from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,39,32,33]))
pari: [g,chi] = znchar(Mod(619,9792))
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.nc
\(\chi_{9792}(619,\cdot)\) \(\chi_{9792}(1363,\cdot)\) \(\chi_{9792}(1915,\cdot)\) \(\chi_{9792}(2851,\cdot)\) \(\chi_{9792}(3019,\cdot)\) \(\chi_{9792}(3355,\cdot)\) \(\chi_{9792}(3883,\cdot)\) \(\chi_{9792}(4723,\cdot)\) \(\chi_{9792}(4867,\cdot)\) \(\chi_{9792}(5179,\cdot)\) \(\chi_{9792}(6115,\cdot)\) \(\chi_{9792}(6619,\cdot)\) \(\chi_{9792}(7891,\cdot)\) \(\chi_{9792}(7987,\cdot)\) \(\chi_{9792}(8131,\cdot)\) \(\chi_{9792}(9547,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{11}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(619, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)