![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(2400))
M = H._module
chi = DirichletCharacter(H, M([0,225,336,1600]))
![Copy content]()
gp:[g,chi] = znchar(Mod(9, 416000))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.9");
| Modulus: | \(416000\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(208000\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(2400\) |
![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: | no, induced from \(\chi_{208000}(32509,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{416000}(9,\cdot)\)
\(\chi_{416000}(809,\cdot)\)
\(\chi_{416000}(2089,\cdot)\)
\(\chi_{416000}(2889,\cdot)\)
\(\chi_{416000}(3129,\cdot)\)
\(\chi_{416000}(3929,\cdot)\)
\(\chi_{416000}(4169,\cdot)\)
\(\chi_{416000}(4969,\cdot)\)
\(\chi_{416000}(5209,\cdot)\)
\(\chi_{416000}(6009,\cdot)\)
\(\chi_{416000}(7289,\cdot)\)
\(\chi_{416000}(8089,\cdot)\)
\(\chi_{416000}(8329,\cdot)\)
\(\chi_{416000}(9129,\cdot)\)
\(\chi_{416000}(9369,\cdot)\)
\(\chi_{416000}(10169,\cdot)\)
\(\chi_{416000}(10409,\cdot)\)
\(\chi_{416000}(11209,\cdot)\)
\(\chi_{416000}(12489,\cdot)\)
\(\chi_{416000}(13289,\cdot)\)
\(\chi_{416000}(13529,\cdot)\)
\(\chi_{416000}(14329,\cdot)\)
\(\chi_{416000}(14569,\cdot)\)
\(\chi_{416000}(15369,\cdot)\)
\(\chi_{416000}(15609,\cdot)\)
\(\chi_{416000}(16409,\cdot)\)
\(\chi_{416000}(17689,\cdot)\)
\(\chi_{416000}(18489,\cdot)\)
\(\chi_{416000}(18729,\cdot)\)
\(\chi_{416000}(19529,\cdot)\)
...
![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: |
$\Q(\zeta_{2400})$ |
![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 2400 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((1,e\left(\frac{3}{32}\right),e\left(\frac{7}{50}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2227}{2400}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{1027}{1200}\right)\) | \(e\left(\frac{661}{2400}\right)\) | \(e\left(\frac{107}{600}\right)\) | \(e\left(\frac{23}{2400}\right)\) | \(e\left(\frac{79}{800}\right)\) | \(e\left(\frac{383}{1200}\right)\) | \(e\left(\frac{627}{800}\right)\) | \(e\left(\frac{2107}{2400}\right)\) |
![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)
