from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,130,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1617))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1617\) | |
| Conductor: | \(1617\) | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1617}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{70}\right)\) |
| \(\chi_{1617}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{70}\right)\) |
| \(\chi_{1617}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{70}\right)\) |
| \(\chi_{1617}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{70}\right)\) |
| \(\chi_{1617}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{70}\right)\) |
| \(\chi_{1617}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{27}{70}\right)\) |
| \(\chi_{1617}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{57}{70}\right)\) |
| \(\chi_{1617}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{70}\right)\) |
| \(\chi_{1617}(326,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{70}\right)\) |
| \(\chi_{1617}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{51}{70}\right)\) |
| \(\chi_{1617}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{70}\right)\) |
| \(\chi_{1617}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{70}\right)\) |
| \(\chi_{1617}(380,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{43}{70}\right)\) |
| \(\chi_{1617}(431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{47}{70}\right)\) |
| \(\chi_{1617}(464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{27}{70}\right)\) |
| \(\chi_{1617}(536,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{51}{70}\right)\) |
| \(\chi_{1617}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{33}{70}\right)\) |
| \(\chi_{1617}(590,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{59}{70}\right)\) |
| \(\chi_{1617}(611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{70}\right)\) |
| \(\chi_{1617}(662,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{67}{70}\right)\) |
| \(\chi_{1617}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{67}{70}\right)\) |
| \(\chi_{1617}(767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{70}\right)\) |
| \(\chi_{1617}(788,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{69}{70}\right)\) |
| \(\chi_{1617}(800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{61}{70}\right)\) |
| \(\chi_{1617}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{53}{70}\right)\) |
| \(\chi_{1617}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{70}\right)\) |
| \(\chi_{1617}(842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{53}{70}\right)\) |
| \(\chi_{1617}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{70}\right)\) |
| \(\chi_{1617}(926,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{37}{70}\right)\) |
| \(\chi_{1617}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{70}\right)\) |
| \(\chi_{1617}(1031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{70}\right)\) |