# Properties

 Modulus $49$ Structure $$C_{42}$$ Order $42$

# Learn more

Show commands: PariGP / SageMath

sage: H = DirichletGroup(49)

pari: g = idealstar(,49,2)

## Character group

 sage: G.order()  pari: g.no Order = 42 sage: H.invariants()  pari: g.cyc Structure = $$C_{42}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{49}(3,\cdot)$

## First 32 of 42 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{49}(1,\cdot)$$ 49.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{49}(2,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$
$$\chi_{49}(3,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$
$$\chi_{49}(4,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{49}(5,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$
$$\chi_{49}(6,\cdot)$$ 49.f 14 yes $$-1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{49}(8,\cdot)$$ 49.e 7 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{49}(9,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{49}(10,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{49}(11,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{49}(12,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{49}(13,\cdot)$$ 49.f 14 yes $$-1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{49}(15,\cdot)$$ 49.e 7 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{49}(16,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$
$$\chi_{49}(17,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{49}(18,\cdot)$$ 49.c 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{49}(19,\cdot)$$ 49.d 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{49}(20,\cdot)$$ 49.f 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{49}(22,\cdot)$$ 49.e 7 yes $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{49}(23,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{49}(24,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{49}(25,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{49}(26,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$
$$\chi_{49}(27,\cdot)$$ 49.f 14 yes $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{49}(29,\cdot)$$ 49.e 7 yes $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{49}(30,\cdot)$$ 49.c 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{49}(31,\cdot)$$ 49.d 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{49}(32,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$
$$\chi_{49}(33,\cdot)$$ 49.h 42 yes $$-1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$
$$\chi_{49}(34,\cdot)$$ 49.f 14 yes $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{49}(36,\cdot)$$ 49.e 7 yes $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{49}(37,\cdot)$$ 49.g 21 yes $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$
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