from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,36]))
pari: [g,chi] = znchar(Mod(451,8021))
Basic properties
| Modulus: | \(8021\) | |
| Conductor: | \(8021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | 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 8021.bd
\(\chi_{8021}(451,\cdot)\) \(\chi_{8021}(555,\cdot)\) \(\chi_{8021}(1108,\cdot)\) \(\chi_{8021}(1654,\cdot)\) \(\chi_{8021}(1725,\cdot)\) \(\chi_{8021}(2271,\cdot)\) \(\chi_{8021}(2876,\cdot)\) \(\chi_{8021}(3227,\cdot)\) \(\chi_{8021}(3493,\cdot)\) \(\chi_{8021}(3844,\cdot)\) \(\chi_{8021}(7855,\cdot)\) \(\chi_{8021}(7959,\cdot)\)
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 21 polynomial |
Values on generators
\((6788,2471)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{6}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(451, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)