from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,12,10]))
pari: [g,chi] = znchar(Mod(406,4275))
Basic properties
Modulus: | \(4275\) | |
Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{475}(406,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.cw
\(\chi_{4275}(406,\cdot)\) \(\chi_{4275}(1261,\cdot)\) \(\chi_{4275}(1531,\cdot)\) \(\chi_{4275}(2116,\cdot)\) \(\chi_{4275}(2386,\cdot)\) \(\chi_{4275}(2971,\cdot)\) \(\chi_{4275}(3241,\cdot)\) \(\chi_{4275}(4096,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((1901,1027,1351)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 4275 }(406, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)