from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,20,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,4800))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4800\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2400.dw | 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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4800}(41,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{4800}(281,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{4800}(521,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{4800}(761,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{4800}(1241,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{4800}(1481,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{4800}(1721,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{4800}(1961,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{4800}(2441,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{4800}(2681,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{4800}(2921,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{4800}(3161,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{4800}(3641,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{4800}(3881,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{4800}(4121,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{4800}(4361,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) |