from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,1,8,4,0]))
pari: [g,chi] = znchar(Mod(197,6720))
Basic properties
Modulus: | \(6720\) | |
Conductor: | \(960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{960}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6720.hk
\(\chi_{6720}(197,\cdot)\) \(\chi_{6720}(1373,\cdot)\) \(\chi_{6720}(1877,\cdot)\) \(\chi_{6720}(3053,\cdot)\) \(\chi_{6720}(3557,\cdot)\) \(\chi_{6720}(4733,\cdot)\) \(\chi_{6720}(5237,\cdot)\) \(\chi_{6720}(6413,\cdot)\)
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: | 16.16.968232702940866945220608000000000000.2 |
Values on generators
\((1471,3781,4481,5377,1921)\) → \((1,e\left(\frac{1}{16}\right),-1,i,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 6720 }(197, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)