![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3200, base_ring=CyclotomicField(160))
M = H._module
chi = DirichletCharacter(H, M([0,125,72]))
![Copy content]()
gp:[g,chi] = znchar(Mod(37, 3200))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3200.37");
| Modulus: | \(3200\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(3200\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(160\) |
![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);
|
\(\chi_{3200}(13,\cdot)\)
\(\chi_{3200}(37,\cdot)\)
\(\chi_{3200}(117,\cdot)\)
\(\chi_{3200}(173,\cdot)\)
\(\chi_{3200}(197,\cdot)\)
\(\chi_{3200}(253,\cdot)\)
\(\chi_{3200}(277,\cdot)\)
\(\chi_{3200}(333,\cdot)\)
\(\chi_{3200}(413,\cdot)\)
\(\chi_{3200}(437,\cdot)\)
\(\chi_{3200}(517,\cdot)\)
\(\chi_{3200}(573,\cdot)\)
\(\chi_{3200}(597,\cdot)\)
\(\chi_{3200}(653,\cdot)\)
\(\chi_{3200}(677,\cdot)\)
\(\chi_{3200}(733,\cdot)\)
\(\chi_{3200}(813,\cdot)\)
\(\chi_{3200}(837,\cdot)\)
\(\chi_{3200}(917,\cdot)\)
\(\chi_{3200}(973,\cdot)\)
\(\chi_{3200}(997,\cdot)\)
\(\chi_{3200}(1053,\cdot)\)
\(\chi_{3200}(1077,\cdot)\)
\(\chi_{3200}(1133,\cdot)\)
\(\chi_{3200}(1213,\cdot)\)
\(\chi_{3200}(1237,\cdot)\)
\(\chi_{3200}(1317,\cdot)\)
\(\chi_{3200}(1373,\cdot)\)
\(\chi_{3200}(1397,\cdot)\)
\(\chi_{3200}(1453,\cdot)\)
...
![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: |
$\Q(\zeta_{160})$ |
![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 160 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((1151,901,2177)\) → \((1,e\left(\frac{25}{32}\right),e\left(\frac{9}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3200 }(37, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{160}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{97}{160}\right)\) | \(e\left(\frac{43}{160}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{160}\right)\) | \(e\left(\frac{89}{160}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{77}{160}\right)\) |
![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)
