from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1254, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,3,5]))
pari: [g,chi] = znchar(Mod(569,1254))
Basic properties
| Modulus: | \(1254\) | |
| Conductor: | \(627\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{627}(569,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1254.x
\(\chi_{1254}(227,\cdot)\) \(\chi_{1254}(569,\cdot)\) \(\chi_{1254}(1025,\cdot)\) \(\chi_{1254}(1139,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.0.1418758396496190387.1 |
Values on generators
\((419,343,1123)\) → \((-1,e\left(\frac{3}{10}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1254 }(569, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)