from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2003, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([103]))
chi.galois_orbit()
[g,chi] = znchar(Mod(50,2003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2003\) | |
| Conductor: | \(2003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(154\) | 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: | $\Q(\zeta_{77})$ |
| Fixed field: | Number field defined by a degree 154 polynomial (not computed) |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2003}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2003}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{2003}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2003}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2003}(124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2003}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{89}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2003}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2003}(210,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{154}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{2003}(214,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2003}(234,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{2003}(245,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{154}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2003}(257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2003}(273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2003}(329,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{2003}(371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2003}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2003}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{101}{154}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2003}(566,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2003}(628,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2003}(648,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{71}{154}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2003}(726,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{2003}(756,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2003}(757,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2003}(797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{2003}(814,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2003}(847,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{89}{154}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{2003}(870,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2003}(935,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{149}{154}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{2003}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{2003}(1059,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{83}{154}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{2003}(1086,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{127}{154}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{9}{22}\right)\) |