from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,42,40,33]))
pari: [g,chi] = znchar(Mod(41,9792))
Basic properties
Modulus: | \(9792\) | |
Conductor: | \(4896\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4896}(653,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.lw
\(\chi_{9792}(41,\cdot)\) \(\chi_{9792}(329,\cdot)\) \(\chi_{9792}(1049,\cdot)\) \(\chi_{9792}(1433,\cdot)\) \(\chi_{9792}(3305,\cdot)\) \(\chi_{9792}(3593,\cdot)\) \(\chi_{9792}(3641,\cdot)\) \(\chi_{9792}(3785,\cdot)\) \(\chi_{9792}(4073,\cdot)\) \(\chi_{9792}(4313,\cdot)\) \(\chi_{9792}(5369,\cdot)\) \(\chi_{9792}(6905,\cdot)\) \(\chi_{9792}(7049,\cdot)\) \(\chi_{9792}(7337,\cdot)\) \(\chi_{9792}(7961,\cdot)\) \(\chi_{9792}(8633,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{11}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)