![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5125, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([12,55]))
![Copy content]()
gp:[g,chi] = znchar(Mod(2096, 5125))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5125.2096");
| Modulus: | \(5125\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(5125\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(100\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{5125}(46,\cdot)\)
\(\chi_{5125}(131,\cdot)\)
\(\chi_{5125}(156,\cdot)\)
\(\chi_{5125}(266,\cdot)\)
\(\chi_{5125}(446,\cdot)\)
\(\chi_{5125}(736,\cdot)\)
\(\chi_{5125}(841,\cdot)\)
\(\chi_{5125}(986,\cdot)\)
\(\chi_{5125}(1071,\cdot)\)
\(\chi_{5125}(1156,\cdot)\)
\(\chi_{5125}(1181,\cdot)\)
\(\chi_{5125}(1291,\cdot)\)
\(\chi_{5125}(1471,\cdot)\)
\(\chi_{5125}(1761,\cdot)\)
\(\chi_{5125}(1866,\cdot)\)
\(\chi_{5125}(2011,\cdot)\)
\(\chi_{5125}(2096,\cdot)\)
\(\chi_{5125}(2181,\cdot)\)
\(\chi_{5125}(2206,\cdot)\)
\(\chi_{5125}(2316,\cdot)\)
\(\chi_{5125}(2496,\cdot)\)
\(\chi_{5125}(2786,\cdot)\)
\(\chi_{5125}(2891,\cdot)\)
\(\chi_{5125}(3036,\cdot)\)
\(\chi_{5125}(3121,\cdot)\)
\(\chi_{5125}(3206,\cdot)\)
\(\chi_{5125}(3231,\cdot)\)
\(\chi_{5125}(3341,\cdot)\)
\(\chi_{5125}(3521,\cdot)\)
\(\chi_{5125}(3811,\cdot)\)
...
![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 };
\((2502,2876)\) → \((e\left(\frac{3}{25}\right),e\left(\frac{11}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 5125 }(2096, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{73}{100}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
