from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,11,8]))
pari: [g,chi] = znchar(Mod(221,2368))
Basic properties
Modulus: | \(2368\) | |
Conductor: | \(2368\) | 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 2368.cc
\(\chi_{2368}(221,\cdot)\) \(\chi_{2368}(517,\cdot)\) \(\chi_{2368}(813,\cdot)\) \(\chi_{2368}(1109,\cdot)\) \(\chi_{2368}(1405,\cdot)\) \(\chi_{2368}(1701,\cdot)\) \(\chi_{2368}(1997,\cdot)\) \(\chi_{2368}(2293,\cdot)\)
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: | 16.16.2123163551355495017117431991053058048.1 |
Values on generators
\((1407,1925,705)\) → \((1,e\left(\frac{11}{16}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2368 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)