from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3968, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,15,16]))
pari: [g,chi] = znchar(Mod(1549,3968))
Basic properties
Modulus: | \(3968\) | |
Conductor: | \(3968\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3968.cf
\(\chi_{3968}(61,\cdot)\) \(\chi_{3968}(309,\cdot)\) \(\chi_{3968}(557,\cdot)\) \(\chi_{3968}(805,\cdot)\) \(\chi_{3968}(1053,\cdot)\) \(\chi_{3968}(1301,\cdot)\) \(\chi_{3968}(1549,\cdot)\) \(\chi_{3968}(1797,\cdot)\) \(\chi_{3968}(2045,\cdot)\) \(\chi_{3968}(2293,\cdot)\) \(\chi_{3968}(2541,\cdot)\) \(\chi_{3968}(2789,\cdot)\) \(\chi_{3968}(3037,\cdot)\) \(\chi_{3968}(3285,\cdot)\) \(\chi_{3968}(3533,\cdot)\) \(\chi_{3968}(3781,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | 32.0.2283054469939826689646603434893544205211768622361793362696380065480141214463295488.1 |
Values on generators
\((2047,3845,2049)\) → \((1,e\left(\frac{15}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 3968 }(1549, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) |
sage: chi.jacobi_sum(n)