from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1425, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,57,50]))
pari: [g,chi] = znchar(Mod(88,1425))
Basic properties
Modulus: | \(1425\) | |
Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{475}(88,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1425.cg
\(\chi_{1425}(88,\cdot)\) \(\chi_{1425}(103,\cdot)\) \(\chi_{1425}(202,\cdot)\) \(\chi_{1425}(217,\cdot)\) \(\chi_{1425}(373,\cdot)\) \(\chi_{1425}(388,\cdot)\) \(\chi_{1425}(487,\cdot)\) \(\chi_{1425}(502,\cdot)\) \(\chi_{1425}(658,\cdot)\) \(\chi_{1425}(673,\cdot)\) \(\chi_{1425}(772,\cdot)\) \(\chi_{1425}(787,\cdot)\) \(\chi_{1425}(958,\cdot)\) \(\chi_{1425}(1072,\cdot)\) \(\chi_{1425}(1228,\cdot)\) \(\chi_{1425}(1342,\cdot)\)
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
\((476,1027,1351)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 1425 }(88, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(-i\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) |
sage: chi.jacobi_sum(n)