from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1048, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,22]))
pari: [g,chi] = znchar(Mod(63,1048))
Basic properties
| Modulus: | \(1048\) | |
| Conductor: | \(524\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{524}(63,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1048.u
\(\chi_{1048}(39,\cdot)\) \(\chi_{1048}(63,\cdot)\) \(\chi_{1048}(183,\cdot)\) \(\chi_{1048}(191,\cdot)\) \(\chi_{1048}(215,\cdot)\) \(\chi_{1048}(375,\cdot)\) \(\chi_{1048}(455,\cdot)\) \(\chi_{1048}(623,\cdot)\) \(\chi_{1048}(631,\cdot)\) \(\chi_{1048}(735,\cdot)\) \(\chi_{1048}(767,\cdot)\) \(\chi_{1048}(831,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.0.43781534893766029378911430108761129396314054312752152838144.1 |
Values on generators
\((263,525,657)\) → \((-1,1,e\left(\frac{11}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1048 }(63, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) |
sage: chi.jacobi_sum(n)