from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(639, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,24]))
pari: [g,chi] = znchar(Mod(20,639))
Basic properties
Modulus: | \(639\) | |
Conductor: | \(639\) | 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 639.x
\(\chi_{639}(20,\cdot)\) \(\chi_{639}(32,\cdot)\) \(\chi_{639}(101,\cdot)\) \(\chi_{639}(119,\cdot)\) \(\chi_{639}(245,\cdot)\) \(\chi_{639}(329,\cdot)\) \(\chi_{639}(392,\cdot)\) \(\chi_{639}(446,\cdot)\) \(\chi_{639}(527,\cdot)\) \(\chi_{639}(542,\cdot)\) \(\chi_{639}(545,\cdot)\) \(\chi_{639}(605,\cdot)\)
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: | 42.0.5057543201805281204185275480299354019747516684301056318578115943139480337529655468381622115531467.1 |
Values on generators
\((569,433)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 639 }(20, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{21}\right)\) |
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)