# Properties

 Modulus 4009 Structure $$C_{630}\times C_{6}$$ Order 3780

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(4009)

pari: g = idealstar(,4009,2)

## Character group

 sage: G.order()  pari: g.no Order = 3780 sage: H.invariants()  pari: g.cyc Structure = $$C_{630}\times C_{6}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{4009}(846,\cdot)$, $\chi_{4009}(2547,\cdot)$

## First 32 of 3780 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.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{4009}(1,\cdot)$$ 4009.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{4009}(2,\cdot)$$ 4009.et 630 yes $$1$$ $$1$$ $$e\left(\frac{19}{315}\right)$$ $$e\left(\frac{292}{315}\right)$$ $$e\left(\frac{38}{315}\right)$$ $$e\left(\frac{163}{315}\right)$$ $$e\left(\frac{311}{315}\right)$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{269}{315}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{46}{105}\right)$$
$$\chi_{4009}(3,\cdot)$$ 4009.et 630 yes $$1$$ $$1$$ $$e\left(\frac{292}{315}\right)$$ $$e\left(\frac{61}{315}\right)$$ $$e\left(\frac{269}{315}\right)$$ $$e\left(\frac{184}{315}\right)$$ $$e\left(\frac{38}{315}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{122}{315}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{88}{105}\right)$$
$$\chi_{4009}(4,\cdot)$$ 4009.em 315 yes $$1$$ $$1$$ $$e\left(\frac{38}{315}\right)$$ $$e\left(\frac{269}{315}\right)$$ $$e\left(\frac{76}{315}\right)$$ $$e\left(\frac{11}{315}\right)$$ $$e\left(\frac{307}{315}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{223}{315}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{92}{105}\right)$$
$$\chi_{4009}(5,\cdot)$$ 4009.en 315 yes $$1$$ $$1$$ $$e\left(\frac{163}{315}\right)$$ $$e\left(\frac{184}{315}\right)$$ $$e\left(\frac{11}{315}\right)$$ $$e\left(\frac{61}{315}\right)$$ $$e\left(\frac{32}{315}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{53}{315}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{52}{105}\right)$$
$$\chi_{4009}(6,\cdot)$$ 4009.em 315 yes $$1$$ $$1$$ $$e\left(\frac{311}{315}\right)$$ $$e\left(\frac{38}{315}\right)$$ $$e\left(\frac{307}{315}\right)$$ $$e\left(\frac{32}{315}\right)$$ $$e\left(\frac{34}{315}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{76}{315}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{29}{105}\right)$$
$$\chi_{4009}(7,\cdot)$$ 4009.eh 210 yes $$-1$$ $$1$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{167}{210}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{83}{105}\right)$$ $$e\left(\frac{1}{210}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{8}{35}\right)$$
$$\chi_{4009}(8,\cdot)$$ 4009.ec 210 yes $$1$$ $$1$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{11}{35}\right)$$
$$\chi_{4009}(9,\cdot)$$ 4009.em 315 yes $$1$$ $$1$$ $$e\left(\frac{269}{315}\right)$$ $$e\left(\frac{122}{315}\right)$$ $$e\left(\frac{223}{315}\right)$$ $$e\left(\frac{53}{315}\right)$$ $$e\left(\frac{76}{315}\right)$$ $$e\left(\frac{62}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{244}{315}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{71}{105}\right)$$
$$\chi_{4009}(10,\cdot)$$ 4009.dm 90 yes $$1$$ $$1$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{4009}(11,\cdot)$$ 4009.do 105 yes $$1$$ $$1$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{88}{105}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{34}{35}\right)$$
$$\chi_{4009}(12,\cdot)$$ 4009.cs 42 yes $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{4009}(13,\cdot)$$ 4009.es 630 yes $$-1$$ $$1$$ $$e\left(\frac{607}{630}\right)$$ $$e\left(\frac{61}{630}\right)$$ $$e\left(\frac{292}{315}\right)$$ $$e\left(\frac{302}{315}\right)$$ $$e\left(\frac{19}{315}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{61}{315}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{44}{105}\right)$$
$$\chi_{4009}(14,\cdot)$$ 4009.br 18 yes $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{4009}(15,\cdot)$$ 4009.bj 18 yes $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{4009}(16,\cdot)$$ 4009.em 315 yes $$1$$ $$1$$ $$e\left(\frac{76}{315}\right)$$ $$e\left(\frac{223}{315}\right)$$ $$e\left(\frac{152}{315}\right)$$ $$e\left(\frac{22}{315}\right)$$ $$e\left(\frac{299}{315}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{131}{315}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{79}{105}\right)$$
$$\chi_{4009}(17,\cdot)$$ 4009.ev 630 yes $$-1$$ $$1$$ $$e\left(\frac{317}{630}\right)$$ $$e\left(\frac{611}{630}\right)$$ $$e\left(\frac{2}{315}\right)$$ $$e\left(\frac{307}{315}\right)$$ $$e\left(\frac{149}{315}\right)$$ $$e\left(\frac{11}{210}\right)$$ $$e\left(\frac{107}{210}\right)$$ $$e\left(\frac{296}{315}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{19}{105}\right)$$
$$\chi_{4009}(18,\cdot)$$ 4009.dd 70 yes $$1$$ $$1$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{11}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{35}\right)$$
$$\chi_{4009}(20,\cdot)$$ 4009.dp 105 no $$1$$ $$1$$ $$e\left(\frac{67}{105}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{29}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{8}{105}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{35}\right)$$
$$\chi_{4009}(21,\cdot)$$ 4009.de 90 yes $$-1$$ $$1$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{4009}(22,\cdot)$$ 4009.et 630 yes $$1$$ $$1$$ $$e\left(\frac{157}{315}\right)$$ $$e\left(\frac{241}{315}\right)$$ $$e\left(\frac{314}{315}\right)$$ $$e\left(\frac{4}{315}\right)$$ $$e\left(\frac{83}{315}\right)$$ $$e\left(\frac{47}{210}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{167}{315}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{43}{105}\right)$$
$$\chi_{4009}(23,\cdot)$$ 4009.df 90 yes $$-1$$ $$1$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{4009}(24,\cdot)$$ 4009.em 315 yes $$1$$ $$1$$ $$e\left(\frac{34}{315}\right)$$ $$e\left(\frac{307}{315}\right)$$ $$e\left(\frac{68}{315}\right)$$ $$e\left(\frac{43}{315}\right)$$ $$e\left(\frac{26}{315}\right)$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{34}{105}\right)$$ $$e\left(\frac{299}{315}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{16}{105}\right)$$
$$\chi_{4009}(25,\cdot)$$ 4009.en 315 yes $$1$$ $$1$$ $$e\left(\frac{11}{315}\right)$$ $$e\left(\frac{53}{315}\right)$$ $$e\left(\frac{22}{315}\right)$$ $$e\left(\frac{122}{315}\right)$$ $$e\left(\frac{64}{315}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{106}{315}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{104}{105}\right)$$
$$\chi_{4009}(26,\cdot)$$ 4009.cn 42 yes $$-1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{4009}(27,\cdot)$$ 4009.ec 210 yes $$1$$ $$1$$ $$e\left(\frac{82}{105}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{18}{35}\right)$$
$$\chi_{4009}(28,\cdot)$$ 4009.ew 630 yes $$-1$$ $$1$$ $$e\left(\frac{73}{630}\right)$$ $$e\left(\frac{409}{630}\right)$$ $$e\left(\frac{73}{315}\right)$$ $$e\left(\frac{233}{315}\right)$$ $$e\left(\frac{241}{315}\right)$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{73}{210}\right)$$ $$e\left(\frac{94}{315}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{11}{105}\right)$$
$$\chi_{4009}(29,\cdot)$$ 4009.et 630 yes $$1$$ $$1$$ $$e\left(\frac{251}{315}\right)$$ $$e\left(\frac{293}{315}\right)$$ $$e\left(\frac{187}{315}\right)$$ $$e\left(\frac{197}{315}\right)$$ $$e\left(\frac{229}{315}\right)$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{41}{105}\right)$$ $$e\left(\frac{271}{315}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{44}{105}\right)$$
$$\chi_{4009}(30,\cdot)$$ 4009.dn 105 yes $$1$$ $$1$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{43}{105}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{27}{35}\right)$$
$$\chi_{4009}(31,\cdot)$$ 4009.cr 42 yes $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{4009}(32,\cdot)$$ 4009.dv 126 yes $$1$$ $$1$$ $$e\left(\frac{19}{63}\right)$$ $$e\left(\frac{40}{63}\right)$$ $$e\left(\frac{38}{63}\right)$$ $$e\left(\frac{37}{63}\right)$$ $$e\left(\frac{59}{63}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{17}{63}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{4009}(33,\cdot)$$ 4009.dv 126 yes $$1$$ $$1$$ $$e\left(\frac{23}{63}\right)$$ $$e\left(\frac{2}{63}\right)$$ $$e\left(\frac{46}{63}\right)$$ $$e\left(\frac{5}{63}\right)$$ $$e\left(\frac{25}{63}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{4}{63}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{21}\right)$$