from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,36,35]))
pari: [g,chi] = znchar(Mod(314,4015))
Basic properties
Modulus: | \(4015\) | |
Conductor: | \(4015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | 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.eb
\(\chi_{4015}(314,\cdot)\) \(\chi_{4015}(574,\cdot)\) \(\chi_{4015}(679,\cdot)\) \(\chi_{4015}(854,\cdot)\) \(\chi_{4015}(959,\cdot)\) \(\chi_{4015}(1304,\cdot)\) \(\chi_{4015}(1669,\cdot)\) \(\chi_{4015}(1689,\cdot)\) \(\chi_{4015}(1949,\cdot)\) \(\chi_{4015}(2054,\cdot)\) \(\chi_{4015}(2504,\cdot)\) \(\chi_{4015}(2679,\cdot)\) \(\chi_{4015}(3044,\cdot)\) \(\chi_{4015}(3494,\cdot)\) \(\chi_{4015}(3599,\cdot)\) \(\chi_{4015}(3879,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((1607,2191,881)\) → \((-1,e\left(\frac{9}{10}\right),e\left(\frac{7}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 4015 }(314, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
sage: chi.jacobi_sum(n)