from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81225, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([152,57,100]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,81225))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(81225\) | |
| Conductor: | \(16245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(228\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 16245.gb | 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: | $\Q(\zeta_{228})$ |
| Fixed field: | Number field defined by a degree 228 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{81225}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{161}{228}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{85}{228}\right)\) | \(e\left(\frac{173}{228}\right)\) |
| \(\chi_{81225}(2443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{163}{228}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{228}\right)\) | \(e\left(\frac{127}{228}\right)\) |
| \(\chi_{81225}(2743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{143}{228}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{67}{228}\right)\) | \(e\left(\frac{131}{228}\right)\) |
| \(\chi_{81225}(3982,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{73}{228}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{149}{228}\right)\) | \(e\left(\frac{145}{228}\right)\) |
| \(\chi_{81225}(4282,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{197}{228}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{29}{228}\right)\) |
| \(\chi_{81225}(6718,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{103}{228}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{139}{228}\right)\) |
| \(\chi_{81225}(7018,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{103}{228}\right)\) | \(e\left(\frac{215}{228}\right)\) |
| \(\chi_{81225}(8257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{13}{228}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{89}{228}\right)\) | \(e\left(\frac{157}{228}\right)\) |
| \(\chi_{81225}(8557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{5}{228}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{157}{228}\right)\) | \(e\left(\frac{113}{228}\right)\) |
| \(\chi_{81225}(10993,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{43}{228}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{119}{228}\right)\) | \(e\left(\frac{151}{228}\right)\) |
| \(\chi_{81225}(11293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{215}{228}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{139}{228}\right)\) | \(e\left(\frac{71}{228}\right)\) |
| \(\chi_{81225}(12532,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{181}{228}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{29}{228}\right)\) | \(e\left(\frac{169}{228}\right)\) |
| \(\chi_{81225}(12832,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{41}{228}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{193}{228}\right)\) | \(e\left(\frac{197}{228}\right)\) |
| \(\chi_{81225}(15268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{211}{228}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{59}{228}\right)\) | \(e\left(\frac{163}{228}\right)\) |
| \(\chi_{81225}(15568,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{23}{228}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{155}{228}\right)\) |
| \(\chi_{81225}(16807,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{197}{228}\right)\) | \(e\left(\frac{181}{228}\right)\) |
| \(\chi_{81225}(17107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{77}{228}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{228}\right)\) | \(e\left(\frac{53}{228}\right)\) |
| \(\chi_{81225}(19543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{151}{228}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{227}{228}\right)\) | \(e\left(\frac{175}{228}\right)\) |
| \(\chi_{81225}(19843,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{59}{228}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{211}{228}\right)\) | \(e\left(\frac{11}{228}\right)\) |
| \(\chi_{81225}(21082,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{61}{228}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{137}{228}\right)\) | \(e\left(\frac{193}{228}\right)\) |
| \(\chi_{81225}(21382,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{37}{228}\right)\) | \(e\left(\frac{137}{228}\right)\) |
| \(\chi_{81225}(23818,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{167}{228}\right)\) | \(e\left(\frac{187}{228}\right)\) |
| \(\chi_{81225}(25357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{1}{228}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{77}{228}\right)\) | \(e\left(\frac{205}{228}\right)\) |
| \(\chi_{81225}(25657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{149}{228}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{73}{228}\right)\) | \(e\left(\frac{221}{228}\right)\) |
| \(\chi_{81225}(28093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{199}{228}\right)\) |
| \(\chi_{81225}(28393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{131}{228}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{55}{228}\right)\) | \(e\left(\frac{179}{228}\right)\) |
| \(\chi_{81225}(29632,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{169}{228}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{228}\right)\) | \(e\left(\frac{217}{228}\right)\) |
| \(\chi_{81225}(29932,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{77}{228}\right)\) |
| \(\chi_{81225}(32368,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{199}{228}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{47}{228}\right)\) | \(e\left(\frac{211}{228}\right)\) |
| \(\chi_{81225}(32668,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{167}{228}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{35}{228}\right)\) |
| \(\chi_{81225}(33907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{1}{228}\right)\) |