# Properties

 Modulus $138$ Structure $$C_{2}\times C_{22}$$ Order $44$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(138)

pari: g = idealstar(,138,2)

## Character group

 sage: G.order()  pari: g.no Order = 44 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{22}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{138}(47,\cdot)$, $\chi_{138}(97,\cdot)$

## First 32 of 44 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{138}(1,\cdot)$$ 138.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{138}(5,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{138}(7,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{138}(11,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{138}(13,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{138}(17,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{138}(19,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{138}(25,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{138}(29,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{138}(31,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{138}(35,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{138}(37,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{138}(41,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{138}(43,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{138}(47,\cdot)$$ 138.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{138}(49,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{138}(53,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{138}(55,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{138}(59,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{138}(61,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{138}(65,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{138}(67,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{138}(71,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{138}(73,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{138}(77,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{138}(79,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{138}(83,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{138}(85,\cdot)$$ 138.e 11 no $$1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{138}(89,\cdot)$$ 138.f 22 no $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{138}(91,\cdot)$$ 138.b 2 no $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{138}(95,\cdot)$$ 138.g 22 no $$-1$$ $$1$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{138}(97,\cdot)$$ 138.h 22 no $$-1$$ $$1$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$