from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,15,40,90,100,24]))
chi.galois_orbit()
[g,chi] = znchar(Mod(103,221760))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(110880\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(120\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 110880.ddi | 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 | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{221760}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{221760}(3127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{221760}(25063,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{221760}(28087,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{221760}(30343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{221760}(33367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{221760}(35143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{221760}(40423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{221760}(58327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{221760}(63607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{221760}(68407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{221760}(73687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{221760}(75463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{221760}(80743,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{221760}(105703,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{221760}(108727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{221760}(110983,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{221760}(114007,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{221760}(135943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{221760}(138967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{221760}(141223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{221760}(144247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{221760}(146023,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{221760}(151303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{221760}(169207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{221760}(174487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{221760}(179287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{221760}(184567,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{221760}(186343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{221760}(191623,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{221760}(216583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |