# Properties

 Modulus $441$ Structure $$C_{6}\times C_{42}$$ Order $252$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(441)

pari: g = idealstar(,441,2)

## Character group

 sage: G.order()  pari: g.no Order = 252 sage: H.invariants()  pari: g.cyc Structure = $$C_{6}\times C_{42}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{441}(344,\cdot)$, $\chi_{441}(199,\cdot)$

## First 32 of 252 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{441}(1,\cdot)$$ 441.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{441}(2,\cdot)$$ 441.bi 42 yes $$-1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(4,\cdot)$$ 441.z 21 yes $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(5,\cdot)$$ 441.bn 42 yes $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(8,\cdot)$$ 441.x 14 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$
$$\chi_{441}(10,\cdot)$$ 441.bj 42 no $$-1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(11,\cdot)$$ 441.bm 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(13,\cdot)$$ 441.bk 42 yes $$-1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$
$$\chi_{441}(16,\cdot)$$ 441.z 21 yes $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(17,\cdot)$$ 441.bg 42 no $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(19,\cdot)$$ 441.m 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(20,\cdot)$$ 441.bh 42 yes $$1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$
$$\chi_{441}(22,\cdot)$$ 441.ba 21 yes $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$
$$\chi_{441}(23,\cdot)$$ 441.bm 42 yes $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(25,\cdot)$$ 441.y 21 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(26,\cdot)$$ 441.bg 42 no $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(29,\cdot)$$ 441.be 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$1$$
$$\chi_{441}(31,\cdot)$$ 441.k 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(32,\cdot)$$ 441.bi 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(34,\cdot)$$ 441.bk 42 yes $$-1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$
$$\chi_{441}(37,\cdot)$$ 441.bb 21 no $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(38,\cdot)$$ 441.bn 42 yes $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(40,\cdot)$$ 441.bc 42 yes $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(41,\cdot)$$ 441.bh 42 yes $$1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$
$$\chi_{441}(43,\cdot)$$ 441.ba 21 yes $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$
$$\chi_{441}(44,\cdot)$$ 441.bf 42 no $$-1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(46,\cdot)$$ 441.bb 21 no $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(47,\cdot)$$ 441.bd 42 yes $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(50,\cdot)$$ 441.r 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$
$$\chi_{441}(52,\cdot)$$ 441.bc 42 yes $$-1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(53,\cdot)$$ 441.bf 42 no $$-1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(55,\cdot)$$ 441.v 14 no $$-1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$