from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8027, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([16,0]))
pari: [g,chi] = znchar(Mod(4189,8027))
Basic properties
| Modulus: | \(8027\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{23}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8027.k
\(\chi_{8027}(699,\cdot)\) \(\chi_{8027}(1048,\cdot)\) \(\chi_{8027}(2095,\cdot)\) \(\chi_{8027}(2444,\cdot)\) \(\chi_{8027}(3491,\cdot)\) \(\chi_{8027}(4189,\cdot)\) \(\chi_{8027}(6283,\cdot)\) \(\chi_{8027}(6632,\cdot)\) \(\chi_{8027}(6981,\cdot)\) \(\chi_{8027}(7330,\cdot)\)
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: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((350,5935)\) → \((e\left(\frac{8}{11}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8027 }(4189, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)