from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([0,105,100,0,60,36]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,221760))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(22176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(120\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 22176.we | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{120})$ |
| Fixed field: | Number field defined by a degree 120 polynomial (not computed) |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{221760}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{30}\right)\) |
| \(\chi_{221760}(6761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{30}\right)\) |
| \(\chi_{221760}(21881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{30}\right)\) |
| \(\chi_{221760}(25241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{30}\right)\) |
| \(\chi_{221760}(31961,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{30}\right)\) |
| \(\chi_{221760}(37001,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{30}\right)\) |
| \(\chi_{221760}(40361,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{30}\right)\) |
| \(\chi_{221760}(50441,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{30}\right)\) |
| \(\chi_{221760}(55481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{30}\right)\) |
| \(\chi_{221760}(62201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{30}\right)\) |
| \(\chi_{221760}(77321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{29}{30}\right)\) |
| \(\chi_{221760}(80681,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{30}\right)\) |
| \(\chi_{221760}(87401,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{30}\right)\) |
| \(\chi_{221760}(92441,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{30}\right)\) |
| \(\chi_{221760}(95801,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{30}\right)\) |
| \(\chi_{221760}(105881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{30}\right)\) |
| \(\chi_{221760}(110921,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{30}\right)\) |
| \(\chi_{221760}(117641,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{30}\right)\) |
| \(\chi_{221760}(132761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{29}{30}\right)\) |
| \(\chi_{221760}(136121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{30}\right)\) |
| \(\chi_{221760}(142841,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{30}\right)\) |
| \(\chi_{221760}(147881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{30}\right)\) |
| \(\chi_{221760}(151241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{30}\right)\) |
| \(\chi_{221760}(161321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{30}\right)\) |
| \(\chi_{221760}(166361,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{30}\right)\) |
| \(\chi_{221760}(173081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{23}{30}\right)\) |
| \(\chi_{221760}(188201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{30}\right)\) |
| \(\chi_{221760}(191561,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{30}\right)\) |
| \(\chi_{221760}(198281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{30}\right)\) |
| \(\chi_{221760}(203321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{30}\right)\) |
| \(\chi_{221760}(206681,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{30}\right)\) |