from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6019, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,28]))
pari: [g,chi] = znchar(Mod(77,6019))
Basic properties
| Modulus: | \(6019\) | |
| Conductor: | \(6019\) | 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 6019.cp
\(\chi_{6019}(77,\cdot)\) \(\chi_{6019}(194,\cdot)\) \(\chi_{6019}(558,\cdot)\) \(\chi_{6019}(1299,\cdot)\) \(\chi_{6019}(2222,\cdot)\) \(\chi_{6019}(2443,\cdot)\) \(\chi_{6019}(2846,\cdot)\) \(\chi_{6019}(2872,\cdot)\) \(\chi_{6019}(2911,\cdot)\) \(\chi_{6019}(3093,\cdot)\) \(\chi_{6019}(3470,\cdot)\) \(\chi_{6019}(3743,\cdot)\) \(\chi_{6019}(3847,\cdot)\) \(\chi_{6019}(4289,\cdot)\) \(\chi_{6019}(4666,\cdot)\) \(\chi_{6019}(4809,\cdot)\) \(\chi_{6019}(5225,\cdot)\) \(\chi_{6019}(5550,\cdot)\) \(\chi_{6019}(5784,\cdot)\) \(\chi_{6019}(5849,\cdot)\)
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
\((2316,1392)\) → \((-1,e\left(\frac{14}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6019 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) |
sage: chi.jacobi_sum(n)