from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(12996))
M = H._module
chi = DirichletCharacter(H, M([0,3249,10830,10040]))
pari: [g,chi] = znchar(Mod(5,987696))
Basic properties
Modulus: | \(987696\) | |
Conductor: | \(987696\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12996\) | 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 987696.vr
\(\chi_{987696}(5,\cdot)\) \(\chi_{987696}(101,\cdot)\) \(\chi_{987696}(149,\cdot)\) \(\chi_{987696}(1157,\cdot)\) \(\chi_{987696}(1373,\cdot)\) \(\chi_{987696}(1469,\cdot)\) \(\chi_{987696}(1517,\cdot)\)
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: | $\Q(\zeta_{12996})$ |
Fixed field: | Number field defined by a degree 12996 polynomial (not computed) |
Values on generators
\((617311,740773,438977,857377)\) → \((1,i,e\left(\frac{5}{6}\right),e\left(\frac{2510}{3249}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 987696 }(5, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7727}{12996}\right)\) | \(e\left(\frac{2005}{2166}\right)\) | \(e\left(\frac{667}{1444}\right)\) | \(e\left(\frac{9643}{12996}\right)\) | \(e\left(\frac{5749}{6498}\right)\) | \(e\left(\frac{2515}{3249}\right)\) | \(e\left(\frac{1229}{6498}\right)\) | \(e\left(\frac{9997}{12996}\right)\) | \(e\left(\frac{99}{361}\right)\) | \(e\left(\frac{6761}{12996}\right)\) |
sage: chi.jacobi_sum(n)