from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(385, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,10,18]))
pari: [g,chi] = znchar(Mod(52,385))
Basic properties
Modulus: | \(385\) | |
Conductor: | \(385\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 385.bs
\(\chi_{385}(17,\cdot)\) \(\chi_{385}(52,\cdot)\) \(\chi_{385}(68,\cdot)\) \(\chi_{385}(73,\cdot)\) \(\chi_{385}(117,\cdot)\) \(\chi_{385}(138,\cdot)\) \(\chi_{385}(173,\cdot)\) \(\chi_{385}(178,\cdot)\) \(\chi_{385}(222,\cdot)\) \(\chi_{385}(227,\cdot)\) \(\chi_{385}(248,\cdot)\) \(\chi_{385}(283,\cdot)\) \(\chi_{385}(292,\cdot)\) \(\chi_{385}(327,\cdot)\) \(\chi_{385}(332,\cdot)\) \(\chi_{385}(348,\cdot)\)
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
\((232,276,211)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 385 }(52, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) |
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)