from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([171,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,1805))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1805\) | |
| Conductor: | \(1805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(228\) | 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: | $\Q(\zeta_{228})$ |
| Fixed field: | Number field defined by a degree 228 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1805}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{228}\right)\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{31}{228}\right)\) |
| \(\chi_{1805}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{228}\right)\) | \(e\left(\frac{13}{228}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{89}{228}\right)\) |
| \(\chi_{1805}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{205}{228}\right)\) |
| \(\chi_{1805}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{228}\right)\) | \(e\left(\frac{211}{228}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{59}{228}\right)\) |
| \(\chi_{1805}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{103}{228}\right)\) |
| \(\chi_{1805}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{228}\right)\) | \(e\left(\frac{61}{228}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{137}{228}\right)\) |
| \(\chi_{1805}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{228}\right)\) | \(e\left(\frac{125}{228}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{49}{228}\right)\) |
| \(\chi_{1805}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{228}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{107}{228}\right)\) |
| \(\chi_{1805}(198,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{228}\right)\) | \(e\left(\frac{23}{228}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{175}{228}\right)\) |
| \(\chi_{1805}(202,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{185}{228}\right)\) |
| \(\chi_{1805}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{228}\right)\) | \(e\left(\frac{197}{228}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{121}{228}\right)\) |
| \(\chi_{1805}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{228}\right)\) | \(e\left(\frac{79}{228}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{155}{228}\right)\) |
| \(\chi_{1805}(297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{157}{228}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{5}{228}\right)\) |
| \(\chi_{1805}(312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{228}\right)\) | \(e\left(\frac{41}{228}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{193}{228}\right)\) |
| \(\chi_{1805}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{228}\right)\) | \(e\left(\frac{127}{228}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{203}{228}\right)\) |
| \(\chi_{1805}(388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{228}\right)\) | \(e\left(\frac{167}{228}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{91}{228}\right)\) |
| \(\chi_{1805}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{205}{228}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{53}{228}\right)\) |
| \(\chi_{1805}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{228}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{37}{228}\right)\) |
| \(\chi_{1805}(468,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{175}{228}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{23}{228}\right)\) |
| \(\chi_{1805}(483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{228}\right)\) | \(e\left(\frac{11}{228}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{163}{228}\right)\) |
| \(\chi_{1805}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{228}\right)\) | \(e\left(\frac{25}{228}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{101}{228}\right)\) |
| \(\chi_{1805}(502,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{228}\right)\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{109}{228}\right)\) |
| \(\chi_{1805}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{228}\right)\) | \(e\left(\frac{223}{228}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{71}{228}\right)\) |
| \(\chi_{1805}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{228}\right)\) | \(e\left(\frac{83}{228}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{7}{228}\right)\) |
| \(\chi_{1805}(582,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{228}\right)\) | \(e\left(\frac{73}{228}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{149}{228}\right)\) |
| \(\chi_{1805}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{228}\right)\) | \(e\left(\frac{29}{228}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{181}{228}\right)\) |
| \(\chi_{1805}(658,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{228}\right)\) | \(e\left(\frac{43}{228}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{119}{228}\right)\) |
| \(\chi_{1805}(673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{228}\right)\) | \(e\left(\frac{155}{228}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{79}{228}\right)\) |
| \(\chi_{1805}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{228}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{197}{228}\right)\) |
| \(\chi_{1805}(692,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{228}\right)\) | \(e\left(\frac{101}{228}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{25}{228}\right)\) |
| \(\chi_{1805}(753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{228}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{167}{228}\right)\) |