from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([42,5]))
pari: [g,chi] = znchar(Mod(304,407))
Basic properties
Modulus: | \(407\) | |
Conductor: | \(407\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 407.bd
\(\chi_{407}(8,\cdot)\) \(\chi_{407}(29,\cdot)\) \(\chi_{407}(51,\cdot)\) \(\chi_{407}(134,\cdot)\) \(\chi_{407}(140,\cdot)\) \(\chi_{407}(156,\cdot)\) \(\chi_{407}(162,\cdot)\) \(\chi_{407}(171,\cdot)\) \(\chi_{407}(193,\cdot)\) \(\chi_{407}(282,\cdot)\) \(\chi_{407}(288,\cdot)\) \(\chi_{407}(304,\cdot)\) \(\chi_{407}(310,\cdot)\) \(\chi_{407}(325,\cdot)\) \(\chi_{407}(347,\cdot)\) \(\chi_{407}(393,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((112,298)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 407 }(304, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
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)