from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,16]))
pari: [g,chi] = znchar(Mod(370,8003))
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(8003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | 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.be
\(\chi_{8003}(370,\cdot)\) \(\chi_{8003}(688,\cdot)\) \(\chi_{8003}(1483,\cdot)\) \(\chi_{8003}(1960,\cdot)\) \(\chi_{8003}(2013,\cdot)\) \(\chi_{8003}(2543,\cdot)\) \(\chi_{8003}(2596,\cdot)\) \(\chi_{8003}(2967,\cdot)\) \(\chi_{8003}(4451,\cdot)\) \(\chi_{8003}(4504,\cdot)\) \(\chi_{8003}(5829,\cdot)\) \(\chi_{8003}(6200,\cdot)\) \(\chi_{8003}(6465,\cdot)\) \(\chi_{8003}(6571,\cdot)\) \(\chi_{8003}(6730,\cdot)\) \(\chi_{8003}(6889,\cdot)\) \(\chi_{8003}(7207,\cdot)\) \(\chi_{8003}(7419,\cdot)\) \(\chi_{8003}(7631,\cdot)\) \(\chi_{8003}(7896,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((4984,7103)\) → \((-1,e\left(\frac{8}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8003 }(370, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
sage: chi.jacobi_sum(n)