# Properties

 Modulus $116$ Structure $$C_{28}\times C_{2}$$ Order $56$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(116)

pari: g = idealstar(,116,2)

## Character group

 sage: G.order()  pari: g.no Order = 56 sage: H.invariants()  pari: g.cyc Structure = $$C_{28}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{116}(59,\cdot)$, $\chi_{116}(89,\cdot)$

## First 32 of 56 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{116}(1,\cdot)$$ 116.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{116}(3,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$-i$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$
$$\chi_{116}(5,\cdot)$$ 116.i 14 no $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$-1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{116}(7,\cdot)$$ 116.j 14 yes $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{116}(9,\cdot)$$ 116.i 14 no $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{116}(11,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$-i$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$
$$\chi_{116}(13,\cdot)$$ 116.i 14 no $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{116}(15,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$i$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$
$$\chi_{116}(17,\cdot)$$ 116.f 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-1$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-i$$ $$-i$$
$$\chi_{116}(19,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$-i$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$
$$\chi_{116}(21,\cdot)$$ 116.k 28 no $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$-i$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$
$$\chi_{116}(23,\cdot)$$ 116.j 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{116}(25,\cdot)$$ 116.g 7 no $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{116}(27,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$i$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$
$$\chi_{116}(31,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$-i$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$
$$\chi_{116}(33,\cdot)$$ 116.i 14 no $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{116}(35,\cdot)$$ 116.h 14 yes $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$-1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{116}(37,\cdot)$$ 116.k 28 no $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$i$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$
$$\chi_{116}(39,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$i$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$
$$\chi_{116}(41,\cdot)$$ 116.f 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$-1$$ $$i$$ $$-1$$ $$-i$$ $$i$$ $$i$$ $$i$$
$$\chi_{116}(43,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$-i$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$
$$\chi_{116}(45,\cdot)$$ 116.g 7 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{116}(47,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$i$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$
$$\chi_{116}(49,\cdot)$$ 116.g 7 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{116}(51,\cdot)$$ 116.h 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{116}(53,\cdot)$$ 116.g 7 no $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{116}(55,\cdot)$$ 116.l 28 yes $$1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$i$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$
$$\chi_{116}(57,\cdot)$$ 116.c 2 no $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$
$$\chi_{116}(59,\cdot)$$ 116.b 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{116}(61,\cdot)$$ 116.k 28 no $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$-i$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$
$$\chi_{116}(63,\cdot)$$ 116.h 14 yes $$-1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{116}(65,\cdot)$$ 116.g 7 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$