![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([41]))
![Copy content]()
gp:[g,chi] = znchar(Mod(411, 722))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.411");
| Modulus: | \(722\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(361\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(114\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
gp:charorder(g,chi)
![Copy content]()
magma:Order(chi);
|
| Real: | no |
![Copy content]()
sage:chi.multiplicative_order() <= 2
![Copy content]()
gp:charorder(g,chi) <= 2
![Copy content]()
magma:Order(chi) le 2;
|
| Primitive: | no, induced from \(\chi_{361}(50,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{722}(27,\cdot)\)
\(\chi_{722}(31,\cdot)\)
\(\chi_{722}(65,\cdot)\)
\(\chi_{722}(103,\cdot)\)
\(\chi_{722}(107,\cdot)\)
\(\chi_{722}(141,\cdot)\)
\(\chi_{722}(145,\cdot)\)
\(\chi_{722}(179,\cdot)\)
\(\chi_{722}(183,\cdot)\)
\(\chi_{722}(217,\cdot)\)
\(\chi_{722}(221,\cdot)\)
\(\chi_{722}(255,\cdot)\)
\(\chi_{722}(259,\cdot)\)
\(\chi_{722}(297,\cdot)\)
\(\chi_{722}(331,\cdot)\)
\(\chi_{722}(335,\cdot)\)
\(\chi_{722}(369,\cdot)\)
\(\chi_{722}(373,\cdot)\)
\(\chi_{722}(407,\cdot)\)
\(\chi_{722}(411,\cdot)\)
\(\chi_{722}(445,\cdot)\)
\(\chi_{722}(449,\cdot)\)
\(\chi_{722}(483,\cdot)\)
\(\chi_{722}(487,\cdot)\)
\(\chi_{722}(521,\cdot)\)
\(\chi_{722}(525,\cdot)\)
\(\chi_{722}(559,\cdot)\)
\(\chi_{722}(563,\cdot)\)
\(\chi_{722}(597,\cdot)\)
\(\chi_{722}(601,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
![Copy content]()
magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\(363\) → \(e\left(\frac{41}{114}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 722 }(411, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
gp:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
