from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,50]))
pari: [g,chi] = znchar(Mod(46,6013))
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.bc
\(\chi_{6013}(46,\cdot)\) \(\chi_{6013}(226,\cdot)\) \(\chi_{6013}(1220,\cdot)\) \(\chi_{6013}(1418,\cdot)\) \(\chi_{6013}(1703,\cdot)\) \(\chi_{6013}(1738,\cdot)\) \(\chi_{6013}(1761,\cdot)\) \(\chi_{6013}(1943,\cdot)\) \(\chi_{6013}(2116,\cdot)\) \(\chi_{6013}(2118,\cdot)\) \(\chi_{6013}(2382,\cdot)\) \(\chi_{6013}(2972,\cdot)\) \(\chi_{6013}(3189,\cdot)\) \(\chi_{6013}(3665,\cdot)\) \(\chi_{6013}(3784,\cdot)\) \(\chi_{6013}(4426,\cdot)\) \(\chi_{6013}(5098,\cdot)\) \(\chi_{6013}(5135,\cdot)\) \(\chi_{6013}(5196,\cdot)\) \(\chi_{6013}(5700,\cdot)\)
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
\((5155,3438)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{25}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6013 }(46, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)