from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(723, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,23]))
pari: [g,chi] = znchar(Mod(32,723))
Basic properties
Modulus: | \(723\) | |
Conductor: | \(723\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | 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 723.x
\(\chi_{723}(2,\cdot)\) \(\chi_{723}(32,\cdot)\) \(\chi_{723}(113,\cdot)\) \(\chi_{723}(128,\cdot)\) \(\chi_{723}(209,\cdot)\) \(\chi_{723}(239,\cdot)\) \(\chi_{723}(362,\cdot)\) \(\chi_{723}(602,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | 24.0.3249728175901551484938855642030803617736713314985356987954561.1 |
Values on generators
\((242,7)\) → \((-1,e\left(\frac{23}{24}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 723 }(32, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{23}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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)