from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6018, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,3,8]))
pari: [g,chi] = znchar(Mod(2713,6018))
Basic properties
Modulus: | \(6018\) | |
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 \(\chi_{1003}(707,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6018.t
\(\chi_{6018}(235,\cdot)\) \(\chi_{6018}(589,\cdot)\) \(\chi_{6018}(1297,\cdot)\) \(\chi_{6018}(2713,\cdot)\) \(\chi_{6018}(3067,\cdot)\) \(\chi_{6018}(4483,\cdot)\) \(\chi_{6018}(5191,\cdot)\) \(\chi_{6018}(5545,\cdot)\)
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
\((4013,1771,1123)\) → \((1,e\left(\frac{3}{16}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 6018 }(2713, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)