from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,17]))
pari: [g,chi] = znchar(Mod(115,124))
Basic properties
Modulus: | \(124\) | |
Conductor: | \(124\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | 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 124.p
\(\chi_{124}(3,\cdot)\) \(\chi_{124}(11,\cdot)\) \(\chi_{124}(43,\cdot)\) \(\chi_{124}(55,\cdot)\) \(\chi_{124}(75,\cdot)\) \(\chi_{124}(79,\cdot)\) \(\chi_{124}(83,\cdot)\) \(\chi_{124}(115,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | \(\Q(\zeta_{124})^+\) |
Values on generators
\((63,65)\) → \((-1,e\left(\frac{17}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 124 }(115, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) |
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)