![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47311, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([78,165,10]))
![Copy content]()
gp:[g,chi] = znchar(Mod(11666, 47311))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("47311.11666");
| Modulus: | \(47311\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(47311\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(220\) |
![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);
|
\(\chi_{47311}(2087,\cdot)\)
\(\chi_{47311}(2767,\cdot)\)
\(\chi_{47311}(3319,\cdot)\)
\(\chi_{47311}(3396,\cdot)\)
\(\chi_{47311}(4407,\cdot)\)
\(\chi_{47311}(5793,\cdot)\)
\(\chi_{47311}(5903,\cdot)\)
\(\chi_{47311}(7025,\cdot)\)
\(\chi_{47311}(7059,\cdot)\)
\(\chi_{47311}(7807,\cdot)\)
\(\chi_{47311}(9286,\cdot)\)
\(\chi_{47311}(9312,\cdot)\)
\(\chi_{47311}(9941,\cdot)\)
\(\chi_{47311}(10485,\cdot)\)
\(\chi_{47311}(10502,\cdot)\)
\(\chi_{47311}(10876,\cdot)\)
\(\chi_{47311}(11530,\cdot)\)
\(\chi_{47311}(11666,\cdot)\)
\(\chi_{47311}(11981,\cdot)\)
\(\chi_{47311}(12040,\cdot)\)
\(\chi_{47311}(12669,\cdot)\)
\(\chi_{47311}(13460,\cdot)\)
\(\chi_{47311}(13851,\cdot)\)
\(\chi_{47311}(14845,\cdot)\)
\(\chi_{47311}(15406,\cdot)\)
\(\chi_{47311}(15440,\cdot)\)
\(\chi_{47311}(17123,\cdot)\)
\(\chi_{47311}(17234,\cdot)\)
\(\chi_{47311}(18322,\cdot)\)
\(\chi_{47311}(18398,\cdot)\)
...
![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 };
\((5084,8350,10286)\) → \((e\left(\frac{39}{110}\right),-i,e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 47311 }(11666, a) \) |
\(1\) | \(1\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
