from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,24,5,24]))
pari: [g,chi] = znchar(Mod(311,5775))
Basic properties
Modulus: | \(5775\) | |
Conductor: | \(5775\) | 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 5775.hz
\(\chi_{5775}(311,\cdot)\) \(\chi_{5775}(731,\cdot)\) \(\chi_{5775}(1571,\cdot)\) \(\chi_{5775}(3491,\cdot)\) \(\chi_{5775}(4436,\cdot)\) \(\chi_{5775}(4856,\cdot)\) \(\chi_{5775}(5141,\cdot)\) \(\chi_{5775}(5696,\cdot)\)
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
\((3851,4852,4126,3676)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 5775 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)