from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3963, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([330,551]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,3963))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3963\) | |
| Conductor: | \(3963\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(660\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{660})$ |
| Fixed field: | Number field defined by a degree 660 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3963}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{551}{660}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{133}{660}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{119}{660}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{529}{660}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{563}{660}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{71}{660}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{361}{660}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{553}{660}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{43}{660}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{647}{660}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{169}{660}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(248,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{67}{660}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3963}(284,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{257}{660}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(290,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{131}{660}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(296,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{641}{660}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{421}{660}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(320,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{617}{660}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(344,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{313}{660}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3963}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{41}{660}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(410,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{541}{660}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{413}{660}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{289}{660}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(494,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{631}{660}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3963}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{569}{660}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3963}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{559}{660}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3963}(620,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{497}{660}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3963}(656,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{353}{660}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{281}{660}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(662,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{611}{660}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3963}(665,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{23}{660}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3963}(701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{167}{660}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{14}{15}\right)\) |