from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([258,275]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,2299))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2299\) | |
| Conductor: | \(2299\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2299}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2299}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2299}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2299}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2299}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{2299}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{2299}(236,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2299}(240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2299}(278,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2299}(312,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{2299}(335,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2299}(350,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2299}(388,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{2299}(411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2299}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2299}(449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2299}(521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2299}(544,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{2299}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{2299}(597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{2299}(620,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2299}(654,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{2299}(658,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{2299}(696,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{2299}(730,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{2299}(768,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{2299}(806,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2299}(829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{2299}(863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{2299}(867,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{2299}(905,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) |