from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1321, base_ring=CyclotomicField(1320))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1321))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1321\) | |
| Conductor: | \(1321\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1320\) | 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: | $\Q(\zeta_{1320})$ |
| Fixed field: | Number field defined by a degree 1320 polynomial (not computed) |
First 31 of 320 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1321}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{551}{660}\right)\) | \(e\left(\frac{193}{660}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{1}{165}\right)\) |
| \(\chi_{1321}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{469}{660}\right)\) | \(e\left(\frac{647}{660}\right)\) | \(e\left(\frac{43}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{29}{165}\right)\) |
| \(\chi_{1321}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{599}{660}\right)\) | \(e\left(\frac{217}{660}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{49}{165}\right)\) |
| \(\chi_{1321}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{71}{660}\right)\) | \(e\left(\frac{613}{660}\right)\) | \(e\left(\frac{257}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{16}{165}\right)\) |
| \(\chi_{1321}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{499}{660}\right)\) | \(e\left(\frac{497}{660}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{59}{165}\right)\) |
| \(\chi_{1321}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{457}{660}\right)\) | \(e\left(\frac{311}{660}\right)\) | \(e\left(\frac{139}{264}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{17}{165}\right)\) |
| \(\chi_{1321}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{491}{660}\right)\) | \(e\left(\frac{493}{660}\right)\) | \(e\left(\frac{197}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{106}{165}\right)\) |
| \(\chi_{1321}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{163}{660}\right)\) | \(e\left(\frac{329}{660}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{53}{165}\right)\) |
| \(\chi_{1321}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{89}{660}\right)\) | \(e\left(\frac{127}{660}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{34}{165}\right)\) |
| \(\chi_{1321}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{521}{660}\right)\) | \(e\left(\frac{343}{660}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{136}{165}\right)\) |
| \(\chi_{1321}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{613}{660}\right)\) | \(e\left(\frac{59}{660}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{8}{165}\right)\) |
| \(\chi_{1321}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{83}{660}\right)\) | \(e\left(\frac{289}{660}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{28}{165}\right)\) |
| \(\chi_{1321}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{553}{660}\right)\) | \(e\left(\frac{359}{660}\right)\) | \(e\left(\frac{31}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{113}{165}\right)\) |
| \(\chi_{1321}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{479}{660}\right)\) | \(e\left(\frac{157}{660}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{94}{165}\right)\) |
| \(\chi_{1321}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{367}{660}\right)\) | \(e\left(\frac{101}{660}\right)\) | \(e\left(\frac{133}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{92}{165}\right)\) |
| \(\chi_{1321}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{643}{660}\right)\) | \(e\left(\frac{569}{660}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{38}{165}\right)\) |
| \(\chi_{1321}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{607}{660}\right)\) | \(e\left(\frac{221}{660}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{2}{165}\right)\) |
| \(\chi_{1321}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{331}{660}\right)\) | \(e\left(\frac{413}{660}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{56}{165}\right)\) |
| \(\chi_{1321}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{299}{660}\right)\) | \(e\left(\frac{397}{660}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{79}{165}\right)\) |
| \(\chi_{1321}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{419}{660}\right)\) | \(e\left(\frac{457}{660}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{34}{165}\right)\) |
| \(\chi_{1321}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{617}{660}\right)\) | \(e\left(\frac{391}{660}\right)\) | \(e\left(\frac{47}{264}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{67}{165}\right)\) |
| \(\chi_{1321}(105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{601}{660}\right)\) | \(e\left(\frac{383}{660}\right)\) | \(e\left(\frac{43}{264}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{161}{165}\right)\) |
| \(\chi_{1321}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{269}{660}\right)\) | \(e\left(\frac{547}{660}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{49}{165}\right)\) |
| \(\chi_{1321}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{73}{660}\right)\) | \(e\left(\frac{119}{660}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{128}{165}\right)\) |
| \(\chi_{1321}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{179}{660}\right)\) | \(e\left(\frac{337}{660}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{124}{165}\right)\) |
| \(\chi_{1321}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{283}{660}\right)\) | \(e\left(\frac{389}{660}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{8}{165}\right)\) |
| \(\chi_{1321}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{151}{660}\right)\) | \(e\left(\frac{653}{660}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{41}{165}\right)\) |
| \(\chi_{1321}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{191}{660}\right)\) | \(e\left(\frac{13}{660}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{136}{165}\right)\) |
| \(\chi_{1321}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{307}{660}\right)\) | \(e\left(\frac{401}{660}\right)\) | \(e\left(\frac{217}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{32}{165}\right)\) |
| \(\chi_{1321}(146,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{167}{660}\right)\) | \(e\left(\frac{1}{660}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{112}{165}\right)\) |
| \(\chi_{1321}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{203}{660}\right)\) | \(e\left(\frac{349}{660}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{148}{165}\right)\) |