from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([136,0,476,69]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,338130))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(338130\) | |
| Conductor: | \(33813\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(816\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 33813.nr | 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: | $\Q(\zeta_{816})$ |
| Fixed field: | Number field defined by a degree 816 polynomial (not computed) |
First 31 of 256 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{53}{272}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{19}{272}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{809}{816}\right)\) | \(e\left(\frac{715}{816}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{19}{51}\right)\) |
| \(\chi_{338130}(821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{353}{816}\right)\) | \(e\left(\frac{47}{272}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{625}{816}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{203}{816}\right)\) | \(e\left(\frac{779}{816}\right)\) | \(e\left(\frac{49}{816}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{338130}(1931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{439}{816}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{167}{816}\right)\) | \(e\left(\frac{31}{272}\right)\) | \(e\left(\frac{301}{816}\right)\) | \(e\left(\frac{733}{816}\right)\) | \(e\left(\frac{551}{816}\right)\) | \(e\left(\frac{77}{408}\right)\) | \(e\left(\frac{14}{51}\right)\) |
| \(\chi_{338130}(3101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{816}\right)\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{95}{816}\right)\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{181}{816}\right)\) | \(e\left(\frac{373}{816}\right)\) | \(e\left(\frac{719}{816}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{44}{51}\right)\) |
| \(\chi_{338130}(3191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{816}\right)\) | \(e\left(\frac{263}{272}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{569}{816}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{19}{816}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{89}{816}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) |
| \(\chi_{338130}(6701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{733}{816}\right)\) | \(e\left(\frac{19}{272}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{461}{816}\right)\) | \(e\left(\frac{53}{272}\right)\) | \(e\left(\frac{655}{816}\right)\) | \(e\left(\frac{367}{816}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{338130}(7031,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{455}{816}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{727}{816}\right)\) | \(e\left(\frac{47}{272}\right)\) | \(e\left(\frac{509}{816}\right)\) | \(e\left(\frac{269}{816}\right)\) | \(e\left(\frac{151}{816}\right)\) | \(e\left(\frac{301}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{338130}(8201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{623}{816}\right)\) | \(e\left(\frac{113}{272}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{79}{816}\right)\) | \(e\left(\frac{215}{272}\right)\) | \(e\left(\frac{245}{816}\right)\) | \(e\left(\frac{293}{816}\right)\) | \(e\left(\frac{31}{816}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{28}{51}\right)\) |
| \(\chi_{338130}(9041,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{517}{816}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{245}{816}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{295}{816}\right)\) | \(e\left(\frac{103}{816}\right)\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{338130}(10121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{816}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{755}{816}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{193}{816}\right)\) | \(e\left(\frac{1}{816}\right)\) | \(e\left(\frac{131}{816}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{338130}(11351,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{816}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{131}{408}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{515}{816}\right)\) | \(e\left(\frac{553}{816}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{338130}(12551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{481}{816}\right)\) | \(e\left(\frac{175}{272}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{209}{816}\right)\) | \(e\left(\frac{73}{272}\right)\) | \(e\left(\frac{235}{816}\right)\) | \(e\left(\frac{331}{816}\right)\) | \(e\left(\frac{113}{816}\right)\) | \(e\left(\frac{155}{408}\right)\) | \(e\left(\frac{5}{51}\right)\) |
| \(\chi_{338130}(14801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{667}{816}\right)\) | \(e\left(\frac{21}{272}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{395}{816}\right)\) | \(e\left(\frac{259}{272}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{649}{816}\right)\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{38}{51}\right)\) |
| \(\chi_{338130}(14861,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{365}{816}\right)\) | \(e\left(\frac{195}{272}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{637}{816}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{767}{816}\right)\) | \(e\left(\frac{431}{816}\right)\) | \(e\left(\frac{157}{816}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{25}{51}\right)\) |
| \(\chi_{338130}(15221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{419}{816}\right)\) | \(e\left(\frac{45}{272}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{11}{272}\right)\) | \(e\left(\frac{449}{816}\right)\) | \(e\left(\frac{497}{816}\right)\) | \(e\left(\frac{643}{816}\right)\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{28}{51}\right)\) |
| \(\chi_{338130}(17201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{816}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{613}{816}\right)\) | \(e\left(\frac{205}{272}\right)\) | \(e\left(\frac{455}{816}\right)\) | \(e\left(\frac{311}{816}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{338130}(20711,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{641}{816}\right)\) | \(e\left(\frac{63}{272}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{97}{816}\right)\) | \(e\left(\frac{233}{272}\right)\) | \(e\left(\frac{683}{816}\right)\) | \(e\left(\frac{587}{816}\right)\) | \(e\left(\frac{193}{816}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{338130}(21821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{583}{816}\right)\) | \(e\left(\frac{73}{272}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{311}{816}\right)\) | \(e\left(\frac{175}{272}\right)\) | \(e\left(\frac{541}{816}\right)\) | \(e\left(\frac{637}{816}\right)\) | \(e\left(\frac{215}{816}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{5}{51}\right)\) |
| \(\chi_{338130}(22991,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{816}\right)\) | \(e\left(\frac{257}{272}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{767}{816}\right)\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{469}{816}\right)\) | \(e\left(\frac{239}{816}\right)\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{338130}(23081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{553}{816}\right)\) | \(e\left(\frac{247}{272}\right)\) | \(e\left(\frac{91}{408}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{145}{272}\right)\) | \(e\left(\frac{355}{816}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{761}{816}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{26}{51}\right)\) |
| \(\chi_{338130}(26591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{397}{816}\right)\) | \(e\left(\frac{227}{272}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{125}{816}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{367}{816}\right)\) | \(e\left(\frac{319}{816}\right)\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{23}{51}\right)\) |
| \(\chi_{338130}(26921,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{599}{816}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{55}{816}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{631}{816}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{4}{51}\right)\) |
| \(\chi_{338130}(28091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{816}\right)\) | \(e\left(\frac{241}{272}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{751}{816}\right)\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{5}{816}\right)\) | \(e\left(\frac{389}{816}\right)\) | \(e\left(\frac{367}{816}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{338130}(28931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{816}\right)\) | \(e\left(\frac{139}{272}\right)\) | \(e\left(\frac{223}{408}\right)\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{173}{272}\right)\) | \(e\left(\frac{583}{816}\right)\) | \(e\left(\frac{151}{816}\right)\) | \(e\left(\frac{197}{816}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{20}{51}\right)\) |
| \(\chi_{338130}(30011,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{451}{816}\right)\) | \(e\left(\frac{77}{272}\right)\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{179}{816}\right)\) | \(e\left(\frac{43}{272}\right)\) | \(e\left(\frac{49}{816}\right)\) | \(e\left(\frac{385}{816}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{26}{51}\right)\) |
| \(\chi_{338130}(31241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{665}{816}\right)\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{121}{816}\right)\) | \(e\left(\frac{257}{272}\right)\) | \(e\left(\frac{179}{816}\right)\) | \(e\left(\frac{707}{816}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{19}{51}\right)\) |
| \(\chi_{338130}(32441,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{769}{816}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{497}{816}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{715}{816}\right)\) | \(e\left(\frac{139}{816}\right)\) | \(e\left(\frac{257}{816}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{38}{51}\right)\) |
| \(\chi_{338130}(34691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{53}{272}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{19}{272}\right)\) | \(e\left(\frac{553}{816}\right)\) | \(e\left(\frac{265}{816}\right)\) | \(e\left(\frac{443}{816}\right)\) | \(e\left(\frac{113}{408}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{338130}(34751,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{816}\right)\) | \(e\left(\frac{131}{272}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{301}{816}\right)\) | \(e\left(\frac{165}{272}\right)\) | \(e\left(\frac{479}{816}\right)\) | \(e\left(\frac{383}{816}\right)\) | \(e\left(\frac{397}{816}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{338130}(35111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{13}{272}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{115}{816}\right)\) | \(e\left(\frac{251}{272}\right)\) | \(e\left(\frac{305}{816}\right)\) | \(e\left(\frac{65}{816}\right)\) | \(e\left(\frac{355}{816}\right)\) | \(e\left(\frac{97}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{338130}(37091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{677}{816}\right)\) | \(e\left(\frac{235}{272}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{133}{816}\right)\) | \(e\left(\frac{269}{272}\right)\) | \(e\left(\frac{743}{816}\right)\) | \(e\left(\frac{359}{816}\right)\) | \(e\left(\frac{517}{816}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{31}{51}\right)\) |