from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2112, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,1,0,0]))
pari: [g,chi] = znchar(Mod(133,2112))
Basic properties
| Modulus: | \(2112\) | |
| Conductor: | \(64\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{64}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2112.bz
\(\chi_{2112}(133,\cdot)\) \(\chi_{2112}(397,\cdot)\) \(\chi_{2112}(661,\cdot)\) \(\chi_{2112}(925,\cdot)\) \(\chi_{2112}(1189,\cdot)\) \(\chi_{2112}(1453,\cdot)\) \(\chi_{2112}(1717,\cdot)\) \(\chi_{2112}(1981,\cdot)\)
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: | \(\Q(\zeta_{64})^+\) |
Values on generators
\((2047,133,1409,1729)\) → \((1,e\left(\frac{1}{16}\right),1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2112 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)