# Properties

 Modulus $115$ Structure $$C_{44}\times C_{2}$$ Order $88$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(115)

pari: g = idealstar(,115,2)

## Character group

 sage: G.order()  pari: g.no Order = 88 sage: H.invariants()  pari: g.cyc Structure = $$C_{44}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{115}(47,\cdot)$, $\chi_{115}(51,\cdot)$

## First 32 of 88 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{115}(1,\cdot)$$ 115.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{115}(2,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$
$$\chi_{115}(3,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{115}(4,\cdot)$$ 115.j 22 yes $$1$$ $$1$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{115}(6,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{115}(7,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$
$$\chi_{115}(8,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{115}(9,\cdot)$$ 115.j 22 yes $$1$$ $$1$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{115}(11,\cdot)$$ 115.h 22 no $$-1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{115}(12,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$
$$\chi_{115}(13,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{115}(14,\cdot)$$ 115.i 22 yes $$-1$$ $$1$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{115}(16,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{115}(17,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{115}(18,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{115}(19,\cdot)$$ 115.i 22 yes $$-1$$ $$1$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{115}(21,\cdot)$$ 115.h 22 no $$-1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{115}(22,\cdot)$$ 115.e 4 yes $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$
$$\chi_{115}(24,\cdot)$$ 115.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{115}(26,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{115}(27,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$
$$\chi_{115}(28,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{115}(29,\cdot)$$ 115.j 22 yes $$1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{115}(31,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{115}(32,\cdot)$$ 115.k 44 yes $$-1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{115}(33,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{115}(34,\cdot)$$ 115.i 22 yes $$-1$$ $$1$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{115}(36,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{115}(37,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{115}(38,\cdot)$$ 115.l 44 yes $$1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{115}(39,\cdot)$$ 115.j 22 yes $$1$$ $$1$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{115}(41,\cdot)$$ 115.g 11 no $$1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$