# Properties

 Modulus $344$ Structure $$C_{42}\times C_{2}\times C_{2}$$ Order $168$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(344)

pari: g = idealstar(,344,2)

## Character group

 sage: G.order()  pari: g.no Order = 168 sage: H.invariants()  pari: g.cyc Structure = $$C_{42}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{344}(87,\cdot)$, $\chi_{344}(173,\cdot)$, $\chi_{344}(89,\cdot)$

## First 32 of 168 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{344}(1,\cdot)$$ 344.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{344}(3,\cdot)$$ 344.bb 42 yes $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{344}(5,\cdot)$$ 344.bf 42 yes $$-1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(7,\cdot)$$ 344.k 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$
$$\chi_{344}(9,\cdot)$$ 344.y 21 no $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{344}(11,\cdot)$$ 344.s 14 yes $$-1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{344}(13,\cdot)$$ 344.z 42 yes $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(15,\cdot)$$ 344.bc 42 no $$-1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{344}(17,\cdot)$$ 344.y 21 no $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{344}(19,\cdot)$$ 344.bb 42 yes $$1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{344}(21,\cdot)$$ 344.x 14 yes $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{344}(23,\cdot)$$ 344.bc 42 no $$-1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{344}(25,\cdot)$$ 344.y 21 no $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{344}(27,\cdot)$$ 344.v 14 yes $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{344}(29,\cdot)$$ 344.bf 42 yes $$-1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{344}(31,\cdot)$$ 344.bc 42 no $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{344}(33,\cdot)$$ 344.ba 42 no $$-1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{344}(35,\cdot)$$ 344.s 14 yes $$-1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(37,\cdot)$$ 344.j 6 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$
$$\chi_{344}(39,\cdot)$$ 344.t 14 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{344}(41,\cdot)$$ 344.q 7 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{344}(45,\cdot)$$ 344.r 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{344}(47,\cdot)$$ 344.u 14 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{344}(49,\cdot)$$ 344.i 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
$$\chi_{344}(51,\cdot)$$ 344.v 14 yes $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(53,\cdot)$$ 344.z 42 yes $$1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{344}(55,\cdot)$$ 344.bd 42 no $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{344}(57,\cdot)$$ 344.y 21 no $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{344}(59,\cdot)$$ 344.s 14 yes $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$-1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{344}(61,\cdot)$$ 344.bf 42 yes $$-1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{344}(63,\cdot)$$ 344.bd 42 no $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{344}(65,\cdot)$$ 344.w 14 no $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$