from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,12]))
pari: [g,chi] = znchar(Mod(1543,6001))
Basic properties
Modulus: | \(6001\) | |
Conductor: | \(6001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | 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.cr
\(\chi_{6001}(140,\cdot)\) \(\chi_{6001}(217,\cdot)\) \(\chi_{6001}(540,\cdot)\) \(\chi_{6001}(837,\cdot)\) \(\chi_{6001}(846,\cdot)\) \(\chi_{6001}(1543,\cdot)\) \(\chi_{6001}(1823,\cdot)\) \(\chi_{6001}(2529,\cdot)\) \(\chi_{6001}(2656,\cdot)\) \(\chi_{6001}(3362,\cdot)\) \(\chi_{6001}(3761,\cdot)\) \(\chi_{6001}(4220,\cdot)\) \(\chi_{6001}(4467,\cdot)\) \(\chi_{6001}(4492,\cdot)\) \(\chi_{6001}(4611,\cdot)\) \(\chi_{6001}(4926,\cdot)\) \(\chi_{6001}(5198,\cdot)\) \(\chi_{6001}(5317,\cdot)\) \(\chi_{6001}(5512,\cdot)\) \(\chi_{6001}(5835,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((2825,3180)\) → \((i,e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6001 }(1543, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(-i\) | \(e\left(\frac{5}{44}\right)\) |
sage: chi.jacobi_sum(n)