from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(3384,6001))
Basic properties
| Modulus: | \(6001\) | |
| Conductor: | \(353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{353}(207,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6001.co
\(\chi_{6001}(35,\cdot)\) \(\chi_{6001}(171,\cdot)\) \(\chi_{6001}(868,\cdot)\) \(\chi_{6001}(1055,\cdot)\) \(\chi_{6001}(1123,\cdot)\) \(\chi_{6001}(1378,\cdot)\) \(\chi_{6001}(1446,\cdot)\) \(\chi_{6001}(1701,\cdot)\) \(\chi_{6001}(1769,\cdot)\) \(\chi_{6001}(1956,\cdot)\) \(\chi_{6001}(2653,\cdot)\) \(\chi_{6001}(2789,\cdot)\) \(\chi_{6001}(2959,\cdot)\) \(\chi_{6001}(3265,\cdot)\) \(\chi_{6001}(3384,\cdot)\) \(\chi_{6001}(4115,\cdot)\) \(\chi_{6001}(4710,\cdot)\) \(\chi_{6001}(5441,\cdot)\) \(\chi_{6001}(5560,\cdot)\) \(\chi_{6001}(5866,\cdot)\)
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
\((2825,3180)\) → \((1,e\left(\frac{1}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6001 }(3384, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)