from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8976, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,4,8,0,13]))
pari: [g,chi] = znchar(Mod(2069,8976))
Basic properties
| Modulus: | \(8976\) | |
| Conductor: | \(816\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{816}(437,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8976.gp
\(\chi_{8976}(2069,\cdot)\) \(\chi_{8976}(3125,\cdot)\) \(\chi_{8976}(3389,\cdot)\) \(\chi_{8976}(3917,\cdot)\) \(\chi_{8976}(5501,\cdot)\) \(\chi_{8976}(6029,\cdot)\) \(\chi_{8976}(6293,\cdot)\) \(\chi_{8976}(7349,\cdot)\)
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: | 16.16.330387545600365800521582857754247168.1 |
Values on generators
\((7855,2245,2993,4897,1057)\) → \((1,i,-1,1,e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8976 }(2069, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)