from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8005, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,13]))
pari: [g,chi] = znchar(Mod(109,8005))
Basic properties
Modulus: | \(8005\) | |
Conductor: | \(8005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 8005.bf
\(\chi_{8005}(109,\cdot)\) \(\chi_{8005}(169,\cdot)\) \(\chi_{8005}(2044,\cdot)\) \(\chi_{8005}(2759,\cdot)\) \(\chi_{8005}(4634,\cdot)\) \(\chi_{8005}(4694,\cdot)\) \(\chi_{8005}(5009,\cdot)\) \(\chi_{8005}(5159,\cdot)\) \(\chi_{8005}(5599,\cdot)\) \(\chi_{8005}(6169,\cdot)\) \(\chi_{8005}(6224,\cdot)\) \(\chi_{8005}(6584,\cdot)\) \(\chi_{8005}(6639,\cdot)\) \(\chi_{8005}(7209,\cdot)\) \(\chi_{8005}(7649,\cdot)\) \(\chi_{8005}(7799,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((1602,4806)\) → \((-1,e\left(\frac{13}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 8005 }(109, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(i\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) |
sage: chi.jacobi_sum(n)