from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,10,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,3600))
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: | odd | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3600}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{3600}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3600}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3600}(587,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3600}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3600}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3600}(1283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{3600}(1523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3600}(1787,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{3600}(2003,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{3600}(2027,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{3600}(2723,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3600}(2747,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{3600}(2963,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{3600}(3227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{3600}(3467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |