![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([0,50,23]))
![Copy content]()
gp:[g,chi] = znchar(Mod(733, 1000))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1000.733");
| Modulus: | \(1000\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1000\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{1000}(13,\cdot)\)
\(\chi_{1000}(37,\cdot)\)
\(\chi_{1000}(53,\cdot)\)
\(\chi_{1000}(77,\cdot)\)
\(\chi_{1000}(117,\cdot)\)
\(\chi_{1000}(133,\cdot)\)
\(\chi_{1000}(173,\cdot)\)
\(\chi_{1000}(197,\cdot)\)
\(\chi_{1000}(213,\cdot)\)
\(\chi_{1000}(237,\cdot)\)
\(\chi_{1000}(253,\cdot)\)
\(\chi_{1000}(277,\cdot)\)
\(\chi_{1000}(317,\cdot)\)
\(\chi_{1000}(333,\cdot)\)
\(\chi_{1000}(373,\cdot)\)
\(\chi_{1000}(397,\cdot)\)
\(\chi_{1000}(413,\cdot)\)
\(\chi_{1000}(437,\cdot)\)
\(\chi_{1000}(453,\cdot)\)
\(\chi_{1000}(477,\cdot)\)
\(\chi_{1000}(517,\cdot)\)
\(\chi_{1000}(533,\cdot)\)
\(\chi_{1000}(573,\cdot)\)
\(\chi_{1000}(597,\cdot)\)
\(\chi_{1000}(613,\cdot)\)
\(\chi_{1000}(637,\cdot)\)
\(\chi_{1000}(653,\cdot)\)
\(\chi_{1000}(677,\cdot)\)
\(\chi_{1000}(717,\cdot)\)
\(\chi_{1000}(733,\cdot)\)
...
![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 };
\((751,501,377)\) → \((1,-1,e\left(\frac{23}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(733, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{33}{100}\right)\) |
![Copy content]()
sage:chi(x) # x integer
![Copy content]()
gp:chareval(g,chi,x) \\ x integer, value in Q/Z
![Copy content]()
magma:chi(x)
