from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,40]))
pari: [g,chi] = znchar(Mod(1167,1792))
Basic properties
Modulus: | \(1792\) | |
Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{448}(187,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bq
\(\chi_{1792}(47,\cdot)\) \(\chi_{1792}(143,\cdot)\) \(\chi_{1792}(271,\cdot)\) \(\chi_{1792}(367,\cdot)\) \(\chi_{1792}(495,\cdot)\) \(\chi_{1792}(591,\cdot)\) \(\chi_{1792}(719,\cdot)\) \(\chi_{1792}(815,\cdot)\) \(\chi_{1792}(943,\cdot)\) \(\chi_{1792}(1039,\cdot)\) \(\chi_{1792}(1167,\cdot)\) \(\chi_{1792}(1263,\cdot)\) \(\chi_{1792}(1391,\cdot)\) \(\chi_{1792}(1487,\cdot)\) \(\chi_{1792}(1615,\cdot)\) \(\chi_{1792}(1711,\cdot)\)
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
\((1023,1541,1025)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1792 }(1167, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) |
sage: chi.jacobi_sum(n)