from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([120,105,0,120,200,72]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,221760))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(24640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(240\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 24640.vn | 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: | $\Q(\zeta_{240})$ |
| Fixed field: | Number field defined by a degree 240 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{221760}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{49}{60}\right)\) |
| \(\chi_{221760}(1459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{41}{60}\right)\) |
| \(\chi_{221760}(3979,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{59}{60}\right)\) |
| \(\chi_{221760}(12619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{43}{60}\right)\) |
| \(\chi_{221760}(16579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{53}{60}\right)\) |
| \(\chi_{221760}(22699,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{60}\right)\) |
| \(\chi_{221760}(25219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{60}\right)\) |
| \(\chi_{221760}(26659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{60}\right)\) |
| \(\chi_{221760}(27739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{19}{60}\right)\) |
| \(\chi_{221760}(29179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{60}\right)\) |
| \(\chi_{221760}(31699,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{29}{60}\right)\) |
| \(\chi_{221760}(40339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{60}\right)\) |
| \(\chi_{221760}(44299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{60}\right)\) |
| \(\chi_{221760}(50419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{37}{60}\right)\) |
| \(\chi_{221760}(52939,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{60}\right)\) |
| \(\chi_{221760}(54379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{47}{60}\right)\) |
| \(\chi_{221760}(55459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{49}{60}\right)\) |
| \(\chi_{221760}(56899,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{41}{60}\right)\) |
| \(\chi_{221760}(59419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{59}{60}\right)\) |
| \(\chi_{221760}(68059,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{43}{60}\right)\) |
| \(\chi_{221760}(72019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{233}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{53}{60}\right)\) |
| \(\chi_{221760}(78139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{60}\right)\) |
| \(\chi_{221760}(80659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{60}\right)\) |
| \(\chi_{221760}(82099,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{17}{60}\right)\) |
| \(\chi_{221760}(83179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{60}\right)\) |
| \(\chi_{221760}(84619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{60}\right)\) |
| \(\chi_{221760}(87139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{60}\right)\) |
| \(\chi_{221760}(95779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{60}\right)\) |
| \(\chi_{221760}(99739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{60}\right)\) |
| \(\chi_{221760}(105859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{37}{60}\right)\) |
| \(\chi_{221760}(108379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{60}\right)\) |