from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([55,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1089))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1089\) | |
| Conductor: | \(1089\) | 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: | even | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1089}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) |
| \(\chi_{1089}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) |
| \(\chi_{1089}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) |
| \(\chi_{1089}(50,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) |
| \(\chi_{1089}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) |
| \(\chi_{1089}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) |
| \(\chi_{1089}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{1089}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) |
| \(\chi_{1089}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{1089}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) |
| \(\chi_{1089}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) |
| \(\chi_{1089}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) |
| \(\chi_{1089}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{7}{55}\right)\) |
| \(\chi_{1089}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{21}{55}\right)\) |
| \(\chi_{1089}(182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{1089}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) |
| \(\chi_{1089}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{1089}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{1089}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{1089}(266,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{2}{55}\right)\) |
| \(\chi_{1089}(272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{1089}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{38}{55}\right)\) |
| \(\chi_{1089}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) |
| \(\chi_{1089}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{1089}(326,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{1089}(338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{1089}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{1089}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) |
| \(\chi_{1089}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) |
| \(\chi_{1089}(380,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) |
| \(\chi_{1089}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) |