from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3895, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([10,20,17]))
pari: [g,chi] = znchar(Mod(3552,3895))
Basic properties
| Modulus: | \(3895\) | |
| Conductor: | \(3895\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3895.ea
\(\chi_{3895}(227,\cdot)\) \(\chi_{3895}(322,\cdot)\) \(\chi_{3895}(398,\cdot)\) \(\chi_{3895}(873,\cdot)\) \(\chi_{3895}(1633,\cdot)\) \(\chi_{3895}(2203,\cdot)\) \(\chi_{3895}(2488,\cdot)\) \(\chi_{3895}(2507,\cdot)\) \(\chi_{3895}(2602,\cdot)\) \(\chi_{3895}(2678,\cdot)\) \(\chi_{3895}(2887,\cdot)\) \(\chi_{3895}(2963,\cdot)\) \(\chi_{3895}(3172,\cdot)\) \(\chi_{3895}(3533,\cdot)\) \(\chi_{3895}(3552,\cdot)\) \(\chi_{3895}(3837,\cdot)\)
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
\((3117,2871,1236)\) → \((i,-1,e\left(\frac{17}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3895 }(3552, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
sage: chi.jacobi_sum(n)