from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,8]))
pari: [g,chi] = znchar(Mod(9,5185))
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(5185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 5185.gt
\(\chi_{5185}(9,\cdot)\) \(\chi_{5185}(264,\cdot)\) \(\chi_{5185}(314,\cdot)\) \(\chi_{5185}(569,\cdot)\) \(\chi_{5185}(644,\cdot)\) \(\chi_{5185}(729,\cdot)\) \(\chi_{5185}(2144,\cdot)\) \(\chi_{5185}(2399,\cdot)\) \(\chi_{5185}(2474,\cdot)\) \(\chi_{5185}(2559,\cdot)\) \(\chi_{5185}(2779,\cdot)\) \(\chi_{5185}(2864,\cdot)\) \(\chi_{5185}(3364,\cdot)\) \(\chi_{5185}(3619,\cdot)\) \(\chi_{5185}(4609,\cdot)\) \(\chi_{5185}(4694,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((3112,4576,2381)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 5185 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)