from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1254, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,6,20]))
pari: [g,chi] = znchar(Mod(125,1254))
Basic properties
| Modulus: | \(1254\) | |
| Conductor: | \(627\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{627}(125,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1254.bi
\(\chi_{1254}(125,\cdot)\) \(\chi_{1254}(311,\cdot)\) \(\chi_{1254}(467,\cdot)\) \(\chi_{1254}(581,\cdot)\) \(\chi_{1254}(653,\cdot)\) \(\chi_{1254}(995,\cdot)\) \(\chi_{1254}(1037,\cdot)\) \(\chi_{1254}(1109,\cdot)\)
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 30 polynomial |
Values on generators
\((419,343,1123)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1254 }(125, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)