from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,21]))
pari: [g,chi] = znchar(Mod(3299,6001))
Basic properties
Modulus: | \(6001\) | |
Conductor: | \(353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{353}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6001.bw
\(\chi_{6001}(222,\cdot)\) \(\chi_{6001}(919,\cdot)\) \(\chi_{6001}(1225,\cdot)\) \(\chi_{6001}(1548,\cdot)\) \(\chi_{6001}(2449,\cdot)\) \(\chi_{6001}(2568,\cdot)\) \(\chi_{6001}(2840,\cdot)\) \(\chi_{6001}(3299,\cdot)\) \(\chi_{6001}(4404,\cdot)\) \(\chi_{6001}(5237,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((2825,3180)\) → \((1,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6001 }(3299, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(-1\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)