from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,17]))
pari: [g,chi] = znchar(Mod(3976,6025))
Basic properties
Modulus: | \(6025\) | |
Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{241}(120,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6025.dr
\(\chi_{6025}(851,\cdot)\) \(\chi_{6025}(1326,\cdot)\) \(\chi_{6025}(1926,\cdot)\) \(\chi_{6025}(2201,\cdot)\) \(\chi_{6025}(3101,\cdot)\) \(\chi_{6025}(3376,\cdot)\) \(\chi_{6025}(3976,\cdot)\) \(\chi_{6025}(4451,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((2652,2176)\) → \((1,e\left(\frac{17}{24}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(3976, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{17}{24}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) |
sage: chi.jacobi_sum(n)