from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,24,55]))
pari: [g,chi] = znchar(Mod(49,4015))
Basic properties
Modulus: | \(4015\) | |
Conductor: | \(4015\) | 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 4015.ej
\(\chi_{4015}(49,\cdot)\) \(\chi_{4015}(289,\cdot)\) \(\chi_{4015}(389,\cdot)\) \(\chi_{4015}(654,\cdot)\) \(\chi_{4015}(779,\cdot)\) \(\chi_{4015}(1384,\cdot)\) \(\chi_{4015}(1609,\cdot)\) \(\chi_{4015}(1874,\cdot)\) \(\chi_{4015}(1974,\cdot)\) \(\chi_{4015}(2214,\cdot)\) \(\chi_{4015}(2479,\cdot)\) \(\chi_{4015}(2579,\cdot)\) \(\chi_{4015}(2704,\cdot)\) \(\chi_{4015}(3309,\cdot)\) \(\chi_{4015}(3699,\cdot)\) \(\chi_{4015}(3799,\cdot)\)
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
\((1607,2191,881)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{11}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 4015 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)