![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([17]))
![Copy content]()
pari:[g,chi] = znchar(Mod(29,121))
| Modulus: | \(121\) | |
| Conductor: | \(121\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(110\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{121}(2,\cdot)\)
\(\chi_{121}(6,\cdot)\)
\(\chi_{121}(7,\cdot)\)
\(\chi_{121}(8,\cdot)\)
\(\chi_{121}(13,\cdot)\)
\(\chi_{121}(17,\cdot)\)
\(\chi_{121}(18,\cdot)\)
\(\chi_{121}(19,\cdot)\)
\(\chi_{121}(24,\cdot)\)
\(\chi_{121}(28,\cdot)\)
\(\chi_{121}(29,\cdot)\)
\(\chi_{121}(30,\cdot)\)
\(\chi_{121}(35,\cdot)\)
\(\chi_{121}(39,\cdot)\)
\(\chi_{121}(41,\cdot)\)
\(\chi_{121}(46,\cdot)\)
\(\chi_{121}(50,\cdot)\)
\(\chi_{121}(51,\cdot)\)
\(\chi_{121}(52,\cdot)\)
\(\chi_{121}(57,\cdot)\)
\(\chi_{121}(61,\cdot)\)
\(\chi_{121}(62,\cdot)\)
\(\chi_{121}(63,\cdot)\)
\(\chi_{121}(68,\cdot)\)
\(\chi_{121}(72,\cdot)\)
\(\chi_{121}(73,\cdot)\)
\(\chi_{121}(74,\cdot)\)
\(\chi_{121}(79,\cdot)\)
\(\chi_{121}(83,\cdot)\)
\(\chi_{121}(84,\cdot)\)
...
![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 ]
\(2\) → \(e\left(\frac{17}{110}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 121 }(29, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) |
![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)
