from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,9]))
pari: [g,chi] = znchar(Mod(152,405))
Basic properties
Modulus: | \(405\) | |
Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{135}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 405.r
\(\chi_{405}(8,\cdot)\) \(\chi_{405}(17,\cdot)\) \(\chi_{405}(62,\cdot)\) \(\chi_{405}(98,\cdot)\) \(\chi_{405}(143,\cdot)\) \(\chi_{405}(152,\cdot)\) \(\chi_{405}(197,\cdot)\) \(\chi_{405}(233,\cdot)\) \(\chi_{405}(278,\cdot)\) \(\chi_{405}(287,\cdot)\) \(\chi_{405}(332,\cdot)\) \(\chi_{405}(368,\cdot)\)
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: | \(\Q(\zeta_{135})^+\) |
Values on generators
\((326,82)\) → \((e\left(\frac{17}{18}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 405 }(152, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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)