from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([54,41]))
pari: [g,chi] = znchar(Mod(144,6025))
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.gs
\(\chi_{6025}(144,\cdot)\) \(\chi_{6025}(364,\cdot)\) \(\chi_{6025}(1109,\cdot)\) \(\chi_{6025}(1564,\cdot)\) \(\chi_{6025}(1919,\cdot)\) \(\chi_{6025}(2259,\cdot)\) \(\chi_{6025}(2419,\cdot)\) \(\chi_{6025}(3284,\cdot)\) \(\chi_{6025}(3939,\cdot)\) \(\chi_{6025}(4014,\cdot)\) \(\chi_{6025}(4179,\cdot)\) \(\chi_{6025}(4194,\cdot)\) \(\chi_{6025}(4204,\cdot)\) \(\chi_{6025}(4434,\cdot)\) \(\chi_{6025}(4954,\cdot)\) \(\chi_{6025}(4979,\cdot)\)
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
\((2652,2176)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{41}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 6025 }(144, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) |
sage: chi.jacobi_sum(n)