from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(291, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,7]))
pari: [g,chi] = znchar(Mod(77,291))
Basic properties
Modulus: | \(291\) | |
Conductor: | \(291\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 291.s
\(\chi_{291}(20,\cdot)\) \(\chi_{291}(77,\cdot)\) \(\chi_{291}(116,\cdot)\) \(\chi_{291}(125,\cdot)\) \(\chi_{291}(131,\cdot)\) \(\chi_{291}(143,\cdot)\) \(\chi_{291}(149,\cdot)\) \(\chi_{291}(152,\cdot)\) \(\chi_{291}(164,\cdot)\) \(\chi_{291}(224,\cdot)\) \(\chi_{291}(236,\cdot)\) \(\chi_{291}(239,\cdot)\) \(\chi_{291}(245,\cdot)\) \(\chi_{291}(257,\cdot)\) \(\chi_{291}(263,\cdot)\) \(\chi_{291}(272,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | 32.32.1674417821664703472295706925363164706188610888296487588786531799142113.1 |
Values on generators
\((98,199)\) → \((-1,e\left(\frac{7}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 291 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(-i\) |
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)