from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1441, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([117,84]))
chi.galois_orbit()
[g,chi] = znchar(Mod(105,1441))
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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1441}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{53}{65}\right)\) |
| \(\chi_{1441}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{14}{65}\right)\) |
| \(\chi_{1441}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) |
| \(\chi_{1441}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{7}{65}\right)\) |
| \(\chi_{1441}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) |
| \(\chi_{1441}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{2}{65}\right)\) |
| \(\chi_{1441}(239,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) |
| \(\chi_{1441}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) |
| \(\chi_{1441}(310,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) |
| \(\chi_{1441}(398,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) |
| \(\chi_{1441}(402,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{63}{65}\right)\) |
| \(\chi_{1441}(420,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{62}{65}\right)\) |
| \(\chi_{1441}(426,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) |
| \(\chi_{1441}(431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{32}{65}\right)\) |
| \(\chi_{1441}(437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) |
| \(\chi_{1441}(468,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{23}{65}\right)\) |
| \(\chi_{1441}(502,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) |
| \(\chi_{1441}(567,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) |
| \(\chi_{1441}(579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) |
| \(\chi_{1441}(589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) |
| \(\chi_{1441}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) |
| \(\chi_{1441}(618,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) |
| \(\chi_{1441}(645,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{49}{65}\right)\) |
| \(\chi_{1441}(662,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) |
| \(\chi_{1441}(668,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) |
| \(\chi_{1441}(778,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) |
| \(\chi_{1441}(798,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) |
| \(\chi_{1441}(820,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) |
| \(\chi_{1441}(821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{54}{65}\right)\) |
| \(\chi_{1441}(827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) |
| \(\chi_{1441}(886,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) |