from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,24]))
pari: [g,chi] = znchar(Mod(133,460))
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 \(\chi_{115}(18,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 460.u
\(\chi_{460}(13,\cdot)\) \(\chi_{460}(73,\cdot)\) \(\chi_{460}(77,\cdot)\) \(\chi_{460}(117,\cdot)\) \(\chi_{460}(133,\cdot)\) \(\chi_{460}(173,\cdot)\) \(\chi_{460}(177,\cdot)\) \(\chi_{460}(193,\cdot)\) \(\chi_{460}(197,\cdot)\) \(\chi_{460}(213,\cdot)\) \(\chi_{460}(233,\cdot)\) \(\chi_{460}(257,\cdot)\) \(\chi_{460}(317,\cdot)\) \(\chi_{460}(353,\cdot)\) \(\chi_{460}(357,\cdot)\) \(\chi_{460}(377,\cdot)\) \(\chi_{460}(393,\cdot)\) \(\chi_{460}(397,\cdot)\) \(\chi_{460}(417,\cdot)\) \(\chi_{460}(453,\cdot)\)
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: | 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1 |
Values on generators
\((231,277,281)\) → \((1,-i,e\left(\frac{6}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 460 }(133, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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)