from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3549, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,52,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,3549))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3549\) | |
| Conductor: | \(3549\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(156\) | 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: | $\Q(\zeta_{156})$ |
| Fixed field: | Number field defined by a degree 156 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3549}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{89}{156}\right)\) |
| \(\chi_{3549}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{133}{156}\right)\) |
| \(\chi_{3549}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{155}{156}\right)\) |
| \(\chi_{3549}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{55}{156}\right)\) |
| \(\chi_{3549}(275,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{156}\right)\) |
| \(\chi_{3549}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{156}\right)\) |
| \(\chi_{3549}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{83}{156}\right)\) |
| \(\chi_{3549}(410,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{31}{156}\right)\) |
| \(\chi_{3549}(548,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{77}{156}\right)\) |
| \(\chi_{3549}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{25}{156}\right)\) |
| \(\chi_{3549}(674,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{156}\right)\) |
| \(\chi_{3549}(683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{156}\right)\) |
| \(\chi_{3549}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{149}{156}\right)\) |
| \(\chi_{3549}(851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{49}{156}\right)\) |
| \(\chi_{3549}(947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{95}{156}\right)\) |
| \(\chi_{3549}(956,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{139}{156}\right)\) |
| \(\chi_{3549}(1124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{73}{156}\right)\) |
| \(\chi_{3549}(1220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{156}\right)\) |
| \(\chi_{3549}(1229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{115}{156}\right)\) |
| \(\chi_{3549}(1367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{137}{156}\right)\) |
| \(\chi_{3549}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{97}{156}\right)\) |
| \(\chi_{3549}(1493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{107}{156}\right)\) |
| \(\chi_{3549}(1640,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{53}{156}\right)\) |
| \(\chi_{3549}(1670,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{121}{156}\right)\) |
| \(\chi_{3549}(1766,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{156}\right)\) |
| \(\chi_{3549}(1775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{67}{156}\right)\) |
| \(\chi_{3549}(1913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{125}{156}\right)\) |
| \(\chi_{3549}(1943,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{145}{156}\right)\) |
| \(\chi_{3549}(2039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{119}{156}\right)\) |
| \(\chi_{3549}(2048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{156}\right)\) |
| \(\chi_{3549}(2186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{156}\right)\) |