# Properties

 Modulus $53$ Structure $$C_{52}$$ Order $52$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(53)

pari: g = idealstar(,53,2)

## Character group

 sage: G.order()  pari: g.no Order = 52 sage: H.invariants()  pari: g.cyc Structure = $$C_{52}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{53}(2,\cdot)$

## First 32 of 52 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{53}(1,\cdot)$$ 53.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{53}(2,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$
$$\chi_{53}(3,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$
$$\chi_{53}(4,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$
$$\chi_{53}(5,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$
$$\chi_{53}(6,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$
$$\chi_{53}(7,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$
$$\chi_{53}(8,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$
$$\chi_{53}(9,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$
$$\chi_{53}(10,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$
$$\chi_{53}(11,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$
$$\chi_{53}(12,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{19}{52}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$
$$\chi_{53}(13,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$
$$\chi_{53}(14,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{19}{26}\right)$$
$$\chi_{53}(15,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$
$$\chi_{53}(16,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{53}(17,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$
$$\chi_{53}(18,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$
$$\chi_{53}(19,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{53}(20,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$
$$\chi_{53}(21,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{53}(22,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{17}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$
$$\chi_{53}(23,\cdot)$$ 53.c 4 yes $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{53}(24,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{53}(25,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{53}(26,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{53}(27,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$
$$\chi_{53}(28,\cdot)$$ 53.d 13 yes $$1$$ $$1$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{53}(29,\cdot)$$ 53.e 26 yes $$1$$ $$1$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{53}(30,\cdot)$$ 53.c 4 yes $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{53}(31,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$
$$\chi_{53}(32,\cdot)$$ 53.f 52 yes $$-1$$ $$1$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{33}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$