from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3520, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,5,8,0]))
pari: [g,chi] = znchar(Mod(309,3520))
Basic properties
Modulus: | \(3520\) | |
Conductor: | \(320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{320}(309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3520.cz
\(\chi_{3520}(309,\cdot)\) \(\chi_{3520}(749,\cdot)\) \(\chi_{3520}(1189,\cdot)\) \(\chi_{3520}(1629,\cdot)\) \(\chi_{3520}(2069,\cdot)\) \(\chi_{3520}(2509,\cdot)\) \(\chi_{3520}(2949,\cdot)\) \(\chi_{3520}(3389,\cdot)\)
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: | 16.16.236118324143482260684800000000.1 |
Values on generators
\((2751,1541,2817,321)\) → \((1,e\left(\frac{5}{16}\right),-1,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3520 }(309, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)