![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35700, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,40,32,0,65]))
![Copy content]()
gp:[g,chi] = znchar(Mod(30731, 35700))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35700.30731");
| Modulus: | \(35700\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(5100\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(80\) |
![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 \(\chi_{5100}(131,\cdot)\) |
![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);
|
\(\chi_{35700}(71,\cdot)\)
\(\chi_{35700}(911,\cdot)\)
\(\chi_{35700}(1331,\cdot)\)
\(\chi_{35700}(2171,\cdot)\)
\(\chi_{35700}(2591,\cdot)\)
\(\chi_{35700}(3431,\cdot)\)
\(\chi_{35700}(5111,\cdot)\)
\(\chi_{35700}(5531,\cdot)\)
\(\chi_{35700}(7211,\cdot)\)
\(\chi_{35700}(8471,\cdot)\)
\(\chi_{35700}(9311,\cdot)\)
\(\chi_{35700}(9731,\cdot)\)
\(\chi_{35700}(10571,\cdot)\)
\(\chi_{35700}(12671,\cdot)\)
\(\chi_{35700}(15191,\cdot)\)
\(\chi_{35700}(15611,\cdot)\)
\(\chi_{35700}(16871,\cdot)\)
\(\chi_{35700}(17711,\cdot)\)
\(\chi_{35700}(19391,\cdot)\)
\(\chi_{35700}(19811,\cdot)\)
\(\chi_{35700}(21491,\cdot)\)
\(\chi_{35700}(22331,\cdot)\)
\(\chi_{35700}(23591,\cdot)\)
\(\chi_{35700}(24011,\cdot)\)
\(\chi_{35700}(26531,\cdot)\)
\(\chi_{35700}(28631,\cdot)\)
\(\chi_{35700}(29471,\cdot)\)
\(\chi_{35700}(29891,\cdot)\)
\(\chi_{35700}(30731,\cdot)\)
\(\chi_{35700}(31991,\cdot)\)
...
![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 };
\((17851,23801,24277,20401,8401)\) → \((-1,-1,e\left(\frac{2}{5}\right),1,e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 35700 }(30731, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
