![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,63]))
![Copy content]()
pari:[g,chi] = znchar(Mod(313,828))
\(\chi_{828}(61,\cdot)\)
\(\chi_{828}(97,\cdot)\)
\(\chi_{828}(157,\cdot)\)
\(\chi_{828}(205,\cdot)\)
\(\chi_{828}(241,\cdot)\)
\(\chi_{828}(313,\cdot)\)
\(\chi_{828}(337,\cdot)\)
\(\chi_{828}(373,\cdot)\)
\(\chi_{828}(385,\cdot)\)
\(\chi_{828}(421,\cdot)\)
\(\chi_{828}(457,\cdot)\)
\(\chi_{828}(481,\cdot)\)
\(\chi_{828}(493,\cdot)\)
\(\chi_{828}(517,\cdot)\)
\(\chi_{828}(589,\cdot)\)
\(\chi_{828}(661,\cdot)\)
\(\chi_{828}(697,\cdot)\)
\(\chi_{828}(709,\cdot)\)
\(\chi_{828}(733,\cdot)\)
\(\chi_{828}(769,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((415,461,649)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(313, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
