from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([15,14]))
pari: [g,chi] = znchar(Mod(394,8003))
Basic properties
Modulus: | \(8003\) | |
Conductor: | \(8003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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)
|
Galois orbit 8003.s
\(\chi_{8003}(394,\cdot)\) \(\chi_{8003}(1189,\cdot)\) \(\chi_{8003}(1295,\cdot)\) \(\chi_{8003}(2408,\cdot)\) \(\chi_{8003}(3263,\cdot)\) \(\chi_{8003}(4058,\cdot)\) \(\chi_{8003}(4164,\cdot)\) \(\chi_{8003}(5277,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((4984,7103)\) → \((-i,e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8003 }(394, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)