from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,4]))
pari: [g,chi] = znchar(Mod(5455,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(8041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | 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 8041.cc
\(\chi_{8041}(824,\cdot)\) \(\chi_{8041}(1198,\cdot)\) \(\chi_{8041}(1770,\cdot)\) \(\chi_{8041}(2144,\cdot)\) \(\chi_{8041}(5081,\cdot)\) \(\chi_{8041}(5455,\cdot)\) \(\chi_{8041}(6027,\cdot)\) \(\chi_{8041}(6401,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((6580,2366,562)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(5455, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) |
sage: chi.jacobi_sum(n)