from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([0,7,12]))
pari: [g,chi] = znchar(Mod(49,435))
Basic properties
Modulus: | \(435\) | |
Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{145}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 435.ba
\(\chi_{435}(49,\cdot)\) \(\chi_{435}(94,\cdot)\) \(\chi_{435}(139,\cdot)\) \(\chi_{435}(169,\cdot)\) \(\chi_{435}(199,\cdot)\) \(\chi_{435}(364,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | Number field defined by a degree 14 polynomial |
Values on generators
\((146,262,31)\) → \((1,-1,e\left(\frac{6}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 435 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) | \(e\left(\frac{5}{7}\right)\) |
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)