from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([9,8]))
pari: [g,chi] = znchar(Mod(235,1003))
Basic properties
| Modulus: | \(1003\) | |
| Conductor: | \(1003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1003.i
\(\chi_{1003}(58,\cdot)\) \(\chi_{1003}(176,\cdot)\) \(\chi_{1003}(235,\cdot)\) \(\chi_{1003}(294,\cdot)\) \(\chi_{1003}(471,\cdot)\) \(\chi_{1003}(530,\cdot)\) \(\chi_{1003}(589,\cdot)\) \(\chi_{1003}(707,\cdot)\)
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
\((768,120)\) → \((e\left(\frac{9}{16}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1003 }(235, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)