![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10470, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([58,58,65]))
![Copy content]()
gp:[g,chi] = znchar(Mod(3479, 10470))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10470.3479");
| Modulus: | \(10470\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(5235\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(116\) |
![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 \(\chi_{5235}(3479,\cdot)\) |
![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);
|
\(\chi_{10470}(179,\cdot)\)
\(\chi_{10470}(359,\cdot)\)
\(\chi_{10470}(659,\cdot)\)
\(\chi_{10470}(719,\cdot)\)
\(\chi_{10470}(989,\cdot)\)
\(\chi_{10470}(1019,\cdot)\)
\(\chi_{10470}(1229,\cdot)\)
\(\chi_{10470}(1349,\cdot)\)
\(\chi_{10470}(1499,\cdot)\)
\(\chi_{10470}(1529,\cdot)\)
\(\chi_{10470}(1559,\cdot)\)
\(\chi_{10470}(1739,\cdot)\)
\(\chi_{10470}(2129,\cdot)\)
\(\chi_{10470}(2159,\cdot)\)
\(\chi_{10470}(3089,\cdot)\)
\(\chi_{10470}(3149,\cdot)\)
\(\chi_{10470}(3179,\cdot)\)
\(\chi_{10470}(3239,\cdot)\)
\(\chi_{10470}(3359,\cdot)\)
\(\chi_{10470}(3389,\cdot)\)
\(\chi_{10470}(3479,\cdot)\)
\(\chi_{10470}(3569,\cdot)\)
\(\chi_{10470}(4109,\cdot)\)
\(\chi_{10470}(4199,\cdot)\)
\(\chi_{10470}(4289,\cdot)\)
\(\chi_{10470}(4319,\cdot)\)
\(\chi_{10470}(4439,\cdot)\)
\(\chi_{10470}(4499,\cdot)\)
\(\chi_{10470}(4529,\cdot)\)
\(\chi_{10470}(4589,\cdot)\)
...
![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: |
$\Q(\zeta_{116})$ |
![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 116 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((3491,8377,3841)\) → \((-1,-1,e\left(\frac{65}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 10470 }(3479, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{31}{116}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) |
![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)
