from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,10,0,4]))
pari: [g,chi] = znchar(Mod(323,3696))
Basic properties
Modulus: | \(3696\) | |
Conductor: | \(528\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{528}(323,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.fj
\(\chi_{3696}(323,\cdot)\) \(\chi_{3696}(995,\cdot)\) \(\chi_{3696}(1499,\cdot)\) \(\chi_{3696}(1835,\cdot)\) \(\chi_{3696}(2171,\cdot)\) \(\chi_{3696}(2843,\cdot)\) \(\chi_{3696}(3347,\cdot)\) \(\chi_{3696}(3683,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((463,2773,2465,1585,673)\) → \((-1,-i,-1,1,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3696 }(323, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)