sage: H = DirichletGroup(1875)
pari: g = idealstar(,1875,2)
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 | = | $\chi_{1875}(626,\cdot)$, $\chi_{1875}(1252,\cdot)$ |
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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1875}(1,\cdot)\) | 1875.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(\chi_{1875}(2,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{251}{500}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{139}{500}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{209}{250}\right)\) |
| \(\chi_{1875}(4,\cdot)\) | 1875.v | 250 | no | \(1\) | \(1\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{84}{125}\right)\) |
| \(\chi_{1875}(7,\cdot)\) | 1875.q | 100 | no | \(-1\) | \(1\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{23}{50}\right)\) |
| \(\chi_{1875}(8,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{259}{500}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{127}{250}\right)\) |
| \(\chi_{1875}(11,\cdot)\) | 1875.u | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{117}{125}\right)\) |
| \(\chi_{1875}(13,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{139}{500}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{321}{500}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{51}{250}\right)\) |
| \(\chi_{1875}(14,\cdot)\) | 1875.t | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{37}{125}\right)\) |
| \(\chi_{1875}(16,\cdot)\) | 1875.s | 125 | no | \(1\) | \(1\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) |
| \(\chi_{1875}(17,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{157}{250}\right)\) |
| \(\chi_{1875}(19,\cdot)\) | 1875.v | 250 | no | \(1\) | \(1\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{56}{125}\right)\) |
| \(\chi_{1875}(22,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{477}{500}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{303}{500}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{21}{500}\right)\) | \(e\left(\frac{193}{250}\right)\) |
| \(\chi_{1875}(23,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{181}{500}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{43}{500}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{409}{500}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{313}{500}\right)\) | \(e\left(\frac{79}{250}\right)\) |
| \(\chi_{1875}(26,\cdot)\) | 1875.n | 50 | no | \(-1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) |
| \(\chi_{1875}(28,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{461}{500}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{193}{500}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{251}{500}\right)\) | \(e\left(\frac{33}{250}\right)\) |
| \(\chi_{1875}(29,\cdot)\) | 1875.t | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{54}{125}\right)\) |
| \(\chi_{1875}(31,\cdot)\) | 1875.s | 125 | no | \(1\) | \(1\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{16}{125}\right)\) |
| \(\chi_{1875}(32,\cdot)\) | 1875.r | 100 | no | \(1\) | \(1\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) |
| \(\chi_{1875}(34,\cdot)\) | 1875.v | 250 | no | \(1\) | \(1\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{58}{125}\right)\) |
| \(\chi_{1875}(37,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{329}{500}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{487}{500}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{231}{500}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{11}{250}\right)\) |
| \(\chi_{1875}(38,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{169}{500}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{7}{500}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{237}{500}\right)\) | \(e\left(\frac{71}{250}\right)\) |
| \(\chi_{1875}(41,\cdot)\) | 1875.u | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) |
| \(\chi_{1875}(43,\cdot)\) | 1875.q | 100 | no | \(-1\) | \(1\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) |
| \(\chi_{1875}(44,\cdot)\) | 1875.t | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{76}{125}\right)\) |
| \(\chi_{1875}(46,\cdot)\) | 1875.s | 125 | no | \(1\) | \(1\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) |
| \(\chi_{1875}(47,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{347}{500}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{41}{500}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{31}{500}\right)\) | \(e\left(\frac{23}{250}\right)\) |
| \(\chi_{1875}(49,\cdot)\) | 1875.o | 50 | no | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) |
| \(\chi_{1875}(52,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{141}{500}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{99}{500}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{393}{500}\right)\) | \(e\left(\frac{219}{250}\right)\) |
| \(\chi_{1875}(53,\cdot)\) | 1875.x | 500 | yes | \(1\) | \(1\) | \(e\left(\frac{117}{500}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{351}{500}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{13}{500}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{203}{250}\right)\) |
| \(\chi_{1875}(56,\cdot)\) | 1875.u | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) |
| \(\chi_{1875}(58,\cdot)\) | 1875.w | 500 | no | \(-1\) | \(1\) | \(e\left(\frac{163}{500}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{489}{500}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{67}{250}\right)\) |
| \(\chi_{1875}(59,\cdot)\) | 1875.t | 250 | yes | \(-1\) | \(1\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{38}{125}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) |