# Properties

 Modulus $58$ Structure $$C_{28}$$ Order $28$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(58)

pari: g = idealstar(,58,2)

## Character group

 sage: G.order()  pari: g.no Order = 28 sage: H.invariants()  pari: g.cyc Structure = $$C_{28}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{58}(31,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{58}(1,\cdot)$$ 58.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{58}(3,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$-i$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$
$$\chi_{58}(5,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$-1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{58}(7,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{58}(9,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{58}(11,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$-i$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$
$$\chi_{58}(13,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{58}(15,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$i$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$
$$\chi_{58}(17,\cdot)$$ 58.c 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-1$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-i$$ $$-i$$
$$\chi_{58}(19,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$-i$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$
$$\chi_{58}(21,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$-i$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$
$$\chi_{58}(23,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{58}(25,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{58}(27,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$i$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$
$$\chi_{58}(31,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$-i$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$
$$\chi_{58}(33,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{58}(35,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$-1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{58}(37,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$i$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$
$$\chi_{58}(39,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$i$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$
$$\chi_{58}(41,\cdot)$$ 58.c 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$-1$$ $$i$$ $$-1$$ $$-i$$ $$i$$ $$i$$ $$i$$
$$\chi_{58}(43,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$-i$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$
$$\chi_{58}(45,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{58}(47,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$i$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$
$$\chi_{58}(49,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{58}(51,\cdot)$$ 58.e 14 no $$1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{58}(53,\cdot)$$ 58.d 7 no $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{58}(55,\cdot)$$ 58.f 28 no $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$i$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$
$$\chi_{58}(57,\cdot)$$ 58.b 2 no $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$