from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8023, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([12,15]))
pari: [g,chi] = znchar(Mod(128,8023))
Basic properties
Modulus: | \(8023\) | |
Conductor: | \(8023\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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 8023.bu
\(\chi_{8023}(128,\cdot)\) \(\chi_{8023}(693,\cdot)\) \(\chi_{8023}(806,\cdot)\) \(\chi_{8023}(1567,\cdot)\) \(\chi_{8023}(2471,\cdot)\) \(\chi_{8023}(3036,\cdot)\) \(\chi_{8023}(3149,\cdot)\) \(\chi_{8023}(7247,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((6894,3054)\) → \((e\left(\frac{3}{5}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8023 }(128, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)