from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1584, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,10,12]))
pari: [g,chi] = znchar(Mod(49,1584))
Basic properties
| Modulus: | \(1584\) | |
| Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{99}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1584.cm
\(\chi_{1584}(49,\cdot)\) \(\chi_{1584}(97,\cdot)\) \(\chi_{1584}(625,\cdot)\) \(\chi_{1584}(817,\cdot)\) \(\chi_{1584}(961,\cdot)\) \(\chi_{1584}(1105,\cdot)\) \(\chi_{1584}(1345,\cdot)\) \(\chi_{1584}(1489,\cdot)\)
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: | 15.15.10943023107606534329121.1 |
Values on generators
\((991,1189,353,145)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1584 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)