from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(527, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([45,16]))
pari: [g,chi] = znchar(Mod(397,527))
Basic properties
| Modulus: | \(527\) | |
| Conductor: | \(527\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 527.be
\(\chi_{527}(5,\cdot)\) \(\chi_{527}(56,\cdot)\) \(\chi_{527}(129,\cdot)\) \(\chi_{527}(160,\cdot)\) \(\chi_{527}(180,\cdot)\) \(\chi_{527}(211,\cdot)\) \(\chi_{527}(284,\cdot)\) \(\chi_{527}(335,\cdot)\) \(\chi_{527}(346,\cdot)\) \(\chi_{527}(377,\cdot)\) \(\chi_{527}(397,\cdot)\) \(\chi_{527}(428,\cdot)\) \(\chi_{527}(439,\cdot)\) \(\chi_{527}(470,\cdot)\) \(\chi_{527}(490,\cdot)\) \(\chi_{527}(521,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((156,375)\) → \((e\left(\frac{15}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 527 }(397, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) |
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)