![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2061, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,68]))
![Copy content]()
gp:[g,chi] = znchar(Mod(217, 2061))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2061.217");
| Modulus: | \(2061\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(229\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(57\) |
![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 \(\chi_{229}(217,\cdot)\) |
![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);
|
\(\chi_{2061}(19,\cdot)\)
\(\chi_{2061}(37,\cdot)\)
\(\chi_{2061}(55,\cdot)\)
\(\chi_{2061}(82,\cdot)\)
\(\chi_{2061}(91,\cdot)\)
\(\chi_{2061}(217,\cdot)\)
\(\chi_{2061}(280,\cdot)\)
\(\chi_{2061}(361,\cdot)\)
\(\chi_{2061}(388,\cdot)\)
\(\chi_{2061}(478,\cdot)\)
\(\chi_{2061}(541,\cdot)\)
\(\chi_{2061}(631,\cdot)\)
\(\chi_{2061}(712,\cdot)\)
\(\chi_{2061}(838,\cdot)\)
\(\chi_{2061}(883,\cdot)\)
\(\chi_{2061}(919,\cdot)\)
\(\chi_{2061}(964,\cdot)\)
\(\chi_{2061}(991,\cdot)\)
\(\chi_{2061}(1027,\cdot)\)
\(\chi_{2061}(1045,\cdot)\)
\(\chi_{2061}(1099,\cdot)\)
\(\chi_{2061}(1369,\cdot)\)
\(\chi_{2061}(1504,\cdot)\)
\(\chi_{2061}(1558,\cdot)\)
\(\chi_{2061}(1567,\cdot)\)
\(\chi_{2061}(1612,\cdot)\)
\(\chi_{2061}(1684,\cdot)\)
\(\chi_{2061}(1729,\cdot)\)
\(\chi_{2061}(1747,\cdot)\)
\(\chi_{2061}(1756,\cdot)\)
...
![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 };
\((1604,235)\) → \((1,e\left(\frac{34}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2061 }(217, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
