from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,32,39]))
pari: [g,chi] = znchar(Mod(403,9792))
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.ob
\(\chi_{9792}(403,\cdot)\) \(\chi_{9792}(2203,\cdot)\) \(\chi_{9792}(2275,\cdot)\) \(\chi_{9792}(2419,\cdot)\) \(\chi_{9792}(3067,\cdot)\) \(\chi_{9792}(3667,\cdot)\) \(\chi_{9792}(4363,\cdot)\) \(\chi_{9792}(5443,\cdot)\) \(\chi_{9792}(5467,\cdot)\) \(\chi_{9792}(5539,\cdot)\) \(\chi_{9792}(5683,\cdot)\) \(\chi_{9792}(5803,\cdot)\) \(\chi_{9792}(6331,\cdot)\) \(\chi_{9792}(7627,\cdot)\) \(\chi_{9792}(8707,\cdot)\) \(\chi_{9792}(9067,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{13}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(403, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)