from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,28]))
pari: [g,chi] = znchar(Mod(879,6013))
Basic properties
Modulus: | \(6013\) | |
Conductor: | \(6013\) | 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 6013.bd
\(\chi_{6013}(361,\cdot)\) \(\chi_{6013}(844,\cdot)\) \(\chi_{6013}(879,\cdot)\) \(\chi_{6013}(905,\cdot)\) \(\chi_{6013}(1257,\cdot)\) \(\chi_{6013}(1523,\cdot)\) \(\chi_{6013}(2277,\cdot)\) \(\chi_{6013}(2620,\cdot)\) \(\chi_{6013}(2802,\cdot)\) \(\chi_{6013}(2977,\cdot)\) \(\chi_{6013}(3567,\cdot)\) \(\chi_{6013}(3831,\cdot)\) \(\chi_{6013}(4048,\cdot)\) \(\chi_{6013}(4239,\cdot)\) \(\chi_{6013}(4337,\cdot)\) \(\chi_{6013}(4524,\cdot)\) \(\chi_{6013}(4643,\cdot)\) \(\chi_{6013}(4841,\cdot)\) \(\chi_{6013}(5380,\cdot)\) \(\chi_{6013}(5994,\cdot)\)
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
\((5155,3438)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{14}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 6013 }(879, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)