from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,1,8,4]))
pari: [g,chi] = znchar(Mod(251,9792))
Basic properties
Modulus: | \(9792\) | |
Conductor: | \(3264\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3264}(251,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.gv
\(\chi_{9792}(251,\cdot)\) \(\chi_{9792}(1619,\cdot)\) \(\chi_{9792}(2699,\cdot)\) \(\chi_{9792}(4067,\cdot)\) \(\chi_{9792}(5147,\cdot)\) \(\chi_{9792}(6515,\cdot)\) \(\chi_{9792}(7595,\cdot)\) \(\chi_{9792}(8963,\cdot)\)
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: | 16.16.2310610548926162912887536114323161557762048.2 |
Values on generators
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{1}{16}\right),-1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)