from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(291, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(25,291))
Basic properties
Modulus: | \(291\) | |
Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{97}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 291.u
\(\chi_{291}(25,\cdot)\) \(\chi_{291}(31,\cdot)\) \(\chi_{291}(49,\cdot)\) \(\chi_{291}(94,\cdot)\) \(\chi_{291}(100,\cdot)\) \(\chi_{291}(145,\cdot)\) \(\chi_{291}(163,\cdot)\) \(\chi_{291}(169,\cdot)\) \(\chi_{291}(196,\cdot)\) \(\chi_{291}(205,\cdot)\) \(\chi_{291}(226,\cdot)\) \(\chi_{291}(238,\cdot)\) \(\chi_{291}(247,\cdot)\) \(\chi_{291}(259,\cdot)\) \(\chi_{291}(280,\cdot)\) \(\chi_{291}(289,\cdot)\)
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
\((98,199)\) → \((1,e\left(\frac{1}{48}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 291 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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)