from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,50,54]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,1232))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1232\) | |
| Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1232}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1232}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1232}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1232}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1232}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{1232}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1232}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{1232}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1232}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{1232}(717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{1232}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1232}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{1232}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1232}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1232}(1053,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1232}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) |