from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,39,16]))
pari: [g,chi] = znchar(Mod(79,1792))
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 \(\chi_{448}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bp
\(\chi_{1792}(79,\cdot)\) \(\chi_{1792}(207,\cdot)\) \(\chi_{1792}(303,\cdot)\) \(\chi_{1792}(431,\cdot)\) \(\chi_{1792}(527,\cdot)\) \(\chi_{1792}(655,\cdot)\) \(\chi_{1792}(751,\cdot)\) \(\chi_{1792}(879,\cdot)\) \(\chi_{1792}(975,\cdot)\) \(\chi_{1792}(1103,\cdot)\) \(\chi_{1792}(1199,\cdot)\) \(\chi_{1792}(1327,\cdot)\) \(\chi_{1792}(1423,\cdot)\) \(\chi_{1792}(1551,\cdot)\) \(\chi_{1792}(1647,\cdot)\) \(\chi_{1792}(1775,\cdot)\)
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
\((1023,1541,1025)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1792 }(79, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)