![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2564, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,57]))
![Copy content]()
gp:[g,chi] = znchar(Mod(2501, 2564))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2564.2501");
| Modulus: | \(2564\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(641\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(80\) |
![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 \(\chi_{641}(578,\cdot)\) |
![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);
|
\(\chi_{2564}(45,\cdot)\)
\(\chi_{2564}(57,\cdot)\)
\(\chi_{2564}(121,\cdot)\)
\(\chi_{2564}(433,\cdot)\)
\(\chi_{2564}(445,\cdot)\)
\(\chi_{2564}(501,\cdot)\)
\(\chi_{2564}(585,\cdot)\)
\(\chi_{2564}(697,\cdot)\)
\(\chi_{2564}(781,\cdot)\)
\(\chi_{2564}(837,\cdot)\)
\(\chi_{2564}(849,\cdot)\)
\(\chi_{2564}(1161,\cdot)\)
\(\chi_{2564}(1225,\cdot)\)
\(\chi_{2564}(1237,\cdot)\)
\(\chi_{2564}(1345,\cdot)\)
\(\chi_{2564}(1369,\cdot)\)
\(\chi_{2564}(1385,\cdot)\)
\(\chi_{2564}(1433,\cdot)\)
\(\chi_{2564}(1445,\cdot)\)
\(\chi_{2564}(1573,\cdot)\)
\(\chi_{2564}(1689,\cdot)\)
\(\chi_{2564}(1745,\cdot)\)
\(\chi_{2564}(1905,\cdot)\)
\(\chi_{2564}(1941,\cdot)\)
\(\chi_{2564}(2101,\cdot)\)
\(\chi_{2564}(2157,\cdot)\)
\(\chi_{2564}(2273,\cdot)\)
\(\chi_{2564}(2401,\cdot)\)
\(\chi_{2564}(2413,\cdot)\)
\(\chi_{2564}(2461,\cdot)\)
...
![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 };
\((1283,1285)\) → \((1,e\left(\frac{57}{80}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2564 }(2501, a) \) |
\(1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{3}{16}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
