from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6615, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([140,63,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(52,6615))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6615\) | |
| Conductor: | \(6615\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(252\) | 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: | $\Q(\zeta_{252})$ |
| Fixed field: | Number field defined by a degree 252 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6615}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{247}{252}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{151}{252}\right)\) | \(e\left(\frac{193}{252}\right)\) |
| \(\chi_{6615}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{65}{252}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{11}{252}\right)\) |
| \(\chi_{6615}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{227}{252}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(1\) | \(e\left(\frac{251}{252}\right)\) | \(e\left(\frac{209}{252}\right)\) |
| \(\chi_{6615}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{252}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{151}{252}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{169}{252}\right)\) |
| \(\chi_{6615}(418,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{252}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{41}{252}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{173}{252}\right)\) | \(e\left(\frac{131}{252}\right)\) |
| \(\chi_{6615}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{252}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{235}{252}\right)\) |
| \(\chi_{6615}(682,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{55}{252}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{103}{252}\right)\) | \(e\left(\frac{145}{252}\right)\) |
| \(\chi_{6615}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{252}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(1\) | \(e\left(\frac{41}{252}\right)\) | \(e\left(\frac{251}{252}\right)\) |
| \(\chi_{6615}(808,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{169}{252}\right)\) | \(e\left(\frac{211}{252}\right)\) |
| \(\chi_{6615}(922,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{252}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{239}{252}\right)\) | \(e\left(\frac{197}{252}\right)\) |
| \(\chi_{6615}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{252}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{211}{252}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{121}{252}\right)\) |
| \(\chi_{6615}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{252}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{145}{252}\right)\) | \(e\left(\frac{187}{252}\right)\) |
| \(\chi_{6615}(1237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{252}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{65}{252}\right)\) |
| \(\chi_{6615}(1312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{252}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{115}{252}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(1\) | \(e\left(\frac{55}{252}\right)\) | \(e\left(\frac{97}{252}\right)\) |
| \(\chi_{6615}(1363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{252}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{221}{252}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{29}{252}\right)\) | \(e\left(\frac{239}{252}\right)\) |
| \(\chi_{6615}(1438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{252}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{1}{252}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(1\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{163}{252}\right)\) |
| \(\chi_{6615}(1552,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{252}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{131}{252}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(1\) | \(e\left(\frac{227}{252}\right)\) | \(e\left(\frac{185}{252}\right)\) |
| \(\chi_{6615}(1627,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{252}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{19}{252}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(1\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{73}{252}\right)\) |
| \(\chi_{6615}(1678,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{252}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{197}{252}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(1\) | \(e\left(\frac{149}{252}\right)\) | \(e\left(\frac{107}{252}\right)\) |
| \(\chi_{6615}(1753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{157}{252}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(1\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{139}{252}\right)\) |
| \(\chi_{6615}(1867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{95}{252}\right)\) | \(e\left(\frac{53}{252}\right)\) |
| \(\chi_{6615}(1993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{252}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{173}{252}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(1\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{227}{252}\right)\) |
| \(\chi_{6615}(2068,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{61}{252}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(1\) | \(e\left(\frac{73}{252}\right)\) | \(e\left(\frac{115}{252}\right)\) |
| \(\chi_{6615}(2182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{252}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{215}{252}\right)\) | \(e\left(\frac{173}{252}\right)\) |
| \(\chi_{6615}(2257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{252}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{235}{252}\right)\) | \(e\left(\frac{25}{252}\right)\) |
| \(\chi_{6615}(2308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{149}{252}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{137}{252}\right)\) | \(e\left(\frac{95}{252}\right)\) |
| \(\chi_{6615}(2497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{59}{252}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(1\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{41}{252}\right)\) |
| \(\chi_{6615}(2572,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{252}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{235}{252}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{211}{252}\right)\) | \(e\left(\frac{1}{252}\right)\) |
| \(\chi_{6615}(2623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{125}{252}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{215}{252}\right)\) |
| \(\chi_{6615}(2698,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{252}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{67}{252}\right)\) |
| \(\chi_{6615}(2887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{252}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{139}{252}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{187}{252}\right)\) | \(e\left(\frac{229}{252}\right)\) |