from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([11,0]))
pari: [g,chi] = znchar(Mod(23,891))
Basic properties
| Modulus: | \(891\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 891.x
\(\chi_{891}(23,\cdot)\) \(\chi_{891}(56,\cdot)\) \(\chi_{891}(122,\cdot)\) \(\chi_{891}(155,\cdot)\) \(\chi_{891}(221,\cdot)\) \(\chi_{891}(254,\cdot)\) \(\chi_{891}(320,\cdot)\) \(\chi_{891}(353,\cdot)\) \(\chi_{891}(419,\cdot)\) \(\chi_{891}(452,\cdot)\) \(\chi_{891}(518,\cdot)\) \(\chi_{891}(551,\cdot)\) \(\chi_{891}(617,\cdot)\) \(\chi_{891}(650,\cdot)\) \(\chi_{891}(716,\cdot)\) \(\chi_{891}(749,\cdot)\) \(\chi_{891}(815,\cdot)\) \(\chi_{891}(848,\cdot)\)
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
\((650,244)\) → \((e\left(\frac{11}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 891 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)