from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,31]))
pari: [g,chi] = znchar(Mod(1495,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(8041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 8041.cy
\(\chi_{8041}(373,\cdot)\) \(\chi_{8041}(1121,\cdot)\) \(\chi_{8041}(1308,\cdot)\) \(\chi_{8041}(1495,\cdot)\) \(\chi_{8041}(1682,\cdot)\) \(\chi_{8041}(1869,\cdot)\) \(\chi_{8041}(3552,\cdot)\) \(\chi_{8041}(4113,\cdot)\) \(\chi_{8041}(4674,\cdot)\) \(\chi_{8041}(5609,\cdot)\) \(\chi_{8041}(5796,\cdot)\) \(\chi_{8041}(7666,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((6580,2366,562)\) → \((-1,-1,e\left(\frac{31}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(1495, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)