from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1137, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,31]))
pari: [g,chi] = znchar(Mod(11,1137))
Basic properties
| Modulus: | \(1137\) | |
| Conductor: | \(1137\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1137.x
\(\chi_{1137}(11,\cdot)\) \(\chi_{1137}(128,\cdot)\) \(\chi_{1137}(140,\cdot)\) \(\chi_{1137}(182,\cdot)\) \(\chi_{1137}(272,\cdot)\) \(\chi_{1137}(464,\cdot)\) \(\chi_{1137}(545,\cdot)\) \(\chi_{1137}(581,\cdot)\) \(\chi_{1137}(671,\cdot)\) \(\chi_{1137}(734,\cdot)\) \(\chi_{1137}(827,\cdot)\) \(\chi_{1137}(866,\cdot)\) \(\chi_{1137}(869,\cdot)\) \(\chi_{1137}(944,\cdot)\) \(\chi_{1137}(986,\cdot)\) \(\chi_{1137}(1016,\cdot)\) \(\chi_{1137}(1058,\cdot)\) \(\chi_{1137}(1076,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((380,760)\) → \((-1,e\left(\frac{31}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1137 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) |
sage: chi.jacobi_sum(n)