# Properties

 Modulus $99$ Structure $$C_{2}\times C_{30}$$ Order $60$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(99)

pari: g = idealstar(,99,2)

## Character group

 sage: G.order()  pari: g.no Order = 60 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{30}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{99}(56,\cdot)$, $\chi_{99}(46,\cdot)$

## First 32 of 60 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{99}(1,\cdot)$$ 99.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{99}(2,\cdot)$$ 99.p 30 yes $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{99}(4,\cdot)$$ 99.m 15 yes $$1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{99}(5,\cdot)$$ 99.n 30 yes $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{99}(7,\cdot)$$ 99.o 30 yes $$-1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{99}(8,\cdot)$$ 99.j 10 no $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{99}(10,\cdot)$$ 99.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{99}(13,\cdot)$$ 99.o 30 yes $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{99}(14,\cdot)$$ 99.n 30 yes $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{99}(16,\cdot)$$ 99.m 15 yes $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{99}(17,\cdot)$$ 99.j 10 no $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{99}(19,\cdot)$$ 99.k 10 no $$-1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{99}(20,\cdot)$$ 99.n 30 yes $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{99}(23,\cdot)$$ 99.i 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$
$$\chi_{99}(25,\cdot)$$ 99.m 15 yes $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{99}(26,\cdot)$$ 99.l 10 no $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{99}(28,\cdot)$$ 99.k 10 no $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{99}(29,\cdot)$$ 99.p 30 yes $$1$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{99}(31,\cdot)$$ 99.m 15 yes $$1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{99}(32,\cdot)$$ 99.g 6 yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$
$$\chi_{99}(34,\cdot)$$ 99.e 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
$$\chi_{99}(35,\cdot)$$ 99.j 10 no $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{99}(37,\cdot)$$ 99.f 5 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{99}(38,\cdot)$$ 99.n 30 yes $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{99}(40,\cdot)$$ 99.o 30 yes $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{99}(41,\cdot)$$ 99.p 30 yes $$1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{99}(43,\cdot)$$ 99.h 6 yes $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$
$$\chi_{99}(46,\cdot)$$ 99.k 10 no $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{99}(47,\cdot)$$ 99.n 30 yes $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{99}(49,\cdot)$$ 99.m 15 yes $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{99}(50,\cdot)$$ 99.p 30 yes $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{99}(52,\cdot)$$ 99.o 30 yes $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$