from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(609, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,28,24]))
pari: [g,chi] = znchar(Mod(431,609))
Basic properties
Modulus: | \(609\) | |
Conductor: | \(609\) | 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 609.bl
\(\chi_{609}(23,\cdot)\) \(\chi_{609}(53,\cdot)\) \(\chi_{609}(65,\cdot)\) \(\chi_{609}(74,\cdot)\) \(\chi_{609}(107,\cdot)\) \(\chi_{609}(170,\cdot)\) \(\chi_{609}(284,\cdot)\) \(\chi_{609}(326,\cdot)\) \(\chi_{609}(368,\cdot)\) \(\chi_{609}(401,\cdot)\) \(\chi_{609}(422,\cdot)\) \(\chi_{609}(431,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((407,262,379)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{4}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 609 }(431, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{21}\right)\) |
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)