from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([544,0,544,777]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,338130))
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.nh | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{191}{816}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{93}{272}\right)\) | \(e\left(\frac{293}{816}\right)\) | \(e\left(\frac{737}{816}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{225}{272}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{181}{204}\right)\) |
| \(\chi_{338130}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{272}\right)\) | \(e\left(\frac{727}{816}\right)\) | \(e\left(\frac{239}{408}\right)\) | \(e\left(\frac{213}{272}\right)\) | \(e\left(\frac{13}{816}\right)\) | \(e\left(\frac{793}{816}\right)\) | \(e\left(\frac{305}{816}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{125}{204}\right)\) |
| \(\chi_{338130}(4891,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{272}\right)\) | \(e\left(\frac{145}{816}\right)\) | \(e\left(\frac{257}{408}\right)\) | \(e\left(\frac{35}{272}\right)\) | \(e\left(\frac{43}{816}\right)\) | \(e\left(\frac{175}{816}\right)\) | \(e\left(\frac{695}{816}\right)\) | \(e\left(\frac{111}{272}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{131}{204}\right)\) |
| \(\chi_{338130}(7231,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{667}{816}\right)\) | \(e\left(\frac{203}{408}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{361}{816}\right)\) | \(e\left(\frac{805}{816}\right)\) | \(e\left(\frac{749}{816}\right)\) | \(e\left(\frac{21}{272}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{113}{204}\right)\) |
| \(\chi_{338130}(8251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{683}{816}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{193}{272}\right)\) | \(e\left(\frac{377}{816}\right)\) | \(e\left(\frac{149}{816}\right)\) | \(e\left(\frac{685}{816}\right)\) | \(e\left(\frac{37}{272}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{157}{204}\right)\) |
| \(\chi_{338130}(8401,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{272}\right)\) | \(e\left(\frac{133}{816}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{79}{272}\right)\) | \(e\left(\frac{439}{816}\right)\) | \(e\left(\frac{667}{816}\right)\) | \(e\left(\frac{131}{816}\right)\) | \(e\left(\frac{235}{272}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{47}{204}\right)\) |
| \(\chi_{338130}(9421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{143}{272}\right)\) | \(e\left(\frac{743}{816}\right)\) | \(e\left(\frac{443}{816}\right)\) | \(e\left(\frac{547}{816}\right)\) | \(e\left(\frac{267}{272}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{67}{204}\right)\) |
| \(\chi_{338130}(10741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{272}\right)\) | \(e\left(\frac{13}{816}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{247}{272}\right)\) | \(e\left(\frac{319}{816}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{203}{816}\right)\) | \(e\left(\frac{115}{272}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{23}{204}\right)\) |
| \(\chi_{338130}(11761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{272}\right)\) | \(e\left(\frac{797}{816}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{183}{272}\right)\) | \(e\left(\frac{287}{816}\right)\) | \(e\left(\frac{371}{816}\right)\) | \(e\left(\frac{331}{816}\right)\) | \(e\left(\frac{83}{272}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{139}{204}\right)\) |
| \(\chi_{338130}(11911,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{403}{816}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{41}{272}\right)\) | \(e\left(\frac{97}{816}\right)\) | \(e\left(\frac{205}{816}\right)\) | \(e\left(\frac{581}{816}\right)\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{101}{204}\right)\) |
| \(\chi_{338130}(12931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{272}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{9}{272}\right)\) | \(e\left(\frac{353}{816}\right)\) | \(e\left(\frac{317}{816}\right)\) | \(e\left(\frac{373}{816}\right)\) | \(e\left(\frac{13}{272}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{193}{204}\right)\) |
| \(\chi_{338130}(14251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{272}\right)\) | \(e\left(\frac{217}{816}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{43}{272}\right)\) | \(e\left(\frac{115}{816}\right)\) | \(e\left(\frac{487}{816}\right)\) | \(e\left(\frac{815}{816}\right)\) | \(e\left(\frac{183}{272}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{23}{204}\right)\) |
| \(\chi_{338130}(15271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{617}{816}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{27}{272}\right)\) | \(e\left(\frac{515}{816}\right)\) | \(e\left(\frac{407}{816}\right)\) | \(e\left(\frac{31}{816}\right)\) | \(e\left(\frac{39}{272}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{103}{204}\right)\) |
| \(\chi_{338130}(18931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{272}\right)\) | \(e\left(\frac{559}{816}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{13}{272}\right)\) | \(e\left(\frac{661}{816}\right)\) | \(e\left(\frac{337}{816}\right)\) | \(e\left(\frac{569}{816}\right)\) | \(e\left(\frac{49}{272}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{173}{204}\right)\) |
| \(\chi_{338130}(19951,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{272}\right)\) | \(e\left(\frac{623}{816}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{725}{816}\right)\) | \(e\left(\frac{161}{816}\right)\) | \(e\left(\frac{313}{816}\right)\) | \(e\left(\frac{113}{272}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{145}{204}\right)\) |
| \(\chi_{338130}(20101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{272}\right)\) | \(e\left(\frac{295}{816}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{165}{272}\right)\) | \(e\left(\frac{397}{816}\right)\) | \(e\left(\frac{553}{816}\right)\) | \(e\left(\frac{401}{816}\right)\) | \(e\left(\frac{57}{272}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{161}{204}\right)\) |
| \(\chi_{338130}(21121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{272}\right)\) | \(e\left(\frac{503}{816}\right)\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{37}{272}\right)\) | \(e\left(\frac{605}{816}\right)\) | \(e\left(\frac{185}{816}\right)\) | \(e\left(\frac{385}{816}\right)\) | \(e\left(\frac{265}{272}\right)\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{121}{204}\right)\) |
| \(\chi_{338130}(24781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{272}\right)\) | \(e\left(\frac{97}{816}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{211}{272}\right)\) | \(e\left(\frac{811}{816}\right)\) | \(e\left(\frac{511}{816}\right)\) | \(e\left(\frac{71}{816}\right)\) | \(e\left(\frac{63}{272}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{203}{204}\right)\) |
| \(\chi_{338130}(25801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{272}\right)\) | \(e\left(\frac{785}{816}\right)\) | \(e\left(\frac{97}{408}\right)\) | \(e\left(\frac{227}{272}\right)\) | \(e\left(\frac{683}{816}\right)\) | \(e\left(\frac{47}{816}\right)\) | \(e\left(\frac{583}{816}\right)\) | \(e\left(\frac{207}{272}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{55}{204}\right)\) |
| \(\chi_{338130}(27121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{272}\right)\) | \(e\left(\frac{571}{816}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{241}{272}\right)\) | \(e\left(\frac{265}{816}\right)\) | \(e\left(\frac{661}{816}\right)\) | \(e\left(\frac{317}{816}\right)\) | \(e\left(\frac{197}{272}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{53}{204}\right)\) |
| \(\chi_{338130}(28141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{272}\right)\) | \(e\left(\frac{587}{816}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{5}{816}\right)\) | \(e\left(\frac{253}{816}\right)\) | \(e\left(\frac{213}{272}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{97}{204}\right)\) |
| \(\chi_{338130}(28291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{239}{272}\right)\) | \(e\left(\frac{247}{816}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{83}{816}\right)\) | \(e\left(\frac{43}{272}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{131}{204}\right)\) |
| \(\chi_{338130}(29311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{272}\right)\) | \(e\left(\frac{245}{816}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{31}{272}\right)\) | \(e\left(\frac{551}{816}\right)\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{151}{204}\right)\) |
| \(\chi_{338130}(30631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{272}\right)\) | \(e\left(\frac{205}{816}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{511}{816}\right)\) | \(e\left(\frac{163}{816}\right)\) | \(e\left(\frac{251}{816}\right)\) | \(e\left(\frac{35}{272}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{143}{204}\right)\) |
| \(\chi_{338130}(31651,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{272}\right)\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{301}{408}\right)\) | \(e\left(\frac{23}{272}\right)\) | \(e\left(\frac{479}{816}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{3}{272}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{55}{204}\right)\) |
| \(\chi_{338130}(31801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{233}{272}\right)\) | \(e\left(\frac{193}{816}\right)\) | \(e\left(\frac{349}{816}\right)\) | \(e\left(\frac{197}{816}\right)\) | \(e\left(\frac{125}{272}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{161}{204}\right)\) |
| \(\chi_{338130}(32821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{272}\right)\) | \(e\left(\frac{755}{816}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{449}{816}\right)\) | \(e\left(\frac{461}{816}\right)\) | \(e\left(\frac{805}{816}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{49}{204}\right)\) |
| \(\chi_{338130}(34141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{265}{816}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{139}{272}\right)\) | \(e\left(\frac{163}{816}\right)\) | \(e\left(\frac{151}{816}\right)\) | \(e\left(\frac{623}{816}\right)\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{155}{204}\right)\) |
| \(\chi_{338130}(35161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{272}\right)\) | \(e\left(\frac{665}{816}\right)\) | \(e\left(\frac{25}{408}\right)\) | \(e\left(\frac{123}{272}\right)\) | \(e\left(\frac{563}{816}\right)\) | \(e\left(\frac{71}{816}\right)\) | \(e\left(\frac{655}{816}\right)\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{31}{204}\right)\) |
| \(\chi_{338130}(38821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{175}{816}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{277}{816}\right)\) | \(e\left(\frac{577}{816}\right)\) | \(e\left(\frac{473}{816}\right)\) | \(e\left(\frac{209}{272}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{137}{204}\right)\) |
| \(\chi_{338130}(39841,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{272}\right)\) | \(e\left(\frac{239}{816}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{189}{272}\right)\) | \(e\left(\frac{341}{816}\right)\) | \(e\left(\frac{401}{816}\right)\) | \(e\left(\frac{217}{816}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{109}{204}\right)\) |