from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,32,21]))
pari: [g,chi] = znchar(Mod(691,9792))
Basic properties
| Modulus: | \(9792\) | |
| Conductor: | \(9792\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 9792.lm
\(\chi_{9792}(691,\cdot)\) \(\chi_{9792}(1195,\cdot)\) \(\chi_{9792}(2131,\cdot)\) \(\chi_{9792}(2443,\cdot)\) \(\chi_{9792}(2587,\cdot)\) \(\chi_{9792}(3427,\cdot)\) \(\chi_{9792}(3955,\cdot)\) \(\chi_{9792}(4291,\cdot)\) \(\chi_{9792}(4459,\cdot)\) \(\chi_{9792}(5395,\cdot)\) \(\chi_{9792}(5947,\cdot)\) \(\chi_{9792}(6691,\cdot)\) \(\chi_{9792}(7555,\cdot)\) \(\chi_{9792}(8971,\cdot)\) \(\chi_{9792}(9115,\cdot)\) \(\chi_{9792}(9211,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(691, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)