from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1441, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([117,98]))
chi.galois_orbit()
[g,chi] = znchar(Mod(28,1441))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1441\) | |
| Conductor: | \(1441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(130\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{65})$ |
| Fixed field: | Number field defined by a degree 130 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1441}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{51}{65}\right)\) |
| \(\chi_{1441}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{54}{65}\right)\) |
| \(\chi_{1441}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) |
| \(\chi_{1441}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{37}{65}\right)\) |
| \(\chi_{1441}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{64}{65}\right)\) |
| \(\chi_{1441}(156,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{24}{65}\right)\) |
| \(\chi_{1441}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{27}{65}\right)\) |
| \(\chi_{1441}(266,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{9}{65}\right)\) |
| \(\chi_{1441}(271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{63}{65}\right)\) |
| \(\chi_{1441}(321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{44}{65}\right)\) |
| \(\chi_{1441}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{23}{65}\right)\) |
| \(\chi_{1441}(404,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) |
| \(\chi_{1441}(413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) |
| \(\chi_{1441}(436,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) |
| \(\chi_{1441}(442,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) |
| \(\chi_{1441}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) |
| \(\chi_{1441}(501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) |
| \(\chi_{1441}(545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) |
| \(\chi_{1441}(667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) |
| \(\chi_{1441}(688,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) |
| \(\chi_{1441}(689,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) |
| \(\chi_{1441}(699,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) |
| \(\chi_{1441}(755,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) |
| \(\chi_{1441}(772,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{14}{65}\right)\) |
| \(\chi_{1441}(822,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{7}{65}\right)\) |
| \(\chi_{1441}(832,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) |
| \(\chi_{1441}(860,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) |
| \(\chi_{1441}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) |
| \(\chi_{1441}(888,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{2}{65}\right)\) |
| \(\chi_{1441}(930,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) |
| \(\chi_{1441}(965,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) |