# Properties

 Modulus $750$ Structure $$C_{100}\times C_{2}$$ Order $200$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(750)

pari: g = idealstar(,750,2)

## Character group

 sage: G.order()  pari: g.no Order = 200 sage: H.invariants()  pari: g.cyc Structure = $$C_{100}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{750}(251,\cdot)$, $\chi_{750}(127,\cdot)$

## 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 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$
$$\chi_{750}(1,\cdot)$$ 750.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{750}(7,\cdot)$$ 750.k 20 no $$-1$$ $$1$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{750}(11,\cdot)$$ 750.n 50 no $$-1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{47}{50}\right)$$
$$\chi_{750}(13,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{4}{25}\right)$$
$$\chi_{750}(17,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{31}{50}\right)$$
$$\chi_{750}(19,\cdot)$$ 750.o 50 no $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$
$$\chi_{750}(23,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$
$$\chi_{750}(29,\cdot)$$ 750.p 50 no $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{39}{50}\right)$$
$$\chi_{750}(31,\cdot)$$ 750.m 25 no $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$
$$\chi_{750}(37,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{41}{100}\right)$$ $$e\left(\frac{19}{25}\right)$$
$$\chi_{750}(41,\cdot)$$ 750.n 50 no $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{43}{50}\right)$$
$$\chi_{750}(43,\cdot)$$ 750.k 20 no $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{750}(47,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{57}{100}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$
$$\chi_{750}(49,\cdot)$$ 750.h 10 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{750}(53,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{41}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$
$$\chi_{750}(59,\cdot)$$ 750.p 50 no $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{23}{50}\right)$$
$$\chi_{750}(61,\cdot)$$ 750.m 25 no $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$
$$\chi_{750}(67,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{77}{100}\right)$$ $$e\left(\frac{18}{25}\right)$$
$$\chi_{750}(71,\cdot)$$ 750.n 50 no $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$
$$\chi_{750}(73,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{1}{25}\right)$$
$$\chi_{750}(77,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{41}{100}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{17}{50}\right)$$
$$\chi_{750}(79,\cdot)$$ 750.o 50 no $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$
$$\chi_{750}(83,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{77}{100}\right)$$ $$e\left(\frac{89}{100}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$
$$\chi_{750}(89,\cdot)$$ 750.p 50 no $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{27}{50}\right)$$
$$\chi_{750}(91,\cdot)$$ 750.m 25 no $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$
$$\chi_{750}(97,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{43}{100}\right)$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{7}{25}\right)$$
$$\chi_{750}(101,\cdot)$$ 750.j 10 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{750}(103,\cdot)$$ 750.q 100 no $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{53}{100}\right)$$ $$e\left(\frac{71}{100}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{37}{100}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{22}{25}\right)$$
$$\chi_{750}(107,\cdot)$$ 750.l 20 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{750}(109,\cdot)$$ 750.o 50 no $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$
$$\chi_{750}(113,\cdot)$$ 750.r 100 no $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{57}{100}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{23}{50}\right)$$
$$\chi_{750}(119,\cdot)$$ 750.p 50 no $$-1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{1}{50}\right)$$