from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,11]))
pari: [g,chi] = znchar(Mod(73,309))
Basic properties
Modulus: | \(309\) | |
Conductor: | \(103\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{103}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 309.j
\(\chi_{309}(10,\cdot)\) \(\chi_{309}(22,\cdot)\) \(\chi_{309}(31,\cdot)\) \(\chi_{309}(37,\cdot)\) \(\chi_{309}(73,\cdot)\) \(\chi_{309}(94,\cdot)\) \(\chi_{309}(106,\cdot)\) \(\chi_{309}(127,\cdot)\) \(\chi_{309}(130,\cdot)\) \(\chi_{309}(142,\cdot)\) \(\chi_{309}(145,\cdot)\) \(\chi_{309}(172,\cdot)\) \(\chi_{309}(193,\cdot)\) \(\chi_{309}(286,\cdot)\) \(\chi_{309}(295,\cdot)\) \(\chi_{309}(301,\cdot)\)
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 34 polynomial |
Values on generators
\((104,211)\) → \((1,e\left(\frac{11}{34}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 309 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) |
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)