from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,16]))
pari: [g,chi] = znchar(Mod(625,1792))
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 \(\chi_{448}(261,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.br
\(\chi_{1792}(81,\cdot)\) \(\chi_{1792}(177,\cdot)\) \(\chi_{1792}(305,\cdot)\) \(\chi_{1792}(401,\cdot)\) \(\chi_{1792}(529,\cdot)\) \(\chi_{1792}(625,\cdot)\) \(\chi_{1792}(753,\cdot)\) \(\chi_{1792}(849,\cdot)\) \(\chi_{1792}(977,\cdot)\) \(\chi_{1792}(1073,\cdot)\) \(\chi_{1792}(1201,\cdot)\) \(\chi_{1792}(1297,\cdot)\) \(\chi_{1792}(1425,\cdot)\) \(\chi_{1792}(1521,\cdot)\) \(\chi_{1792}(1649,\cdot)\) \(\chi_{1792}(1745,\cdot)\)
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
\((1023,1541,1025)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1792 }(625, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) |
sage: chi.jacobi_sum(n)