from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([12,41]))
pari: [g,chi] = znchar(Mod(2632,6025))
Basic properties
Modulus: | \(6025\) | |
Conductor: | \(1205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1205}(222,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6025.fm
\(\chi_{6025}(393,\cdot)\) \(\chi_{6025}(1243,\cdot)\) \(\chi_{6025}(1293,\cdot)\) \(\chi_{6025}(1457,\cdot)\) \(\chi_{6025}(1468,\cdot)\) \(\chi_{6025}(2232,\cdot)\) \(\chi_{6025}(2632,\cdot)\) \(\chi_{6025}(2957,\cdot)\) \(\chi_{6025}(3593,\cdot)\) \(\chi_{6025}(3768,\cdot)\) \(\chi_{6025}(3818,\cdot)\) \(\chi_{6025}(4032,\cdot)\) \(\chi_{6025}(4357,\cdot)\) \(\chi_{6025}(4668,\cdot)\) \(\chi_{6025}(4757,\cdot)\) \(\chi_{6025}(5532,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2652,2176)\) → \((i,e\left(\frac{41}{48}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(2632, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) |
sage: chi.jacobi_sum(n)