from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3920, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,7,24]))
pari: [g,chi] = znchar(Mod(57,3920))
Basic properties
Modulus: | \(3920\) | |
Conductor: | \(1960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1960}(1037,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3920.el
\(\chi_{3920}(57,\cdot)\) \(\chi_{3920}(617,\cdot)\) \(\chi_{3920}(953,\cdot)\) \(\chi_{3920}(1513,\cdot)\) \(\chi_{3920}(1737,\cdot)\) \(\chi_{3920}(2073,\cdot)\) \(\chi_{3920}(2297,\cdot)\) \(\chi_{3920}(2633,\cdot)\) \(\chi_{3920}(2857,\cdot)\) \(\chi_{3920}(3193,\cdot)\) \(\chi_{3920}(3417,\cdot)\) \(\chi_{3920}(3753,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((1471,981,3137,3041)\) → \((1,-1,i,e\left(\frac{6}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 3920 }(57, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)