from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,45,160,0,200,216]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,221760))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(44352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(240\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 44352.bel | 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: | $\Q(\zeta_{240})$ |
| Fixed field: | Number field defined by a degree 240 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{221760}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{17}{60}\right)\) |
| \(\chi_{221760}(2581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{60}\right)\) |
| \(\chi_{221760}(3181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{13}{60}\right)\) |
| \(\chi_{221760}(5101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{29}{60}\right)\) |
| \(\chi_{221760}(13261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{37}{60}\right)\) |
| \(\chi_{221760}(15781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{31}{60}\right)\) |
| \(\chi_{221760}(17701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{60}\right)\) |
| \(\chi_{221760}(18301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{49}{60}\right)\) |
| \(\chi_{221760}(27781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{47}{60}\right)\) |
| \(\chi_{221760}(30301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{41}{60}\right)\) |
| \(\chi_{221760}(30901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{43}{60}\right)\) |
| \(\chi_{221760}(32821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{59}{60}\right)\) |
| \(\chi_{221760}(40981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{53}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{60}\right)\) |
| \(\chi_{221760}(43501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{60}\right)\) |
| \(\chi_{221760}(45421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{53}{60}\right)\) |
| \(\chi_{221760}(46021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{19}{60}\right)\) |
| \(\chi_{221760}(55501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{17}{60}\right)\) |
| \(\chi_{221760}(58021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{181}{240}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{60}\right)\) |
| \(\chi_{221760}(58621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{53}{240}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{60}\right)\) |
| \(\chi_{221760}(60541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{29}{60}\right)\) |
| \(\chi_{221760}(68701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{37}{60}\right)\) |
| \(\chi_{221760}(71221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{31}{60}\right)\) |
| \(\chi_{221760}(73141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{23}{60}\right)\) |
| \(\chi_{221760}(73741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{49}{60}\right)\) |
| \(\chi_{221760}(83221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{47}{60}\right)\) |
| \(\chi_{221760}(85741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{240}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{60}\right)\) |
| \(\chi_{221760}(86341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{43}{60}\right)\) |
| \(\chi_{221760}(88261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{59}{60}\right)\) |
| \(\chi_{221760}(96421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{60}\right)\) |
| \(\chi_{221760}(98941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{60}\right)\) |
| \(\chi_{221760}(100861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{53}{60}\right)\) |