from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(656, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,10,23]))
pari: [g,chi] = znchar(Mod(235,656))
Basic properties
Modulus: | \(656\) | |
Conductor: | \(656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | 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 656.cd
\(\chi_{656}(11,\cdot)\) \(\chi_{656}(19,\cdot)\) \(\chi_{656}(35,\cdot)\) \(\chi_{656}(67,\cdot)\) \(\chi_{656}(75,\cdot)\) \(\chi_{656}(99,\cdot)\) \(\chi_{656}(147,\cdot)\) \(\chi_{656}(171,\cdot)\) \(\chi_{656}(179,\cdot)\) \(\chi_{656}(211,\cdot)\) \(\chi_{656}(227,\cdot)\) \(\chi_{656}(235,\cdot)\) \(\chi_{656}(299,\cdot)\) \(\chi_{656}(315,\cdot)\) \(\chi_{656}(587,\cdot)\) \(\chi_{656}(603,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.1027708468267178047292394722862044397918868556644399912781578154071083295594368567462835848740864.1 |
Values on generators
\((575,165,129)\) → \((-1,i,e\left(\frac{23}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 656 }(235, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(-i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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)