from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,31]))
pari: [g,chi] = znchar(Mod(195,6013))
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.bp
\(\chi_{6013}(195,\cdot)\) \(\chi_{6013}(300,\cdot)\) \(\chi_{6013}(461,\cdot)\) \(\chi_{6013}(839,\cdot)\) \(\chi_{6013}(874,\cdot)\) \(\chi_{6013}(1357,\cdot)\) \(\chi_{6013}(1672,\cdot)\) \(\chi_{6013}(2351,\cdot)\) \(\chi_{6013}(2596,\cdot)\) \(\chi_{6013}(2890,\cdot)\) \(\chi_{6013}(3394,\cdot)\) \(\chi_{6013}(3492,\cdot)\) \(\chi_{6013}(3947,\cdot)\) \(\chi_{6013}(4066,\cdot)\) \(\chi_{6013}(4164,\cdot)\) \(\chi_{6013}(4542,\cdot)\) \(\chi_{6013}(4759,\cdot)\) \(\chi_{6013}(5613,\cdot)\) \(\chi_{6013}(5788,\cdot)\) \(\chi_{6013}(5970,\cdot)\)
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
\((5155,3438)\) → \((-1,e\left(\frac{31}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6013 }(195, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)