# Properties

 Modulus $350$ Structure $$C_{2}\times C_{60}$$ Order $120$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(350)

pari: g = idealstar(,350,2)

## Character group

 sage: G.order()  pari: g.no Order = 120 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{60}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{350}(127,\cdot)$, $\chi_{350}(101,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$
$$\chi_{350}(1,\cdot)$$ 350.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{350}(3,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{350}(9,\cdot)$$ 350.u 30 no $$1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{350}(11,\cdot)$$ 350.q 15 no $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{350}(13,\cdot)$$ 350.r 20 no $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{350}(17,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{350}(19,\cdot)$$ 350.v 30 no $$-1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{350}(23,\cdot)$$ 350.w 60 no $$-1$$ $$1$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{350}(27,\cdot)$$ 350.r 20 no $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{350}(29,\cdot)$$ 350.m 10 no $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{350}(31,\cdot)$$ 350.t 30 no $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{350}(33,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{350}(37,\cdot)$$ 350.w 60 no $$-1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{350}(39,\cdot)$$ 350.u 30 no $$1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{350}(41,\cdot)$$ 350.n 10 no $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{350}(43,\cdot)$$ 350.f 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$1$$
$$\chi_{350}(47,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{350}(51,\cdot)$$ 350.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{350}(53,\cdot)$$ 350.w 60 no $$-1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$
$$\chi_{350}(57,\cdot)$$ 350.f 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$1$$
$$\chi_{350}(59,\cdot)$$ 350.v 30 no $$-1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{350}(61,\cdot)$$ 350.t 30 no $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{350}(67,\cdot)$$ 350.w 60 no $$-1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{350}(69,\cdot)$$ 350.l 10 no $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{350}(71,\cdot)$$ 350.h 5 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{350}(73,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{350}(79,\cdot)$$ 350.u 30 no $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$
$$\chi_{350}(81,\cdot)$$ 350.q 15 no $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{350}(83,\cdot)$$ 350.r 20 no $$1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{350}(87,\cdot)$$ 350.x 60 no $$1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{350}(89,\cdot)$$ 350.v 30 no $$-1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{350}(93,\cdot)$$ 350.p 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$