![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2366, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,151]))
![Copy content]()
gp:[g,chi] = znchar(Mod(713, 2366))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2366.713");
| Modulus: | \(2366\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1183\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(156\) |
![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 \(\chi_{1183}(713,\cdot)\) |
![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);
|
\(\chi_{2366}(41,\cdot)\)
\(\chi_{2366}(97,\cdot)\)
\(\chi_{2366}(111,\cdot)\)
\(\chi_{2366}(167,\cdot)\)
\(\chi_{2366}(223,\cdot)\)
\(\chi_{2366}(279,\cdot)\)
\(\chi_{2366}(293,\cdot)\)
\(\chi_{2366}(349,\cdot)\)
\(\chi_{2366}(405,\cdot)\)
\(\chi_{2366}(461,\cdot)\)
\(\chi_{2366}(475,\cdot)\)
\(\chi_{2366}(531,\cdot)\)
\(\chi_{2366}(643,\cdot)\)
\(\chi_{2366}(713,\cdot)\)
\(\chi_{2366}(769,\cdot)\)
\(\chi_{2366}(825,\cdot)\)
\(\chi_{2366}(839,\cdot)\)
\(\chi_{2366}(895,\cdot)\)
\(\chi_{2366}(951,\cdot)\)
\(\chi_{2366}(1007,\cdot)\)
\(\chi_{2366}(1021,\cdot)\)
\(\chi_{2366}(1077,\cdot)\)
\(\chi_{2366}(1133,\cdot)\)
\(\chi_{2366}(1189,\cdot)\)
\(\chi_{2366}(1203,\cdot)\)
\(\chi_{2366}(1259,\cdot)\)
\(\chi_{2366}(1315,\cdot)\)
\(\chi_{2366}(1385,\cdot)\)
\(\chi_{2366}(1497,\cdot)\)
\(\chi_{2366}(1553,\cdot)\)
...
![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: |
$\Q(\zeta_{156})$ |
![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 156 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((339,2199)\) → \((-1,e\left(\frac{151}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2366 }(713, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
![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)
