from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1048, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,12]))
pari: [g,chi] = znchar(Mod(45,1048))
Basic properties
| Modulus: | \(1048\) | |
| Conductor: | \(1048\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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 1048.w
\(\chi_{1048}(45,\cdot)\) \(\chi_{1048}(301,\cdot)\) \(\chi_{1048}(325,\cdot)\) \(\chi_{1048}(445,\cdot)\) \(\chi_{1048}(453,\cdot)\) \(\chi_{1048}(477,\cdot)\) \(\chi_{1048}(637,\cdot)\) \(\chi_{1048}(717,\cdot)\) \(\chi_{1048}(885,\cdot)\) \(\chi_{1048}(893,\cdot)\) \(\chi_{1048}(997,\cdot)\) \(\chi_{1048}(1029,\cdot)\)
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: | Number field defined by a degree 26 polynomial |
Values on generators
\((263,525,657)\) → \((1,-1,e\left(\frac{6}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1048 }(45, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
sage: chi.jacobi_sum(n)