from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8325, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,9,25]))
pari: [g,chi] = znchar(Mod(1639,8325))
Basic properties
| Modulus: | \(8325\) | |
| Conductor: | \(925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{925}(714,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8325.gl
\(\chi_{8325}(64,\cdot)\) \(\chi_{8325}(1639,\cdot)\) \(\chi_{8325}(1729,\cdot)\) \(\chi_{8325}(3304,\cdot)\) \(\chi_{8325}(3394,\cdot)\) \(\chi_{8325}(4969,\cdot)\) \(\chi_{8325}(5059,\cdot)\) \(\chi_{8325}(6634,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.712054643298888889330514089285546802246873454578235396184027194976806640625.1 |
Values on generators
\((3701,7327,5626)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8325 }(1639, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)