![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,2]))
![Copy content]()
pari:[g,chi] = znchar(Mod(235,368))
| Modulus: | \(368\) | |
| Conductor: | \(368\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{368}(11,\cdot)\)
\(\chi_{368}(19,\cdot)\)
\(\chi_{368}(43,\cdot)\)
\(\chi_{368}(51,\cdot)\)
\(\chi_{368}(67,\cdot)\)
\(\chi_{368}(83,\cdot)\)
\(\chi_{368}(99,\cdot)\)
\(\chi_{368}(107,\cdot)\)
\(\chi_{368}(155,\cdot)\)
\(\chi_{368}(171,\cdot)\)
\(\chi_{368}(195,\cdot)\)
\(\chi_{368}(203,\cdot)\)
\(\chi_{368}(227,\cdot)\)
\(\chi_{368}(235,\cdot)\)
\(\chi_{368}(251,\cdot)\)
\(\chi_{368}(267,\cdot)\)
\(\chi_{368}(283,\cdot)\)
\(\chi_{368}(291,\cdot)\)
\(\chi_{368}(339,\cdot)\)
\(\chi_{368}(355,\cdot)\)
![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 ]
\((47,277,97)\) → \((-1,i,e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(235, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) |
![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)
