from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,0,16,21]))
pari: [g,chi] = znchar(Mod(895,9792))
Basic properties
| Modulus: | \(9792\) | |
| Conductor: | \(612\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{612}(283,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.lt
\(\chi_{9792}(895,\cdot)\) \(\chi_{9792}(1663,\cdot)\) \(\chi_{9792}(2047,\cdot)\) \(\chi_{9792}(2239,\cdot)\) \(\chi_{9792}(2623,\cdot)\) \(\chi_{9792}(2815,\cdot)\) \(\chi_{9792}(3199,\cdot)\) \(\chi_{9792}(3967,\cdot)\) \(\chi_{9792}(4927,\cdot)\) \(\chi_{9792}(5503,\cdot)\) \(\chi_{9792}(6079,\cdot)\) \(\chi_{9792}(7231,\cdot)\) \(\chi_{9792}(7423,\cdot)\) \(\chi_{9792}(8575,\cdot)\) \(\chi_{9792}(9151,\cdot)\) \(\chi_{9792}(9727,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((7039,5509,8705,9217)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(895, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)