from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(6498))
M = H._module
chi = DirichletCharacter(H, M([0,3249,4332,3542]))
pari: [g,chi] = znchar(Mod(25,987696))
Basic properties
| Modulus: | \(987696\) | |
| Conductor: | \(493848\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6498\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{493848}(246949,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 987696.ui
\(\chi_{987696}(25,\cdot)\) \(\chi_{987696}(313,\cdot)\) \(\chi_{987696}(745,\cdot)\) \(\chi_{987696}(841,\cdot)\) \(\chi_{987696}(985,\cdot)\)
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: | $\Q(\zeta_{3249})$ |
| Fixed field: | Number field defined by a degree 6498 polynomial (not computed) |
Values on generators
\((617311,740773,438977,857377)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{1771}{3249}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 987696 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{1229}{6498}\right)\) | \(e\left(\frac{922}{1083}\right)\) | \(e\left(\frac{667}{722}\right)\) | \(e\left(\frac{3145}{6498}\right)\) | \(e\left(\frac{2500}{3249}\right)\) | \(e\left(\frac{1781}{3249}\right)\) | \(e\left(\frac{1229}{3249}\right)\) | \(e\left(\frac{3499}{6498}\right)\) | \(e\left(\frac{198}{361}\right)\) | \(e\left(\frac{263}{6498}\right)\) |
sage: chi.jacobi_sum(n)