from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([6,13]))
pari: [g,chi] = znchar(Mod(3393,6001))
Basic properties
Modulus: | \(6001\) | |
Conductor: | \(6001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 6001.cb
\(\chi_{6001}(252,\cdot)\) \(\chi_{6001}(420,\cdot)\) \(\chi_{6001}(700,\cdot)\) \(\chi_{6001}(843,\cdot)\) \(\chi_{6001}(1405,\cdot)\) \(\chi_{6001}(1706,\cdot)\) \(\chi_{6001}(1771,\cdot)\) \(\chi_{6001}(3167,\cdot)\) \(\chi_{6001}(3236,\cdot)\) \(\chi_{6001}(3278,\cdot)\) \(\chi_{6001}(3393,\cdot)\) \(\chi_{6001}(3424,\cdot)\) \(\chi_{6001}(3463,\cdot)\) \(\chi_{6001}(4342,\cdot)\) \(\chi_{6001}(4952,\cdot)\) \(\chi_{6001}(5655,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((2825,3180)\) → \((e\left(\frac{3}{16}\right),e\left(\frac{13}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6001 }(3393, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{19}{32}\right)\) | \(-1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)