from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,30]))
pari: [g,chi] = znchar(Mod(433,460))
Basic properties
Modulus: | \(460\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(88,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 460.x
\(\chi_{460}(17,\cdot)\) \(\chi_{460}(33,\cdot)\) \(\chi_{460}(37,\cdot)\) \(\chi_{460}(53,\cdot)\) \(\chi_{460}(57,\cdot)\) \(\chi_{460}(97,\cdot)\) \(\chi_{460}(113,\cdot)\) \(\chi_{460}(153,\cdot)\) \(\chi_{460}(157,\cdot)\) \(\chi_{460}(217,\cdot)\) \(\chi_{460}(237,\cdot)\) \(\chi_{460}(273,\cdot)\) \(\chi_{460}(293,\cdot)\) \(\chi_{460}(297,\cdot)\) \(\chi_{460}(313,\cdot)\) \(\chi_{460}(333,\cdot)\) \(\chi_{460}(337,\cdot)\) \(\chi_{460}(373,\cdot)\) \(\chi_{460}(433,\cdot)\) \(\chi_{460}(457,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((231,277,281)\) → \((1,-i,e\left(\frac{15}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 460 }(433, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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)