from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([24,53]))
pari: [g,chi] = znchar(Mod(331,6025))
Basic properties
Modulus: | \(6025\) | |
Conductor: | \(6025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 6025.fp
\(\chi_{6025}(331,\cdot)\) \(\chi_{6025}(616,\cdot)\) \(\chi_{6025}(641,\cdot)\) \(\chi_{6025}(846,\cdot)\) \(\chi_{6025}(1061,\cdot)\) \(\chi_{6025}(1356,\cdot)\) \(\chi_{6025}(2046,\cdot)\) \(\chi_{6025}(2506,\cdot)\) \(\chi_{6025}(3036,\cdot)\) \(\chi_{6025}(4421,\cdot)\) \(\chi_{6025}(4496,\cdot)\) \(\chi_{6025}(4811,\cdot)\) \(\chi_{6025}(5206,\cdot)\) \(\chi_{6025}(5311,\cdot)\) \(\chi_{6025}(5866,\cdot)\) \(\chi_{6025}(5891,\cdot)\)
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
\((2652,2176)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{53}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(331, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)