from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,9,7]))
pari: [g,chi] = znchar(Mod(3191,3724))
Basic properties
Modulus: | \(3724\) | |
Conductor: | \(3724\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | 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 3724.by
\(\chi_{3724}(531,\cdot)\) \(\chi_{3724}(1063,\cdot)\) \(\chi_{3724}(1595,\cdot)\) \(\chi_{3724}(2127,\cdot)\) \(\chi_{3724}(2659,\cdot)\) \(\chi_{3724}(3191,\cdot)\)
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: | 14.0.19640210869081831858492466681823232.1 |
Values on generators
\((1863,3041,3137)\) → \((-1,e\left(\frac{9}{14}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 3724 }(3191, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)