from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9310, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([63,138,56]))
chi.galois_orbit()
[g,chi] = znchar(Mod(187,9310))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9310\) | |
| Conductor: | \(4655\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(252\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4655.hb | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9310}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{252}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{235}{252}\right)\) | \(e\left(\frac{41}{252}\right)\) | \(e\left(\frac{1}{252}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{191}{252}\right)\) |
| \(\chi_{9310}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{252}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{113}{252}\right)\) | \(e\left(\frac{187}{252}\right)\) | \(e\left(\frac{23}{252}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{109}{252}\right)\) |
| \(\chi_{9310}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{252}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{13}{252}\right)\) | \(e\left(\frac{197}{252}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{252}\right)\) |
| \(\chi_{9310}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{252}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{239}{252}\right)\) |
| \(\chi_{9310}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{252}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{252}\right)\) | \(e\left(\frac{131}{252}\right)\) | \(e\left(\frac{163}{252}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{137}{252}\right)\) |
| \(\chi_{9310}(537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{29}{252}\right)\) | \(e\left(\frac{13}{252}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{215}{252}\right)\) |
| \(\chi_{9310}(633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{125}{252}\right)\) | \(e\left(\frac{247}{252}\right)\) | \(e\left(\frac{215}{252}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{241}{252}\right)\) |
| \(\chi_{9310}(663,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{252}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{187}{252}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{185}{252}\right)\) |
| \(\chi_{9310}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{145}{252}\right)\) | \(e\left(\frac{65}{252}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{252}\right)\) |
| \(\chi_{9310}(1263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{101}{252}\right)\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{229}{252}\right)\) |
| \(\chi_{9310}(1277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{252}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{252}\right)\) | \(e\left(\frac{241}{252}\right)\) | \(e\left(\frac{221}{252}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{127}{252}\right)\) |
| \(\chi_{9310}(1517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{252}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{113}{252}\right)\) | \(e\left(\frac{181}{252}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{47}{252}\right)\) |
| \(\chi_{9310}(1543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{252}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{221}{252}\right)\) | \(e\left(\frac{223}{252}\right)\) | \(e\left(\frac{239}{252}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{252}\right)\) |
| \(\chi_{9310}(1727,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{252}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{43}{252}\right)\) | \(e\left(\frac{89}{252}\right)\) | \(e\left(\frac{205}{252}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{95}{252}\right)\) |
| \(\chi_{9310}(1867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{252}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{247}{252}\right)\) | \(e\left(\frac{101}{252}\right)\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{71}{252}\right)\) |
| \(\chi_{9310}(1963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{233}{252}\right)\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{169}{252}\right)\) |
| \(\chi_{9310}(1993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{252}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{61}{252}\right)\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{115}{252}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{252}\right)\) |
| \(\chi_{9310}(2133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{252}\right)\) | \(e\left(\frac{191}{252}\right)\) | \(e\left(\frac{103}{252}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{252}\right)\) |
| \(\chi_{9310}(2327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{252}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{252}\right)\) | \(e\left(\frac{181}{252}\right)\) | \(e\left(\frac{29}{252}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{247}{252}\right)\) |
| \(\chi_{9310}(2593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{209}{252}\right)\) | \(e\left(\frac{163}{252}\right)\) | \(e\left(\frac{47}{252}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{157}{252}\right)\) |
| \(\chi_{9310}(2607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{131}{252}\right)\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{185}{252}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{55}{252}\right)\) |
| \(\chi_{9310}(2847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{163}{252}\right)\) | \(e\left(\frac{185}{252}\right)\) | \(e\left(\frac{109}{252}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{155}{252}\right)\) |
| \(\chi_{9310}(3027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{143}{252}\right)\) | \(e\left(\frac{85}{252}\right)\) | \(e\left(\frac{125}{252}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{187}{252}\right)\) |
| \(\chi_{9310}(3113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{252}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{181}{252}\right)\) | \(e\left(\frac{23}{252}\right)\) | \(e\left(\frac{19}{252}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{101}{252}\right)\) |
| \(\chi_{9310}(3197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{252}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{211}{252}\right)\) | \(e\left(\frac{173}{252}\right)\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{179}{252}\right)\) |
| \(\chi_{9310}(3293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{252}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{89}{252}\right)\) | \(e\left(\frac{67}{252}\right)\) | \(e\left(\frac{143}{252}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{97}{252}\right)\) |
| \(\chi_{9310}(3323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{251}{252}\right)\) | \(e\left(\frac{43}{252}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{149}{252}\right)\) |
| \(\chi_{9310}(3463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{252}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{229}{252}\right)\) | \(e\left(\frac{11}{252}\right)\) | \(e\left(\frac{31}{252}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{125}{252}\right)\) |
| \(\chi_{9310}(3923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{252}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{65}{252}\right)\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{11}{252}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{85}{252}\right)\) |
| \(\chi_{9310}(3937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{239}{252}\right)\) | \(e\left(\frac{61}{252}\right)\) | \(e\left(\frac{149}{252}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{235}{252}\right)\) |
| \(\chi_{9310}(4177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{252}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{252}\right)\) |