from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,37]))
pari: [g,chi] = znchar(Mod(191,648))
Basic properties
Modulus: | \(648\) | |
Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{324}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 648.be
\(\chi_{648}(23,\cdot)\) \(\chi_{648}(47,\cdot)\) \(\chi_{648}(95,\cdot)\) \(\chi_{648}(119,\cdot)\) \(\chi_{648}(167,\cdot)\) \(\chi_{648}(191,\cdot)\) \(\chi_{648}(239,\cdot)\) \(\chi_{648}(263,\cdot)\) \(\chi_{648}(311,\cdot)\) \(\chi_{648}(335,\cdot)\) \(\chi_{648}(383,\cdot)\) \(\chi_{648}(407,\cdot)\) \(\chi_{648}(455,\cdot)\) \(\chi_{648}(479,\cdot)\) \(\chi_{648}(527,\cdot)\) \(\chi_{648}(551,\cdot)\) \(\chi_{648}(599,\cdot)\) \(\chi_{648}(623,\cdot)\)
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
\((487,325,569)\) → \((-1,1,e\left(\frac{37}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 648 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) |
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)