from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,18,10]))
pari: [g,chi] = znchar(Mod(121,1575))
Basic properties
| Modulus: | \(1575\) | |
| Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1575.cq
\(\chi_{1575}(121,\cdot)\) \(\chi_{1575}(436,\cdot)\) \(\chi_{1575}(466,\cdot)\) \(\chi_{1575}(781,\cdot)\) \(\chi_{1575}(1066,\cdot)\) \(\chi_{1575}(1096,\cdot)\) \(\chi_{1575}(1381,\cdot)\) \(\chi_{1575}(1411,\cdot)\)
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
\((1226,127,451)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 1575 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)