from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,57]))
pari: [g,chi] = znchar(Mod(254,861))
Basic properties
Modulus: | \(861\) | |
Conductor: | \(861\) | 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 861.ch
\(\chi_{861}(2,\cdot)\) \(\chi_{861}(74,\cdot)\) \(\chi_{861}(128,\cdot)\) \(\chi_{861}(200,\cdot)\) \(\chi_{861}(254,\cdot)\) \(\chi_{861}(326,\cdot)\) \(\chi_{861}(389,\cdot)\) \(\chi_{861}(431,\cdot)\) \(\chi_{861}(443,\cdot)\) \(\chi_{861}(494,\cdot)\) \(\chi_{861}(569,\cdot)\) \(\chi_{861}(620,\cdot)\) \(\chi_{861}(695,\cdot)\) \(\chi_{861}(746,\cdot)\) \(\chi_{861}(758,\cdot)\) \(\chi_{861}(800,\cdot)\)
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
\((575,493,211)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 861 }(254, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) |
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)