from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,7,45]))
pari: [g,chi] = znchar(Mod(47,2268))
Basic properties
Modulus: | \(2268\) | |
Conductor: | \(2268\) | 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 2268.da
\(\chi_{2268}(47,\cdot)\) \(\chi_{2268}(59,\cdot)\) \(\chi_{2268}(299,\cdot)\) \(\chi_{2268}(311,\cdot)\) \(\chi_{2268}(551,\cdot)\) \(\chi_{2268}(563,\cdot)\) \(\chi_{2268}(803,\cdot)\) \(\chi_{2268}(815,\cdot)\) \(\chi_{2268}(1055,\cdot)\) \(\chi_{2268}(1067,\cdot)\) \(\chi_{2268}(1307,\cdot)\) \(\chi_{2268}(1319,\cdot)\) \(\chi_{2268}(1559,\cdot)\) \(\chi_{2268}(1571,\cdot)\) \(\chi_{2268}(1811,\cdot)\) \(\chi_{2268}(1823,\cdot)\) \(\chi_{2268}(2063,\cdot)\) \(\chi_{2268}(2075,\cdot)\)
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
\((1135,1541,325)\) → \((-1,e\left(\frac{7}{54}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 2268 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)