# Properties

 Modulus $135$ Structure $$C_{2}\times C_{36}$$ Order $72$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(135)

pari: g = idealstar(,135,2)

## Character group

 sage: G.order()  pari: g.no Order = 72 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{135}(56,\cdot)$, $\chi_{135}(82,\cdot)$

## First 32 of 72 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{135}(1,\cdot)$$ 135.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{135}(2,\cdot)$$ 135.q 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(4,\cdot)$$ 135.p 18 yes $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{135}(7,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(8,\cdot)$$ 135.m 12 no $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$-1$$
$$\chi_{135}(11,\cdot)$$ 135.o 18 no $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{135}(13,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{135}(14,\cdot)$$ 135.n 18 yes $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{135}(16,\cdot)$$ 135.k 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{135}(17,\cdot)$$ 135.m 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$-1$$
$$\chi_{135}(19,\cdot)$$ 135.j 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$
$$\chi_{135}(22,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{135}(23,\cdot)$$ 135.q 36 yes $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{135}(26,\cdot)$$ 135.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{135}(28,\cdot)$$ 135.g 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$-1$$
$$\chi_{135}(29,\cdot)$$ 135.n 18 yes $$-1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{135}(31,\cdot)$$ 135.k 9 no $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{135}(32,\cdot)$$ 135.q 36 yes $$1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{135}(34,\cdot)$$ 135.p 18 yes $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{135}(37,\cdot)$$ 135.l 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$-1$$
$$\chi_{135}(38,\cdot)$$ 135.q 36 yes $$1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(41,\cdot)$$ 135.o 18 no $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{135}(43,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(44,\cdot)$$ 135.h 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$
$$\chi_{135}(46,\cdot)$$ 135.e 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$
$$\chi_{135}(47,\cdot)$$ 135.q 36 yes $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(49,\cdot)$$ 135.p 18 yes $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{135}(52,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{135}(53,\cdot)$$ 135.f 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-1$$
$$\chi_{135}(56,\cdot)$$ 135.o 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{135}(58,\cdot)$$ 135.r 36 yes $$-1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{135}(59,\cdot)$$ 135.n 18 yes $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$