from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5488, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,6]))
pari: [g,chi] = znchar(Mod(3331,5488))
Basic properties
Modulus: | \(5488\) | |
Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{784}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5488.bk
\(\chi_{5488}(195,\cdot)\) \(\chi_{5488}(587,\cdot)\) \(\chi_{5488}(979,\cdot)\) \(\chi_{5488}(1763,\cdot)\) \(\chi_{5488}(2155,\cdot)\) \(\chi_{5488}(2547,\cdot)\) \(\chi_{5488}(2939,\cdot)\) \(\chi_{5488}(3331,\cdot)\) \(\chi_{5488}(3723,\cdot)\) \(\chi_{5488}(4507,\cdot)\) \(\chi_{5488}(4899,\cdot)\) \(\chi_{5488}(5291,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.28.271776353216347717810469630450516372938858574109997048774397001728.1 |
Values on generators
\((687,4117,689)\) → \((-1,-i,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 5488 }(3331, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)