![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8815, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([210,819,740]))
![Copy content]()
gp:[g,chi] = znchar(Mod(622, 8815))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8815.622");
| Modulus: | \(8815\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(8815\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(840\) |
![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: | yes |
![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);
|
\(\chi_{8815}(12,\cdot)\)
\(\chi_{8815}(63,\cdot)\)
\(\chi_{8815}(112,\cdot)\)
\(\chi_{8815}(157,\cdot)\)
\(\chi_{8815}(158,\cdot)\)
\(\chi_{8815}(177,\cdot)\)
\(\chi_{8815}(192,\cdot)\)
\(\chi_{8815}(263,\cdot)\)
\(\chi_{8815}(313,\cdot)\)
\(\chi_{8815}(362,\cdot)\)
\(\chi_{8815}(363,\cdot)\)
\(\chi_{8815}(417,\cdot)\)
\(\chi_{8815}(562,\cdot)\)
\(\chi_{8815}(587,\cdot)\)
\(\chi_{8815}(593,\cdot)\)
\(\chi_{8815}(622,\cdot)\)
\(\chi_{8815}(673,\cdot)\)
\(\chi_{8815}(678,\cdot)\)
\(\chi_{8815}(792,\cdot)\)
\(\chi_{8815}(803,\cdot)\)
\(\chi_{8815}(807,\cdot)\)
\(\chi_{8815}(872,\cdot)\)
\(\chi_{8815}(878,\cdot)\)
\(\chi_{8815}(908,\cdot)\)
\(\chi_{8815}(932,\cdot)\)
\(\chi_{8815}(958,\cdot)\)
\(\chi_{8815}(972,\cdot)\)
\(\chi_{8815}(1008,\cdot)\)
\(\chi_{8815}(1037,\cdot)\)
\(\chi_{8815}(1137,\cdot)\)
...
![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 };
\((3527,4516,3486)\) → \((i,e\left(\frac{39}{40}\right),e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 8815 }(622, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{99}{280}\right)\) | \(e\left(\frac{23}{840}\right)\) | \(e\left(\frac{139}{840}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
