from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1048, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,21]))
pari: [g,chi] = znchar(Mod(47,1048))
Basic properties
Modulus: | \(1048\) | |
Conductor: | \(524\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{524}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1048.v
\(\chi_{1048}(47,\cdot)\) \(\chi_{1048}(71,\cdot)\) \(\chi_{1048}(79,\cdot)\) \(\chi_{1048}(199,\cdot)\) \(\chi_{1048}(223,\cdot)\) \(\chi_{1048}(479,\cdot)\) \(\chi_{1048}(543,\cdot)\) \(\chi_{1048}(575,\cdot)\) \(\chi_{1048}(679,\cdot)\) \(\chi_{1048}(687,\cdot)\) \(\chi_{1048}(855,\cdot)\) \(\chi_{1048}(935,\cdot)\)
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: | 26.26.5735381071083349848637397344247707950917141114970532021796864.1 |
Values on generators
\((263,525,657)\) → \((-1,1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 1048 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
sage: chi.jacobi_sum(n)