from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3520, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,35,0,12]))
pari: [g,chi] = znchar(Mod(151,3520))
Basic properties
| Modulus: | \(3520\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{352}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3520.eo
\(\chi_{3520}(151,\cdot)\) \(\chi_{3520}(391,\cdot)\) \(\chi_{3520}(711,\cdot)\) \(\chi_{3520}(871,\cdot)\) \(\chi_{3520}(1031,\cdot)\) \(\chi_{3520}(1271,\cdot)\) \(\chi_{3520}(1591,\cdot)\) \(\chi_{3520}(1751,\cdot)\) \(\chi_{3520}(1911,\cdot)\) \(\chi_{3520}(2151,\cdot)\) \(\chi_{3520}(2471,\cdot)\) \(\chi_{3520}(2631,\cdot)\) \(\chi_{3520}(2791,\cdot)\) \(\chi_{3520}(3031,\cdot)\) \(\chi_{3520}(3351,\cdot)\) \(\chi_{3520}(3511,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.1411841662908675517629776705295515492024702234241930698046194396081616318012166504448.1 |
Values on generators
\((2751,1541,2817,321)\) → \((-1,e\left(\frac{7}{8}\right),1,e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3520 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
sage: chi.jacobi_sum(n)