from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([0,88,69]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,3204))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3204\) | |
| Conductor: | \(801\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(264\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 801.bf | 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: | $\Q(\zeta_{264})$ |
| Fixed field: | Number field defined by a degree 264 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3204}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{133}{264}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{203}{264}\right)\) |
| \(\chi_{3204}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{47}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{97}{264}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{7}{264}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{245}{264}\right)\) | \(e\left(\frac{169}{264}\right)\) |
| \(\chi_{3204}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{181}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{59}{264}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{173}{264}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{247}{264}\right)\) | \(e\left(\frac{179}{264}\right)\) |
| \(\chi_{3204}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{31}{264}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{103}{264}\right)\) |
| \(\chi_{3204}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{131}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{53}{264}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{7}{264}\right)\) | \(e\left(\frac{35}{264}\right)\) |
| \(\chi_{3204}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{7}{264}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{83}{264}\right)\) | \(e\left(\frac{151}{264}\right)\) |
| \(\chi_{3204}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{227}{264}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{109}{264}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{163}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{13}{264}\right)\) |
| \(\chi_{3204}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{83}{264}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{221}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{79}{264}\right)\) | \(e\left(\frac{131}{264}\right)\) |
| \(\chi_{3204}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{79}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{97}{264}\right)\) |
| \(\chi_{3204}(385,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{257}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{59}{264}\right)\) | \(e\left(\frac{31}{264}\right)\) |
| \(\chi_{3204}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{71}{264}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{151}{264}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{264}\right)\) | \(e\left(\frac{25}{264}\right)\) |
| \(\chi_{3204}(493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{131}{264}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{85}{264}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{65}{264}\right)\) | \(e\left(\frac{61}{264}\right)\) |
| \(\chi_{3204}(553,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{215}{264}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{263}{264}\right)\) |
| \(\chi_{3204}(565,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{53}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{109}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{119}{264}\right)\) | \(e\left(\frac{67}{264}\right)\) |
| \(\chi_{3204}(637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{251}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{181}{264}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{43}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{133}{264}\right)\) |
| \(\chi_{3204}(661,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{263}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{139}{264}\right)\) | \(e\left(\frac{167}{264}\right)\) |
| \(\chi_{3204}(697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{47}{264}\right)\) |
| \(\chi_{3204}(709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{23}{264}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{127}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{221}{264}\right)\) | \(e\left(\frac{49}{264}\right)\) |
| \(\chi_{3204}(745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{103}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{173}{264}\right)\) | \(e\left(\frac{73}{264}\right)\) |
| \(\chi_{3204}(877,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{47}{264}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{67}{264}\right)\) | \(e\left(\frac{71}{264}\right)\) |
| \(\chi_{3204}(913,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{197}{264}\right)\) |
| \(\chi_{3204}(925,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{173}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{37}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(e\left(\frac{139}{264}\right)\) |
| \(\chi_{3204}(949,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{241}{264}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{137}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{43}{264}\right)\) | \(e\left(\frac{215}{264}\right)\) |
| \(\chi_{3204}(985,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(e\left(\frac{173}{264}\right)\) |
| \(\chi_{3204}(1033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{79}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{107}{264}\right)\) | \(e\left(\frac{7}{264}\right)\) |
| \(\chi_{3204}(1129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{223}{264}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{95}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{157}{264}\right)\) | \(e\left(\frac{257}{264}\right)\) |
| \(\chi_{3204}(1213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{107}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{235}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{205}{264}\right)\) |
| \(\chi_{3204}(1249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{157}{264}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{181}{264}\right)\) |
| \(\chi_{3204}(1273,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{119}{264}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{91}{264}\right)\) | \(e\left(\frac{191}{264}\right)\) |
| \(\chi_{3204}(1309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{95}{264}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{239}{264}\right)\) |
| \(\chi_{3204}(1321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{119}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{53}{264}\right)\) | \(e\left(\frac{1}{264}\right)\) |