from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,25,24]))
pari: [g,chi] = znchar(Mod(775,1148))
Basic properties
Modulus: | \(1148\) | |
Conductor: | \(1148\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | 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 1148.bu
\(\chi_{1148}(59,\cdot)\) \(\chi_{1148}(215,\cdot)\) \(\chi_{1148}(283,\cdot)\) \(\chi_{1148}(467,\cdot)\) \(\chi_{1148}(551,\cdot)\) \(\chi_{1148}(775,\cdot)\) \(\chi_{1148}(871,\cdot)\) \(\chi_{1148}(1123,\cdot)\)
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 30 polynomial |
Values on generators
\((575,493,785)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1148 }(775, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)