# Properties

 Modulus $225$ Structure $$C_{2}\times C_{60}$$ Order $120$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(225)

pari: g = idealstar(,225,2)

## 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 = $\chi_{225}(101,\cdot)$, $\chi_{225}(127,\cdot)$

## 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{225}(1,\cdot)$$ 225.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{225}(2,\cdot)$$ 225.w 60 yes $$1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{225}(4,\cdot)$$ 225.u 30 yes $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{225}(7,\cdot)$$ 225.o 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$-1$$
$$\chi_{225}(8,\cdot)$$ 225.s 20 no $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{225}(11,\cdot)$$ 225.t 30 yes $$-1$$ $$1$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{225}(13,\cdot)$$ 225.x 60 yes $$-1$$ $$1$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{225}(14,\cdot)$$ 225.v 30 yes $$-1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{225}(16,\cdot)$$ 225.q 15 yes $$1$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{225}(17,\cdot)$$ 225.s 20 no $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{225}(19,\cdot)$$ 225.m 10 no $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{225}(22,\cdot)$$ 225.x 60 yes $$-1$$ $$1$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{225}(23,\cdot)$$ 225.w 60 yes $$1$$ $$1$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{225}(26,\cdot)$$ 225.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{225}(28,\cdot)$$ 225.r 20 no $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{225}(29,\cdot)$$ 225.v 30 yes $$-1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{225}(31,\cdot)$$ 225.q 15 yes $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{225}(32,\cdot)$$ 225.p 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$-1$$
$$\chi_{225}(34,\cdot)$$ 225.u 30 yes $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{225}(37,\cdot)$$ 225.r 20 no $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{225}(38,\cdot)$$ 225.w 60 yes $$1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{225}(41,\cdot)$$ 225.t 30 yes $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{225}(43,\cdot)$$ 225.o 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$-1$$
$$\chi_{225}(44,\cdot)$$ 225.l 10 no $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{225}(46,\cdot)$$ 225.h 5 no $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{225}(47,\cdot)$$ 225.w 60 yes $$1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{225}(49,\cdot)$$ 225.k 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$
$$\chi_{225}(52,\cdot)$$ 225.x 60 yes $$-1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{225}(53,\cdot)$$ 225.s 20 no $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{225}(56,\cdot)$$ 225.t 30 yes $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{225}(58,\cdot)$$ 225.x 60 yes $$-1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{225}(59,\cdot)$$ 225.v 30 yes $$-1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$