from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([0,204,476,387]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,338130))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(338130\) | |
| Conductor: | \(18785\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(816\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 18785.jz | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{816}\right)\) | \(e\left(\frac{809}{816}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{281}{816}\right)\) | \(e\left(\frac{95}{816}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{419}{816}\right)\) | \(e\left(\frac{557}{816}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{29}{68}\right)\) |
| \(\chi_{338130}(1567,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{816}\right)\) | \(e\left(\frac{313}{816}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{25}{816}\right)\) | \(e\left(\frac{415}{816}\right)\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{157}{816}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{5}{68}\right)\) |
| \(\chi_{338130}(3763,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{816}\right)\) | \(e\left(\frac{55}{816}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{7}{816}\right)\) | \(e\left(\frac{769}{816}\right)\) | \(e\left(\frac{19}{272}\right)\) | \(e\left(\frac{205}{816}\right)\) | \(e\left(\frac{403}{816}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{15}{68}\right)\) |
| \(\chi_{338130}(3907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{816}\right)\) | \(e\left(\frac{241}{816}\right)\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{1}{816}\right)\) | \(e\left(\frac{343}{816}\right)\) | \(e\left(\frac{197}{272}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{41}{68}\right)\) |
| \(\chi_{338130}(5293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{559}{816}\right)\) | \(e\left(\frac{215}{816}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{695}{816}\right)\) | \(e\left(\frac{113}{816}\right)\) | \(e\left(\frac{99}{272}\right)\) | \(e\left(\frac{653}{816}\right)\) | \(e\left(\frac{611}{816}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{3}{68}\right)\) |
| \(\chi_{338130}(5887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{816}\right)\) | \(e\left(\frac{269}{816}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{509}{816}\right)\) | \(e\left(\frac{779}{816}\right)\) | \(e\left(\frac{177}{272}\right)\) | \(e\left(\frac{335}{816}\right)\) | \(e\left(\frac{161}{816}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{61}{68}\right)\) |
| \(\chi_{338130}(7417,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{733}{816}\right)\) | \(e\left(\frac{331}{408}\right)\) | \(e\left(\frac{301}{816}\right)\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{655}{816}\right)\) | \(e\left(\frac{193}{816}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{33}{68}\right)\) |
| \(\chi_{338130}(8443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{599}{816}\right)\) | \(e\left(\frac{607}{816}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{463}{816}\right)\) | \(e\left(\frac{505}{816}\right)\) | \(e\left(\frac{91}{272}\right)\) | \(e\left(\frac{37}{816}\right)\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{11}{68}\right)\) |
| \(\chi_{338130}(9973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{727}{816}\right)\) | \(e\left(\frac{719}{816}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{47}{816}\right)\) | \(e\left(\frac{617}{816}\right)\) | \(e\left(\frac{11}{272}\right)\) | \(e\left(\frac{677}{816}\right)\) | \(e\left(\frac{491}{816}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{338130}(15463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{515}{816}\right)\) | \(e\left(\frac{763}{816}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{253}{816}\right)\) | \(e\left(\frac{135}{272}\right)\) | \(e\left(\frac{25}{816}\right)\) | \(e\left(\frac{487}{816}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{35}{68}\right)\) |
| \(\chi_{338130}(16417,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{816}\right)\) | \(e\left(\frac{725}{816}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{389}{816}\right)\) | \(e\left(\frac{419}{816}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{551}{816}\right)\) | \(e\left(\frac{713}{816}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{37}{68}\right)\) |
| \(\chi_{338130}(16633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{816}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{19}{816}\right)\) | \(e\left(\frac{805}{816}\right)\) | \(e\left(\frac{207}{272}\right)\) | \(e\left(\frac{673}{816}\right)\) | \(e\left(\frac{511}{816}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{31}{68}\right)\) |
| \(\chi_{338130}(16993,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{547}{816}\right)\) | \(e\left(\frac{587}{816}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{683}{816}\right)\) | \(e\left(\frac{77}{816}\right)\) | \(e\left(\frac{183}{272}\right)\) | \(e\left(\frac{185}{816}\right)\) | \(e\left(\frac{503}{816}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{55}{68}\right)\) |
| \(\chi_{338130}(17947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{816}\right)\) | \(e\left(\frac{805}{816}\right)\) | \(e\left(\frac{211}{408}\right)\) | \(e\left(\frac{325}{816}\right)\) | \(e\left(\frac{499}{816}\right)\) | \(e\left(\frac{105}{272}\right)\) | \(e\left(\frac{775}{816}\right)\) | \(e\left(\frac{409}{816}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{65}{68}\right)\) |
| \(\chi_{338130}(18163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{667}{816}\right)\) | \(e\left(\frac{131}{816}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{803}{816}\right)\) | \(e\left(\frac{437}{816}\right)\) | \(e\left(\frac{159}{272}\right)\) | \(e\left(\frac{785}{816}\right)\) | \(e\left(\frac{767}{816}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{11}{68}\right)\) |
| \(\chi_{338130}(19927,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{625}{816}\right)\) | \(e\left(\frac{617}{816}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{761}{816}\right)\) | \(e\left(\frac{719}{816}\right)\) | \(e\left(\frac{45}{272}\right)\) | \(e\left(\frac{371}{816}\right)\) | \(e\left(\frac{797}{816}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{57}{68}\right)\) |
| \(\chi_{338130}(21457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{641}{816}\right)\) | \(e\left(\frac{121}{816}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{505}{816}\right)\) | \(e\left(\frac{223}{816}\right)\) | \(e\left(\frac{205}{272}\right)\) | \(e\left(\frac{451}{816}\right)\) | \(e\left(\frac{397}{816}\right)\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{33}{68}\right)\) |
| \(\chi_{338130}(22267,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{816}\right)\) | \(e\left(\frac{209}{816}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{353}{816}\right)\) | \(e\left(\frac{311}{816}\right)\) | \(e\left(\frac{181}{272}\right)\) | \(e\left(\frac{779}{816}\right)\) | \(e\left(\frac{389}{816}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{57}{68}\right)\) |
| \(\chi_{338130}(23653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{383}{816}\right)\) | \(e\left(\frac{775}{816}\right)\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{247}{816}\right)\) | \(e\left(\frac{673}{816}\right)\) | \(e\left(\frac{243}{272}\right)\) | \(e\left(\frac{589}{816}\right)\) | \(e\left(\frac{115}{816}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{63}{68}\right)\) |
| \(\chi_{338130}(23797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{473}{816}\right)\) | \(e\left(\frac{433}{816}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{337}{816}\right)\) | \(e\left(\frac{535}{816}\right)\) | \(e\left(\frac{21}{272}\right)\) | \(e\left(\frac{427}{816}\right)\) | \(e\left(\frac{517}{816}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{13}{68}\right)\) |
| \(\chi_{338130}(25777,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{661}{816}\right)\) | \(e\left(\frac{317}{816}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{797}{816}\right)\) | \(e\left(\frac{11}{816}\right)\) | \(e\left(\frac{65}{272}\right)\) | \(e\left(\frac{143}{816}\right)\) | \(e\left(\frac{305}{816}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{37}{68}\right)\) |
| \(\chi_{338130}(27307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{725}{816}\right)\) | \(e\left(\frac{781}{816}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{589}{816}\right)\) | \(e\left(\frac{475}{816}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{463}{816}\right)\) | \(e\left(\frac{337}{816}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{9}{68}\right)\) |
| \(\chi_{338130}(28333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{816}\right)\) | \(e\left(\frac{703}{816}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{223}{816}\right)\) | \(e\left(\frac{601}{816}\right)\) | \(e\left(\frac{139}{272}\right)\) | \(e\left(\frac{469}{816}\right)\) | \(e\left(\frac{715}{816}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{31}{68}\right)\) |
| \(\chi_{338130}(29863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{487}{816}\right)\) | \(e\left(\frac{815}{816}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{623}{816}\right)\) | \(e\left(\frac{713}{816}\right)\) | \(e\left(\frac{59}{272}\right)\) | \(e\left(\frac{293}{816}\right)\) | \(e\left(\frac{779}{816}\right)\) | \(e\left(\frac{239}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) |
| \(\chi_{338130}(35353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{379}{816}\right)\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{523}{816}\right)\) | \(e\left(\frac{685}{816}\right)\) | \(e\left(\frac{215}{272}\right)\) | \(e\left(\frac{745}{816}\right)\) | \(e\left(\frac{151}{816}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{338130}(36307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{781}{816}\right)\) | \(e\left(\frac{677}{816}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{101}{816}\right)\) | \(e\left(\frac{371}{816}\right)\) | \(e\left(\frac{41}{272}\right)\) | \(e\left(\frac{743}{816}\right)\) | \(e\left(\frac{569}{816}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{61}{68}\right)\) |
| \(\chi_{338130}(36523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{816}\right)\) | \(e\left(\frac{67}{816}\right)\) | \(e\left(\frac{13}{408}\right)\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{373}{816}\right)\) | \(e\left(\frac{127}{272}\right)\) | \(e\left(\frac{769}{816}\right)\) | \(e\left(\frac{31}{816}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) |
| \(\chi_{338130}(36883,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{691}{816}\right)\) | \(e\left(\frac{203}{816}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{11}{816}\right)\) | \(e\left(\frac{509}{816}\right)\) | \(e\left(\frac{263}{272}\right)\) | \(e\left(\frac{89}{816}\right)\) | \(e\left(\frac{167}{816}\right)\) | \(e\left(\frac{35}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) |
| \(\chi_{338130}(37837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{816}\right)\) | \(e\left(\frac{757}{816}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{37}{816}\right)\) | \(e\left(\frac{451}{816}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{151}{816}\right)\) | \(e\left(\frac{265}{816}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{21}{68}\right)\) |
| \(\chi_{338130}(38053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{523}{816}\right)\) | \(e\left(\frac{515}{816}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{659}{816}\right)\) | \(e\left(\frac{5}{816}\right)\) | \(e\left(\frac{79}{272}\right)\) | \(e\left(\frac{65}{816}\right)\) | \(e\left(\frac{287}{816}\right)\) | \(e\left(\frac{131}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{338130}(41347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{305}{816}\right)\) | \(e\left(\frac{745}{816}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{169}{816}\right)\) | \(e\left(\frac{31}{816}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{403}{816}\right)\) | \(e\left(\frac{637}{816}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{61}{68}\right)\) |