![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30025, base_ring=CyclotomicField(600))
M = H._module
chi = DirichletCharacter(H, M([300,221]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1949, 30025))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30025.1949");
| Modulus: | \(30025\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(6005\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(600\) |
![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 \(\chi_{6005}(1949,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{30025}(774,\cdot)\)
\(\chi_{30025}(799,\cdot)\)
\(\chi_{30025}(899,\cdot)\)
\(\chi_{30025}(1599,\cdot)\)
\(\chi_{30025}(1949,\cdot)\)
\(\chi_{30025}(2024,\cdot)\)
\(\chi_{30025}(2374,\cdot)\)
\(\chi_{30025}(2574,\cdot)\)
\(\chi_{30025}(2599,\cdot)\)
\(\chi_{30025}(2799,\cdot)\)
\(\chi_{30025}(2999,\cdot)\)
\(\chi_{30025}(3024,\cdot)\)
\(\chi_{30025}(3099,\cdot)\)
\(\chi_{30025}(3174,\cdot)\)
\(\chi_{30025}(3299,\cdot)\)
\(\chi_{30025}(3524,\cdot)\)
\(\chi_{30025}(3624,\cdot)\)
\(\chi_{30025}(3724,\cdot)\)
\(\chi_{30025}(3774,\cdot)\)
\(\chi_{30025}(3824,\cdot)\)
\(\chi_{30025}(4499,\cdot)\)
\(\chi_{30025}(4774,\cdot)\)
\(\chi_{30025}(4799,\cdot)\)
\(\chi_{30025}(4824,\cdot)\)
\(\chi_{30025}(4849,\cdot)\)
\(\chi_{30025}(5049,\cdot)\)
\(\chi_{30025}(5099,\cdot)\)
\(\chi_{30025}(5124,\cdot)\)
\(\chi_{30025}(5249,\cdot)\)
\(\chi_{30025}(5499,\cdot)\)
...
![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 };
\((1202,18026)\) → \((-1,e\left(\frac{221}{600}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 30025 }(1949, a) \) |
\(1\) | \(1\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(i\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{221}{600}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{103}{200}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
