from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(239, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([12]))
pari: [g,chi] = znchar(Mod(101,239))
Basic properties
Modulus: | \(239\) | |
Conductor: | \(239\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(17\) | 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 239.e
\(\chi_{239}(6,\cdot)\) \(\chi_{239}(22,\cdot)\) \(\chi_{239}(36,\cdot)\) \(\chi_{239}(40,\cdot)\) \(\chi_{239}(51,\cdot)\) \(\chi_{239}(67,\cdot)\) \(\chi_{239}(71,\cdot)\) \(\chi_{239}(75,\cdot)\) \(\chi_{239}(101,\cdot)\) \(\chi_{239}(128,\cdot)\) \(\chi_{239}(132,\cdot)\) \(\chi_{239}(163,\cdot)\) \(\chi_{239}(166,\cdot)\) \(\chi_{239}(187,\cdot)\) \(\chi_{239}(211,\cdot)\) \(\chi_{239}(216,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | Number field defined by a degree 17 polynomial |
Values on generators
\(7\) → \(e\left(\frac{6}{17}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 239 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(1\) | \(e\left(\frac{7}{17}\right)\) |
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)