from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9128, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,36,34]))
pari: [g,chi] = znchar(Mod(8971,9128))
Basic properties
Modulus: | \(9128\) | |
Conductor: | \(9128\) | 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 9128.fg
\(\chi_{9128}(347,\cdot)\) \(\chi_{9128}(387,\cdot)\) \(\chi_{9128}(403,\cdot)\) \(\chi_{9128}(947,\cdot)\) \(\chi_{9128}(2307,\cdot)\) \(\chi_{9128}(2571,\cdot)\) \(\chi_{9128}(2907,\cdot)\) \(\chi_{9128}(3651,\cdot)\) \(\chi_{9128}(4027,\cdot)\) \(\chi_{9128}(4139,\cdot)\) \(\chi_{9128}(4547,\cdot)\) \(\chi_{9128}(5203,\cdot)\) \(\chi_{9128}(6003,\cdot)\) \(\chi_{9128}(6675,\cdot)\) \(\chi_{9128}(7683,\cdot)\) \(\chi_{9128}(8675,\cdot)\) \(\chi_{9128}(8971,\cdot)\) \(\chi_{9128}(9123,\cdot)\)
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
\((6847,4565,1305,7337)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{17}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 9128 }(8971, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(-1\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)