from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([2,0]))
pari: [g,chi] = znchar(Mod(1691,8450))
Basic properties
Modulus: | \(8450\) | |
Conductor: | \(25\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(5\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{25}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8450.l
\(\chi_{8450}(1691,\cdot)\) \(\chi_{8450}(3381,\cdot)\) \(\chi_{8450}(5071,\cdot)\) \(\chi_{8450}(6761,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
Fixed field: | 5.5.390625.1 |
Values on generators
\((677,3551)\) → \((e\left(\frac{1}{5}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 8450 }(1691, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)