from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,0,30]))
pari: [g,chi] = znchar(Mod(617,805))
Basic properties
| Modulus: | \(805\) | |
| 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 \(\chi_{115}(42,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 805.bi
\(\chi_{805}(43,\cdot)\) \(\chi_{805}(57,\cdot)\) \(\chi_{805}(113,\cdot)\) \(\chi_{805}(148,\cdot)\) \(\chi_{805}(218,\cdot)\) \(\chi_{805}(267,\cdot)\) \(\chi_{805}(337,\cdot)\) \(\chi_{805}(428,\cdot)\) \(\chi_{805}(442,\cdot)\) \(\chi_{805}(477,\cdot)\) \(\chi_{805}(498,\cdot)\) \(\chi_{805}(582,\cdot)\) \(\chi_{805}(603,\cdot)\) \(\chi_{805}(617,\cdot)\) \(\chi_{805}(638,\cdot)\) \(\chi_{805}(687,\cdot)\) \(\chi_{805}(743,\cdot)\) \(\chi_{805}(757,\cdot)\) \(\chi_{805}(778,\cdot)\) \(\chi_{805}(792,\cdot)\)
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
\((162,346,281)\) → \((i,1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(617, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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)