![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([91,93]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1172, 1323))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.1172");
| Modulus: | \(1323\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1323\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(126\) |
![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);
|
\(\chi_{1323}(5,\cdot)\)
\(\chi_{1323}(38,\cdot)\)
\(\chi_{1323}(101,\cdot)\)
\(\chi_{1323}(131,\cdot)\)
\(\chi_{1323}(164,\cdot)\)
\(\chi_{1323}(194,\cdot)\)
\(\chi_{1323}(257,\cdot)\)
\(\chi_{1323}(290,\cdot)\)
\(\chi_{1323}(320,\cdot)\)
\(\chi_{1323}(353,\cdot)\)
\(\chi_{1323}(383,\cdot)\)
\(\chi_{1323}(416,\cdot)\)
\(\chi_{1323}(446,\cdot)\)
\(\chi_{1323}(479,\cdot)\)
\(\chi_{1323}(542,\cdot)\)
\(\chi_{1323}(572,\cdot)\)
\(\chi_{1323}(605,\cdot)\)
\(\chi_{1323}(635,\cdot)\)
\(\chi_{1323}(698,\cdot)\)
\(\chi_{1323}(731,\cdot)\)
\(\chi_{1323}(761,\cdot)\)
\(\chi_{1323}(794,\cdot)\)
\(\chi_{1323}(824,\cdot)\)
\(\chi_{1323}(857,\cdot)\)
\(\chi_{1323}(887,\cdot)\)
\(\chi_{1323}(920,\cdot)\)
\(\chi_{1323}(983,\cdot)\)
\(\chi_{1323}(1013,\cdot)\)
\(\chi_{1323}(1046,\cdot)\)
\(\chi_{1323}(1076,\cdot)\)
...
![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: |
$\Q(\zeta_{63})$ |
![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 126 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((785,1081)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(1172, a) \) |
\(1\) | \(1\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |
![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)
