![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,12]))
![Copy content]()
gp:[g,chi] = znchar(Mod(224, 605))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.224");
| Modulus: | \(605\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(605\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(110\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{605}(4,\cdot)\)
\(\chi_{605}(14,\cdot)\)
\(\chi_{605}(49,\cdot)\)
\(\chi_{605}(59,\cdot)\)
\(\chi_{605}(64,\cdot)\)
\(\chi_{605}(69,\cdot)\)
\(\chi_{605}(104,\cdot)\)
\(\chi_{605}(114,\cdot)\)
\(\chi_{605}(119,\cdot)\)
\(\chi_{605}(159,\cdot)\)
\(\chi_{605}(169,\cdot)\)
\(\chi_{605}(174,\cdot)\)
\(\chi_{605}(179,\cdot)\)
\(\chi_{605}(214,\cdot)\)
\(\chi_{605}(224,\cdot)\)
\(\chi_{605}(229,\cdot)\)
\(\chi_{605}(234,\cdot)\)
\(\chi_{605}(279,\cdot)\)
\(\chi_{605}(284,\cdot)\)
\(\chi_{605}(289,\cdot)\)
\(\chi_{605}(324,\cdot)\)
\(\chi_{605}(334,\cdot)\)
\(\chi_{605}(339,\cdot)\)
\(\chi_{605}(344,\cdot)\)
\(\chi_{605}(379,\cdot)\)
\(\chi_{605}(389,\cdot)\)
\(\chi_{605}(394,\cdot)\)
\(\chi_{605}(399,\cdot)\)
\(\chi_{605}(434,\cdot)\)
\(\chi_{605}(449,\cdot)\)
...
![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 };
| Field of values: |
$\Q(\zeta_{55})$ |
![Copy content]()
sage:CyclotomicField(chi.multiplicative_order())
![Copy content]()
gp:nfinit(polcyclo(charorder(g,chi)))
![Copy content]()
magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 110 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((122,486)\) → \((-1,e\left(\frac{6}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 605 }(224, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) |
![Copy content]()
sage:chi(x) # x integer
![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)
