from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([18,0,20]))
pari: [g,chi] = znchar(Mod(1769,8041))
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(473\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{473}(350,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8041.bs
\(\chi_{8041}(1038,\cdot)\) \(\chi_{8041}(1412,\cdot)\) \(\chi_{8041}(1769,\cdot)\) \(\chi_{8041}(2143,\cdot)\) \(\chi_{8041}(2500,\cdot)\) \(\chi_{8041}(2874,\cdot)\) \(\chi_{8041}(7617,\cdot)\) \(\chi_{8041}(7991,\cdot)\)
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: | Number field defined by a degree 15 polynomial |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{3}{5}\right),1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(1769, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)