from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(801, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,3]))
pari: [g,chi] = znchar(Mod(704,801))
Basic properties
| Modulus: | \(801\) | |
| Conductor: | \(801\) | 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 801.x
\(\chi_{801}(11,\cdot)\) \(\chi_{801}(50,\cdot)\) \(\chi_{801}(146,\cdot)\) \(\chi_{801}(176,\cdot)\) \(\chi_{801}(200,\cdot)\) \(\chi_{801}(203,\cdot)\) \(\chi_{801}(263,\cdot)\) \(\chi_{801}(311,\cdot)\) \(\chi_{801}(317,\cdot)\) \(\chi_{801}(437,\cdot)\) \(\chi_{801}(443,\cdot)\) \(\chi_{801}(470,\cdot)\) \(\chi_{801}(518,\cdot)\) \(\chi_{801}(545,\cdot)\) \(\chi_{801}(578,\cdot)\) \(\chi_{801}(680,\cdot)\) \(\chi_{801}(704,\cdot)\) \(\chi_{801}(734,\cdot)\) \(\chi_{801}(785,\cdot)\) \(\chi_{801}(797,\cdot)\)
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
\((713,181)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 801 }(704, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) |
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)