from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([12,5]))
pari: [g,chi] = znchar(Mod(4493,7225))
Basic properties
| Modulus: | \(7225\) | |
| Conductor: | \(85\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{85}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7225.s
\(\chi_{7225}(643,\cdot)\) \(\chi_{7225}(907,\cdot)\) \(\chi_{7225}(2443,\cdot)\) \(\chi_{7225}(3682,\cdot)\) \(\chi_{7225}(3832,\cdot)\) \(\chi_{7225}(4493,\cdot)\) \(\chi_{7225}(6293,\cdot)\) \(\chi_{7225}(6607,\cdot)\)
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: | 16.16.698833752810013621337890625.2 |
Values on generators
\((2602,2026)\) → \((-i,e\left(\frac{5}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 7225 }(4493, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)