from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2997, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([53,33]))
pari: [g,chi] = znchar(Mod(1094,2997))
Basic properties
Modulus: | \(2997\) | |
Conductor: | \(2997\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2997.ej
\(\chi_{2997}(95,\cdot)\) \(\chi_{2997}(326,\cdot)\) \(\chi_{2997}(617,\cdot)\) \(\chi_{2997}(842,\cdot)\) \(\chi_{2997}(929,\cdot)\) \(\chi_{2997}(965,\cdot)\) \(\chi_{2997}(1094,\cdot)\) \(\chi_{2997}(1325,\cdot)\) \(\chi_{2997}(1616,\cdot)\) \(\chi_{2997}(1841,\cdot)\) \(\chi_{2997}(1928,\cdot)\) \(\chi_{2997}(1964,\cdot)\) \(\chi_{2997}(2093,\cdot)\) \(\chi_{2997}(2324,\cdot)\) \(\chi_{2997}(2615,\cdot)\) \(\chi_{2997}(2840,\cdot)\) \(\chi_{2997}(2927,\cdot)\) \(\chi_{2997}(2963,\cdot)\)
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
\((1703,1297)\) → \((e\left(\frac{53}{54}\right),e\left(\frac{11}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2997 }(1094, a) \) | \(-1\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) |
sage: chi.jacobi_sum(n)