from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8027, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([42,55]))
pari: [g,chi] = znchar(Mod(2566,8027))
Basic properties
| Modulus: | \(8027\) | |
| Conductor: | \(8027\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 8027.y
\(\chi_{8027}(123,\cdot)\) \(\chi_{8027}(472,\cdot)\) \(\chi_{8027}(821,\cdot)\) \(\chi_{8027}(1274,\cdot)\) \(\chi_{8027}(1623,\cdot)\) \(\chi_{8027}(2217,\cdot)\) \(\chi_{8027}(2566,\cdot)\) \(\chi_{8027}(2670,\cdot)\) \(\chi_{8027}(3019,\cdot)\) \(\chi_{8027}(3613,\cdot)\) \(\chi_{8027}(3962,\cdot)\) \(\chi_{8027}(4066,\cdot)\) \(\chi_{8027}(4764,\cdot)\) \(\chi_{8027}(5009,\cdot)\) \(\chi_{8027}(5707,\cdot)\) \(\chi_{8027}(6858,\cdot)\) \(\chi_{8027}(7207,\cdot)\) \(\chi_{8027}(7556,\cdot)\) \(\chi_{8027}(7801,\cdot)\) \(\chi_{8027}(7905,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((350,5935)\) → \((e\left(\frac{7}{11}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8027 }(2566, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)