![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5082, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,55,261]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1151, 5082))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5082.1151");
| Modulus: | \(5082\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2541\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(330\) |
![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 \(\chi_{2541}(1151,\cdot)\) |
![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);
|
\(\chi_{5082}(17,\cdot)\)
\(\chi_{5082}(101,\cdot)\)
\(\chi_{5082}(173,\cdot)\)
\(\chi_{5082}(227,\cdot)\)
\(\chi_{5082}(299,\cdot)\)
\(\chi_{5082}(425,\cdot)\)
\(\chi_{5082}(437,\cdot)\)
\(\chi_{5082}(479,\cdot)\)
\(\chi_{5082}(563,\cdot)\)
\(\chi_{5082}(635,\cdot)\)
\(\chi_{5082}(677,\cdot)\)
\(\chi_{5082}(689,\cdot)\)
\(\chi_{5082}(761,\cdot)\)
\(\chi_{5082}(899,\cdot)\)
\(\chi_{5082}(1025,\cdot)\)
\(\chi_{5082}(1097,\cdot)\)
\(\chi_{5082}(1139,\cdot)\)
\(\chi_{5082}(1151,\cdot)\)
\(\chi_{5082}(1223,\cdot)\)
\(\chi_{5082}(1349,\cdot)\)
\(\chi_{5082}(1361,\cdot)\)
\(\chi_{5082}(1403,\cdot)\)
\(\chi_{5082}(1487,\cdot)\)
\(\chi_{5082}(1559,\cdot)\)
\(\chi_{5082}(1601,\cdot)\)
\(\chi_{5082}(1811,\cdot)\)
\(\chi_{5082}(1823,\cdot)\)
\(\chi_{5082}(1865,\cdot)\)
\(\chi_{5082}(1949,\cdot)\)
\(\chi_{5082}(2021,\cdot)\)
...
![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 };
| Field of values: |
$\Q(\zeta_{165})$ |
![Copy content]()
sage:CyclotomicField(chi.multiplicative_order())
![Copy content]()
gp:nfinit(polcyclo(charorder(g,chi)))
![Copy content]()
magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 330 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((3389,4357,2059)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{87}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5082 }(1151, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{21}{110}\right)\) |
![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)
