from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(223, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,223))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(223\) | |
| Conductor: | \(223\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(222\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{111})$ |
| Fixed field: | Number field defined by a degree 222 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{223}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{1}{222}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{89}{222}\right)\) | \(e\left(\frac{181}{222}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{1}{111}\right)\) | \(e\left(\frac{47}{222}\right)\) | \(e\left(\frac{107}{222}\right)\) |
| \(\chi_{223}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{89}{222}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{151}{222}\right)\) | \(e\left(\frac{125}{222}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{89}{111}\right)\) | \(e\left(\frac{187}{222}\right)\) | \(e\left(\frac{199}{222}\right)\) |
| \(\chi_{223}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{181}{222}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{125}{222}\right)\) | \(e\left(\frac{127}{222}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{70}{111}\right)\) | \(e\left(\frac{71}{222}\right)\) | \(e\left(\frac{53}{222}\right)\) |
| \(\chi_{223}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{47}{222}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{187}{222}\right)\) | \(e\left(\frac{71}{222}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{47}{111}\right)\) | \(e\left(\frac{211}{222}\right)\) | \(e\left(\frac{145}{222}\right)\) |
| \(\chi_{223}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{107}{222}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{199}{222}\right)\) | \(e\left(\frac{53}{222}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{145}{222}\right)\) | \(e\left(\frac{127}{222}\right)\) |
| \(\chi_{223}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{139}{222}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{161}{222}\right)\) | \(e\left(\frac{73}{222}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{28}{111}\right)\) | \(e\left(\frac{95}{222}\right)\) | \(e\left(\frac{221}{222}\right)\) |
| \(\chi_{223}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{5}{222}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{1}{222}\right)\) | \(e\left(\frac{17}{222}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{13}{222}\right)\) | \(e\left(\frac{91}{222}\right)\) |
| \(\chi_{223}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{211}{222}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{131}{222}\right)\) | \(e\left(\frac{7}{222}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{100}{111}\right)\) | \(e\left(\frac{149}{222}\right)\) | \(e\left(\frac{155}{222}\right)\) |
| \(\chi_{223}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{65}{222}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{13}{222}\right)\) | \(e\left(\frac{221}{222}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{65}{111}\right)\) | \(e\left(\frac{169}{222}\right)\) | \(e\left(\frac{73}{222}\right)\) |
| \(\chi_{223}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{163}{222}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{77}{222}\right)\) | \(e\left(\frac{199}{222}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{113}{222}\right)\) | \(e\left(\frac{125}{222}\right)\) |
| \(\chi_{223}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{97}{222}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{197}{222}\right)\) | \(e\left(\frac{19}{222}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{97}{111}\right)\) | \(e\left(\frac{119}{222}\right)\) | \(e\left(\frac{167}{222}\right)\) |
| \(\chi_{223}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{77}{222}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{193}{222}\right)\) | \(e\left(\frac{173}{222}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{77}{111}\right)\) | \(e\left(\frac{67}{222}\right)\) | \(e\left(\frac{25}{222}\right)\) |
| \(\chi_{223}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{169}{222}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{167}{222}\right)\) | \(e\left(\frac{175}{222}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{58}{111}\right)\) | \(e\left(\frac{173}{222}\right)\) | \(e\left(\frac{101}{222}\right)\) |
| \(\chi_{223}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{23}{222}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{49}{222}\right)\) | \(e\left(\frac{167}{222}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{23}{111}\right)\) | \(e\left(\frac{193}{222}\right)\) | \(e\left(\frac{19}{222}\right)\) |
| \(\chi_{223}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{91}{222}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{107}{222}\right)\) | \(e\left(\frac{43}{222}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{91}{111}\right)\) | \(e\left(\frac{59}{222}\right)\) | \(e\left(\frac{191}{222}\right)\) |
| \(\chi_{223}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{121}{222}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{113}{222}\right)\) | \(e\left(\frac{145}{222}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{10}{111}\right)\) | \(e\left(\frac{137}{222}\right)\) | \(e\left(\frac{71}{222}\right)\) |
| \(\chi_{223}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{55}{222}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{11}{222}\right)\) | \(e\left(\frac{187}{222}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{55}{111}\right)\) | \(e\left(\frac{143}{222}\right)\) | \(e\left(\frac{113}{222}\right)\) |
| \(\chi_{223}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{145}{222}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{29}{222}\right)\) | \(e\left(\frac{49}{222}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{34}{111}\right)\) | \(e\left(\frac{155}{222}\right)\) | \(e\left(\frac{197}{222}\right)\) |
| \(\chi_{223}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{173}{222}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{79}{222}\right)\) | \(e\left(\frac{11}{222}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{62}{111}\right)\) | \(e\left(\frac{139}{222}\right)\) | \(e\left(\frac{85}{222}\right)\) |
| \(\chi_{223}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{73}{222}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{59}{222}\right)\) | \(e\left(\frac{115}{222}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{73}{111}\right)\) | \(e\left(\frac{101}{222}\right)\) | \(e\left(\frac{41}{222}\right)\) |
| \(\chi_{223}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{175}{222}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{35}{222}\right)\) | \(e\left(\frac{151}{222}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{64}{111}\right)\) | \(e\left(\frac{11}{222}\right)\) | \(e\left(\frac{77}{222}\right)\) |
| \(\chi_{223}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{35}{222}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{7}{222}\right)\) | \(e\left(\frac{119}{222}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{35}{111}\right)\) | \(e\left(\frac{91}{222}\right)\) | \(e\left(\frac{193}{222}\right)\) |
| \(\chi_{223}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{157}{222}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{209}{222}\right)\) | \(e\left(\frac{1}{222}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{53}{222}\right)\) | \(e\left(\frac{149}{222}\right)\) |
| \(\chi_{223}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{179}{222}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{169}{222}\right)\) | \(e\left(\frac{209}{222}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{68}{111}\right)\) | \(e\left(\frac{199}{222}\right)\) | \(e\left(\frac{61}{222}\right)\) |
| \(\chi_{223}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{95}{222}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{19}{222}\right)\) | \(e\left(\frac{101}{222}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{95}{111}\right)\) | \(e\left(\frac{25}{222}\right)\) | \(e\left(\frac{175}{222}\right)\) |
| \(\chi_{223}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{167}{222}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{211}{222}\right)\) | \(e\left(\frac{35}{222}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{56}{111}\right)\) | \(e\left(\frac{79}{222}\right)\) | \(e\left(\frac{109}{222}\right)\) |
| \(\chi_{223}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{143}{222}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{73}{222}\right)\) | \(e\left(\frac{131}{222}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{32}{111}\right)\) | \(e\left(\frac{61}{222}\right)\) | \(e\left(\frac{205}{222}\right)\) |
| \(\chi_{223}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{127}{222}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{203}{222}\right)\) | \(e\left(\frac{121}{222}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{16}{111}\right)\) | \(e\left(\frac{197}{222}\right)\) | \(e\left(\frac{47}{222}\right)\) |
| \(\chi_{223}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{11}{222}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{91}{222}\right)\) | \(e\left(\frac{215}{222}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{11}{111}\right)\) | \(e\left(\frac{73}{222}\right)\) | \(e\left(\frac{67}{222}\right)\) |
| \(\chi_{223}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{203}{222}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{85}{222}\right)\) | \(e\left(\frac{113}{222}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{92}{111}\right)\) | \(e\left(\frac{217}{222}\right)\) | \(e\left(\frac{187}{222}\right)\) |
| \(\chi_{223}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{49}{222}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{143}{222}\right)\) | \(e\left(\frac{211}{222}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{49}{111}\right)\) | \(e\left(\frac{83}{222}\right)\) | \(e\left(\frac{137}{222}\right)\) |