from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([12,15,20]))
pari: [g,chi] = znchar(Mod(5252,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(8041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | 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.ck
\(\chi_{8041}(135,\cdot)\) \(\chi_{8041}(509,\cdot)\) \(\chi_{8041}(5252,\cdot)\) \(\chi_{8041}(5626,\cdot)\) \(\chi_{8041}(6714,\cdot)\) \(\chi_{8041}(7088,\cdot)\) \(\chi_{8041}(7445,\cdot)\) \(\chi_{8041}(7819,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{2}{5}\right),-1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(5252, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)