# Properties

 Modulus $294$ Structure $$C_{2}\times C_{42}$$ Order $84$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(294)

pari: g = idealstar(,294,2)

## Character group

 sage: G.order()  pari: g.no Order = 84 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{42}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{294}(197,\cdot)$, $\chi_{294}(199,\cdot)$

## First 32 of 84 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 $$-1$$ $$1$$ $$5$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$37$$
$$\chi_{294}(1,\cdot)$$ 294.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{294}(5,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{294}(11,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{294}(13,\cdot)$$ 294.k 14 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$-1$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{294}(17,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{21}\right)$$
$$\chi_{294}(19,\cdot)$$ 294.g 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{294}(23,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{294}(25,\cdot)$$ 294.m 21 no $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{294}(29,\cdot)$$ 294.l 14 no $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{294}(31,\cdot)$$ 294.g 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{294}(37,\cdot)$$ 294.m 21 no $$1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$
$$\chi_{294}(41,\cdot)$$ 294.j 14 no $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{294}(43,\cdot)$$ 294.i 7 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{294}(47,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{21}\right)$$
$$\chi_{294}(53,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{294}(55,\cdot)$$ 294.k 14 no $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{294}(59,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{294}(61,\cdot)$$ 294.o 42 no $$-1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{21}\right)$$
$$\chi_{294}(65,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{21}\right)$$
$$\chi_{294}(67,\cdot)$$ 294.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{294}(71,\cdot)$$ 294.l 14 no $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{294}(73,\cdot)$$ 294.o 42 no $$-1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{294}(79,\cdot)$$ 294.e 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{294}(83,\cdot)$$ 294.j 14 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{294}(85,\cdot)$$ 294.i 7 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{294}(89,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{294}(95,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{294}(97,\cdot)$$ 294.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{294}(101,\cdot)$$ 294.p 42 no $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{294}(103,\cdot)$$ 294.o 42 no $$-1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{294}(107,\cdot)$$ 294.n 42 no $$-1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{294}(109,\cdot)$$ 294.m 21 no $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$