# Properties

 Modulus $342$ Structure $$C_{6}\times C_{18}$$ Order $108$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(342)

pari: g = idealstar(,342,2)

## Character group

 sage: G.order()  pari: g.no Order = 108 sage: H.invariants()  pari: g.cyc Structure = $$C_{6}\times C_{18}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{342}(191,\cdot)$, $\chi_{342}(325,\cdot)$

## First 32 of 108 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{342}(1,\cdot)$$ 342.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{342}(5,\cdot)$$ 342.y 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$1$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{342}(7,\cdot)$$ 342.f 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{342}(11,\cdot)$$ 342.i 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{342}(13,\cdot)$$ 342.bd 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{342}(17,\cdot)$$ 342.ba 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{342}(23,\cdot)$$ 342.be 18 no $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{342}(25,\cdot)$$ 342.v 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{342}(29,\cdot)$$ 342.x 18 no $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$-1$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{342}(31,\cdot)$$ 342.t 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{342}(35,\cdot)$$ 342.ba 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{342}(37,\cdot)$$ 342.d 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{342}(41,\cdot)$$ 342.x 18 no $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{342}(43,\cdot)$$ 342.w 9 no $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{342}(47,\cdot)$$ 342.y 18 no $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$1$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{342}(49,\cdot)$$ 342.f 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{342}(53,\cdot)$$ 342.bb 18 no $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{342}(55,\cdot)$$ 342.u 9 no $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{342}(59,\cdot)$$ 342.x 18 no $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$-1$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{342}(61,\cdot)$$ 342.v 9 no $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$1$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{342}(65,\cdot)$$ 342.j 6 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{342}(67,\cdot)$$ 342.bd 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{342}(71,\cdot)$$ 342.bb 18 no $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{342}(73,\cdot)$$ 342.u 9 no $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{342}(77,\cdot)$$ 342.o 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$
$$\chi_{342}(79,\cdot)$$ 342.bd 18 no $$-1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{342}(83,\cdot)$$ 342.q 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{342}(85,\cdot)$$ 342.w 9 no $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{342}(89,\cdot)$$ 342.bb 18 no $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{342}(91,\cdot)$$ 342.z 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{342}(97,\cdot)$$ 342.bd 18 no $$-1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{342}(101,\cdot)$$ 342.y 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$1$$ $$e\left(\frac{11}{18}\right)$$
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