from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([24,20,12]))
pari: [g,chi] = znchar(Mod(1061,1925))
Basic properties
Modulus: | \(1925\) | |
Conductor: | \(1925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(15\) | 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 1925.cz
\(\chi_{1925}(81,\cdot)\) \(\chi_{1925}(786,\cdot)\) \(\chi_{1925}(1061,\cdot)\) \(\chi_{1925}(1241,\cdot)\) \(\chi_{1925}(1516,\cdot)\) \(\chi_{1925}(1521,\cdot)\) \(\chi_{1925}(1731,\cdot)\) \(\chi_{1925}(1796,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((1002,276,1751)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 1925 }(1061, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)