from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,37]))
pari: [g,chi] = znchar(Mod(106,387))
Basic properties
Modulus: | \(387\) | |
Conductor: | \(387\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 387.bh
\(\chi_{387}(106,\cdot)\) \(\chi_{387}(112,\cdot)\) \(\chi_{387}(157,\cdot)\) \(\chi_{387}(202,\cdot)\) \(\chi_{387}(205,\cdot)\) \(\chi_{387}(277,\cdot)\) \(\chi_{387}(292,\cdot)\) \(\chi_{387}(304,\cdot)\) \(\chi_{387}(313,\cdot)\) \(\chi_{387}(319,\cdot)\) \(\chi_{387}(349,\cdot)\) \(\chi_{387}(373,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((173,46)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{37}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 387 }(106, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)