from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100016, base_ring=CyclotomicField(276))
M = H._module
chi = DirichletCharacter(H, M([0,69,92,138,252]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,100016))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(100016\) | |
| Conductor: | \(100016\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(276\) | 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: | $\Q(\zeta_{276})$ |
| Fixed field: | Number field defined by a degree 276 polynomial (not computed) |
First 31 of 88 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{100016}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{276}\right)\) | \(e\left(\frac{229}{276}\right)\) | \(e\left(\frac{95}{138}\right)\) | \(e\left(\frac{269}{276}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{101}{138}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{49}{92}\right)\) |
| \(\chi_{100016}(949,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{276}\right)\) | \(e\left(\frac{125}{276}\right)\) | \(e\left(\frac{85}{138}\right)\) | \(e\left(\frac{1}{276}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{25}{138}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{85}{92}\right)\) |
| \(\chi_{100016}(2165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{276}\right)\) | \(e\left(\frac{97}{276}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{173}{276}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{97}{138}\right)\) | \(e\left(\frac{77}{92}\right)\) |
| \(\chi_{100016}(4293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{276}\right)\) | \(e\left(\frac{133}{276}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{149}{276}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{133}{138}\right)\) | \(e\left(\frac{61}{92}\right)\) |
| \(\chi_{100016}(6269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{276}\right)\) | \(e\left(\frac{95}{276}\right)\) | \(e\left(\frac{37}{138}\right)\) | \(e\left(\frac{67}{276}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{19}{138}\right)\) | \(e\left(\frac{95}{138}\right)\) | \(e\left(\frac{83}{92}\right)\) |
| \(\chi_{100016}(7485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{276}\right)\) | \(e\left(\frac{175}{276}\right)\) | \(e\left(\frac{119}{138}\right)\) | \(e\left(\frac{167}{276}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{35}{138}\right)\) | \(e\left(\frac{37}{138}\right)\) | \(e\left(\frac{27}{92}\right)\) |
| \(\chi_{100016}(8549,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{276}\right)\) | \(e\left(\frac{121}{276}\right)\) | \(e\left(\frac{5}{138}\right)\) | \(e\left(\frac{65}{276}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{121}{138}\right)\) | \(e\left(\frac{5}{92}\right)\) |
| \(\chi_{100016}(9461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{276}\right)\) | \(e\left(\frac{185}{276}\right)\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{145}{276}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{37}{138}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{89}{92}\right)\) |
| \(\chi_{100016}(9613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{276}\right)\) | \(e\left(\frac{127}{276}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{107}{276}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{79}{92}\right)\) |
| \(\chi_{100016}(10677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{276}\right)\) | \(e\left(\frac{25}{276}\right)\) | \(e\left(\frac{17}{138}\right)\) | \(e\left(\frac{221}{276}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{5}{138}\right)\) | \(e\left(\frac{25}{138}\right)\) | \(e\left(\frac{17}{92}\right)\) |
| \(\chi_{100016}(11589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{276}\right)\) | \(e\left(\frac{245}{276}\right)\) | \(e\left(\frac{1}{138}\right)\) | \(e\left(\frac{13}{276}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{49}{138}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{1}{92}\right)\) |
| \(\chi_{100016}(12805,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{276}\right)\) | \(e\left(\frac{13}{276}\right)\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{137}{276}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{113}{138}\right)\) | \(e\left(\frac{13}{138}\right)\) | \(e\left(\frac{53}{92}\right)\) |
| \(\chi_{100016}(13869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{276}\right)\) | \(e\left(\frac{55}{276}\right)\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{155}{276}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{11}{138}\right)\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{19}{92}\right)\) |
| \(\chi_{100016}(14933,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{276}\right)\) | \(e\left(\frac{181}{276}\right)\) | \(e\left(\frac{101}{138}\right)\) | \(e\left(\frac{209}{276}\right)\) | \(e\left(\frac{35}{92}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{119}{138}\right)\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{9}{92}\right)\) |
| \(\chi_{100016}(15845,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{276}\right)\) | \(e\left(\frac{113}{276}\right)\) | \(e\left(\frac{121}{138}\right)\) | \(e\left(\frac{193}{276}\right)\) | \(e\left(\frac{31}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{133}{138}\right)\) | \(e\left(\frac{113}{138}\right)\) | \(e\left(\frac{29}{92}\right)\) |
| \(\chi_{100016}(15997,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{276}\right)\) | \(e\left(\frac{211}{276}\right)\) | \(e\left(\frac{11}{138}\right)\) | \(e\left(\frac{143}{276}\right)\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{73}{138}\right)\) | \(e\left(\frac{11}{92}\right)\) |
| \(\chi_{100016}(16909,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{276}\right)\) | \(e\left(\frac{203}{276}\right)\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{271}{276}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{13}{138}\right)\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{35}{92}\right)\) |
| \(\chi_{100016}(19037,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{276}\right)\) | \(e\left(\frac{131}{276}\right)\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{43}{276}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{109}{138}\right)\) | \(e\left(\frac{131}{138}\right)\) | \(e\left(\frac{67}{92}\right)\) |
| \(\chi_{100016}(20101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{276}\right)\) | \(e\left(\frac{149}{276}\right)\) | \(e\left(\frac{13}{138}\right)\) | \(e\left(\frac{169}{276}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{85}{138}\right)\) | \(e\left(\frac{11}{138}\right)\) | \(e\left(\frac{13}{92}\right)\) |
| \(\chi_{100016}(22381,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{276}\right)\) | \(e\left(\frac{79}{276}\right)\) | \(e\left(\frac{131}{138}\right)\) | \(e\left(\frac{47}{276}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{71}{138}\right)\) | \(e\left(\frac{79}{138}\right)\) | \(e\left(\frac{39}{92}\right)\) |
| \(\chi_{100016}(23293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{276}\right)\) | \(e\left(\frac{155}{276}\right)\) | \(e\left(\frac{133}{138}\right)\) | \(e\left(\frac{211}{276}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{17}{138}\right)\) | \(e\left(\frac{87}{92}\right)\) |
| \(\chi_{100016}(26485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{276}\right)\) | \(e\left(\frac{53}{276}\right)\) | \(e\left(\frac{25}{138}\right)\) | \(e\left(\frac{49}{276}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{121}{138}\right)\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{25}{92}\right)\) |
| \(\chi_{100016}(27549,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{276}\right)\) | \(e\left(\frac{215}{276}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{79}{276}\right)\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{91}{92}\right)\) |
| \(\chi_{100016}(27701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{276}\right)\) | \(e\left(\frac{49}{276}\right)\) | \(e\left(\frac{83}{138}\right)\) | \(e\left(\frac{113}{276}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{49}{138}\right)\) | \(e\left(\frac{37}{92}\right)\) |
| \(\chi_{100016}(28613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{276}\right)\) | \(e\left(\frac{137}{276}\right)\) | \(e\left(\frac{49}{138}\right)\) | \(e\left(\frac{85}{276}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{49}{92}\right)\) |
| \(\chi_{100016}(30741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{276}\right)\) | \(e\left(\frac{5}{276}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{265}{276}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{1}{138}\right)\) | \(e\left(\frac{5}{138}\right)\) | \(e\left(\frac{77}{92}\right)\) |
| \(\chi_{100016}(30893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{276}\right)\) | \(e\left(\frac{139}{276}\right)\) | \(e\left(\frac{89}{138}\right)\) | \(e\left(\frac{191}{276}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{83}{138}\right)\) | \(e\left(\frac{1}{138}\right)\) | \(e\left(\frac{43}{92}\right)\) |
| \(\chi_{100016}(32869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{276}\right)\) | \(e\left(\frac{41}{276}\right)\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{241}{276}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{41}{138}\right)\) | \(e\left(\frac{61}{92}\right)\) |
| \(\chi_{100016}(33021,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{276}\right)\) | \(e\left(\frac{199}{276}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{59}{276}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{95}{138}\right)\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{47}{92}\right)\) |
| \(\chi_{100016}(36061,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{276}\right)\) | \(e\left(\frac{83}{276}\right)\) | \(e\left(\frac{73}{138}\right)\) | \(e\left(\frac{259}{276}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{83}{138}\right)\) | \(e\left(\frac{27}{92}\right)\) |
| \(\chi_{100016}(37125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{276}\right)\) | \(e\left(\frac{29}{276}\right)\) | \(e\left(\frac{97}{138}\right)\) | \(e\left(\frac{157}{276}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{61}{138}\right)\) | \(e\left(\frac{29}{138}\right)\) | \(e\left(\frac{5}{92}\right)\) |