from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,8]))
pari: [g,chi] = znchar(Mod(443,576))
Basic properties
Modulus: | \(576\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 576.bl
\(\chi_{576}(11,\cdot)\) \(\chi_{576}(59,\cdot)\) \(\chi_{576}(83,\cdot)\) \(\chi_{576}(131,\cdot)\) \(\chi_{576}(155,\cdot)\) \(\chi_{576}(203,\cdot)\) \(\chi_{576}(227,\cdot)\) \(\chi_{576}(275,\cdot)\) \(\chi_{576}(299,\cdot)\) \(\chi_{576}(347,\cdot)\) \(\chi_{576}(371,\cdot)\) \(\chi_{576}(419,\cdot)\) \(\chi_{576}(443,\cdot)\) \(\chi_{576}(491,\cdot)\) \(\chi_{576}(515,\cdot)\) \(\chi_{576}(563,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,325,65)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 576 }(443, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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)