from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(841, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([12]))
pari: [g,chi] = znchar(Mod(571,841))
Basic properties
Modulus: | \(841\) | |
Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{29}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 841.d
\(\chi_{841}(190,\cdot)\) \(\chi_{841}(571,\cdot)\) \(\chi_{841}(574,\cdot)\) \(\chi_{841}(605,\cdot)\) \(\chi_{841}(645,\cdot)\) \(\chi_{841}(778,\cdot)\)
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: | 7.7.594823321.1 |
Values on generators
\(2\) → \(e\left(\frac{6}{7}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 841 }(571, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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)