from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(432))
M = H._module
chi = DirichletCharacter(H, M([216,135,104]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,5184))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5184\) | |
| Conductor: | \(5184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(432\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{432})$ |
| Fixed field: | Number field defined by a degree 432 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5184}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{432}\right)\) | \(e\left(\frac{103}{216}\right)\) | \(e\left(\frac{83}{432}\right)\) | \(e\left(\frac{265}{432}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{113}{216}\right)\) | \(e\left(\frac{151}{216}\right)\) | \(e\left(\frac{149}{432}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{5184}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{432}\right)\) | \(e\left(\frac{59}{216}\right)\) | \(e\left(\frac{295}{432}\right)\) | \(e\left(\frac{5}{432}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{157}{216}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{337}{432}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{5184}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{432}\right)\) | \(e\left(\frac{37}{216}\right)\) | \(e\left(\frac{401}{432}\right)\) | \(e\left(\frac{307}{432}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{179}{216}\right)\) | \(e\left(\frac{157}{216}\right)\) | \(e\left(\frac{215}{432}\right)\) | \(e\left(\frac{10}{27}\right)\) |
| \(\chi_{5184}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{432}\right)\) | \(e\left(\frac{65}{216}\right)\) | \(e\left(\frac{325}{432}\right)\) | \(e\left(\frac{335}{432}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{151}{216}\right)\) | \(e\left(\frac{89}{216}\right)\) | \(e\left(\frac{115}{432}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{5184}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{379}{432}\right)\) | \(e\left(\frac{187}{216}\right)\) | \(e\left(\frac{287}{432}\right)\) | \(e\left(\frac{349}{432}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{29}{216}\right)\) | \(e\left(\frac{163}{216}\right)\) | \(e\left(\frac{281}{432}\right)\) | \(e\left(\frac{25}{27}\right)\) |
| \(\chi_{5184}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{432}\right)\) | \(e\left(\frac{71}{216}\right)\) | \(e\left(\frac{355}{432}\right)\) | \(e\left(\frac{233}{432}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{145}{216}\right)\) | \(e\left(\frac{167}{216}\right)\) | \(e\left(\frac{325}{432}\right)\) | \(e\left(\frac{17}{27}\right)\) |
| \(\chi_{5184}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{385}{432}\right)\) | \(e\left(\frac{121}{216}\right)\) | \(e\left(\frac{173}{432}\right)\) | \(e\left(\frac{391}{432}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{95}{216}\right)\) | \(e\left(\frac{169}{216}\right)\) | \(e\left(\frac{347}{432}\right)\) | \(e\left(\frac{13}{27}\right)\) |
| \(\chi_{5184}(275,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{432}\right)\) | \(e\left(\frac{77}{216}\right)\) | \(e\left(\frac{385}{432}\right)\) | \(e\left(\frac{131}{432}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{139}{216}\right)\) | \(e\left(\frac{29}{216}\right)\) | \(e\left(\frac{103}{432}\right)\) | \(e\left(\frac{23}{27}\right)\) |
| \(\chi_{5184}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{432}\right)\) | \(e\left(\frac{55}{216}\right)\) | \(e\left(\frac{59}{432}\right)\) | \(e\left(\frac{1}{432}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{161}{216}\right)\) | \(e\left(\frac{175}{216}\right)\) | \(e\left(\frac{413}{432}\right)\) | \(e\left(\frac{1}{27}\right)\) |
| \(\chi_{5184}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{432}\right)\) | \(e\left(\frac{83}{216}\right)\) | \(e\left(\frac{415}{432}\right)\) | \(e\left(\frac{29}{432}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{133}{216}\right)\) | \(e\left(\frac{107}{216}\right)\) | \(e\left(\frac{313}{432}\right)\) | \(e\left(\frac{2}{27}\right)\) |
| \(\chi_{5184}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{432}\right)\) | \(e\left(\frac{205}{216}\right)\) | \(e\left(\frac{377}{432}\right)\) | \(e\left(\frac{43}{432}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{11}{216}\right)\) | \(e\left(\frac{181}{216}\right)\) | \(e\left(\frac{47}{432}\right)\) | \(e\left(\frac{16}{27}\right)\) |
| \(\chi_{5184}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{432}\right)\) | \(e\left(\frac{89}{216}\right)\) | \(e\left(\frac{13}{432}\right)\) | \(e\left(\frac{359}{432}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{127}{216}\right)\) | \(e\left(\frac{185}{216}\right)\) | \(e\left(\frac{91}{432}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{5184}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{403}{432}\right)\) | \(e\left(\frac{139}{216}\right)\) | \(e\left(\frac{263}{432}\right)\) | \(e\left(\frac{85}{432}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{77}{216}\right)\) | \(e\left(\frac{187}{216}\right)\) | \(e\left(\frac{113}{432}\right)\) | \(e\left(\frac{4}{27}\right)\) |
| \(\chi_{5184}(491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{432}\right)\) | \(e\left(\frac{95}{216}\right)\) | \(e\left(\frac{43}{432}\right)\) | \(e\left(\frac{257}{432}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{121}{216}\right)\) | \(e\left(\frac{47}{216}\right)\) | \(e\left(\frac{301}{432}\right)\) | \(e\left(\frac{14}{27}\right)\) |
| \(\chi_{5184}(515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{409}{432}\right)\) | \(e\left(\frac{73}{216}\right)\) | \(e\left(\frac{149}{432}\right)\) | \(e\left(\frac{127}{432}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{193}{216}\right)\) | \(e\left(\frac{179}{432}\right)\) | \(e\left(\frac{19}{27}\right)\) |
| \(\chi_{5184}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{432}\right)\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{73}{432}\right)\) | \(e\left(\frac{155}{432}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{115}{216}\right)\) | \(e\left(\frac{125}{216}\right)\) | \(e\left(\frac{79}{432}\right)\) | \(e\left(\frac{20}{27}\right)\) |
| \(\chi_{5184}(587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{415}{432}\right)\) | \(e\left(\frac{7}{216}\right)\) | \(e\left(\frac{35}{432}\right)\) | \(e\left(\frac{169}{432}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{209}{216}\right)\) | \(e\left(\frac{199}{216}\right)\) | \(e\left(\frac{245}{432}\right)\) | \(e\left(\frac{7}{27}\right)\) |
| \(\chi_{5184}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{419}{432}\right)\) | \(e\left(\frac{107}{216}\right)\) | \(e\left(\frac{103}{432}\right)\) | \(e\left(\frac{53}{432}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{109}{216}\right)\) | \(e\left(\frac{203}{216}\right)\) | \(e\left(\frac{289}{432}\right)\) | \(e\left(\frac{26}{27}\right)\) |
| \(\chi_{5184}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{432}\right)\) | \(e\left(\frac{157}{216}\right)\) | \(e\left(\frac{353}{432}\right)\) | \(e\left(\frac{211}{432}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{59}{216}\right)\) | \(e\left(\frac{205}{216}\right)\) | \(e\left(\frac{311}{432}\right)\) | \(e\left(\frac{22}{27}\right)\) |
| \(\chi_{5184}(707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{432}\right)\) | \(e\left(\frac{113}{216}\right)\) | \(e\left(\frac{133}{432}\right)\) | \(e\left(\frac{383}{432}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{103}{216}\right)\) | \(e\left(\frac{65}{216}\right)\) | \(e\left(\frac{67}{432}\right)\) | \(e\left(\frac{5}{27}\right)\) |
| \(\chi_{5184}(731,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{427}{432}\right)\) | \(e\left(\frac{91}{216}\right)\) | \(e\left(\frac{239}{432}\right)\) | \(e\left(\frac{253}{432}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{125}{216}\right)\) | \(e\left(\frac{211}{216}\right)\) | \(e\left(\frac{377}{432}\right)\) | \(e\left(\frac{10}{27}\right)\) |
| \(\chi_{5184}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{432}\right)\) | \(e\left(\frac{119}{216}\right)\) | \(e\left(\frac{163}{432}\right)\) | \(e\left(\frac{281}{432}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{97}{216}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{277}{432}\right)\) | \(e\left(\frac{11}{27}\right)\) |
| \(\chi_{5184}(803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{432}\right)\) | \(e\left(\frac{25}{216}\right)\) | \(e\left(\frac{125}{432}\right)\) | \(e\left(\frac{295}{432}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{191}{216}\right)\) | \(e\left(\frac{1}{216}\right)\) | \(e\left(\frac{11}{432}\right)\) | \(e\left(\frac{25}{27}\right)\) |
| \(\chi_{5184}(851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{432}\right)\) | \(e\left(\frac{125}{216}\right)\) | \(e\left(\frac{193}{432}\right)\) | \(e\left(\frac{179}{432}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{91}{216}\right)\) | \(e\left(\frac{5}{216}\right)\) | \(e\left(\frac{55}{432}\right)\) | \(e\left(\frac{17}{27}\right)\) |
| \(\chi_{5184}(875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{432}\right)\) | \(e\left(\frac{175}{216}\right)\) | \(e\left(\frac{11}{432}\right)\) | \(e\left(\frac{337}{432}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{41}{216}\right)\) | \(e\left(\frac{7}{216}\right)\) | \(e\left(\frac{77}{432}\right)\) | \(e\left(\frac{13}{27}\right)\) |
| \(\chi_{5184}(923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{432}\right)\) | \(e\left(\frac{131}{216}\right)\) | \(e\left(\frac{223}{432}\right)\) | \(e\left(\frac{77}{432}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{85}{216}\right)\) | \(e\left(\frac{83}{216}\right)\) | \(e\left(\frac{265}{432}\right)\) | \(e\left(\frac{23}{27}\right)\) |
| \(\chi_{5184}(947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{432}\right)\) | \(e\left(\frac{109}{216}\right)\) | \(e\left(\frac{329}{432}\right)\) | \(e\left(\frac{379}{432}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{107}{216}\right)\) | \(e\left(\frac{13}{216}\right)\) | \(e\left(\frac{143}{432}\right)\) | \(e\left(\frac{1}{27}\right)\) |
| \(\chi_{5184}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{432}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{253}{432}\right)\) | \(e\left(\frac{407}{432}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{79}{216}\right)\) | \(e\left(\frac{161}{216}\right)\) | \(e\left(\frac{43}{432}\right)\) | \(e\left(\frac{2}{27}\right)\) |
| \(\chi_{5184}(1019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{432}\right)\) | \(e\left(\frac{43}{216}\right)\) | \(e\left(\frac{215}{432}\right)\) | \(e\left(\frac{421}{432}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{173}{216}\right)\) | \(e\left(\frac{19}{216}\right)\) | \(e\left(\frac{209}{432}\right)\) | \(e\left(\frac{16}{27}\right)\) |
| \(\chi_{5184}(1067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{432}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{283}{432}\right)\) | \(e\left(\frac{305}{432}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{73}{216}\right)\) | \(e\left(\frac{23}{216}\right)\) | \(e\left(\frac{253}{432}\right)\) | \(e\left(\frac{8}{27}\right)\) |
| \(\chi_{5184}(1091,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{432}\right)\) | \(e\left(\frac{193}{216}\right)\) | \(e\left(\frac{101}{432}\right)\) | \(e\left(\frac{31}{432}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{23}{216}\right)\) | \(e\left(\frac{25}{216}\right)\) | \(e\left(\frac{275}{432}\right)\) | \(e\left(\frac{4}{27}\right)\) |