from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,211))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(211\) | |
| Conductor: | \(211\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(105\) | 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 105 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{211}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{211}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{211}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{211}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{35}\right)\) |
| \(\chi_{211}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{211}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{211}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{211}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{211}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{211}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{211}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{6}{35}\right)\) |
| \(\chi_{211}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{211}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{35}\right)\) |
| \(\chi_{211}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{211}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{211}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{211}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{211}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{211}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{211}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{211}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{211}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{211}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{211}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{211}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{211}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{211}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{211}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{211}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{211}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{211}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{35}\right)\) |