# Properties

 Modulus $275$ Structure $$C_{10}\times C_{20}$$ Order $200$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(275)

pari: g = idealstar(,275,2)

## Character group

 sage: G.order()  pari: g.no Order = 200 sage: H.invariants()  pari: g.cyc Structure = $$C_{10}\times C_{20}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{275}(177,\cdot)$, $\chi_{275}(101,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$
$$\chi_{275}(1,\cdot)$$ 275.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{275}(2,\cdot)$$ 275.bn 20 yes $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{275}(3,\cdot)$$ 275.bp 20 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$-1$$
$$\chi_{275}(4,\cdot)$$ 275.ba 10 yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(6,\cdot)$$ 275.bc 10 yes $$-1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{275}(7,\cdot)$$ 275.bm 20 no $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{275}(8,\cdot)$$ 275.bn 20 yes $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{275}(9,\cdot)$$ 275.bb 10 yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$
$$\chi_{275}(12,\cdot)$$ 275.bi 20 no $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{275}(13,\cdot)$$ 275.bg 20 yes $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-1$$
$$\chi_{275}(14,\cdot)$$ 275.t 10 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(16,\cdot)$$ 275.g 5 yes $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{275}(17,\cdot)$$ 275.bl 20 yes $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$-1$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{275}(18,\cdot)$$ 275.bm 20 no $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{275}(19,\cdot)$$ 275.r 10 yes $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{275}(21,\cdot)$$ 275.v 10 yes $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{275}(23,\cdot)$$ 275.bi 20 no $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{275}(24,\cdot)$$ 275.q 10 no $$-1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{275}(26,\cdot)$$ 275.h 5 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{275}(27,\cdot)$$ 275.bp 20 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-1$$
$$\chi_{275}(28,\cdot)$$ 275.bf 20 yes $$1$$ $$1$$ $$i$$ $$e\left(\frac{13}{20}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{275}(29,\cdot)$$ 275.r 10 yes $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(31,\cdot)$$ 275.l 5 yes $$1$$ $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(32,\cdot)$$ 275.e 4 no $$1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-1$$
$$\chi_{275}(34,\cdot)$$ 275.y 10 no $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(36,\cdot)$$ 275.k 5 yes $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{275}(37,\cdot)$$ 275.bh 20 yes $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-1$$ $$e\left(\frac{1}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{275}(38,\cdot)$$ 275.bj 20 yes $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{275}(39,\cdot)$$ 275.p 10 yes $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$
$$\chi_{275}(41,\cdot)$$ 275.m 10 yes $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{275}(42,\cdot)$$ 275.be 20 yes $$-1$$ $$1$$ $$i$$ $$e\left(\frac{7}{20}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$-i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{275}(43,\cdot)$$ 275.e 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$