# Properties

 Modulus 4001 Structure $$C_{4000}$$ Order 4000

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(4001)

pari: g = idealstar(,4001,2)

## Character group

 sage: G.order()  pari: g.no Order = 4000 sage: H.invariants()  pari: g.cyc Structure = $$C_{4000}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{4001}(3,\cdot)$

## First 32 of 4000 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.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{4001}(1,\cdot)$$ 4001.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{4001}(2,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{29}{250}\right)$$ $$e\left(\frac{523}{1000}\right)$$ $$e\left(\frac{29}{125}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{639}{1000}\right)$$ $$e\left(\frac{61}{250}\right)$$ $$e\left(\frac{87}{250}\right)$$ $$e\left(\frac{23}{500}\right)$$ $$e\left(\frac{32}{125}\right)$$ $$e\left(\frac{11}{40}\right)$$
$$\chi_{4001}(3,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{523}{1000}\right)$$ $$e\left(\frac{1}{4000}\right)$$ $$e\left(\frac{23}{500}\right)$$ $$e\left(\frac{59}{200}\right)$$ $$e\left(\frac{2093}{4000}\right)$$ $$e\left(\frac{57}{1000}\right)$$ $$e\left(\frac{569}{1000}\right)$$ $$e\left(\frac{1}{2000}\right)$$ $$e\left(\frac{409}{500}\right)$$ $$e\left(\frac{57}{160}\right)$$
$$\chi_{4001}(4,\cdot)$$ 4001.t 500 Yes $$1$$ $$1$$ $$e\left(\frac{29}{125}\right)$$ $$e\left(\frac{23}{500}\right)$$ $$e\left(\frac{58}{125}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{139}{500}\right)$$ $$e\left(\frac{61}{125}\right)$$ $$e\left(\frac{87}{125}\right)$$ $$e\left(\frac{23}{250}\right)$$ $$e\left(\frac{64}{125}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{4001}(5,\cdot)$$ 4001.q 200 Yes $$1$$ $$1$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{59}{200}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{87}{200}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{4001}(6,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{639}{1000}\right)$$ $$e\left(\frac{2093}{4000}\right)$$ $$e\left(\frac{139}{500}\right)$$ $$e\left(\frac{87}{200}\right)$$ $$e\left(\frac{649}{4000}\right)$$ $$e\left(\frac{301}{1000}\right)$$ $$e\left(\frac{917}{1000}\right)$$ $$e\left(\frac{93}{2000}\right)$$ $$e\left(\frac{37}{500}\right)$$ $$e\left(\frac{101}{160}\right)$$
$$\chi_{4001}(7,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{61}{250}\right)$$ $$e\left(\frac{57}{1000}\right)$$ $$e\left(\frac{61}{125}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{301}{1000}\right)$$ $$e\left(\frac{249}{250}\right)$$ $$e\left(\frac{183}{250}\right)$$ $$e\left(\frac{57}{500}\right)$$ $$e\left(\frac{63}{125}\right)$$ $$e\left(\frac{9}{40}\right)$$
$$\chi_{4001}(8,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{87}{250}\right)$$ $$e\left(\frac{569}{1000}\right)$$ $$e\left(\frac{87}{125}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{917}{1000}\right)$$ $$e\left(\frac{183}{250}\right)$$ $$e\left(\frac{11}{250}\right)$$ $$e\left(\frac{69}{500}\right)$$ $$e\left(\frac{96}{125}\right)$$ $$e\left(\frac{33}{40}\right)$$
$$\chi_{4001}(9,\cdot)$$ 4001.w 2000 Yes $$1$$ $$1$$ $$e\left(\frac{23}{500}\right)$$ $$e\left(\frac{1}{2000}\right)$$ $$e\left(\frac{23}{250}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{93}{2000}\right)$$ $$e\left(\frac{57}{500}\right)$$ $$e\left(\frac{69}{500}\right)$$ $$e\left(\frac{1}{1000}\right)$$ $$e\left(\frac{159}{250}\right)$$ $$e\left(\frac{57}{80}\right)$$
$$\chi_{4001}(10,\cdot)$$ 4001.t 500 Yes $$1$$ $$1$$ $$e\left(\frac{32}{125}\right)$$ $$e\left(\frac{409}{500}\right)$$ $$e\left(\frac{64}{125}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{37}{500}\right)$$ $$e\left(\frac{63}{125}\right)$$ $$e\left(\frac{96}{125}\right)$$ $$e\left(\frac{159}{250}\right)$$ $$e\left(\frac{62}{125}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{4001}(11,\cdot)$$ 4001.p 160 Yes $$-1$$ $$1$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{57}{160}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{101}{160}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{57}{80}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{21}{32}\right)$$
$$\chi_{4001}(12,\cdot)$$ 4001.u 800 Yes $$-1$$ $$1$$ $$e\left(\frac{151}{200}\right)$$ $$e\left(\frac{37}{800}\right)$$ $$e\left(\frac{51}{100}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{641}{800}\right)$$ $$e\left(\frac{109}{200}\right)$$ $$e\left(\frac{53}{200}\right)$$ $$e\left(\frac{37}{400}\right)$$ $$e\left(\frac{33}{100}\right)$$ $$e\left(\frac{29}{32}\right)$$
$$\chi_{4001}(13,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{193}{250}\right)$$ $$e\left(\frac{541}{1000}\right)$$ $$e\left(\frac{68}{125}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{313}{1000}\right)$$ $$e\left(\frac{87}{250}\right)$$ $$e\left(\frac{79}{250}\right)$$ $$e\left(\frac{41}{500}\right)$$ $$e\left(\frac{19}{125}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{4001}(14,\cdot)$$ 4001.l 50 Yes $$1$$ $$1$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$-1$$
$$\chi_{4001}(15,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{663}{1000}\right)$$ $$e\left(\frac{1181}{4000}\right)$$ $$e\left(\frac{163}{500}\right)$$ $$e\left(\frac{79}{200}\right)$$ $$e\left(\frac{3833}{4000}\right)$$ $$e\left(\frac{317}{1000}\right)$$ $$e\left(\frac{989}{1000}\right)$$ $$e\left(\frac{1181}{2000}\right)$$ $$e\left(\frac{29}{500}\right)$$ $$e\left(\frac{117}{160}\right)$$
$$\chi_{4001}(16,\cdot)$$ 4001.r 250 Yes $$1$$ $$1$$ $$e\left(\frac{58}{125}\right)$$ $$e\left(\frac{23}{250}\right)$$ $$e\left(\frac{116}{125}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{139}{250}\right)$$ $$e\left(\frac{122}{125}\right)$$ $$e\left(\frac{49}{125}\right)$$ $$e\left(\frac{23}{125}\right)$$ $$e\left(\frac{3}{125}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{4001}(17,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{413}{1000}\right)$$ $$e\left(\frac{1431}{4000}\right)$$ $$e\left(\frac{413}{500}\right)$$ $$e\left(\frac{29}{200}\right)$$ $$e\left(\frac{3083}{4000}\right)$$ $$e\left(\frac{567}{1000}\right)$$ $$e\left(\frac{239}{1000}\right)$$ $$e\left(\frac{1431}{2000}\right)$$ $$e\left(\frac{279}{500}\right)$$ $$e\left(\frac{127}{160}\right)$$
$$\chi_{4001}(18,\cdot)$$ 4001.w 2000 Yes $$1$$ $$1$$ $$e\left(\frac{81}{500}\right)$$ $$e\left(\frac{1047}{2000}\right)$$ $$e\left(\frac{81}{250}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{1371}{2000}\right)$$ $$e\left(\frac{179}{500}\right)$$ $$e\left(\frac{243}{500}\right)$$ $$e\left(\frac{47}{1000}\right)$$ $$e\left(\frac{223}{250}\right)$$ $$e\left(\frac{79}{80}\right)$$
$$\chi_{4001}(19,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{63}{250}\right)$$ $$e\left(\frac{981}{1000}\right)$$ $$e\left(\frac{63}{125}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{233}{1000}\right)$$ $$e\left(\frac{167}{250}\right)$$ $$e\left(\frac{189}{250}\right)$$ $$e\left(\frac{481}{500}\right)$$ $$e\left(\frac{104}{125}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{4001}(20,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{93}{250}\right)$$ $$e\left(\frac{341}{1000}\right)$$ $$e\left(\frac{93}{125}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{713}{1000}\right)$$ $$e\left(\frac{187}{250}\right)$$ $$e\left(\frac{29}{250}\right)$$ $$e\left(\frac{341}{500}\right)$$ $$e\left(\frac{94}{125}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{4001}(21,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{767}{1000}\right)$$ $$e\left(\frac{229}{4000}\right)$$ $$e\left(\frac{267}{500}\right)$$ $$e\left(\frac{111}{200}\right)$$ $$e\left(\frac{3297}{4000}\right)$$ $$e\left(\frac{53}{1000}\right)$$ $$e\left(\frac{301}{1000}\right)$$ $$e\left(\frac{229}{2000}\right)$$ $$e\left(\frac{161}{500}\right)$$ $$e\left(\frac{93}{160}\right)$$
$$\chi_{4001}(22,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{391}{1000}\right)$$ $$e\left(\frac{3517}{4000}\right)$$ $$e\left(\frac{391}{500}\right)$$ $$e\left(\frac{103}{200}\right)$$ $$e\left(\frac{1081}{4000}\right)$$ $$e\left(\frac{469}{1000}\right)$$ $$e\left(\frac{173}{1000}\right)$$ $$e\left(\frac{1517}{2000}\right)$$ $$e\left(\frac{453}{500}\right)$$ $$e\left(\frac{149}{160}\right)$$
$$\chi_{4001}(23,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{347}{1000}\right)$$ $$e\left(\frac{3689}{4000}\right)$$ $$e\left(\frac{347}{500}\right)$$ $$e\left(\frac{51}{200}\right)$$ $$e\left(\frac{1077}{4000}\right)$$ $$e\left(\frac{273}{1000}\right)$$ $$e\left(\frac{41}{1000}\right)$$ $$e\left(\frac{1689}{2000}\right)$$ $$e\left(\frac{301}{500}\right)$$ $$e\left(\frac{33}{160}\right)$$
$$\chi_{4001}(24,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{871}{1000}\right)$$ $$e\left(\frac{2277}{4000}\right)$$ $$e\left(\frac{371}{500}\right)$$ $$e\left(\frac{143}{200}\right)$$ $$e\left(\frac{1761}{4000}\right)$$ $$e\left(\frac{789}{1000}\right)$$ $$e\left(\frac{613}{1000}\right)$$ $$e\left(\frac{277}{2000}\right)$$ $$e\left(\frac{293}{500}\right)$$ $$e\left(\frac{29}{160}\right)$$
$$\chi_{4001}(25,\cdot)$$ 4001.n 100 Yes $$1$$ $$1$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{87}{100}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$-i$$
$$\chi_{4001}(26,\cdot)$$ 4001.o 125 Yes $$1$$ $$1$$ $$e\left(\frac{111}{125}\right)$$ $$e\left(\frac{8}{125}\right)$$ $$e\left(\frac{97}{125}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{119}{125}\right)$$ $$e\left(\frac{74}{125}\right)$$ $$e\left(\frac{83}{125}\right)$$ $$e\left(\frac{16}{125}\right)$$ $$e\left(\frac{51}{125}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{4001}(27,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{569}{1000}\right)$$ $$e\left(\frac{3}{4000}\right)$$ $$e\left(\frac{69}{500}\right)$$ $$e\left(\frac{177}{200}\right)$$ $$e\left(\frac{2279}{4000}\right)$$ $$e\left(\frac{171}{1000}\right)$$ $$e\left(\frac{707}{1000}\right)$$ $$e\left(\frac{3}{2000}\right)$$ $$e\left(\frac{227}{500}\right)$$ $$e\left(\frac{11}{160}\right)$$
$$\chi_{4001}(28,\cdot)$$ 4001.v 1000 Yes $$1$$ $$1$$ $$e\left(\frac{119}{250}\right)$$ $$e\left(\frac{103}{1000}\right)$$ $$e\left(\frac{119}{125}\right)$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{579}{1000}\right)$$ $$e\left(\frac{121}{250}\right)$$ $$e\left(\frac{107}{250}\right)$$ $$e\left(\frac{103}{500}\right)$$ $$e\left(\frac{2}{125}\right)$$ $$e\left(\frac{31}{40}\right)$$
$$\chi_{4001}(29,\cdot)$$ 4001.w 2000 Yes $$1$$ $$1$$ $$e\left(\frac{7}{500}\right)$$ $$e\left(\frac{1109}{2000}\right)$$ $$e\left(\frac{7}{250}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{1137}{2000}\right)$$ $$e\left(\frac{213}{500}\right)$$ $$e\left(\frac{21}{500}\right)$$ $$e\left(\frac{109}{1000}\right)$$ $$e\left(\frac{81}{250}\right)$$ $$e\left(\frac{13}{80}\right)$$
$$\chi_{4001}(30,\cdot)$$ 4001.x 4000 Yes $$-1$$ $$1$$ $$e\left(\frac{779}{1000}\right)$$ $$e\left(\frac{3273}{4000}\right)$$ $$e\left(\frac{279}{500}\right)$$ $$e\left(\frac{107}{200}\right)$$ $$e\left(\frac{2389}{4000}\right)$$ $$e\left(\frac{561}{1000}\right)$$ $$e\left(\frac{337}{1000}\right)$$ $$e\left(\frac{1273}{2000}\right)$$ $$e\left(\frac{157}{500}\right)$$ $$e\left(\frac{1}{160}\right)$$
$$\chi_{4001}(31,\cdot)$$ 4001.w 2000 Yes $$1$$ $$1$$ $$e\left(\frac{493}{500}\right)$$ $$e\left(\frac{391}{2000}\right)$$ $$e\left(\frac{243}{250}\right)$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{363}{2000}\right)$$ $$e\left(\frac{287}{500}\right)$$ $$e\left(\frac{479}{500}\right)$$ $$e\left(\frac{391}{1000}\right)$$ $$e\left(\frac{169}{250}\right)$$ $$e\left(\frac{47}{80}\right)$$
$$\chi_{4001}(32,\cdot)$$ 4001.q 200 Yes $$1$$ $$1$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{123}{200}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{39}{200}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{23}{100}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{3}{8}\right)$$