from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([175,101]))
chi.galois_orbit()
[g,chi] = znchar(Mod(145,4009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4009\) | |
| Conductor: | \(4009\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4009}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{4009}(202,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{4009}(240,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{35}\right)\) |
| \(\chi_{4009}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{35}\right)\) |
| \(\chi_{4009}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{4009}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{4009}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{4009}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{4009}(582,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{4009}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{4009}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{4009}(749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{4009}(962,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{4009}(996,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{4009}(1072,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{4009}(1186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{4009}(1323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{4009}(1357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{4009}(1399,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{4009}(1589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{4009}(1604,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{4009}(1642,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{4009}(1684,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{4009}(1760,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{35}\right)\) |
| \(\chi_{4009}(1855,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{4009}(1893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{4009}(1984,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{4009}(2007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{4009}(2041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{4009}(2117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{8}{35}\right)\) |
| \(\chi_{4009}(2216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{8}{35}\right)\) |