from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,9,50]))
pari: [g,chi] = znchar(Mod(1783,2800))
Basic properties
Modulus: | \(2800\) | |
Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1400}(1083,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2800.fy
\(\chi_{2800}(87,\cdot)\) \(\chi_{2800}(103,\cdot)\) \(\chi_{2800}(327,\cdot)\) \(\chi_{2800}(423,\cdot)\) \(\chi_{2800}(647,\cdot)\) \(\chi_{2800}(663,\cdot)\) \(\chi_{2800}(887,\cdot)\) \(\chi_{2800}(983,\cdot)\) \(\chi_{2800}(1223,\cdot)\) \(\chi_{2800}(1447,\cdot)\) \(\chi_{2800}(1767,\cdot)\) \(\chi_{2800}(1783,\cdot)\) \(\chi_{2800}(2103,\cdot)\) \(\chi_{2800}(2327,\cdot)\) \(\chi_{2800}(2567,\cdot)\) \(\chi_{2800}(2663,\cdot)\)
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
\((351,2101,2577,801)\) → \((-1,-1,e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2800 }(1783, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)