from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,14]))
pari: [g,chi] = znchar(Mod(35,2601))
Basic properties
Modulus: | \(2601\) | |
Conductor: | \(867\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{867}(35,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2601.v
\(\chi_{2601}(35,\cdot)\) \(\chi_{2601}(188,\cdot)\) \(\chi_{2601}(341,\cdot)\) \(\chi_{2601}(494,\cdot)\) \(\chi_{2601}(647,\cdot)\) \(\chi_{2601}(800,\cdot)\) \(\chi_{2601}(953,\cdot)\) \(\chi_{2601}(1106,\cdot)\) \(\chi_{2601}(1259,\cdot)\) \(\chi_{2601}(1412,\cdot)\) \(\chi_{2601}(1565,\cdot)\) \(\chi_{2601}(1718,\cdot)\) \(\chi_{2601}(1871,\cdot)\) \(\chi_{2601}(2177,\cdot)\) \(\chi_{2601}(2330,\cdot)\) \(\chi_{2601}(2483,\cdot)\)
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
\((290,2026)\) → \((-1,e\left(\frac{7}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2601 }(35, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) |
sage: chi.jacobi_sum(n)