![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([171,57,103]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1466, 7581))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7581.1466");
| Modulus: | \(7581\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(7581\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(342\) |
![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);
|
\(\chi_{7581}(59,\cdot)\)
\(\chi_{7581}(89,\cdot)\)
\(\chi_{7581}(110,\cdot)\)
\(\chi_{7581}(185,\cdot)\)
\(\chi_{7581}(257,\cdot)\)
\(\chi_{7581}(269,\cdot)\)
\(\chi_{7581}(458,\cdot)\)
\(\chi_{7581}(509,\cdot)\)
\(\chi_{7581}(584,\cdot)\)
\(\chi_{7581}(656,\cdot)\)
\(\chi_{7581}(857,\cdot)\)
\(\chi_{7581}(887,\cdot)\)
\(\chi_{7581}(908,\cdot)\)
\(\chi_{7581}(983,\cdot)\)
\(\chi_{7581}(1067,\cdot)\)
\(\chi_{7581}(1256,\cdot)\)
\(\chi_{7581}(1286,\cdot)\)
\(\chi_{7581}(1307,\cdot)\)
\(\chi_{7581}(1454,\cdot)\)
\(\chi_{7581}(1466,\cdot)\)
\(\chi_{7581}(1655,\cdot)\)
\(\chi_{7581}(1685,\cdot)\)
\(\chi_{7581}(1781,\cdot)\)
\(\chi_{7581}(1853,\cdot)\)
\(\chi_{7581}(1865,\cdot)\)
\(\chi_{7581}(2054,\cdot)\)
\(\chi_{7581}(2084,\cdot)\)
\(\chi_{7581}(2105,\cdot)\)
\(\chi_{7581}(2180,\cdot)\)
\(\chi_{7581}(2252,\cdot)\)
...
![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 };
\((2528,6499,1807)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{103}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(1466, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{9}{19}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
