from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3888, base_ring=CyclotomicField(324))
M = H._module
chi = DirichletCharacter(H, M([0,243,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,3888))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3888\) | |
| Conductor: | \(3888\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(324\) | 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: | $\Q(\zeta_{324})$ |
| Fixed field: | Number field defined by a degree 324 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3888}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{324}\right)\) | \(e\left(\frac{155}{162}\right)\) | \(e\left(\frac{235}{324}\right)\) | \(e\left(\frac{209}{324}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{25}{324}\right)\) | \(e\left(\frac{80}{81}\right)\) |
| \(\chi_{3888}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{324}\right)\) | \(e\left(\frac{103}{162}\right)\) | \(e\left(\frac{83}{324}\right)\) | \(e\left(\frac{49}{324}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{151}{162}\right)\) | \(e\left(\frac{257}{324}\right)\) | \(e\left(\frac{61}{81}\right)\) |
| \(\chi_{3888}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{324}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{25}{324}\right)\) | \(e\left(\frac{167}{324}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{65}{162}\right)\) | \(e\left(\frac{175}{324}\right)\) | \(e\left(\frac{74}{81}\right)\) |
| \(\chi_{3888}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{324}\right)\) | \(e\left(\frac{25}{162}\right)\) | \(e\left(\frac{17}{324}\right)\) | \(e\left(\frac{295}{324}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{109}{162}\right)\) | \(e\left(\frac{119}{324}\right)\) | \(e\left(\frac{73}{81}\right)\) |
| \(\chi_{3888}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{324}\right)\) | \(e\left(\frac{17}{162}\right)\) | \(e\left(\frac{31}{324}\right)\) | \(e\left(\frac{233}{324}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{217}{324}\right)\) | \(e\left(\frac{14}{81}\right)\) |
| \(\chi_{3888}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{324}\right)\) | \(e\left(\frac{55}{162}\right)\) | \(e\left(\frac{167}{324}\right)\) | \(e\left(\frac{1}{324}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{161}{162}\right)\) | \(e\left(\frac{13}{162}\right)\) | \(e\left(\frac{197}{324}\right)\) | \(e\left(\frac{31}{81}\right)\) |
| \(\chi_{3888}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{324}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{253}{324}\right)\) | \(e\left(\frac{83}{324}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{79}{162}\right)\) | \(e\left(\frac{107}{162}\right)\) | \(e\left(\frac{151}{324}\right)\) | \(e\left(\frac{62}{81}\right)\) |
| \(\chi_{3888}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{324}\right)\) | \(e\left(\frac{31}{162}\right)\) | \(e\left(\frac{209}{324}\right)\) | \(e\left(\frac{139}{324}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{23}{162}\right)\) | \(e\left(\frac{25}{162}\right)\) | \(e\left(\frac{167}{324}\right)\) | \(e\left(\frac{16}{81}\right)\) |
| \(\chi_{3888}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{324}\right)\) | \(e\left(\frac{149}{162}\right)\) | \(e\left(\frac{43}{324}\right)\) | \(e\left(\frac{41}{324}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{121}{162}\right)\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{301}{324}\right)\) | \(e\left(\frac{56}{81}\right)\) |
| \(\chi_{3888}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{324}\right)\) | \(e\left(\frac{115}{162}\right)\) | \(e\left(\frac{143}{324}\right)\) | \(e\left(\frac{61}{324}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{29}{324}\right)\) | \(e\left(\frac{28}{81}\right)\) |
| \(\chi_{3888}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{324}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{49}{324}\right)\) | \(e\left(\frac{107}{324}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{55}{162}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{19}{324}\right)\) | \(e\left(\frac{77}{81}\right)\) |
| \(\chi_{3888}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{324}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{293}{324}\right)\) | \(e\left(\frac{91}{324}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{71}{162}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{107}{324}\right)\) | \(e\left(\frac{67}{81}\right)\) |
| \(\chi_{3888}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{324}\right)\) | \(e\left(\frac{65}{162}\right)\) | \(e\left(\frac{271}{324}\right)\) | \(e\left(\frac{281}{324}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{43}{162}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{277}{324}\right)\) | \(e\left(\frac{44}{81}\right)\) |
| \(\chi_{3888}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{324}\right)\) | \(e\left(\frac{121}{162}\right)\) | \(e\left(\frac{11}{324}\right)\) | \(e\left(\frac{229}{324}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{61}{162}\right)\) | \(e\left(\frac{77}{324}\right)\) | \(e\left(\frac{52}{81}\right)\) |
| \(\chi_{3888}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{324}\right)\) | \(e\left(\frac{23}{162}\right)\) | \(e\left(\frac{61}{324}\right)\) | \(e\left(\frac{239}{324}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{85}{162}\right)\) | \(e\left(\frac{29}{162}\right)\) | \(e\left(\frac{103}{324}\right)\) | \(e\left(\frac{38}{81}\right)\) |
| \(\chi_{3888}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{324}\right)\) | \(e\left(\frac{43}{162}\right)\) | \(e\left(\frac{269}{324}\right)\) | \(e\left(\frac{151}{324}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{11}{162}\right)\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{263}{324}\right)\) | \(e\left(\frac{64}{81}\right)\) |
| \(\chi_{3888}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{324}\right)\) | \(e\left(\frac{89}{162}\right)\) | \(e\left(\frac{67}{324}\right)\) | \(e\left(\frac{305}{324}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{19}{162}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{145}{324}\right)\) | \(e\left(\frac{59}{81}\right)\) |
| \(\chi_{3888}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{324}\right)\) | \(e\left(\frac{73}{162}\right)\) | \(e\left(\frac{95}{324}\right)\) | \(e\left(\frac{181}{324}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{143}{162}\right)\) | \(e\left(\frac{85}{162}\right)\) | \(e\left(\frac{17}{324}\right)\) | \(e\left(\frac{22}{81}\right)\) |
| \(\chi_{3888}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{324}\right)\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{289}{324}\right)\) | \(e\left(\frac{155}{324}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{7}{162}\right)\) | \(e\left(\frac{71}{162}\right)\) | \(e\left(\frac{79}{324}\right)\) | \(e\left(\frac{26}{81}\right)\) |
| \(\chi_{3888}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{324}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{137}{324}\right)\) | \(e\left(\frac{319}{324}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{5}{162}\right)\) | \(e\left(\frac{97}{162}\right)\) | \(e\left(\frac{311}{324}\right)\) | \(e\left(\frac{7}{81}\right)\) |
| \(\chi_{3888}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{324}\right)\) | \(e\left(\frac{59}{162}\right)\) | \(e\left(\frac{79}{324}\right)\) | \(e\left(\frac{113}{324}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{11}{162}\right)\) | \(e\left(\frac{229}{324}\right)\) | \(e\left(\frac{20}{81}\right)\) |
| \(\chi_{3888}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{324}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{71}{324}\right)\) | \(e\left(\frac{241}{324}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{83}{162}\right)\) | \(e\left(\frac{55}{162}\right)\) | \(e\left(\frac{173}{324}\right)\) | \(e\left(\frac{19}{81}\right)\) |
| \(\chi_{3888}(805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{324}\right)\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{85}{324}\right)\) | \(e\left(\frac{179}{324}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{145}{162}\right)\) | \(e\left(\frac{59}{162}\right)\) | \(e\left(\frac{271}{324}\right)\) | \(e\left(\frac{41}{81}\right)\) |
| \(\chi_{3888}(853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{324}\right)\) | \(e\left(\frac{1}{162}\right)\) | \(e\left(\frac{221}{324}\right)\) | \(e\left(\frac{271}{324}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{121}{162}\right)\) | \(e\left(\frac{251}{324}\right)\) | \(e\left(\frac{58}{81}\right)\) |
| \(\chi_{3888}(877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{324}\right)\) | \(e\left(\frac{137}{162}\right)\) | \(e\left(\frac{307}{324}\right)\) | \(e\left(\frac{29}{324}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{53}{162}\right)\) | \(e\left(\frac{205}{324}\right)\) | \(e\left(\frac{8}{81}\right)\) |
| \(\chi_{3888}(925,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{324}\right)\) | \(e\left(\frac{139}{162}\right)\) | \(e\left(\frac{263}{324}\right)\) | \(e\left(\frac{85}{324}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{221}{324}\right)\) | \(e\left(\frac{43}{81}\right)\) |
| \(\chi_{3888}(949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{324}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{97}{324}\right)\) | \(e\left(\frac{311}{324}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{13}{162}\right)\) | \(e\left(\frac{155}{162}\right)\) | \(e\left(\frac{31}{324}\right)\) | \(e\left(\frac{2}{81}\right)\) |
| \(\chi_{3888}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{324}\right)\) | \(e\left(\frac{61}{162}\right)\) | \(e\left(\frac{197}{324}\right)\) | \(e\left(\frac{7}{324}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{155}{162}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{83}{324}\right)\) | \(e\left(\frac{55}{81}\right)\) |
| \(\chi_{3888}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{324}\right)\) | \(e\left(\frac{161}{162}\right)\) | \(e\left(\frac{103}{324}\right)\) | \(e\left(\frac{53}{324}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{109}{162}\right)\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{73}{324}\right)\) | \(e\left(\frac{23}{81}\right)\) |
| \(\chi_{3888}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{319}{324}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{23}{324}\right)\) | \(e\left(\frac{37}{324}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{161}{324}\right)\) | \(e\left(\frac{13}{81}\right)\) |
| \(\chi_{3888}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{324}\right)\) | \(e\left(\frac{11}{162}\right)\) | \(e\left(\frac{1}{324}\right)\) | \(e\left(\frac{227}{324}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{97}{162}\right)\) | \(e\left(\frac{35}{162}\right)\) | \(e\left(\frac{7}{324}\right)\) | \(e\left(\frac{71}{81}\right)\) |