from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,24]))
pari: [g,chi] = znchar(Mod(55,6019))
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.bt
\(\chi_{6019}(55,\cdot)\) \(\chi_{6019}(425,\cdot)\) \(\chi_{6019}(1173,\cdot)\) \(\chi_{6019}(1745,\cdot)\) \(\chi_{6019}(1751,\cdot)\) \(\chi_{6019}(1907,\cdot)\) \(\chi_{6019}(2330,\cdot)\) \(\chi_{6019}(2473,\cdot)\) \(\chi_{6019}(3025,\cdot)\) \(\chi_{6019}(3578,\cdot)\) \(\chi_{6019}(3597,\cdot)\) \(\chi_{6019}(3838,\cdot)\) \(\chi_{6019}(3929,\cdot)\) \(\chi_{6019}(4182,\cdot)\) \(\chi_{6019}(4325,\cdot)\) \(\chi_{6019}(4592,\cdot)\) \(\chi_{6019}(5430,\cdot)\) \(\chi_{6019}(5690,\cdot)\) \(\chi_{6019}(5781,\cdot)\) \(\chi_{6019}(5918,\cdot)\)
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
\((2316,1392)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6019 }(55, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) |
sage: chi.jacobi_sum(n)