from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([104,90]))
chi.galois_orbit()
[g,chi] = znchar(Mod(14,1859))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1859\) | |
| Conductor: | \(1859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(65\) | 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 65 polynomial |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1859}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{1859}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{1859}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{1859}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
| \(\chi_{1859}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{1859}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{1859}(300,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(378,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |
| \(\chi_{1859}(456,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{1859}(482,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(586,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{1859}(599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{1859}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{1859}(664,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
| \(\chi_{1859}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{1859}(742,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{1859}(768,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{1859}(807,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(872,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{1859}(885,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{1859}(950,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |
| \(\chi_{1859}(1028,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(1054,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{1859}(1158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(1171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |