from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,7,4]))
pari: [g,chi] = znchar(Mod(11,5185))
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1037}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5185.ec
\(\chi_{5185}(11,\cdot)\) \(\chi_{5185}(316,\cdot)\) \(\chi_{5185}(1231,\cdot)\) \(\chi_{5185}(1331,\cdot)\) \(\chi_{5185}(1536,\cdot)\) \(\chi_{5185}(1941,\cdot)\) \(\chi_{5185}(3771,\cdot)\) \(\chi_{5185}(4381,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.7597869690780846180466826362661244163553.2 |
Values on generators
\((3112,4576,2381)\) → \((1,e\left(\frac{7}{16}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 5185 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)