{13}{100}\right)$	$e\left(\frac{203}{250}\right)$	$e\left(\frac{26}{125}\right)$	$e\left(\frac{231}{500}\right)$	$e\left(\frac{417}{500}\right)$	$e\left(\frac{11}{250}\right)$	$e\left(\frac{67}{125}\right)$	$e\left(\frac{299}{500}\right)$	$e\left(\frac{109}{500}\right)$
$\chi_{2500}(39,\cdot)$	2500.v	250	yes	$-1$	$1$	$e\left(\frac{18}{125}\right)$	$e\left(\frac{3}{25}\right)$	$e\left(\frac{36}{125}\right)$	$e\left(\frac{173}{250}\right)$	$e\left(\frac{97}{250}\right)$	$e\left(\frac{29}{250}\right)$	$e\left(\frac{39}{250}\right)$	$e\left(\frac{33}{125}\right)$	$e\left(\frac{69}{125}\right)$	$e\left(\frac{54}{125}\right)$
$\chi_{2500}(41,\cdot)$	2500.s	125	no	$1$	$1$	$e\left(\frac{52}{125}\right)$	$e\left(\frac{17}{25}\right)$	$e\left(\frac{104}{125}\right)$	$e\left(\frac{111}{125}\right)$	$e\left(\frac{29}{125}\right)$	$e\left(\frac{28}{125}\right)$	$e\left(\frac{98}{125}\right)$	$e\left(\frac{12}{125}\right)$	$e\left(\frac{116}{125}\right)$	$e\left(\frac{31}{125}\right)$
$\chi_{2500}(43,\cdot)$	2500.r	100	no	$1$	$1$	$e\left(\frac{23}{100}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{23}{50}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{21}{100}\right)$	$e\left(\frac{47}{100}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{59}{100}\right)$	$e\left(\frac{69}{100}\right)$
$\chi_{2500}(47,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{129}{500}\right)$	$e\left(\frac{59}{100}\right)$	$e\left(\frac{129}{250}\right)$	$e\left(\frac{211}{250}\right)$	$e\left(\frac{483}{500}\right)$	$e\left(\frac{281}{500}\right)$	$e\left(\frac{74}{125}\right)$	$e\left(\frac{106}{125}\right)$	$e\left(\frac{57}{500}\right)$	$e\left(\frac{387}{500}\right)$
$\chi_{2500}(49,\cdot)$	2500.o	50	no	$1$	$1$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{4}{25}\right)$	$e\left(\frac{11}{25}\right)$	$e\left(\frac{33}{50}\right)$	$e\left(\frac{31}{50}\right)$	$e\left(\frac{23}{25}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{7}{50}\right)$	$e\left(\frac{37}{50}\right)$
$\chi_{2500}(51,\cdot)$	2500.p	50	no	$-1$	$1$	$e\left(\frac{21}{50}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{3}{50}\right)$	$e\left(\frac{21}{25}\right)$	$e\left(\frac{22}{25}\right)$	$e\left(\frac{29}{50}\right)$	$e\left(\frac{13}{25}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{13}{50}\right)$
$\chi_{2500}(53,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{269}{500}\right)$	$e\left(\frac{99}{100}\right)$	$e\left(\frac{19}{250}\right)$	$e\left(\frac{48}{125}\right)$	$e\left(\frac{13}{500}\right)$	$e\left(\frac{491}{500}\right)$	$e\left(\frac{203}{250}\right)$	$e\left(\frac{66}{125}\right)$	$e\left(\frac{177}{500}\right)$	$e\left(\frac{307}{500}\right)$
$\chi_{2500}(57,\cdot)$	2500.k	20	no	$-1$	$1$	$e\left(\frac{7}{20}\right)$	$i$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{1}{20}\right)$
$\chi_{2500}(59,\cdot)$	2500.v	250	yes	$-1$	$1$	$e\left(\frac{22}{125}\right)$	$e\left(\frac{12}{25}\right)$	$e\left(\frac{44}{125}\right)$	$e\left(\frac{17}{250}\right)$	$e\left(\frac{63}{250}\right)$	$e\left(\frac{91}{250}\right)$	$e\left(\frac{131}{250}\right)$	$e\left(\frac{82}{125}\right)$	$e\left(\frac{1}{125}\right)$	$e\left(\frac{66}{125}\right)$
$\chi_{2500}(61,\cdot)$	2500.s	125	no	$1$	$1$	$e\left(\frac{73}{125}\right)$	$e\left(\frac{8}{25}\right)$	$e\left(\frac{21}{125}\right)$	$e\left(\frac{14}{125}\right)$	$e\left(\frac{96}{125}\right)$	$e\left(\frac{97}{125}\right)$	$e\left(\frac{27}{125}\right)$	$e\left(\frac{113}{125}\right)$	$e\left(\frac{9}{125}\right)$	$e\left(\frac{94}{125}\right)$
$\chi_{2500}(63,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{43}{500}\right)$	$e\left(\frac{53}{100}\right)$	$e\left(\frac{43}{250}\right)$	$e\left(\frac{237}{250}\right)$	$e\left(\frac{161}{500}\right)$	$e\left(\frac{427}{500}\right)$	$e\left(\frac{108}{125}\right)$	$e\left(\frac{77}{125}\right)$	$e\left(\frac{19}{500}\right)$	$e\left(\frac{129}{500}\right)$
$\chi_{2500}(67,\cdot)$	2500.x	500	yes	$1$	$1$	$e\left(\frac{141}{500}\right)$	$e\left(\frac{11}{100}\right)$	$e\left(\frac{141}{250}\right)$	$e\left(\frac{219}{250}\right)$	$e\left(\frac{307}{500}\right)$	$e\left(\frac{249}{500}\right)$	$e\left(\frac{46}{125}\right)$	$e\left(\frac{49}{125}\right)$	$e\left(\frac{353}{500}\right)$	$e\left(\frac{423}{500}\right)$
$\chi_{2500}(69,\cdot)$	2500.u	250	no	$1$	$1$	$e\left(\frac{33}{250}\right)$	$e\left(\frac{43}{50}\right)$	$e\left(\frac{33}{125}\right)$	$e\left(\frac{22}{125}\right)$	$e\left(\frac{141}{250}\right)$	$e\left(\frac{37}{250}\right)$	$e\left(\frac{96}{125}\right)$	$e\left(\frac{124}{125}\right)$	$e\left(\frac{189}{250}\right)$	$e\left(\frac{99}{250}\right)$
$\chi_{2500}(71,\cdot)$	2500.t	250	yes	$-1$	$1$	$e\left(\frac{177}{250}\right)$	$e\left(\frac{17}{50}\right)$	$e\left(\frac{52}{125}\right)$	$e\left(\frac{111}{250}\right)$	$e\left(\frac{77}{125}\right)$	$e\left(\frac{14}{125}\right)$	$e\left(\frac{223}{250}\right)$	$e\left(\frac{6}{125}\right)$	$e\left(\frac{241}{250}\right)$	$e\left(\frac{31}{250}\right)$
$\chi_{2500}(73,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{137}{500}\right)$	$e\left(\frac{27}{100}\right)$	$e\left(\frac{137}{250}\right)$	$e\left(\frac{4}{125}\right)$	$e\left(\frac{449}{500}\right)$	$e\left(\frac{343}{500}\right)$	$e\left(\frac{69}{250}\right)$	$e\left(\frac{68}{125}\right)$	$e\left(\frac{421}{500}\right)$	$e\left(\frac{411}{500}\right)$
$\chi_{2500}(77,\cdot)$	2500.w	500	no	$-1$	$1$	$e\left(\frac{327}{500}\right)$	$e\left(\frac{17}{100}\right)$	$e\left(\frac{77}{250}\right)$	$e\left(\frac{109}{125}\right)$	$e\left(\frac{79}{500}\right)$	$e\left(\frac{253}{500}\right)$	$e\left(\frac{99}{250}\right)$	$e\left(\frac{103}{125}\right)$	$e\left(\frac{191}{500}\right)$	$e\left(\frac{481}{500}\right)$
$\chi_{2500}(79,\cdot)$	2500.v	250	yes	$-1$	$1$	$e\left(\frac{81}{125}\right)$	$e\left(\frac{1}{25}\right)$	$e\left(\frac{37}{125}\right)$	$e\left(\frac{91}{250}\right)$	$e\left(\frac{249}{250}\right)$	$e\left(\frac{193}{250}\right)$	$e\left(\frac{113}{250}\right)$	$e\left(\frac{86}{125}\right)$	$e\left(\frac{123}{125}\right)$	$e\left(\frac{118}{125}\right)$

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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	$2500$
Structure	\(C_{2}\times C_{500}\)
Order	$1000$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{2500}(1,\cdot)\)	2500.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{2500}(3,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{199}{500}\right)\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{199}{250}\right)\)	\(e\left(\frac{91}{250}\right)\)	\(e\left(\frac{373}{500}\right)\)	\(e\left(\frac{11}{500}\right)\)	\(e\left(\frac{119}{125}\right)\)	\(e\left(\frac{86}{125}\right)\)	\(e\left(\frac{367}{500}\right)\)	\(e\left(\frac{97}{500}\right)\)
\(\chi_{2500}(7,\cdot)\)	2500.r	100	no	\(1\)	\(1\)	\(e\left(\frac{29}{100}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{83}{100}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{57}{100}\right)\)	\(e\left(\frac{87}{100}\right)\)
\(\chi_{2500}(9,\cdot)\)	2500.u	250	no	\(1\)	\(1\)	\(e\left(\frac{199}{250}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{74}{125}\right)\)	\(e\left(\frac{91}{125}\right)\)	\(e\left(\frac{123}{250}\right)\)	\(e\left(\frac{11}{250}\right)\)	\(e\left(\frac{113}{125}\right)\)	\(e\left(\frac{47}{125}\right)\)	\(e\left(\frac{117}{250}\right)\)	\(e\left(\frac{97}{250}\right)\)
\(\chi_{2500}(11,\cdot)\)	2500.t	250	yes	\(-1\)	\(1\)	\(e\left(\frac{91}{250}\right)\)	\(e\left(\frac{11}{50}\right)\)	\(e\left(\frac{91}{125}\right)\)	\(e\left(\frac{163}{250}\right)\)	\(e\left(\frac{41}{125}\right)\)	\(e\left(\frac{87}{125}\right)\)	\(e\left(\frac{109}{250}\right)\)	\(e\left(\frac{73}{125}\right)\)	\(e\left(\frac{203}{250}\right)\)	\(e\left(\frac{23}{250}\right)\)
\(\chi_{2500}(13,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{373}{500}\right)\)	\(e\left(\frac{83}{100}\right)\)	\(e\left(\frac{123}{250}\right)\)	\(e\left(\frac{41}{125}\right)\)	\(e\left(\frac{321}{500}\right)\)	\(e\left(\frac{47}{500}\right)\)	\(e\left(\frac{51}{250}\right)\)	\(e\left(\frac{72}{125}\right)\)	\(e\left(\frac{409}{500}\right)\)	\(e\left(\frac{119}{500}\right)\)
\(\chi_{2500}(17,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{11}{500}\right)\)	\(e\left(\frac{81}{100}\right)\)	\(e\left(\frac{11}{250}\right)\)	\(e\left(\frac{87}{125}\right)\)	\(e\left(\frac{47}{500}\right)\)	\(e\left(\frac{429}{500}\right)\)	\(e\left(\frac{157}{250}\right)\)	\(e\left(\frac{104}{125}\right)\)	\(e\left(\frac{63}{500}\right)\)	\(e\left(\frac{33}{500}\right)\)
\(\chi_{2500}(19,\cdot)\)	2500.v	250	yes	\(-1\)	\(1\)	\(e\left(\frac{119}{125}\right)\)	\(e\left(\frac{24}{25}\right)\)	\(e\left(\frac{113}{125}\right)\)	\(e\left(\frac{109}{250}\right)\)	\(e\left(\frac{51}{250}\right)\)	\(e\left(\frac{157}{250}\right)\)	\(e\left(\frac{237}{250}\right)\)	\(e\left(\frac{114}{125}\right)\)	\(e\left(\frac{102}{125}\right)\)	\(e\left(\frac{107}{125}\right)\)
\(\chi_{2500}(21,\cdot)\)	2500.s	125	no	\(1\)	\(1\)	\(e\left(\frac{86}{125}\right)\)	\(e\left(\frac{6}{25}\right)\)	\(e\left(\frac{47}{125}\right)\)	\(e\left(\frac{73}{125}\right)\)	\(e\left(\frac{72}{125}\right)\)	\(e\left(\frac{104}{125}\right)\)	\(e\left(\frac{114}{125}\right)\)	\(e\left(\frac{116}{125}\right)\)	\(e\left(\frac{38}{125}\right)\)	\(e\left(\frac{8}{125}\right)\)
\(\chi_{2500}(23,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{367}{500}\right)\)	\(e\left(\frac{57}{100}\right)\)	\(e\left(\frac{117}{250}\right)\)	\(e\left(\frac{203}{250}\right)\)	\(e\left(\frac{409}{500}\right)\)	\(e\left(\frac{63}{500}\right)\)	\(e\left(\frac{102}{125}\right)\)	\(e\left(\frac{38}{125}\right)\)	\(e\left(\frac{11}{500}\right)\)	\(e\left(\frac{101}{500}\right)\)
\(\chi_{2500}(27,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{97}{500}\right)\)	\(e\left(\frac{87}{100}\right)\)	\(e\left(\frac{97}{250}\right)\)	\(e\left(\frac{23}{250}\right)\)	\(e\left(\frac{119}{500}\right)\)	\(e\left(\frac{33}{500}\right)\)	\(e\left(\frac{107}{125}\right)\)	\(e\left(\frac{8}{125}\right)\)	\(e\left(\frac{101}{500}\right)\)	\(e\left(\frac{291}{500}\right)\)
\(\chi_{2500}(29,\cdot)\)	2500.u	250	no	\(1\)	\(1\)	\(e\left(\frac{167}{250}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{42}{125}\right)\)	\(e\left(\frac{28}{125}\right)\)	\(e\left(\frac{9}{250}\right)\)	\(e\left(\frac{13}{250}\right)\)	\(e\left(\frac{54}{125}\right)\)	\(e\left(\frac{101}{125}\right)\)	\(e\left(\frac{161}{250}\right)\)	\(e\left(\frac{1}{250}\right)\)
\(\chi_{2500}(31,\cdot)\)	2500.t	250	yes	\(-1\)	\(1\)	\(e\left(\frac{193}{250}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{68}{125}\right)\)	\(e\left(\frac{49}{250}\right)\)	\(e\left(\frac{43}{125}\right)\)	\(e\left(\frac{76}{125}\right)\)	\(e\left(\frac{157}{250}\right)\)	\(e\left(\frac{104}{125}\right)\)	\(e\left(\frac{219}{250}\right)\)	\(e\left(\frac{79}{250}\right)\)
\(\chi_{2500}(33,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{381}{500}\right)\)	\(e\left(\frac{51}{100}\right)\)	\(e\left(\frac{131}{250}\right)\)	\(e\left(\frac{2}{125}\right)\)	\(e\left(\frac{37}{500}\right)\)	\(e\left(\frac{359}{500}\right)\)	\(e\left(\frac{97}{250}\right)\)	\(e\left(\frac{34}{125}\right)\)	\(e\left(\frac{273}{500}\right)\)	\(e\left(\frac{143}{500}\right)\)
\(\chi_{2500}(37,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{203}{500}\right)\)	\(e\left(\frac{13}{100}\right)\)	\(e\left(\frac{203}{250}\right)\)	\(e\left(\frac{26}{125}\right)\)	\(e\left(\frac{231}{500}\right)\)	\(e\left(\frac{417}{500}\right)\)	\(e\left(\frac{11}{250}\right)\)	\(e\left(\frac{67}{125}\right)\)	\(e\left(\frac{299}{500}\right)\)	\(e\left(\frac{109}{500}\right)\)
\(\chi_{2500}(39,\cdot)\)	2500.v	250	yes	\(-1\)	\(1\)	\(e\left(\frac{18}{125}\right)\)	\(e\left(\frac{3}{25}\right)\)	\(e\left(\frac{36}{125}\right)\)	\(e\left(\frac{173}{250}\right)\)	\(e\left(\frac{97}{250}\right)\)	\(e\left(\frac{29}{250}\right)\)	\(e\left(\frac{39}{250}\right)\)	\(e\left(\frac{33}{125}\right)\)	\(e\left(\frac{69}{125}\right)\)	\(e\left(\frac{54}{125}\right)\)
\(\chi_{2500}(41,\cdot)\)	2500.s	125	no	\(1\)	\(1\)	\(e\left(\frac{52}{125}\right)\)	\(e\left(\frac{17}{25}\right)\)	\(e\left(\frac{104}{125}\right)\)	\(e\left(\frac{111}{125}\right)\)	\(e\left(\frac{29}{125}\right)\)	\(e\left(\frac{28}{125}\right)\)	\(e\left(\frac{98}{125}\right)\)	\(e\left(\frac{12}{125}\right)\)	\(e\left(\frac{116}{125}\right)\)	\(e\left(\frac{31}{125}\right)\)
\(\chi_{2500}(43,\cdot)\)	2500.r	100	no	\(1\)	\(1\)	\(e\left(\frac{23}{100}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{23}{50}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{21}{100}\right)\)	\(e\left(\frac{47}{100}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{59}{100}\right)\)	\(e\left(\frac{69}{100}\right)\)
\(\chi_{2500}(47,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{129}{500}\right)\)	\(e\left(\frac{59}{100}\right)\)	\(e\left(\frac{129}{250}\right)\)	\(e\left(\frac{211}{250}\right)\)	\(e\left(\frac{483}{500}\right)\)	\(e\left(\frac{281}{500}\right)\)	\(e\left(\frac{74}{125}\right)\)	\(e\left(\frac{106}{125}\right)\)	\(e\left(\frac{57}{500}\right)\)	\(e\left(\frac{387}{500}\right)\)
\(\chi_{2500}(49,\cdot)\)	2500.o	50	no	\(1\)	\(1\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{4}{25}\right)\)	\(e\left(\frac{11}{25}\right)\)	\(e\left(\frac{33}{50}\right)\)	\(e\left(\frac{31}{50}\right)\)	\(e\left(\frac{23}{25}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{7}{50}\right)\)	\(e\left(\frac{37}{50}\right)\)
\(\chi_{2500}(51,\cdot)\)	2500.p	50	no	\(-1\)	\(1\)	\(e\left(\frac{21}{50}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{3}{50}\right)\)	\(e\left(\frac{21}{25}\right)\)	\(e\left(\frac{22}{25}\right)\)	\(e\left(\frac{29}{50}\right)\)	\(e\left(\frac{13}{25}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{13}{50}\right)\)
\(\chi_{2500}(53,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{269}{500}\right)\)	\(e\left(\frac{99}{100}\right)\)	\(e\left(\frac{19}{250}\right)\)	\(e\left(\frac{48}{125}\right)\)	\(e\left(\frac{13}{500}\right)\)	\(e\left(\frac{491}{500}\right)\)	\(e\left(\frac{203}{250}\right)\)	\(e\left(\frac{66}{125}\right)\)	\(e\left(\frac{177}{500}\right)\)	\(e\left(\frac{307}{500}\right)\)
\(\chi_{2500}(57,\cdot)\)	2500.k	20	no	\(-1\)	\(1\)	\(e\left(\frac{7}{20}\right)\)	\(i\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)
\(\chi_{2500}(59,\cdot)\)	2500.v	250	yes	\(-1\)	\(1\)	\(e\left(\frac{22}{125}\right)\)	\(e\left(\frac{12}{25}\right)\)	\(e\left(\frac{44}{125}\right)\)	\(e\left(\frac{17}{250}\right)\)	\(e\left(\frac{63}{250}\right)\)	\(e\left(\frac{91}{250}\right)\)	\(e\left(\frac{131}{250}\right)\)	\(e\left(\frac{82}{125}\right)\)	\(e\left(\frac{1}{125}\right)\)	\(e\left(\frac{66}{125}\right)\)
\(\chi_{2500}(61,\cdot)\)	2500.s	125	no	\(1\)	\(1\)	\(e\left(\frac{73}{125}\right)\)	\(e\left(\frac{8}{25}\right)\)	\(e\left(\frac{21}{125}\right)\)	\(e\left(\frac{14}{125}\right)\)	\(e\left(\frac{96}{125}\right)\)	\(e\left(\frac{97}{125}\right)\)	\(e\left(\frac{27}{125}\right)\)	\(e\left(\frac{113}{125}\right)\)	\(e\left(\frac{9}{125}\right)\)	\(e\left(\frac{94}{125}\right)\)
\(\chi_{2500}(63,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{43}{500}\right)\)	\(e\left(\frac{53}{100}\right)\)	\(e\left(\frac{43}{250}\right)\)	\(e\left(\frac{237}{250}\right)\)	\(e\left(\frac{161}{500}\right)\)	\(e\left(\frac{427}{500}\right)\)	\(e\left(\frac{108}{125}\right)\)	\(e\left(\frac{77}{125}\right)\)	\(e\left(\frac{19}{500}\right)\)	\(e\left(\frac{129}{500}\right)\)
\(\chi_{2500}(67,\cdot)\)	2500.x	500	yes	\(1\)	\(1\)	\(e\left(\frac{141}{500}\right)\)	\(e\left(\frac{11}{100}\right)\)	\(e\left(\frac{141}{250}\right)\)	\(e\left(\frac{219}{250}\right)\)	\(e\left(\frac{307}{500}\right)\)	\(e\left(\frac{249}{500}\right)\)	\(e\left(\frac{46}{125}\right)\)	\(e\left(\frac{49}{125}\right)\)	\(e\left(\frac{353}{500}\right)\)	\(e\left(\frac{423}{500}\right)\)
\(\chi_{2500}(69,\cdot)\)	2500.u	250	no	\(1\)	\(1\)	\(e\left(\frac{33}{250}\right)\)	\(e\left(\frac{43}{50}\right)\)	\(e\left(\frac{33}{125}\right)\)	\(e\left(\frac{22}{125}\right)\)	\(e\left(\frac{141}{250}\right)\)	\(e\left(\frac{37}{250}\right)\)	\(e\left(\frac{96}{125}\right)\)	\(e\left(\frac{124}{125}\right)\)	\(e\left(\frac{189}{250}\right)\)	\(e\left(\frac{99}{250}\right)\)
\(\chi_{2500}(71,\cdot)\)	2500.t	250	yes	\(-1\)	\(1\)	\(e\left(\frac{177}{250}\right)\)	\(e\left(\frac{17}{50}\right)\)	\(e\left(\frac{52}{125}\right)\)	\(e\left(\frac{111}{250}\right)\)	\(e\left(\frac{77}{125}\right)\)	\(e\left(\frac{14}{125}\right)\)	\(e\left(\frac{223}{250}\right)\)	\(e\left(\frac{6}{125}\right)\)	\(e\left(\frac{241}{250}\right)\)	\(e\left(\frac{31}{250}\right)\)
\(\chi_{2500}(73,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{137}{500}\right)\)	\(e\left(\frac{27}{100}\right)\)	\(e\left(\frac{137}{250}\right)\)	\(e\left(\frac{4}{125}\right)\)	\(e\left(\frac{449}{500}\right)\)	\(e\left(\frac{343}{500}\right)\)	\(e\left(\frac{69}{250}\right)\)	\(e\left(\frac{68}{125}\right)\)	\(e\left(\frac{421}{500}\right)\)	\(e\left(\frac{411}{500}\right)\)
\(\chi_{2500}(77,\cdot)\)	2500.w	500	no	\(-1\)	\(1\)	\(e\left(\frac{327}{500}\right)\)	\(e\left(\frac{17}{100}\right)\)	\(e\left(\frac{77}{250}\right)\)	\(e\left(\frac{109}{125}\right)\)	\(e\left(\frac{79}{500}\right)\)	\(e\left(\frac{253}{500}\right)\)	\(e\left(\frac{99}{250}\right)\)	\(e\left(\frac{103}{125}\right)\)	\(e\left(\frac{191}{500}\right)\)	\(e\left(\frac{481}{500}\right)\)
\(\chi_{2500}(79,\cdot)\)	2500.v	250	yes	\(-1\)	\(1\)	\(e\left(\frac{81}{125}\right)\)	\(e\left(\frac{1}{25}\right)\)	\(e\left(\frac{37}{125}\right)\)	\(e\left(\frac{91}{250}\right)\)	\(e\left(\frac{249}{250}\right)\)	\(e\left(\frac{193}{250}\right)\)	\(e\left(\frac{113}{250}\right)\)	\(e\left(\frac{86}{125}\right)\)	\(e\left(\frac{123}{125}\right)\)	\(e\left(\frac{118}{125}\right)\)

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.