from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,21,40,36]))
pari: [g,chi] = znchar(Mod(53,5775))
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.lk
\(\chi_{5775}(53,\cdot)\) \(\chi_{5775}(1292,\cdot)\) \(\chi_{5775}(1313,\cdot)\) \(\chi_{5775}(1472,\cdot)\) \(\chi_{5775}(1598,\cdot)\) \(\chi_{5775}(1703,\cdot)\) \(\chi_{5775}(2027,\cdot)\) \(\chi_{5775}(2237,\cdot)\) \(\chi_{5775}(2942,\cdot)\) \(\chi_{5775}(2963,\cdot)\) \(\chi_{5775}(3677,\cdot)\) \(\chi_{5775}(3833,\cdot)\) \(\chi_{5775}(3887,\cdot)\) \(\chi_{5775}(5483,\cdot)\) \(\chi_{5775}(5597,\cdot)\) \(\chi_{5775}(5723,\cdot)\)
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
\((3851,4852,4126,3676)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 5775 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)