# Properties

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

# Learn more

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 $$-1$$ $$1$$ $$e\left(\frac{81}{125}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{37}{125}\right)$$ $$e\left(\frac{91}{250}\right)$$ $$e\left(\frac{249}{250}\right)$$ $$e\left(\frac{193}{250}\right)$$ $$e\left(\frac{113}{250}\right)$$ $$e\left(\frac{86}{125}\right)$$ $$e\left(\frac{123}{125}\right)$$ $$e\left(\frac{118}{125}\right)$$
Click here to search among the remaining 968 characters.