from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([272,0,612,27]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,338130))
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.mq | 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: | $\Q(\zeta_{816})$ |
| Fixed field: | Number field defined by a degree 816 polynomial (not computed) |
First 31 of 256 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{445}{816}\right)\) | \(e\left(\frac{383}{816}\right)\) | \(e\left(\frac{583}{816}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{61}{816}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{91}{102}\right)\) |
| \(\chi_{338130}(2101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{503}{816}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{367}{816}\right)\) | \(e\left(\frac{149}{816}\right)\) | \(e\left(\frac{589}{816}\right)\) | \(e\left(\frac{79}{272}\right)\) | \(e\left(\frac{175}{816}\right)\) | \(e\left(\frac{157}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) |
| \(\chi_{338130}(3271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{575}{816}\right)\) | \(e\left(\frac{59}{816}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{439}{816}\right)\) | \(e\left(\frac{365}{816}\right)\) | \(e\left(\frac{709}{816}\right)\) | \(e\left(\frac{199}{272}\right)\) | \(e\left(\frac{7}{816}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) |
| \(\chi_{338130}(3541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{497}{816}\right)\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{361}{816}\right)\) | \(e\left(\frac{131}{816}\right)\) | \(e\left(\frac{715}{816}\right)\) | \(e\left(\frac{137}{272}\right)\) | \(e\left(\frac{121}{816}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{25}{102}\right)\) |
| \(\chi_{338130}(3661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{271}{816}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{11}{816}\right)\) | \(e\left(\frac{169}{816}\right)\) | \(e\left(\frac{449}{816}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{779}{816}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{5}{102}\right)\) |
| \(\chi_{338130}(7441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{361}{816}\right)\) | \(e\left(\frac{301}{816}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{497}{816}\right)\) | \(e\left(\frac{811}{816}\right)\) | \(e\left(\frac{35}{816}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{257}{816}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{59}{102}\right)\) |
| \(\chi_{338130}(10291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{816}\right)\) | \(e\left(\frac{239}{816}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{91}{816}\right)\) | \(e\left(\frac{137}{816}\right)\) | \(e\left(\frac{673}{816}\right)\) | \(e\left(\frac{27}{272}\right)\) | \(e\left(\frac{139}{816}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{97}{102}\right)\) |
| \(\chi_{338130}(10951,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{445}{816}\right)\) | \(e\left(\frac{145}{816}\right)\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{247}{816}\right)\) | \(e\left(\frac{719}{816}\right)\) | \(e\left(\frac{5}{272}\right)\) | \(e\left(\frac{197}{816}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) |
| \(\chi_{338130}(13291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{805}{816}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{125}{816}\right)\) | \(e\left(\frac{511}{816}\right)\) | \(e\left(\frac{503}{816}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{29}{102}\right)\) |
| \(\chi_{338130}(14071,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{809}{816}\right)\) | \(e\left(\frac{557}{816}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{673}{816}\right)\) | \(e\left(\frac{251}{816}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{113}{272}\right)\) | \(e\left(\frac{481}{816}\right)\) | \(e\left(\frac{259}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) |
| \(\chi_{338130}(14971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{816}\right)\) | \(e\left(\frac{695}{816}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{787}{816}\right)\) | \(e\left(\frac{593}{816}\right)\) | \(e\left(\frac{745}{816}\right)\) | \(e\left(\frac{99}{272}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{61}{102}\right)\) |
| \(\chi_{338130}(15361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{816}\right)\) | \(e\left(\frac{403}{816}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{191}{816}\right)\) | \(e\left(\frac{709}{816}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{239}{272}\right)\) | \(e\left(\frac{767}{816}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{59}{102}\right)\) |
| \(\chi_{338130}(16531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{751}{816}\right)\) | \(e\left(\frac{43}{816}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{71}{816}\right)\) | \(e\left(\frac{349}{816}\right)\) | \(e\left(\frac{5}{816}\right)\) | \(e\left(\frac{39}{272}\right)\) | \(e\left(\frac{503}{816}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) |
| \(\chi_{338130}(16801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{816}\right)\) | \(e\left(\frac{469}{816}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{163}{816}\right)\) | \(e\left(\frac{491}{816}\right)\) | \(e\left(\frac{185}{272}\right)\) | \(e\left(\frac{761}{816}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{35}{102}\right)\) |
| \(\chi_{338130}(17581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{605}{816}\right)\) | \(e\left(\frac{353}{816}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{469}{816}\right)\) | \(e\left(\frac{455}{816}\right)\) | \(e\left(\frac{79}{816}\right)\) | \(e\left(\frac{181}{272}\right)\) | \(e\left(\frac{277}{816}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) |
| \(\chi_{338130}(19921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{816}\right)\) | \(e\left(\frac{473}{816}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{781}{816}\right)\) | \(e\left(\frac{575}{816}\right)\) | \(e\left(\frac{55}{816}\right)\) | \(e\left(\frac{157}{272}\right)\) | \(e\left(\frac{637}{816}\right)\) | \(e\left(\frac{343}{408}\right)\) | \(e\left(\frac{49}{102}\right)\) |
| \(\chi_{338130}(21991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{647}{816}\right)\) | \(e\left(\frac{275}{816}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{511}{816}\right)\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{13}{816}\right)\) | \(e\left(\frac{47}{272}\right)\) | \(e\left(\frac{655}{816}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{19}{102}\right)\) |
| \(\chi_{338130}(23161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{431}{816}\right)\) | \(e\left(\frac{443}{816}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{295}{816}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{469}{816}\right)\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{343}{816}\right)\) | \(e\left(\frac{373}{408}\right)\) | \(e\left(\frac{97}{102}\right)\) |
| \(\chi_{338130}(23431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{785}{816}\right)\) | \(e\left(\frac{485}{816}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{649}{816}\right)\) | \(e\left(\frac{179}{816}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{73}{272}\right)\) | \(e\left(\frac{265}{816}\right)\) | \(e\left(\frac{331}{408}\right)\) | \(e\left(\frac{91}{102}\right)\) |
| \(\chi_{338130}(23551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{816}\right)\) | \(e\left(\frac{175}{816}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{251}{816}\right)\) | \(e\left(\frac{73}{816}\right)\) | \(e\left(\frac{305}{816}\right)\) | \(e\left(\frac{203}{272}\right)\) | \(e\left(\frac{491}{816}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{77}{102}\right)\) |
| \(\chi_{338130}(27331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{816}\right)\) | \(e\left(\frac{253}{816}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{209}{816}\right)\) | \(e\left(\frac{763}{816}\right)\) | \(e\left(\frac{371}{816}\right)\) | \(e\left(\frac{65}{272}\right)\) | \(e\left(\frac{113}{816}\right)\) | \(e\left(\frac{155}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) |
| \(\chi_{338130}(28231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{763}{816}\right)\) | \(e\left(\frac{487}{816}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{83}{816}\right)\) | \(e\left(\frac{385}{816}\right)\) | \(e\left(\frac{569}{816}\right)\) | \(e\left(\frac{195}{272}\right)\) | \(e\left(\frac{611}{816}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{47}{102}\right)\) |
| \(\chi_{338130}(30181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{816}\right)\) | \(e\left(\frac{143}{816}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{331}{816}\right)\) | \(e\left(\frac{41}{816}\right)\) | \(e\left(\frac{529}{816}\right)\) | \(e\left(\frac{155}{272}\right)\) | \(e\left(\frac{667}{816}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{67}{102}\right)\) |
| \(\chi_{338130}(30841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{816}\right)\) | \(e\left(\frac{769}{816}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{245}{816}\right)\) | \(e\left(\frac{55}{816}\right)\) | \(e\left(\frac{431}{816}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{65}{102}\right)\) |
| \(\chi_{338130}(33181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{325}{816}\right)\) | \(e\left(\frac{601}{816}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{461}{816}\right)\) | \(e\left(\frac{703}{816}\right)\) | \(e\left(\frac{791}{816}\right)\) | \(e\left(\frac{77}{272}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{89}{102}\right)\) |
| \(\chi_{338130}(33961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{521}{816}\right)\) | \(e\left(\frac{509}{816}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{385}{816}\right)\) | \(e\left(\frac{203}{816}\right)\) | \(e\left(\frac{211}{816}\right)\) | \(e\left(\frac{177}{272}\right)\) | \(e\left(\frac{337}{816}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{73}{102}\right)\) |
| \(\chi_{338130}(34861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{683}{816}\right)\) | \(e\left(\frac{791}{816}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{547}{816}\right)\) | \(e\left(\frac{689}{816}\right)\) | \(e\left(\frac{73}{816}\right)\) | \(e\left(\frac{243}{272}\right)\) | \(e\left(\frac{163}{816}\right)\) | \(e\left(\frac{25}{408}\right)\) | \(e\left(\frac{91}{102}\right)\) |
| \(\chi_{338130}(35251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{816}\right)\) | \(e\left(\frac{19}{816}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{335}{816}\right)\) | \(e\left(\frac{325}{816}\right)\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{207}{272}\right)\) | \(e\left(\frac{431}{816}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{41}{102}\right)\) |
| \(\chi_{338130}(36421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{607}{816}\right)\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{743}{816}\right)\) | \(e\left(\frac{733}{816}\right)\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{23}{816}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{41}{102}\right)\) |
| \(\chi_{338130}(36691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{433}{816}\right)\) | \(e\left(\frac{517}{816}\right)\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{569}{816}\right)\) | \(e\left(\frac{211}{816}\right)\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{121}{272}\right)\) | \(e\left(\frac{89}{816}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{101}{102}\right)\) |
| \(\chi_{338130}(37471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{816}\right)\) | \(e\left(\frac{161}{816}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{133}{816}\right)\) | \(e\left(\frac{263}{816}\right)\) | \(e\left(\frac{607}{816}\right)\) | \(e\left(\frac{165}{272}\right)\) | \(e\left(\frac{517}{816}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) |