from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(29,41))
Basic properties
Modulus: | \(41\) | |
Conductor: | \(41\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 41.h
\(\chi_{41}(6,\cdot)\) \(\chi_{41}(7,\cdot)\) \(\chi_{41}(11,\cdot)\) \(\chi_{41}(12,\cdot)\) \(\chi_{41}(13,\cdot)\) \(\chi_{41}(15,\cdot)\) \(\chi_{41}(17,\cdot)\) \(\chi_{41}(19,\cdot)\) \(\chi_{41}(22,\cdot)\) \(\chi_{41}(24,\cdot)\) \(\chi_{41}(26,\cdot)\) \(\chi_{41}(28,\cdot)\) \(\chi_{41}(29,\cdot)\) \(\chi_{41}(30,\cdot)\) \(\chi_{41}(34,\cdot)\) \(\chi_{41}(35,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\(6\) → \(e\left(\frac{7}{40}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 41 }(29, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) |
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)