from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,25]))
pari: [g,chi] = znchar(Mod(95,804))
Basic properties
Modulus: | \(804\) | |
Conductor: | \(804\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 804.bb
\(\chi_{804}(11,\cdot)\) \(\chi_{804}(95,\cdot)\) \(\chi_{804}(191,\cdot)\) \(\chi_{804}(203,\cdot)\) \(\chi_{804}(251,\cdot)\) \(\chi_{804}(275,\cdot)\) \(\chi_{804}(299,\cdot)\) \(\chi_{804}(347,\cdot)\) \(\chi_{804}(383,\cdot)\) \(\chi_{804}(443,\cdot)\) \(\chi_{804}(503,\cdot)\) \(\chi_{804}(515,\cdot)\) \(\chi_{804}(587,\cdot)\) \(\chi_{804}(599,\cdot)\) \(\chi_{804}(623,\cdot)\) \(\chi_{804}(635,\cdot)\) \(\chi_{804}(647,\cdot)\) \(\chi_{804}(683,\cdot)\) \(\chi_{804}(731,\cdot)\) \(\chi_{804}(755,\cdot)\)
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
\((403,269,337)\) → \((-1,-1,e\left(\frac{25}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 804 }(95, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{33}\right)\) |
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)