from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([8,28]))
pari: [g,chi] = znchar(Mod(191,8281))
Basic properties
Modulus: | \(8281\) | |
Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{637}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8281.bk
\(\chi_{8281}(191,\cdot)\) \(\chi_{8281}(991,\cdot)\) \(\chi_{8281}(1374,\cdot)\) \(\chi_{8281}(2557,\cdot)\) \(\chi_{8281}(3357,\cdot)\) \(\chi_{8281}(3740,\cdot)\) \(\chi_{8281}(4540,\cdot)\) \(\chi_{8281}(4923,\cdot)\) \(\chi_{8281}(5723,\cdot)\) \(\chi_{8281}(6906,\cdot)\) \(\chi_{8281}(7289,\cdot)\) \(\chi_{8281}(8089,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1522,3382)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 8281 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)