from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,12,12,20,21]))
pari: [g,chi] = znchar(Mod(971,2856))
Basic properties
| Modulus: | \(2856\) | |
| Conductor: | \(2856\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2856.ez
\(\chi_{2856}(59,\cdot)\) \(\chi_{2856}(467,\cdot)\) \(\chi_{2856}(563,\cdot)\) \(\chi_{2856}(899,\cdot)\) \(\chi_{2856}(971,\cdot)\) \(\chi_{2856}(1307,\cdot)\) \(\chi_{2856}(1403,\cdot)\) \(\chi_{2856}(1811,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.0.201336748083306095961389068891776734916463808834202859732992.1 |
Values on generators
\((2143,1429,953,409,2689)\) → \((-1,-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(971, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)