from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110880, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,45,40,30,100,72]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4387,110880))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(110880\) | |
| Conductor: | \(110880\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(120\) | 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: | $\Q(\zeta_{120})$ |
| Fixed field: | Number field defined by a degree 120 polynomial (not computed) |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{110880}(4387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{110880}(8683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{110880}(13963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{110880}(14227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{110880}(19507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{110880}(33883,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{110880}(38923,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{110880}(39163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{110880}(39427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{110880}(44203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{110880}(44467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{110880}(44707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{110880}(49003,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{110880}(49747,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{110880}(54283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{110880}(54547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{110880}(59827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{110880}(64123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{110880}(69403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{110880}(69667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{110880}(74947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{110880}(89323,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{110880}(94363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{110880}(94603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{110880}(94867,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{110880}(99643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{110880}(99907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{110880}(100147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{110880}(104443,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{110880}(105187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{110880}(109723,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{20}\right)\) |