from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13]))
pari: [g,chi] = znchar(Mod(998,7803))
Basic properties
Modulus: | \(7803\) | |
Conductor: | \(51\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{51}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7803.p
\(\chi_{7803}(998,\cdot)\) \(\chi_{7803}(1025,\cdot)\) \(\chi_{7803}(3104,\cdot)\) \(\chi_{7803}(3428,\cdot)\) \(\chi_{7803}(4400,\cdot)\) \(\chi_{7803}(5426,\cdot)\) \(\chi_{7803}(6398,\cdot)\) \(\chi_{7803}(6722,\cdot)\)
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: | \(\Q(\zeta_{51})^+\) |
Values on generators
\((2891,2026)\) → \((-1,e\left(\frac{13}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 7803 }(998, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)