from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(687, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,6]))
pari: [g,chi] = znchar(Mod(61,687))
Basic properties
Modulus: | \(687\) | |
Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{229}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 687.m
\(\chi_{687}(16,\cdot)\) \(\chi_{687}(43,\cdot)\) \(\chi_{687}(61,\cdot)\) \(\chi_{687}(121,\cdot)\) \(\chi_{687}(214,\cdot)\) \(\chi_{687}(256,\cdot)\) \(\chi_{687}(271,\cdot)\) \(\chi_{687}(286,\cdot)\) \(\chi_{687}(289,\cdot)\) \(\chi_{687}(394,\cdot)\) \(\chi_{687}(454,\cdot)\) \(\chi_{687}(475,\cdot)\) \(\chi_{687}(502,\cdot)\) \(\chi_{687}(511,\cdot)\) \(\chi_{687}(562,\cdot)\) \(\chi_{687}(619,\cdot)\) \(\chi_{687}(661,\cdot)\) \(\chi_{687}(676,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 19 polynomial |
Values on generators
\((230,235)\) → \((1,e\left(\frac{3}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 687 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) |
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)