from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,8]))
pari: [g,chi] = znchar(Mod(1106,8021))
Basic properties
| Modulus: | \(8021\) | |
| Conductor: | \(617\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{617}(489,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8021.t
\(\chi_{8021}(1106,\cdot)\) \(\chi_{8021}(1626,\cdot)\) \(\chi_{8021}(1652,\cdot)\) \(\chi_{8021}(1964,\cdot)\) \(\chi_{8021}(4044,\cdot)\) \(\chi_{8021}(4967,\cdot)\) \(\chi_{8021}(6345,\cdot)\) \(\chi_{8021}(6514,\cdot)\) \(\chi_{8021}(7138,\cdot)\) \(\chi_{8021}(7216,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 11.11.7995610696842977761957082449.1 |
Values on generators
\((6788,2471)\) → \((1,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(1106, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)