from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,39]))
pari: [g,chi] = znchar(Mod(66,6013))
Basic properties
| Modulus: | \(6013\) | |
| Conductor: | \(6013\) | 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 6013.bo
\(\chi_{6013}(66,\cdot)\) \(\chi_{6013}(654,\cdot)\) \(\chi_{6013}(1433,\cdot)\) \(\chi_{6013}(1657,\cdot)\) \(\chi_{6013}(2098,\cdot)\) \(\chi_{6013}(2292,\cdot)\) \(\chi_{6013}(2308,\cdot)\) \(\chi_{6013}(2516,\cdot)\) \(\chi_{6013}(2957,\cdot)\) \(\chi_{6013}(3167,\cdot)\) \(\chi_{6013}(3267,\cdot)\) \(\chi_{6013}(4126,\cdot)\) \(\chi_{6013}(4282,\cdot)\) \(\chi_{6013}(4940,\cdot)\) \(\chi_{6013}(5066,\cdot)\) \(\chi_{6013}(5141,\cdot)\) \(\chi_{6013}(5220,\cdot)\) \(\chi_{6013}(5799,\cdot)\) \(\chi_{6013}(5808,\cdot)\) \(\chi_{6013}(5925,\cdot)\)
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
\((5155,3438)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6013 }(66, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)