from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
pari: [g,chi] = znchar(Mod(833,6019))
Basic properties
| Modulus: | \(6019\) | |
| Conductor: | \(463\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{463}(370,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6019.bu
\(\chi_{6019}(833,\cdot)\) \(\chi_{6019}(1054,\cdot)\) \(\chi_{6019}(1457,\cdot)\) \(\chi_{6019}(1483,\cdot)\) \(\chi_{6019}(1522,\cdot)\) \(\chi_{6019}(1704,\cdot)\) \(\chi_{6019}(2081,\cdot)\) \(\chi_{6019}(2354,\cdot)\) \(\chi_{6019}(2458,\cdot)\) \(\chi_{6019}(2900,\cdot)\) \(\chi_{6019}(3277,\cdot)\) \(\chi_{6019}(3420,\cdot)\) \(\chi_{6019}(3836,\cdot)\) \(\chi_{6019}(4161,\cdot)\) \(\chi_{6019}(4395,\cdot)\) \(\chi_{6019}(4460,\cdot)\) \(\chi_{6019}(4707,\cdot)\) \(\chi_{6019}(4824,\cdot)\) \(\chi_{6019}(5188,\cdot)\) \(\chi_{6019}(5929,\cdot)\)
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
\((2316,1392)\) → \((1,e\left(\frac{10}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6019 }(833, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)