from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81225, base_ring=CyclotomicField(570))
M = H._module
chi = DirichletCharacter(H, M([0,342,245]))
chi.galois_orbit()
[g,chi] = znchar(Mod(46,81225))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(81225\) | |
| Conductor: | \(9025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(570\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 9025.ci | 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: | $\Q(\zeta_{285})$ |
| Fixed field: | Number field defined by a degree 570 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{81225}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{570}\right)\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{190}\right)\) | \(e\left(\frac{42}{95}\right)\) | \(e\left(\frac{463}{570}\right)\) | \(e\left(\frac{287}{570}\right)\) | \(e\left(\frac{34}{285}\right)\) | \(e\left(\frac{268}{285}\right)\) | \(e\left(\frac{269}{570}\right)\) |
| \(\chi_{81225}(316,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{570}\right)\) | \(e\left(\frac{109}{285}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{521}{570}\right)\) | \(e\left(\frac{499}{570}\right)\) | \(e\left(\frac{218}{285}\right)\) | \(e\left(\frac{176}{285}\right)\) | \(e\left(\frac{283}{570}\right)\) |
| \(\chi_{81225}(1171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{517}{570}\right)\) | \(e\left(\frac{232}{285}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{137}{190}\right)\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{233}{570}\right)\) | \(e\left(\frac{547}{570}\right)\) | \(e\left(\frac{179}{285}\right)\) | \(e\left(\frac{53}{285}\right)\) | \(e\left(\frac{469}{570}\right)\) |
| \(\chi_{81225}(1756,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{443}{570}\right)\) | \(e\left(\frac{158}{285}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{63}{190}\right)\) | \(e\left(\frac{83}{95}\right)\) | \(e\left(\frac{397}{570}\right)\) | \(e\left(\frac{203}{570}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{127}{285}\right)\) | \(e\left(\frac{371}{570}\right)\) |
| \(\chi_{81225}(2611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{371}{570}\right)\) | \(e\left(\frac{86}{285}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{181}{190}\right)\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{79}{570}\right)\) | \(e\left(\frac{161}{570}\right)\) | \(e\left(\frac{172}{285}\right)\) | \(e\left(\frac{199}{285}\right)\) | \(e\left(\frac{137}{570}\right)\) |
| \(\chi_{81225}(2881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{570}\right)\) | \(e\left(\frac{193}{285}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{3}{190}\right)\) | \(e\left(\frac{13}{95}\right)\) | \(e\left(\frac{227}{570}\right)\) | \(e\left(\frac{73}{570}\right)\) | \(e\left(\frac{101}{285}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{271}{570}\right)\) |
| \(\chi_{81225}(3466,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{570}\right)\) | \(e\left(\frac{14}{285}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{331}{570}\right)\) | \(e\left(\frac{119}{570}\right)\) | \(e\left(\frac{28}{285}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{473}{570}\right)\) |
| \(\chi_{81225}(3736,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{570}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{509}{570}\right)\) | \(e\left(\frac{121}{570}\right)\) | \(e\left(\frac{62}{285}\right)\) | \(e\left(\frac{254}{285}\right)\) | \(e\left(\frac{457}{570}\right)\) |
| \(\chi_{81225}(4321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{570}\right)\) | \(e\left(\frac{227}{285}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{2}{95}\right)\) | \(e\left(\frac{13}{570}\right)\) | \(e\left(\frac{77}{570}\right)\) | \(e\left(\frac{169}{285}\right)\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{239}{570}\right)\) |
| \(\chi_{81225}(4591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{439}{570}\right)\) | \(e\left(\frac{154}{285}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{59}{190}\right)\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{221}{570}\right)\) | \(e\left(\frac{169}{570}\right)\) | \(e\left(\frac{23}{285}\right)\) | \(e\left(\frac{131}{285}\right)\) | \(e\left(\frac{73}{570}\right)\) |
| \(\chi_{81225}(5446,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{277}{570}\right)\) | \(e\left(\frac{277}{285}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{87}{190}\right)\) | \(e\left(\frac{92}{95}\right)\) | \(e\left(\frac{503}{570}\right)\) | \(e\left(\frac{217}{570}\right)\) | \(e\left(\frac{269}{285}\right)\) | \(e\left(\frac{8}{285}\right)\) | \(e\left(\frac{259}{570}\right)\) |
| \(\chi_{81225}(6031,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{570}\right)\) | \(e\left(\frac{83}{285}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{83}{190}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{517}{570}\right)\) | \(e\left(\frac{563}{570}\right)\) | \(e\left(\frac{166}{285}\right)\) | \(e\left(\frac{202}{285}\right)\) | \(e\left(\frac{341}{570}\right)\) |
| \(\chi_{81225}(6886,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{570}\right)\) | \(e\left(\frac{11}{285}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{190}\right)\) | \(e\left(\frac{16}{95}\right)\) | \(e\left(\frac{199}{570}\right)\) | \(e\left(\frac{521}{570}\right)\) | \(e\left(\frac{22}{285}\right)\) | \(e\left(\frac{274}{285}\right)\) | \(e\left(\frac{107}{570}\right)\) |
| \(\chi_{81225}(7156,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{523}{570}\right)\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{18}{95}\right)\) | \(e\left(\frac{497}{570}\right)\) | \(e\left(\frac{313}{570}\right)\) | \(e\left(\frac{191}{285}\right)\) | \(e\left(\frac{47}{285}\right)\) | \(e\left(\frac{61}{570}\right)\) |
| \(\chi_{81225}(7741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{509}{570}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{129}{190}\right)\) | \(e\left(\frac{84}{95}\right)\) | \(e\left(\frac{451}{570}\right)\) | \(e\left(\frac{479}{570}\right)\) | \(e\left(\frac{163}{285}\right)\) | \(e\left(\frac{61}{285}\right)\) | \(e\left(\frac{443}{570}\right)\) |
| \(\chi_{81225}(8866,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{570}\right)\) | \(e\left(\frac{199}{285}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{39}{95}\right)\) | \(e\left(\frac{491}{570}\right)\) | \(e\left(\frac{409}{570}\right)\) | \(e\left(\frac{113}{285}\right)\) | \(e\left(\frac{86}{285}\right)\) | \(e\left(\frac{433}{570}\right)\) |
| \(\chi_{81225}(9721,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{570}\right)\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{2}{95}\right)\) | \(e\left(\frac{203}{570}\right)\) | \(e\left(\frac{457}{570}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{248}{285}\right)\) | \(e\left(\frac{49}{570}\right)\) |
| \(\chi_{81225}(10306,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{570}\right)\) | \(e\left(\frac{8}{285}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{67}{570}\right)\) | \(e\left(\frac{353}{570}\right)\) | \(e\left(\frac{16}{285}\right)\) | \(e\left(\frac{277}{285}\right)\) | \(e\left(\frac{311}{570}\right)\) |
| \(\chi_{81225}(11161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{570}\right)\) | \(e\left(\frac{221}{285}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{319}{570}\right)\) | \(e\left(\frac{311}{570}\right)\) | \(e\left(\frac{157}{285}\right)\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{77}{570}\right)\) |
| \(\chi_{81225}(11431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{570}\right)\) | \(e\left(\frac{283}{285}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{93}{190}\right)\) | \(e\left(\frac{23}{95}\right)\) | \(e\left(\frac{197}{570}\right)\) | \(e\left(\frac{553}{570}\right)\) | \(e\left(\frac{281}{285}\right)\) | \(e\left(\frac{2}{285}\right)\) | \(e\left(\frac{421}{570}\right)\) |
| \(\chi_{81225}(12016,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{570}\right)\) | \(e\left(\frac{149}{285}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{1}{570}\right)\) | \(e\left(\frac{269}{570}\right)\) | \(e\left(\frac{13}{285}\right)\) | \(e\left(\frac{136}{285}\right)\) | \(e\left(\frac{413}{570}\right)\) |
| \(\chi_{81225}(12286,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{570}\right)\) | \(e\left(\frac{121}{285}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{81}{95}\right)\) | \(e\left(\frac{479}{570}\right)\) | \(e\left(\frac{31}{570}\right)\) | \(e\left(\frac{242}{285}\right)\) | \(e\left(\frac{164}{285}\right)\) | \(e\left(\frac{37}{570}\right)\) |
| \(\chi_{81225}(12871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{570}\right)\) | \(e\left(\frac{77}{285}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{77}{190}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{253}{570}\right)\) | \(e\left(\frac{227}{570}\right)\) | \(e\left(\frac{154}{285}\right)\) | \(e\left(\frac{208}{285}\right)\) | \(e\left(\frac{179}{570}\right)\) |
| \(\chi_{81225}(13141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{529}{570}\right)\) | \(e\left(\frac{244}{285}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{191}{570}\right)\) | \(e\left(\frac{79}{570}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{223}{570}\right)\) |
| \(\chi_{81225}(13996,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{570}\right)\) | \(e\left(\frac{82}{285}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{177}{190}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{473}{570}\right)\) | \(e\left(\frac{127}{570}\right)\) | \(e\left(\frac{164}{285}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{409}{570}\right)\) |
| \(\chi_{81225}(14581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{570}\right)\) | \(e\left(\frac{218}{285}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{123}{190}\right)\) | \(e\left(\frac{58}{95}\right)\) | \(e\left(\frac{187}{570}\right)\) | \(e\left(\frac{143}{570}\right)\) | \(e\left(\frac{151}{285}\right)\) | \(e\left(\frac{67}{285}\right)\) | \(e\left(\frac{281}{570}\right)\) |
| \(\chi_{81225}(15436,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{431}{570}\right)\) | \(e\left(\frac{146}{285}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{51}{190}\right)\) | \(e\left(\frac{31}{95}\right)\) | \(e\left(\frac{439}{570}\right)\) | \(e\left(\frac{101}{570}\right)\) | \(e\left(\frac{7}{285}\right)\) | \(e\left(\frac{139}{285}\right)\) | \(e\left(\frac{47}{570}\right)\) |
| \(\chi_{81225}(15706,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{570}\right)\) | \(e\left(\frac{43}{285}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{28}{95}\right)\) | \(e\left(\frac{467}{570}\right)\) | \(e\left(\frac{223}{570}\right)\) | \(e\left(\frac{86}{285}\right)\) | \(e\left(\frac{242}{285}\right)\) | \(e\left(\frac{211}{570}\right)\) |
| \(\chi_{81225}(16291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{570}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{169}{190}\right)\) | \(e\left(\frac{4}{95}\right)\) | \(e\left(\frac{121}{570}\right)\) | \(e\left(\frac{59}{570}\right)\) | \(e\left(\frac{148}{285}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{383}{570}\right)\) |
| \(\chi_{81225}(16561,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{451}{570}\right)\) | \(e\left(\frac{166}{285}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{71}{190}\right)\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{179}{570}\right)\) | \(e\left(\frac{271}{570}\right)\) | \(e\left(\frac{47}{285}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{397}{570}\right)\) |
| \(\chi_{81225}(17146,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{570}\right)\) | \(e\left(\frac{2}{285}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{97}{190}\right)\) | \(e\left(\frac{72}{95}\right)\) | \(e\left(\frac{373}{570}\right)\) | \(e\left(\frac{17}{570}\right)\) | \(e\left(\frac{4}{285}\right)\) | \(e\left(\frac{283}{285}\right)\) | \(e\left(\frac{149}{570}\right)\) |