from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1021, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([21]))
pari: [g,chi] = znchar(Mod(586,1021))
Basic properties
| Modulus: | \(1021\) | |
| Conductor: | \(1021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | 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 1021.m
\(\chi_{1021}(3,\cdot)\) \(\chi_{1021}(27,\cdot)\) \(\chi_{1021}(145,\cdot)\) \(\chi_{1021}(169,\cdot)\) \(\chi_{1021}(243,\cdot)\) \(\chi_{1021}(284,\cdot)\) \(\chi_{1021}(292,\cdot)\) \(\chi_{1021}(416,\cdot)\) \(\chi_{1021}(500,\cdot)\) \(\chi_{1021}(514,\cdot)\) \(\chi_{1021}(542,\cdot)\) \(\chi_{1021}(586,\cdot)\) \(\chi_{1021}(681,\cdot)\) \(\chi_{1021}(794,\cdot)\) \(\chi_{1021}(940,\cdot)\) \(\chi_{1021}(1012,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\(10\) → \(e\left(\frac{21}{34}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1021 }(586, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) |
sage: chi.jacobi_sum(n)