from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,7]))
pari: [g,chi] = znchar(Mod(113,6042))
Basic properties
Modulus: | \(6042\) | |
Conductor: | \(3021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3021}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6042.bj
\(\chi_{6042}(113,\cdot)\) \(\chi_{6042}(1937,\cdot)\) \(\chi_{6042}(2051,\cdot)\) \(\chi_{6042}(2849,\cdot)\) \(\chi_{6042}(3191,\cdot)\) \(\chi_{6042}(3647,\cdot)\) \(\chi_{6042}(3875,\cdot)\) \(\chi_{6042}(4331,\cdot)\) \(\chi_{6042}(4673,\cdot)\) \(\chi_{6042}(4787,\cdot)\) \(\chi_{6042}(4901,\cdot)\) \(\chi_{6042}(5357,\cdot)\)
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 26 polynomial |
Values on generators
\((2015,4771,2281)\) → \((-1,-1,e\left(\frac{7}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 6042 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
sage: chi.jacobi_sum(n)