from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,40]))
pari: [g,chi] = znchar(Mod(151,324))
Basic properties
Modulus: | \(324\) | |
Conductor: | \(324\) | 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 324.n
\(\chi_{324}(7,\cdot)\) \(\chi_{324}(31,\cdot)\) \(\chi_{324}(43,\cdot)\) \(\chi_{324}(67,\cdot)\) \(\chi_{324}(79,\cdot)\) \(\chi_{324}(103,\cdot)\) \(\chi_{324}(115,\cdot)\) \(\chi_{324}(139,\cdot)\) \(\chi_{324}(151,\cdot)\) \(\chi_{324}(175,\cdot)\) \(\chi_{324}(187,\cdot)\) \(\chi_{324}(211,\cdot)\) \(\chi_{324}(223,\cdot)\) \(\chi_{324}(247,\cdot)\) \(\chi_{324}(259,\cdot)\) \(\chi_{324}(283,\cdot)\) \(\chi_{324}(295,\cdot)\) \(\chi_{324}(319,\cdot)\)
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
\((163,245)\) → \((-1,e\left(\frac{20}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 324 }(151, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) |
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)