from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,19]))
pari: [g,chi] = znchar(Mod(1470,8021))
Basic properties
| Modulus: | \(8021\) | |
| Conductor: | \(617\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{617}(236,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8021.bl
\(\chi_{8021}(1470,\cdot)\) \(\chi_{8021}(2068,\cdot)\) \(\chi_{8021}(2432,\cdot)\) \(\chi_{8021}(2588,\cdot)\) \(\chi_{8021}(2965,\cdot)\) \(\chi_{8021}(3121,\cdot)\) \(\chi_{8021}(3485,\cdot)\) \(\chi_{8021}(4083,\cdot)\) \(\chi_{8021}(5994,\cdot)\) \(\chi_{8021}(6475,\cdot)\) \(\chi_{8021}(7099,\cdot)\) \(\chi_{8021}(7580,\cdot)\)
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
\((6788,2471)\) → \((1,e\left(\frac{19}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(1470, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(i\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)