from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([54,7]))
pari: [g,chi] = znchar(Mod(6,671))
Basic properties
Modulus: | \(671\) | |
Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 671.dc
\(\chi_{671}(6,\cdot)\) \(\chi_{671}(17,\cdot)\) \(\chi_{671}(30,\cdot)\) \(\chi_{671}(79,\cdot)\) \(\chi_{671}(105,\cdot)\) \(\chi_{671}(112,\cdot)\) \(\chi_{671}(116,\cdot)\) \(\chi_{671}(129,\cdot)\) \(\chi_{671}(376,\cdot)\) \(\chi_{671}(392,\cdot)\) \(\chi_{671}(409,\cdot)\) \(\chi_{671}(425,\cdot)\) \(\chi_{671}(481,\cdot)\) \(\chi_{671}(580,\cdot)\) \(\chi_{671}(612,\cdot)\) \(\chi_{671}(645,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((123,551)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{7}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 671 }(6, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) |
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)