from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1343, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,4]))
pari: [g,chi] = znchar(Mod(798,1343))
Basic properties
| Modulus: | \(1343\) | |
| Conductor: | \(1343\) | 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 1343.v
\(\chi_{1343}(67,\cdot)\) \(\chi_{1343}(101,\cdot)\) \(\chi_{1343}(220,\cdot)\) \(\chi_{1343}(441,\cdot)\) \(\chi_{1343}(492,\cdot)\) \(\chi_{1343}(526,\cdot)\) \(\chi_{1343}(696,\cdot)\) \(\chi_{1343}(798,\cdot)\) \(\chi_{1343}(934,\cdot)\) \(\chi_{1343}(1206,\cdot)\) \(\chi_{1343}(1223,\cdot)\) \(\chi_{1343}(1274,\cdot)\)
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
\((870,477)\) → \((-1,e\left(\frac{2}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1343 }(798, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
sage: chi.jacobi_sum(n)