from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([26,27,12]))
pari: [g,chi] = znchar(Mod(443,945))
Basic properties
Modulus: | \(945\) | |
Conductor: | \(945\) | 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 945.do
\(\chi_{945}(2,\cdot)\) \(\chi_{945}(32,\cdot)\) \(\chi_{945}(128,\cdot)\) \(\chi_{945}(158,\cdot)\) \(\chi_{945}(317,\cdot)\) \(\chi_{945}(347,\cdot)\) \(\chi_{945}(443,\cdot)\) \(\chi_{945}(473,\cdot)\) \(\chi_{945}(632,\cdot)\) \(\chi_{945}(662,\cdot)\) \(\chi_{945}(758,\cdot)\) \(\chi_{945}(788,\cdot)\)
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
\((596,757,136)\) → \((e\left(\frac{13}{18}\right),-i,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 945 }(443, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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)