from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([147,145]))
chi.galois_orbit()
[g,chi] = znchar(Mod(18,473))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(473\) | |
| Conductor: | \(473\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(210\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{105})$ |
| Fixed field: | Number field defined by a degree 210 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{473}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{473}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{473}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{473}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{473}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{473}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{473}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{473}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{473}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{473}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{473}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{473}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{473}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{473}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{473}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{473}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{473}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{473}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{473}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{473}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{473}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{473}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{473}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{473}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{473}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{473}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{473}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{473}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{473}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{473}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{473}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) |