from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,12,25,3]))
pari: [g,chi] = znchar(Mod(1531,5775))
Basic properties
| Modulus: | \(5775\) | |
| Conductor: | \(1925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1925}(1531,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5775.kw
\(\chi_{5775}(1531,\cdot)\) \(\chi_{5775}(2371,\cdot)\) \(\chi_{5775}(2791,\cdot)\) \(\chi_{5775}(3181,\cdot)\) \(\chi_{5775}(3736,\cdot)\) \(\chi_{5775}(4021,\cdot)\) \(\chi_{5775}(4441,\cdot)\) \(\chi_{5775}(5386,\cdot)\)
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
\((3851,4852,4126,3676)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 5775 }(1531, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)