from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1027, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,24]))
pari: [g,chi] = znchar(Mod(417,1027))
Basic properties
| Modulus: | \(1027\) | |
| Conductor: | \(79\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{79}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1027.be
\(\chi_{1027}(131,\cdot)\) \(\chi_{1027}(144,\cdot)\) \(\chi_{1027}(196,\cdot)\) \(\chi_{1027}(222,\cdot)\) \(\chi_{1027}(326,\cdot)\) \(\chi_{1027}(378,\cdot)\) \(\chi_{1027}(417,\cdot)\) \(\chi_{1027}(482,\cdot)\) \(\chi_{1027}(495,\cdot)\) \(\chi_{1027}(599,\cdot)\) \(\chi_{1027}(729,\cdot)\) \(\chi_{1027}(1015,\cdot)\)
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 13 polynomial |
Values on generators
\((80,872)\) → \((1,e\left(\frac{12}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1027 }(417, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
sage: chi.jacobi_sum(n)