from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,15]))
pari: [g,chi] = znchar(Mod(1078,8021))
Basic properties
| Modulus: | \(8021\) | |
| Conductor: | \(8021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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.ca
\(\chi_{8021}(1078,\cdot)\) \(\chi_{8021}(1390,\cdot)\) \(\chi_{8021}(2625,\cdot)\) \(\chi_{8021}(2820,\cdot)\) \(\chi_{8021}(3717,\cdot)\) \(\chi_{8021}(3990,\cdot)\) \(\chi_{8021}(4029,\cdot)\) \(\chi_{8021}(4250,\cdot)\) \(\chi_{8021}(4419,\cdot)\) \(\chi_{8021}(4471,\cdot)\) \(\chi_{8021}(5160,\cdot)\) \(\chi_{8021}(5329,\cdot)\) \(\chi_{8021}(6018,\cdot)\) \(\chi_{8021}(6070,\cdot)\) \(\chi_{8021}(6239,\cdot)\) \(\chi_{8021}(6460,\cdot)\) \(\chi_{8021}(6499,\cdot)\) \(\chi_{8021}(6772,\cdot)\) \(\chi_{8021}(7669,\cdot)\) \(\chi_{8021}(7864,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((6788,2471)\) → \((-1,e\left(\frac{15}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8021 }(1078, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)