from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,51]))
pari: [g,chi] = znchar(Mod(61,805))
Basic properties
Modulus: | \(805\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 805.bp
\(\chi_{805}(61,\cdot)\) \(\chi_{805}(66,\cdot)\) \(\chi_{805}(136,\cdot)\) \(\chi_{805}(166,\cdot)\) \(\chi_{805}(171,\cdot)\) \(\chi_{805}(201,\cdot)\) \(\chi_{805}(241,\cdot)\) \(\chi_{805}(306,\cdot)\) \(\chi_{805}(341,\cdot)\) \(\chi_{805}(411,\cdot)\) \(\chi_{805}(451,\cdot)\) \(\chi_{805}(481,\cdot)\) \(\chi_{805}(516,\cdot)\) \(\chi_{805}(521,\cdot)\) \(\chi_{805}(586,\cdot)\) \(\chi_{805}(626,\cdot)\) \(\chi_{805}(661,\cdot)\) \(\chi_{805}(766,\cdot)\) \(\chi_{805}(796,\cdot)\) \(\chi_{805}(801,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((162,346,281)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 805 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) |
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)