from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([34,75]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,2601))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(2601\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(204\) | 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: | $\Q(\zeta_{204})$ |
| Fixed field: | Number field defined by a degree 204 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2601}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{4}{51}\right)\) |
| \(\chi_{2601}(140,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{79}{204}\right)\) | \(e\left(\frac{143}{204}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{2601}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{2601}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{107}{204}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{2601}(200,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{149}{204}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{2601}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{5}{51}\right)\) |
| \(\chi_{2601}(302,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{50}{51}\right)\) |
| \(\chi_{2601}(344,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{203}{204}\right)\) | \(e\left(\frac{55}{204}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{40}{51}\right)\) |
| \(\chi_{2601}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{125}{204}\right)\) | \(e\left(\frac{61}{204}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{2601}(446,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{8}{51}\right)\) |
| \(\chi_{2601}(455,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{169}{204}\right)\) | \(e\left(\frac{47}{51}\right)\) |
| \(\chi_{2601}(497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{143}{204}\right)\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{43}{51}\right)\) |
| \(\chi_{2601}(506,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{2601}(599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{11}{51}\right)\) |
| \(\chi_{2601}(608,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{71}{204}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{44}{51}\right)\) |
| \(\chi_{2601}(650,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{109}{204}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{71}{204}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{2601}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{43}{51}\right)\) |
| \(\chi_{2601}(752,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{113}{204}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{14}{51}\right)\) |
| \(\chi_{2601}(761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{167}{204}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{2601}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{2601}(812,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{40}{51}\right)\) |
| \(\chi_{2601}(914,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{73}{204}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{38}{51}\right)\) |
| \(\chi_{2601}(956,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{167}{204}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{2601}(965,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{2601}(1058,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{125}{204}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{20}{51}\right)\) |
| \(\chi_{2601}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{35}{51}\right)\) |
| \(\chi_{2601}(1109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{107}{204}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{4}{51}\right)\) |
| \(\chi_{2601}(1211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{67}{204}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{23}{51}\right)\) |
| \(\chi_{2601}(1220,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{32}{51}\right)\) |
| \(\chi_{2601}(1262,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{67}{204}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{7}{51}\right)\) |
| \(\chi_{2601}(1271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{79}{204}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{31}{51}\right)\) |