from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([36,35,20]))
pari: [g,chi] = znchar(Mod(5374,8041))
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(8041\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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.cs
\(\chi_{8041}(128,\cdot)\) \(\chi_{8041}(644,\cdot)\) \(\chi_{8041}(1504,\cdot)\) \(\chi_{8041}(1590,\cdot)\) \(\chi_{8041}(2235,\cdot)\) \(\chi_{8041}(2450,\cdot)\) \(\chi_{8041}(2966,\cdot)\) \(\chi_{8041}(3181,\cdot)\) \(\chi_{8041}(3912,\cdot)\) \(\chi_{8041}(4428,\cdot)\) \(\chi_{8041}(5374,\cdot)\) \(\chi_{8041}(5761,\cdot)\) \(\chi_{8041}(6492,\cdot)\) \(\chi_{8041}(6707,\cdot)\) \(\chi_{8041}(7223,\cdot)\) \(\chi_{8041}(7438,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{7}{8}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(5374, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)