from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,17]))
pari: [g,chi] = znchar(Mod(419,648))
Basic properties
Modulus: | \(648\) | |
Conductor: | \(648\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 648.bb
\(\chi_{648}(11,\cdot)\) \(\chi_{648}(59,\cdot)\) \(\chi_{648}(83,\cdot)\) \(\chi_{648}(131,\cdot)\) \(\chi_{648}(155,\cdot)\) \(\chi_{648}(203,\cdot)\) \(\chi_{648}(227,\cdot)\) \(\chi_{648}(275,\cdot)\) \(\chi_{648}(299,\cdot)\) \(\chi_{648}(347,\cdot)\) \(\chi_{648}(371,\cdot)\) \(\chi_{648}(419,\cdot)\) \(\chi_{648}(443,\cdot)\) \(\chi_{648}(491,\cdot)\) \(\chi_{648}(515,\cdot)\) \(\chi_{648}(563,\cdot)\) \(\chi_{648}(587,\cdot)\) \(\chi_{648}(635,\cdot)\)
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
\((487,325,569)\) → \((-1,-1,e\left(\frac{17}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 648 }(419, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) |
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)