# Properties

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

# Learn more about

Show commands for: Pari/GP / SageMath

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

sage: H = DirichletGroup_conrey(4000)

pari: g = idealstar(,4000,2)

## Character group

 sage: G.order()  pari: g.no Order = 1600 sage: H.invariants()  pari: g.cyc Structure = $$C_{200}\times C_{4}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{4000}(877,\cdot)$, $\chi_{4000}(1057,\cdot)$, $\chi_{4000}(2751,\cdot)$

## 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 $$-1$$ $$1$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{11}{50}\right)$$