from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([13]))
pari: [g,chi] = znchar(Mod(43,47))
Basic properties
Modulus: | \(47\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | 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 47.d
\(\chi_{47}(5,\cdot)\) \(\chi_{47}(10,\cdot)\) \(\chi_{47}(11,\cdot)\) \(\chi_{47}(13,\cdot)\) \(\chi_{47}(15,\cdot)\) \(\chi_{47}(19,\cdot)\) \(\chi_{47}(20,\cdot)\) \(\chi_{47}(22,\cdot)\) \(\chi_{47}(23,\cdot)\) \(\chi_{47}(26,\cdot)\) \(\chi_{47}(29,\cdot)\) \(\chi_{47}(30,\cdot)\) \(\chi_{47}(31,\cdot)\) \(\chi_{47}(33,\cdot)\) \(\chi_{47}(35,\cdot)\) \(\chi_{47}(38,\cdot)\) \(\chi_{47}(39,\cdot)\) \(\chi_{47}(40,\cdot)\) \(\chi_{47}(41,\cdot)\) \(\chi_{47}(43,\cdot)\) \(\chi_{47}(44,\cdot)\) \(\chi_{47}(45,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | Number field defined by a degree 46 polynomial |
Values on generators
\(5\) → \(e\left(\frac{13}{46}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 47 }(43, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{45}{46}\right)\) |
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)