from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,52]))
pari: [g,chi] = znchar(Mod(68,6019))
Basic properties
Modulus: | \(6019\) | |
Conductor: | \(6019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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 6019.bv
\(\chi_{6019}(68,\cdot)\) \(\chi_{6019}(94,\cdot)\) \(\chi_{6019}(315,\cdot)\) \(\chi_{6019}(373,\cdot)\) \(\chi_{6019}(692,\cdot)\) \(\chi_{6019}(1296,\cdot)\) \(\chi_{6019}(1888,\cdot)\) \(\chi_{6019}(1985,\cdot)\) \(\chi_{6019}(2447,\cdot)\) \(\chi_{6019}(2817,\cdot)\) \(\chi_{6019}(2921,\cdot)\) \(\chi_{6019}(3318,\cdot)\) \(\chi_{6019}(3363,\cdot)\) \(\chi_{6019}(3435,\cdot)\) \(\chi_{6019}(3799,\cdot)\) \(\chi_{6019}(3883,\cdot)\) \(\chi_{6019}(4624,\cdot)\) \(\chi_{6019}(4858,\cdot)\) \(\chi_{6019}(4923,\cdot)\) \(\chi_{6019}(5684,\cdot)\)
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 33 polynomial |
Values on generators
\((2316,1392)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{26}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6019 }(68, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) |
sage: chi.jacobi_sum(n)