from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1011, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,13]))
pari: [g,chi] = znchar(Mod(164,1011))
Basic properties
| Modulus: | \(1011\) | |
| Conductor: | \(1011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 1011.y
\(\chi_{1011}(164,\cdot)\) \(\chi_{1011}(173,\cdot)\) \(\chi_{1011}(386,\cdot)\) \(\chi_{1011}(392,\cdot)\) \(\chi_{1011}(440,\cdot)\) \(\chi_{1011}(524,\cdot)\) \(\chi_{1011}(638,\cdot)\) \(\chi_{1011}(710,\cdot)\) \(\chi_{1011}(824,\cdot)\) \(\chi_{1011}(908,\cdot)\) \(\chi_{1011}(956,\cdot)\) \(\chi_{1011}(962,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((338,10)\) → \((-1,e\left(\frac{13}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1011 }(164, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)