from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,0,8,0,7]))
pari: [g,chi] = znchar(Mod(113,2856))
Basic properties
| Modulus: | \(2856\) | |
| Conductor: | \(51\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{51}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2856.ep
\(\chi_{2856}(113,\cdot)\) \(\chi_{2856}(449,\cdot)\) \(\chi_{2856}(617,\cdot)\) \(\chi_{2856}(785,\cdot)\) \(\chi_{2856}(1289,\cdot)\) \(\chi_{2856}(1457,\cdot)\) \(\chi_{2856}(1625,\cdot)\) \(\chi_{2856}(1961,\cdot)\)
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: | \(\Q(\zeta_{51})^+\) |
Values on generators
\((2143,1429,953,409,2689)\) → \((1,1,-1,1,e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)