from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(71, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([29]))
pari: [g,chi] = znchar(Mod(35,71))
Basic properties
Modulus: | \(71\) | |
Conductor: | \(71\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(70\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 71.h
\(\chi_{71}(7,\cdot)\) \(\chi_{71}(11,\cdot)\) \(\chi_{71}(13,\cdot)\) \(\chi_{71}(21,\cdot)\) \(\chi_{71}(22,\cdot)\) \(\chi_{71}(28,\cdot)\) \(\chi_{71}(31,\cdot)\) \(\chi_{71}(33,\cdot)\) \(\chi_{71}(35,\cdot)\) \(\chi_{71}(42,\cdot)\) \(\chi_{71}(44,\cdot)\) \(\chi_{71}(47,\cdot)\) \(\chi_{71}(52,\cdot)\) \(\chi_{71}(53,\cdot)\) \(\chi_{71}(55,\cdot)\) \(\chi_{71}(56,\cdot)\) \(\chi_{71}(59,\cdot)\) \(\chi_{71}(61,\cdot)\) \(\chi_{71}(62,\cdot)\) \(\chi_{71}(63,\cdot)\) \(\chi_{71}(65,\cdot)\) \(\chi_{71}(67,\cdot)\) \(\chi_{71}(68,\cdot)\) \(\chi_{71}(69,\cdot)\)
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: | $\Q(\zeta_{35})$ |
Fixed field: | Number field defined by a degree 70 polynomial |
Values on generators
\(7\) → \(e\left(\frac{29}{70}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 71 }(35, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)