from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,5]))
pari: [g,chi] = znchar(Mod(32,407))
Basic properties
Modulus: | \(407\) | |
Conductor: | \(407\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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.bb
\(\chi_{407}(32,\cdot)\) \(\chi_{407}(54,\cdot)\) \(\chi_{407}(76,\cdot)\) \(\chi_{407}(87,\cdot)\) \(\chi_{407}(98,\cdot)\) \(\chi_{407}(109,\cdot)\) \(\chi_{407}(131,\cdot)\) \(\chi_{407}(153,\cdot)\) \(\chi_{407}(241,\cdot)\) \(\chi_{407}(274,\cdot)\) \(\chi_{407}(318,\cdot)\) \(\chi_{407}(351,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | 36.36.42867551195672722495351149174628212645190717310766431451792643783547492533.1 |
Values on generators
\((112,298)\) → \((-1,e\left(\frac{5}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 407 }(32, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(i\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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)