from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,52,27]))
pari: [g,chi] = znchar(Mod(1519,1620))
Basic properties
Modulus: | \(1620\) | |
Conductor: | \(1620\) | 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 1620.bp
\(\chi_{1620}(79,\cdot)\) \(\chi_{1620}(139,\cdot)\) \(\chi_{1620}(259,\cdot)\) \(\chi_{1620}(319,\cdot)\) \(\chi_{1620}(439,\cdot)\) \(\chi_{1620}(499,\cdot)\) \(\chi_{1620}(619,\cdot)\) \(\chi_{1620}(679,\cdot)\) \(\chi_{1620}(799,\cdot)\) \(\chi_{1620}(859,\cdot)\) \(\chi_{1620}(979,\cdot)\) \(\chi_{1620}(1039,\cdot)\) \(\chi_{1620}(1159,\cdot)\) \(\chi_{1620}(1219,\cdot)\) \(\chi_{1620}(1339,\cdot)\) \(\chi_{1620}(1399,\cdot)\) \(\chi_{1620}(1519,\cdot)\) \(\chi_{1620}(1579,\cdot)\)
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
\((811,1541,1297)\) → \((-1,e\left(\frac{26}{27}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1620 }(1519, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) |
sage: chi.jacobi_sum(n)