from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8002, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([49]))
pari: [g,chi] = znchar(Mod(4347,8002))
Basic properties
Modulus: | \(8002\) | |
Conductor: | \(4001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4001}(346,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8002.l
\(\chi_{8002}(625,\cdot)\) \(\chi_{8002}(1429,\cdot)\) \(\chi_{8002}(1473,\cdot)\) \(\chi_{8002}(1529,\cdot)\) \(\chi_{8002}(3253,\cdot)\) \(\chi_{8002}(3407,\cdot)\) \(\chi_{8002}(3805,\cdot)\) \(\chi_{8002}(4015,\cdot)\) \(\chi_{8002}(4315,\cdot)\) \(\chi_{8002}(4347,\cdot)\) \(\chi_{8002}(4637,\cdot)\) \(\chi_{8002}(5595,\cdot)\) \(\chi_{8002}(6471,\cdot)\) \(\chi_{8002}(6745,\cdot)\) \(\chi_{8002}(6815,\cdot)\) \(\chi_{8002}(7159,\cdot)\) \(\chi_{8002}(7385,\cdot)\) \(\chi_{8002}(7607,\cdot)\) \(\chi_{8002}(7611,\cdot)\) \(\chi_{8002}(7801,\cdot)\)
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
\(3\) → \(e\left(\frac{49}{50}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 8002 }(4347, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(-1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) |
sage: chi.jacobi_sum(n)