from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243675, base_ring=CyclotomicField(684))
M = H._module
chi = DirichletCharacter(H, M([152,513,104]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,243675))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(243675\) | |
| Conductor: | \(48735\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(684\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 48735.ph | 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: | $\Q(\zeta_{684})$ |
| Fixed field: | Number field defined by a degree 684 polynomial (not computed) |
First 31 of 216 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{243675}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{684}\right)\) | \(e\left(\frac{85}{342}\right)\) | \(e\left(\frac{77}{684}\right)\) | \(e\left(\frac{85}{228}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{35}{684}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{305}{684}\right)\) | \(e\left(\frac{119}{228}\right)\) |
| \(\chi_{243675}(682,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{684}\right)\) | \(e\left(\frac{223}{342}\right)\) | \(e\left(\frac{371}{684}\right)\) | \(e\left(\frac{223}{228}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{293}{684}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{599}{684}\right)\) | \(e\left(\frac{221}{228}\right)\) |
| \(\chi_{243675}(1582,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{684}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{167}{684}\right)\) | \(e\left(\frac{211}{228}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{449}{684}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{395}{684}\right)\) | \(e\left(\frac{113}{228}\right)\) |
| \(\chi_{243675}(6343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{684}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{197}{684}\right)\) | \(e\left(\frac{25}{228}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{587}{684}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{425}{684}\right)\) | \(e\left(\frac{35}{228}\right)\) |
| \(\chi_{243675}(7882,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{684}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{503}{684}\right)\) | \(e\left(\frac{43}{228}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{353}{684}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{47}{684}\right)\) | \(e\left(\frac{197}{228}\right)\) |
| \(\chi_{243675}(8293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{684}\right)\) | \(e\left(\frac{29}{342}\right)\) | \(e\left(\frac{493}{684}\right)\) | \(e\left(\frac{29}{228}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{535}{684}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{265}{684}\right)\) | \(e\left(\frac{223}{228}\right)\) |
| \(\chi_{243675}(8518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{684}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{289}{684}\right)\) | \(e\left(\frac{17}{228}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{7}{684}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{61}{684}\right)\) | \(e\left(\frac{115}{228}\right)\) |
| \(\chi_{243675}(9832,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{684}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{619}{684}\right)\) | \(e\left(\frac{23}{228}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{157}{684}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{391}{684}\right)\) | \(e\left(\frac{169}{228}\right)\) |
| \(\chi_{243675}(10057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{684}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{307}{684}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{637}{684}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{79}{684}\right)\) | \(e\left(\frac{205}{228}\right)\) |
| \(\chi_{243675}(10093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{684}\right)\) | \(e\left(\frac{77}{342}\right)\) | \(e\left(\frac{625}{684}\right)\) | \(e\left(\frac{77}{228}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{595}{684}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{397}{684}\right)\) | \(e\left(\frac{199}{228}\right)\) |
| \(\chi_{243675}(11632,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{491}{684}\right)\) | \(e\left(\frac{149}{342}\right)\) | \(e\left(\frac{139}{684}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{1}{684}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{595}{684}\right)\) | \(e\left(\frac{49}{228}\right)\) |
| \(\chi_{243675}(11968,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{421}{684}\right)\) | \(e\left(\frac{79}{342}\right)\) | \(e\left(\frac{317}{684}\right)\) | \(e\left(\frac{193}{228}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{455}{684}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{545}{684}\right)\) | \(e\left(\frac{179}{228}\right)\) |
| \(\chi_{243675}(12868,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{684}\right)\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{257}{684}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{179}{684}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{485}{684}\right)\) | \(e\left(\frac{107}{228}\right)\) |
| \(\chi_{243675}(13507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{619}{684}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{263}{684}\right)\) | \(e\left(\frac{163}{228}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{617}{684}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{491}{684}\right)\) | \(e\left(\frac{137}{228}\right)\) |
| \(\chi_{243675}(14407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{463}{684}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{347}{684}\right)\) | \(e\left(\frac{7}{228}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{593}{684}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{575}{684}\right)\) | \(e\left(\frac{101}{228}\right)\) |
| \(\chi_{243675}(19168,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{684}\right)\) | \(e\left(\frac{289}{342}\right)\) | \(e\left(\frac{125}{684}\right)\) | \(e\left(\frac{61}{228}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{119}{684}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{353}{684}\right)\) | \(e\left(\frac{131}{228}\right)\) |
| \(\chi_{243675}(20707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{684}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{431}{684}\right)\) | \(e\left(\frac{79}{228}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{569}{684}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{659}{684}\right)\) | \(e\left(\frac{65}{228}\right)\) |
| \(\chi_{243675}(21118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{684}\right)\) | \(e\left(\frac{245}{342}\right)\) | \(e\left(\frac{61}{684}\right)\) | \(e\left(\frac{17}{228}\right)\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{463}{684}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{517}{684}\right)\) | \(e\left(\frac{115}{228}\right)\) |
| \(\chi_{243675}(21343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{684}\right)\) | \(e\left(\frac{341}{342}\right)\) | \(e\left(\frac{325}{684}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{583}{684}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{97}{684}\right)\) | \(e\left(\frac{67}{228}\right)\) |
| \(\chi_{243675}(22657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{684}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{187}{684}\right)\) | \(e\left(\frac{11}{228}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{85}{684}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{643}{684}\right)\) | \(e\left(\frac{61}{228}\right)\) |
| \(\chi_{243675}(22882,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{684}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{343}{684}\right)\) | \(e\left(\frac{47}{228}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{529}{684}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{115}{684}\right)\) | \(e\left(\frac{157}{228}\right)\) |
| \(\chi_{243675}(22918,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{684}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{337}{684}\right)\) | \(e\left(\frac{221}{228}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{91}{684}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{109}{684}\right)\) | \(e\left(\frac{127}{228}\right)\) |
| \(\chi_{243675}(24457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{635}{684}\right)\) | \(e\left(\frac{293}{342}\right)\) | \(e\left(\frac{535}{684}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{181}{684}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{307}{684}\right)\) | \(e\left(\frac{205}{228}\right)\) |
| \(\chi_{243675}(26332,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{684}\right)\) | \(e\left(\frac{331}{342}\right)\) | \(e\left(\frac{155}{684}\right)\) | \(e\left(\frac{103}{228}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{257}{684}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{383}{684}\right)\) | \(e\left(\frac{53}{228}\right)\) |
| \(\chi_{243675}(27232,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{684}\right)\) | \(e\left(\frac{31}{342}\right)\) | \(e\left(\frac{527}{684}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{53}{684}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{71}{684}\right)\) | \(e\left(\frac{89}{228}\right)\) |
| \(\chi_{243675}(31993,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{325}{684}\right)\) | \(e\left(\frac{325}{342}\right)\) | \(e\left(\frac{53}{684}\right)\) | \(e\left(\frac{97}{228}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{335}{684}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{281}{684}\right)\) | \(e\left(\frac{227}{228}\right)\) |
| \(\chi_{243675}(33532,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{343}{684}\right)\) | \(e\left(\frac{1}{342}\right)\) | \(e\left(\frac{359}{684}\right)\) | \(e\left(\frac{115}{228}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{101}{684}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{587}{684}\right)\) | \(e\left(\frac{161}{228}\right)\) |
| \(\chi_{243675}(33943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{461}{684}\right)\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{313}{684}\right)\) | \(e\left(\frac{5}{228}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{391}{684}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{85}{684}\right)\) | \(e\left(\frac{7}{228}\right)\) |
| \(\chi_{243675}(35482,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{684}\right)\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{439}{684}\right)\) | \(e\left(\frac{227}{228}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{13}{684}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{211}{684}\right)\) | \(e\left(\frac{181}{228}\right)\) |
| \(\chi_{243675}(35707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{684}\right)\) | \(e\left(\frac{143}{342}\right)\) | \(e\left(\frac{379}{684}\right)\) | \(e\left(\frac{143}{228}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{421}{684}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{151}{684}\right)\) | \(e\left(\frac{109}{228}\right)\) |
| \(\chi_{243675}(35743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{365}{684}\right)\) | \(e\left(\frac{23}{342}\right)\) | \(e\left(\frac{49}{684}\right)\) | \(e\left(\frac{137}{228}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{271}{684}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{505}{684}\right)\) | \(e\left(\frac{55}{228}\right)\) |