# Properties

 Modulus $500$ Structure $$C_{2}\times C_{100}$$ Order $200$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(500)

pari: g = idealstar(,500,2)

## Character group

 sage: G.order()  pari: g.no Order = 200 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{100}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{500}(251,\cdot)$, $\chi_{500}(377,\cdot)$

## First 32 of 200 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 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$
$$\chi_{500}(1,\cdot)$$ 500.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{500}(3,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{67}{100}\right)$$ $$e\left(\frac{97}{100}\right)$$
$$\chi_{500}(7,\cdot)$$ 500.l 20 no $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$-i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$
$$\chi_{500}(9,\cdot)$$ 500.o 50 no $$1$$ $$1$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{47}{50}\right)$$
$$\chi_{500}(11,\cdot)$$ 500.p 50 yes $$-1$$ $$1$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{23}{50}\right)$$
$$\chi_{500}(13,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{19}{100}\right)$$
$$\chi_{500}(17,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{29}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{33}{100}\right)$$
$$\chi_{500}(19,\cdot)$$ 500.n 50 yes $$-1$$ $$1$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$
$$\chi_{500}(21,\cdot)$$ 500.m 25 no $$1$$ $$1$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$
$$\chi_{500}(23,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{67}{100}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{1}{100}\right)$$
$$\chi_{500}(27,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{97}{100}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{33}{100}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{91}{100}\right)$$
$$\chi_{500}(29,\cdot)$$ 500.o 50 no $$1$$ $$1$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{1}{50}\right)$$
$$\chi_{500}(31,\cdot)$$ 500.p 50 yes $$-1$$ $$1$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{29}{50}\right)$$
$$\chi_{500}(33,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{37}{100}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{43}{100}\right)$$
$$\chi_{500}(37,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{9}{100}\right)$$
$$\chi_{500}(39,\cdot)$$ 500.n 50 yes $$-1$$ $$1$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$
$$\chi_{500}(41,\cdot)$$ 500.m 25 no $$1$$ $$1$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$
$$\chi_{500}(43,\cdot)$$ 500.l 20 no $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$
$$\chi_{500}(47,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{29}{100}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{57}{100}\right)$$ $$e\left(\frac{87}{100}\right)$$
$$\chi_{500}(49,\cdot)$$ 500.i 10 no $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{500}(51,\cdot)$$ 500.j 10 no $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{500}(53,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{91}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{77}{100}\right)$$ $$e\left(\frac{7}{100}\right)$$
$$\chi_{500}(57,\cdot)$$ 500.f 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{500}(59,\cdot)$$ 500.n 50 yes $$-1$$ $$1$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$
$$\chi_{500}(61,\cdot)$$ 500.m 25 no $$1$$ $$1$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$
$$\chi_{500}(63,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{61}{100}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{29}{100}\right)$$
$$\chi_{500}(67,\cdot)$$ 500.r 100 yes $$1$$ $$1$$ $$e\left(\frac{41}{100}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{53}{100}\right)$$ $$e\left(\frac{23}{100}\right)$$
$$\chi_{500}(69,\cdot)$$ 500.o 50 no $$1$$ $$1$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{49}{50}\right)$$
$$\chi_{500}(71,\cdot)$$ 500.p 50 yes $$-1$$ $$1$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{31}{50}\right)$$
$$\chi_{500}(73,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{37}{100}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{11}{100}\right)$$
$$\chi_{500}(77,\cdot)$$ 500.q 100 no $$-1$$ $$1$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{53}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{91}{100}\right)$$ $$e\left(\frac{81}{100}\right)$$
$$\chi_{500}(79,\cdot)$$ 500.n 50 yes $$-1$$ $$1$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$