from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2006, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([9,8]))
pari: [g,chi] = znchar(Mod(235,2006))
Basic properties
| Modulus: | \(2006\) | |
| Conductor: | \(1003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1003}(235,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2006.j
\(\chi_{2006}(235,\cdot)\) \(\chi_{2006}(471,\cdot)\) \(\chi_{2006}(589,\cdot)\) \(\chi_{2006}(707,\cdot)\) \(\chi_{2006}(1061,\cdot)\) \(\chi_{2006}(1179,\cdot)\) \(\chi_{2006}(1297,\cdot)\) \(\chi_{2006}(1533,\cdot)\)
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: | 16.16.420290829261882123587154930841553.1 |
Values on generators
\((1771,1123)\) → \((e\left(\frac{9}{16}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2006 }(235, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)