from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,5]))
pari: [g,chi] = znchar(Mod(47,588))
Basic properties
Modulus: | \(588\) | |
Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 588.bf
\(\chi_{588}(47,\cdot)\) \(\chi_{588}(59,\cdot)\) \(\chi_{588}(131,\cdot)\) \(\chi_{588}(143,\cdot)\) \(\chi_{588}(299,\cdot)\) \(\chi_{588}(311,\cdot)\) \(\chi_{588}(383,\cdot)\) \(\chi_{588}(395,\cdot)\) \(\chi_{588}(467,\cdot)\) \(\chi_{588}(479,\cdot)\) \(\chi_{588}(551,\cdot)\) \(\chi_{588}(563,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | 42.0.5436948860695888782893198886016377149049148530413040928765325951335574011833955525853184.1 |
Values on generators
\((295,197,493)\) → \((-1,-1,e\left(\frac{5}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 588 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) |
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)