from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([12,15,20]))
pari: [g,chi] = znchar(Mod(7561,8041))
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(8041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 8041.do
\(\chi_{8041}(251,\cdot)\) \(\chi_{8041}(565,\cdot)\) \(\chi_{8041}(608,\cdot)\) \(\chi_{8041}(939,\cdot)\) \(\chi_{8041}(982,\cdot)\) \(\chi_{8041}(1296,\cdot)\) \(\chi_{8041}(1670,\cdot)\) \(\chi_{8041}(2027,\cdot)\) \(\chi_{8041}(2401,\cdot)\) \(\chi_{8041}(5725,\cdot)\) \(\chi_{8041}(6099,\cdot)\) \(\chi_{8041}(7144,\cdot)\) \(\chi_{8041}(7187,\cdot)\) \(\chi_{8041}(7518,\cdot)\) \(\chi_{8041}(7561,\cdot)\) \(\chi_{8041}(7918,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{1}{5}\right),i,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(7561, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)