# Properties

 Modulus $231$ Structure $$C_{30}\times C_{2}\times C_{2}$$ Order $120$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(231)

pari: g = idealstar(,231,2)

## Character group

 sage: G.order()  pari: g.no Order = 120 sage: H.invariants()  pari: g.cyc Structure = $$C_{30}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{231}(155,\cdot)$, $\chi_{231}(199,\cdot)$, $\chi_{231}(211,\cdot)$

## First 32 of 120 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$13$$ $$16$$ $$17$$ $$19$$ $$20$$
$$\chi_{231}(1,\cdot)$$ 231.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{231}(2,\cdot)$$ 231.be 30 yes $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{231}(4,\cdot)$$ 231.y 15 no $$1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{231}(5,\cdot)$$ 231.bc 30 yes $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{231}(8,\cdot)$$ 231.s 10 no $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{231}(10,\cdot)$$ 231.p 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$
$$\chi_{231}(13,\cdot)$$ 231.w 10 no $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{231}(16,\cdot)$$ 231.y 15 no $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{231}(17,\cdot)$$ 231.bf 30 yes $$-1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{231}(19,\cdot)$$ 231.ba 30 no $$1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{231}(20,\cdot)$$ 231.u 10 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{231}(23,\cdot)$$ 231.q 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$
$$\chi_{231}(25,\cdot)$$ 231.y 15 no $$1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{231}(26,\cdot)$$ 231.bc 30 yes $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{231}(29,\cdot)$$ 231.s 10 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{231}(31,\cdot)$$ 231.bd 30 no $$-1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{231}(32,\cdot)$$ 231.l 6 yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$
$$\chi_{231}(34,\cdot)$$ 231.f 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$
$$\chi_{231}(37,\cdot)$$ 231.y 15 no $$1$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{231}(38,\cdot)$$ 231.bc 30 yes $$1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{231}(40,\cdot)$$ 231.ba 30 no $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{231}(41,\cdot)$$ 231.r 10 yes $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{231}(43,\cdot)$$ 231.d 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{231}(46,\cdot)$$ 231.bb 30 no $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{231}(47,\cdot)$$ 231.bc 30 yes $$1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{231}(50,\cdot)$$ 231.s 10 no $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{231}(52,\cdot)$$ 231.ba 30 no $$1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{231}(53,\cdot)$$ 231.z 30 yes $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{231}(58,\cdot)$$ 231.y 15 no $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{231}(59,\cdot)$$ 231.bc 30 yes $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{231}(61,\cdot)$$ 231.ba 30 no $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{231}(62,\cdot)$$ 231.r 10 yes $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$