![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80275, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([351,438,208]))
![Copy content]()
gp:[g,chi] = znchar(Mod(693, 80275))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80275.693");
| Modulus: | \(80275\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(16055\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(468\) |
![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 \(\chi_{16055}(693,\cdot)\) |
![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);
|
\(\chi_{80275}(82,\cdot)\)
\(\chi_{80275}(693,\cdot)\)
\(\chi_{80275}(1232,\cdot)\)
\(\chi_{80275}(1518,\cdot)\)
\(\chi_{80275}(2532,\cdot)\)
\(\chi_{80275}(2968,\cdot)\)
\(\chi_{80275}(3293,\cdot)\)
\(\chi_{80275}(3657,\cdot)\)
\(\chi_{80275}(4443,\cdot)\)
\(\chi_{80275}(4482,\cdot)\)
\(\chi_{80275}(5743,\cdot)\)
\(\chi_{80275}(5932,\cdot)\)
\(\chi_{80275}(6257,\cdot)\)
\(\chi_{80275}(6868,\cdot)\)
\(\chi_{80275}(7407,\cdot)\)
\(\chi_{80275}(7693,\cdot)\)
\(\chi_{80275}(8707,\cdot)\)
\(\chi_{80275}(9143,\cdot)\)
\(\chi_{80275}(9468,\cdot)\)
\(\chi_{80275}(9832,\cdot)\)
\(\chi_{80275}(10618,\cdot)\)
\(\chi_{80275}(10657,\cdot)\)
\(\chi_{80275}(11918,\cdot)\)
\(\chi_{80275}(12107,\cdot)\)
\(\chi_{80275}(12432,\cdot)\)
\(\chi_{80275}(13043,\cdot)\)
\(\chi_{80275}(13582,\cdot)\)
\(\chi_{80275}(13868,\cdot)\)
\(\chi_{80275}(14882,\cdot)\)
\(\chi_{80275}(15318,\cdot)\)
...
![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 };
\((51377,62701,46476)\) → \((-i,e\left(\frac{73}{78}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 80275 }(693, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{468}\right)\) | \(e\left(\frac{37}{468}\right)\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{161}{234}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
