from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,39]))
pari: [g,chi] = znchar(Mod(135,6001))
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.cp
\(\chi_{6001}(135,\cdot)\) \(\chi_{6001}(441,\cdot)\) \(\chi_{6001}(560,\cdot)\) \(\chi_{6001}(1291,\cdot)\) \(\chi_{6001}(1886,\cdot)\) \(\chi_{6001}(2617,\cdot)\) \(\chi_{6001}(2736,\cdot)\) \(\chi_{6001}(3042,\cdot)\) \(\chi_{6001}(3212,\cdot)\) \(\chi_{6001}(3348,\cdot)\) \(\chi_{6001}(4045,\cdot)\) \(\chi_{6001}(4232,\cdot)\) \(\chi_{6001}(4300,\cdot)\) \(\chi_{6001}(4555,\cdot)\) \(\chi_{6001}(4623,\cdot)\) \(\chi_{6001}(4878,\cdot)\) \(\chi_{6001}(4946,\cdot)\) \(\chi_{6001}(5133,\cdot)\) \(\chi_{6001}(5830,\cdot)\) \(\chi_{6001}(5966,\cdot)\)
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
\((2825,3180)\) → \((-1,e\left(\frac{39}{44}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6001 }(135, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(-i\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)