from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,33]))
pari: [g,chi] = znchar(Mod(62,8021))
Basic properties
| Modulus: | \(8021\) | |
| Conductor: | \(8021\) | 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 8021.bs
\(\chi_{8021}(62,\cdot)\) \(\chi_{8021}(166,\cdot)\) \(\chi_{8021}(4177,\cdot)\) \(\chi_{8021}(4528,\cdot)\) \(\chi_{8021}(4794,\cdot)\) \(\chi_{8021}(5145,\cdot)\) \(\chi_{8021}(5750,\cdot)\) \(\chi_{8021}(6296,\cdot)\) \(\chi_{8021}(6367,\cdot)\) \(\chi_{8021}(6913,\cdot)\) \(\chi_{8021}(7466,\cdot)\) \(\chi_{8021}(7570,\cdot)\)
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
\((6788,2471)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(62, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{42}\right)\) |
sage: chi.jacobi_sum(n)