from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,32,6]))
pari: [g,chi] = znchar(Mod(349,9792))
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.nh
\(\chi_{9792}(349,\cdot)\) \(\chi_{9792}(1141,\cdot)\) \(\chi_{9792}(1453,\cdot)\) \(\chi_{9792}(3085,\cdot)\) \(\chi_{9792}(3109,\cdot)\) \(\chi_{9792}(3613,\cdot)\) \(\chi_{9792}(4405,\cdot)\) \(\chi_{9792}(4741,\cdot)\) \(\chi_{9792}(5245,\cdot)\) \(\chi_{9792}(6037,\cdot)\) \(\chi_{9792}(6349,\cdot)\) \(\chi_{9792}(7981,\cdot)\) \(\chi_{9792}(8005,\cdot)\) \(\chi_{9792}(8509,\cdot)\) \(\chi_{9792}(9301,\cdot)\) \(\chi_{9792}(9637,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(349, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)