from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1568, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,7,10]))
pari: [g,chi] = znchar(Mod(15,1568))
Basic properties
Modulus: | \(1568\) | |
Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{392}(211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1568.be
\(\chi_{1568}(15,\cdot)\) \(\chi_{1568}(239,\cdot)\) \(\chi_{1568}(463,\cdot)\) \(\chi_{1568}(911,\cdot)\) \(\chi_{1568}(1135,\cdot)\) \(\chi_{1568}(1359,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | Number field defined by a degree 14 polynomial |
Values on generators
\((1471,197,1473)\) → \((-1,-1,e\left(\frac{5}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1568 }(15, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)