from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,32]))
pari: [g,chi] = znchar(Mod(133,576))
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.bm
\(\chi_{576}(13,\cdot)\) \(\chi_{576}(61,\cdot)\) \(\chi_{576}(85,\cdot)\) \(\chi_{576}(133,\cdot)\) \(\chi_{576}(157,\cdot)\) \(\chi_{576}(205,\cdot)\) \(\chi_{576}(229,\cdot)\) \(\chi_{576}(277,\cdot)\) \(\chi_{576}(301,\cdot)\) \(\chi_{576}(349,\cdot)\) \(\chi_{576}(373,\cdot)\) \(\chi_{576}(421,\cdot)\) \(\chi_{576}(445,\cdot)\) \(\chi_{576}(493,\cdot)\) \(\chi_{576}(517,\cdot)\) \(\chi_{576}(565,\cdot)\)
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
\((127,325,65)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 576 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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)