# Properties

 Modulus $473$ Structure $$C_{210}\times C_{2}$$ Order $420$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(473)

pari: g = idealstar(,473,2)

## Character group

 sage: G.order()  pari: g.no Order = 420 sage: H.invariants()  pari: g.cyc Structure = $$C_{210}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{473}(431,\cdot)$, $\chi_{473}(89,\cdot)$

## First 32 of 420 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$
$$\chi_{473}(1,\cdot)$$ 473.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{473}(2,\cdot)$$ 473.bb 70 yes $$1$$ $$1$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{31}{70}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{33}{70}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{473}(3,\cdot)$$ 473.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{31}{70}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$
$$\chi_{473}(4,\cdot)$$ 473.v 35 yes $$1$$ $$1$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{473}(5,\cdot)$$ 473.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{33}{70}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{101}{210}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$
$$\chi_{473}(6,\cdot)$$ 473.u 30 yes $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{473}(7,\cdot)$$ 473.s 30 yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{473}(8,\cdot)$$ 473.bb 70 yes $$1$$ $$1$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{473}(9,\cdot)$$ 473.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{473}(10,\cdot)$$ 473.y 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{473}(12,\cdot)$$ 473.x 42 no $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$
$$\chi_{473}(13,\cdot)$$ 473.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{47}{70}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{70}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{473}(14,\cdot)$$ 473.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{473}(15,\cdot)$$ 473.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$
$$\chi_{473}(16,\cdot)$$ 473.v 35 yes $$1$$ $$1$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{473}(17,\cdot)$$ 473.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{473}(18,\cdot)$$ 473.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{61}{210}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{13}{210}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$
$$\chi_{473}(19,\cdot)$$ 473.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{179}{210}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{107}{210}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{473}(20,\cdot)$$ 473.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{11}{70}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$
$$\chi_{473}(21,\cdot)$$ 473.o 14 yes $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{473}(23,\cdot)$$ 473.r 21 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{473}(24,\cdot)$$ 473.bd 210 yes $$-1$$ $$1$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{31}{70}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$
$$\chi_{473}(25,\cdot)$$ 473.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{473}(26,\cdot)$$ 473.be 210 yes $$-1$$ $$1$$ $$e\left(\frac{9}{70}\right)$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{193}{210}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$
$$\chi_{473}(27,\cdot)$$ 473.z 70 yes $$-1$$ $$1$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{19}{70}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{473}(28,\cdot)$$ 473.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{67}{210}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{121}{210}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{473}(29,\cdot)$$ 473.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{121}{210}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{43}{210}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{16}{105}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{473}(30,\cdot)$$ 473.bf 210 yes $$1$$ $$1$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{139}{210}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{157}{210}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{473}(31,\cdot)$$ 473.bc 105 yes $$1$$ $$1$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{473}(32,\cdot)$$ 473.n 14 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{473}(34,\cdot)$$ 473.x 42 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$
$$\chi_{473}(35,\cdot)$$ 473.ba 70 yes $$-1$$ $$1$$ $$e\left(\frac{47}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{70}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$