from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,12]))
pari: [g,chi] = znchar(Mod(1024,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{731}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8041.ci
\(\chi_{8041}(914,\cdot)\) \(\chi_{8041}(1024,\cdot)\) \(\chi_{8041}(1288,\cdot)\) \(\chi_{8041}(2333,\cdot)\) \(\chi_{8041}(2707,\cdot)\) \(\chi_{8041}(2971,\cdot)\) \(\chi_{8041}(3719,\cdot)\) \(\chi_{8041}(4390,\cdot)\) \(\chi_{8041}(5138,\cdot)\) \(\chi_{8041}(6337,\cdot)\) \(\chi_{8041}(7646,\cdot)\) \(\chi_{8041}(7756,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((6580,2366,562)\) → \((1,-i,e\left(\frac{3}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(1024, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(i\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) |
sage: chi.jacobi_sum(n)