from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(351, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,3]))
pari: [g,chi] = znchar(Mod(41,351))
Basic properties
Modulus: | \(351\) | |
Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 351.bq
\(\chi_{351}(20,\cdot)\) \(\chi_{351}(41,\cdot)\) \(\chi_{351}(50,\cdot)\) \(\chi_{351}(110,\cdot)\) \(\chi_{351}(137,\cdot)\) \(\chi_{351}(158,\cdot)\) \(\chi_{351}(167,\cdot)\) \(\chi_{351}(227,\cdot)\) \(\chi_{351}(254,\cdot)\) \(\chi_{351}(275,\cdot)\) \(\chi_{351}(284,\cdot)\) \(\chi_{351}(344,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((326,28)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 351 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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)