from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6004, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,6]))
pari: [g,chi] = znchar(Mod(381,6004))
Basic properties
Modulus: | \(6004\) | |
Conductor: | \(79\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{79}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6004.br
\(\chi_{6004}(381,\cdot)\) \(\chi_{6004}(457,\cdot)\) \(\chi_{6004}(1065,\cdot)\) \(\chi_{6004}(1825,\cdot)\) \(\chi_{6004}(3497,\cdot)\) \(\chi_{6004}(3573,\cdot)\) \(\chi_{6004}(4333,\cdot)\) \(\chi_{6004}(4409,\cdot)\) \(\chi_{6004}(4713,\cdot)\) \(\chi_{6004}(4865,\cdot)\) \(\chi_{6004}(5473,\cdot)\) \(\chi_{6004}(5777,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 13 polynomial |
Values on generators
\((3003,2529,3953)\) → \((1,1,e\left(\frac{3}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
\( \chi_{ 6004 }(381, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)