from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,105,10,126]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,8624))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(4312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(210\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4312.eu | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8624}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{3}{70}\right)\) |
| \(\chi_{8624}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{59}{70}\right)\) |
| \(\chi_{8624}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{37}{70}\right)\) |
| \(\chi_{8624}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{11}{70}\right)\) |
| \(\chi_{8624}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{11}{70}\right)\) |
| \(\chi_{8624}(905,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{29}{70}\right)\) |
| \(\chi_{8624}(1017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{27}{70}\right)\) |
| \(\chi_{8624}(1241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{23}{70}\right)\) |
| \(\chi_{8624}(1257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{69}{70}\right)\) |
| \(\chi_{8624}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{47}{70}\right)\) |
| \(\chi_{8624}(1577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{31}{70}\right)\) |
| \(\chi_{8624}(1593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{3}{70}\right)\) |
| \(\chi_{8624}(2249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{47}{70}\right)\) |
| \(\chi_{8624}(2473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{43}{70}\right)\) |
| \(\chi_{8624}(2489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{9}{70}\right)\) |
| \(\chi_{8624}(2601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{57}{70}\right)\) |
| \(\chi_{8624}(2809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{51}{70}\right)\) |
| \(\chi_{8624}(2825,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{13}{70}\right)\) |
| \(\chi_{8624}(3161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{31}{70}\right)\) |
| \(\chi_{8624}(3369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{69}{70}\right)\) |
| \(\chi_{8624}(3481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{67}{70}\right)\) |
| \(\chi_{8624}(3721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{19}{70}\right)\) |
| \(\chi_{8624}(3833,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{67}{70}\right)\) |
| \(\chi_{8624}(4041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{1}{70}\right)\) |
| \(\chi_{8624}(4057,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{23}{70}\right)\) |
| \(\chi_{8624}(4393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{41}{70}\right)\) |
| \(\chi_{8624}(4601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{19}{70}\right)\) |
| \(\chi_{8624}(4713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{17}{70}\right)\) |
| \(\chi_{8624}(4937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{13}{70}\right)\) |
| \(\chi_{8624}(4953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{29}{70}\right)\) |
| \(\chi_{8624}(5289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{33}{70}\right)\) |