from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1441, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([117,11]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,1441))
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: | even | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1441}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) |
| \(\chi_{1441}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) |
| \(\chi_{1441}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) |
| \(\chi_{1441}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) |
| \(\chi_{1441}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) |
| \(\chi_{1441}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) |
| \(\chi_{1441}(272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) |
| \(\chi_{1441}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) |
| \(\chi_{1441}(316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) |
| \(\chi_{1441}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) |
| \(\chi_{1441}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) |
| \(\chi_{1441}(480,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) |
| \(\chi_{1441}(491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) |
| \(\chi_{1441}(508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) |
| \(\chi_{1441}(634,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) |
| \(\chi_{1441}(644,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) |
| \(\chi_{1441}(678,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) |
| \(\chi_{1441}(695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) |
| \(\chi_{1441}(712,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) |
| \(\chi_{1441}(766,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) |
| \(\chi_{1441}(800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) |
| \(\chi_{1441}(816,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) |
| \(\chi_{1441}(853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) |
| \(\chi_{1441}(871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) |
| \(\chi_{1441}(919,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) |
| \(\chi_{1441}(948,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) |
| \(\chi_{1441}(1007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) |
| \(\chi_{1441}(1014,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) |
| \(\chi_{1441}(1036,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) |
| \(\chi_{1441}(1130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) |
| \(\chi_{1441}(1151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) |