from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([33,220,60]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,4025))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4025\) | |
| Conductor: | \(4025\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4025}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{31}{165}\right)\) |
| \(\chi_{4025}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{17}{165}\right)\) |
| \(\chi_{4025}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{73}{165}\right)\) |
| \(\chi_{4025}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{79}{165}\right)\) |
| \(\chi_{4025}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{16}{165}\right)\) |
| \(\chi_{4025}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{89}{165}\right)\) |
| \(\chi_{4025}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{52}{165}\right)\) |
| \(\chi_{4025}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{158}{165}\right)\) |
| \(\chi_{4025}(354,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{61}{165}\right)\) |
| \(\chi_{4025}(394,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{14}{165}\right)\) |
| \(\chi_{4025}(464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{53}{165}\right)\) |
| \(\chi_{4025}(564,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{133}{165}\right)\) |
| \(\chi_{4025}(604,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{101}{165}\right)\) |
| \(\chi_{4025}(634,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{37}{165}\right)\) |
| \(\chi_{4025}(639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{38}{165}\right)\) |
| \(\chi_{4025}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{139}{165}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{109}{165}\right)\) |
| \(\chi_{4025}(739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{58}{165}\right)\) |
| \(\chi_{4025}(744,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{74}{165}\right)\) |
| \(\chi_{4025}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{97}{165}\right)\) |
| \(\chi_{4025}(814,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{83}{165}\right)\) |
| \(\chi_{4025}(844,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{139}{165}\right)\) |
| \(\chi_{4025}(984,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{82}{165}\right)\) |
| \(\chi_{4025}(1089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{118}{165}\right)\) |
| \(\chi_{4025}(1094,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{59}{165}\right)\) |
| \(\chi_{4025}(1129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{131}{165}\right)\) |
| \(\chi_{4025}(1159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{127}{165}\right)\) |
| \(\chi_{4025}(1269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{119}{165}\right)\) |
| \(\chi_{4025}(1304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{161}{165}\right)\) |
| \(\chi_{4025}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{34}{165}\right)\) |
| \(\chi_{4025}(1409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{2}{165}\right)\) |
| \(\chi_{4025}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{103}{165}\right)\) |