sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,42,42,63,40]))
pari:[g,chi] = znchar(Mod(4013,5160))
Modulus: | \(5160\) | |
Conductor: | \(5160\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(84\) |
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)
|
\(\chi_{5160}(53,\cdot)\)
\(\chi_{5160}(197,\cdot)\)
\(\chi_{5160}(533,\cdot)\)
\(\chi_{5160}(797,\cdot)\)
\(\chi_{5160}(917,\cdot)\)
\(\chi_{5160}(1013,\cdot)\)
\(\chi_{5160}(1133,\cdot)\)
\(\chi_{5160}(1373,\cdot)\)
\(\chi_{5160}(1493,\cdot)\)
\(\chi_{5160}(1733,\cdot)\)
\(\chi_{5160}(1973,\cdot)\)
\(\chi_{5160}(2117,\cdot)\)
\(\chi_{5160}(2597,\cdot)\)
\(\chi_{5160}(2933,\cdot)\)
\(\chi_{5160}(3077,\cdot)\)
\(\chi_{5160}(3197,\cdot)\)
\(\chi_{5160}(3293,\cdot)\)
\(\chi_{5160}(3437,\cdot)\)
\(\chi_{5160}(3557,\cdot)\)
\(\chi_{5160}(3797,\cdot)\)
\(\chi_{5160}(3893,\cdot)\)
\(\chi_{5160}(4013,\cdot)\)
\(\chi_{5160}(4037,\cdot)\)
\(\chi_{5160}(4997,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((3871,2581,1721,3097,4561)\) → \((1,-1,-1,-i,e\left(\frac{10}{21}\right))\)
\(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5160 }(4013, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage:chi.jacobi_sum(n)