from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,43]))
pari: [g,chi] = znchar(Mod(2024,6025))
Basic properties
Modulus: | \(6025\) | |
Conductor: | \(1205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1205}(819,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6025.fo
\(\chi_{6025}(324,\cdot)\) \(\chi_{6025}(399,\cdot)\) \(\chi_{6025}(874,\cdot)\) \(\chi_{6025}(1349,\cdot)\) \(\chi_{6025}(2024,\cdot)\) \(\chi_{6025}(2774,\cdot)\) \(\chi_{6025}(2974,\cdot)\) \(\chi_{6025}(2999,\cdot)\) \(\chi_{6025}(3124,\cdot)\) \(\chi_{6025}(3624,\cdot)\) \(\chi_{6025}(3749,\cdot)\) \(\chi_{6025}(3774,\cdot)\) \(\chi_{6025}(3974,\cdot)\) \(\chi_{6025}(4724,\cdot)\) \(\chi_{6025}(5399,\cdot)\) \(\chi_{6025}(5874,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2652,2176)\) → \((-1,e\left(\frac{43}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(2024, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) |
sage: chi.jacobi_sum(n)