![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(400))
M = H._module
chi = DirichletCharacter(H, M([200,325,292,100]))
![Copy content]()
gp:[g,chi] = znchar(Mod(12267, 104000))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.12267");
| Modulus: | \(104000\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(104000\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(400\) |
![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);
|
\(\chi_{104000}(83,\cdot)\)
\(\chi_{104000}(827,\cdot)\)
\(\chi_{104000}(1123,\cdot)\)
\(\chi_{104000}(1867,\cdot)\)
\(\chi_{104000}(2163,\cdot)\)
\(\chi_{104000}(3203,\cdot)\)
\(\chi_{104000}(3947,\cdot)\)
\(\chi_{104000}(4987,\cdot)\)
\(\chi_{104000}(5283,\cdot)\)
\(\chi_{104000}(6027,\cdot)\)
\(\chi_{104000}(6323,\cdot)\)
\(\chi_{104000}(7067,\cdot)\)
\(\chi_{104000}(7363,\cdot)\)
\(\chi_{104000}(8403,\cdot)\)
\(\chi_{104000}(9147,\cdot)\)
\(\chi_{104000}(10187,\cdot)\)
\(\chi_{104000}(10483,\cdot)\)
\(\chi_{104000}(11227,\cdot)\)
\(\chi_{104000}(11523,\cdot)\)
\(\chi_{104000}(12267,\cdot)\)
\(\chi_{104000}(12563,\cdot)\)
\(\chi_{104000}(13603,\cdot)\)
\(\chi_{104000}(14347,\cdot)\)
\(\chi_{104000}(15387,\cdot)\)
\(\chi_{104000}(15683,\cdot)\)
\(\chi_{104000}(16427,\cdot)\)
\(\chi_{104000}(16723,\cdot)\)
\(\chi_{104000}(17467,\cdot)\)
\(\chi_{104000}(17763,\cdot)\)
\(\chi_{104000}(18803,\cdot)\)
...
![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: |
$\Q(\zeta_{400})$ |
![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 400 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((74751,58501,77377,64001)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{73}{100}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(12267, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{400}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{200}\right)\) | \(e\left(\frac{317}{400}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{231}{400}\right)\) | \(e\left(\frac{189}{400}\right)\) | \(e\left(\frac{1}{200}\right)\) | \(e\left(\frac{57}{400}\right)\) | \(e\left(\frac{79}{400}\right)\) |
![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)
