from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,8]))
pari: [g,chi] = znchar(Mod(6788,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{731}(209,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8041.dg
\(\chi_{8041}(265,\cdot)\) \(\chi_{8041}(639,\cdot)\) \(\chi_{8041}(738,\cdot)\) \(\chi_{8041}(1112,\cdot)\) \(\chi_{8041}(2630,\cdot)\) \(\chi_{8041}(3004,\cdot)\) \(\chi_{8041}(3576,\cdot)\) \(\chi_{8041}(3950,\cdot)\) \(\chi_{8041}(4049,\cdot)\) \(\chi_{8041}(4423,\cdot)\) \(\chi_{8041}(4995,\cdot)\) \(\chi_{8041}(5369,\cdot)\) \(\chi_{8041}(5468,\cdot)\) \(\chi_{8041}(5842,\cdot)\) \(\chi_{8041}(6414,\cdot)\) \(\chi_{8041}(6788,\cdot)\)
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
\((6580,2366,562)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(6788, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(-i\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) |
sage: chi.jacobi_sum(n)