from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,9,40,12]))
pari: [g,chi] = znchar(Mod(158,5775))
Basic properties
| Modulus: | \(5775\) | |
| Conductor: | \(5775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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.lj
\(\chi_{5775}(158,\cdot)\) \(\chi_{5775}(872,\cdot)\) \(\chi_{5775}(1103,\cdot)\) \(\chi_{5775}(1523,\cdot)\) \(\chi_{5775}(1808,\cdot)\) \(\chi_{5775}(2363,\cdot)\) \(\chi_{5775}(2522,\cdot)\) \(\chi_{5775}(2753,\cdot)\) \(\chi_{5775}(3173,\cdot)\) \(\chi_{5775}(3287,\cdot)\) \(\chi_{5775}(3392,\cdot)\) \(\chi_{5775}(3602,\cdot)\) \(\chi_{5775}(4013,\cdot)\) \(\chi_{5775}(4937,\cdot)\) \(\chi_{5775}(5042,\cdot)\) \(\chi_{5775}(5252,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((3851,4852,4126,3676)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 5775 }(158, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)