from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,50,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(77,3600))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3600\) | |
| Conductor: | \(3600\) | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3600}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3600}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3600}(533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3600}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3600}(1013,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3600}(1037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3600}(1253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3600}(1517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3600}(1733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3600}(1973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3600}(2237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3600}(2453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3600}(2477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3600}(3173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3600}(3197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3600}(3413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) |