from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1037, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([3,8]))
pari: [g,chi] = znchar(Mod(13,1037))
Basic properties
Modulus: | \(1037\) | |
Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | 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 1037.bb
\(\chi_{1037}(13,\cdot)\) \(\chi_{1037}(47,\cdot)\) \(\chi_{1037}(684,\cdot)\) \(\chi_{1037}(718,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.12.22734163157293282124804657.1 |
Values on generators
\((428,307)\) → \((i,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 1037 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)