![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2825, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([56,25]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1031, 2825))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2825.1031");
| Modulus: | \(2825\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2825\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(140\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2825}(56,\cdot)\)
\(\chi_{2825}(81,\cdot)\)
\(\chi_{2825}(111,\cdot)\)
\(\chi_{2825}(121,\cdot)\)
\(\chi_{2825}(166,\cdot)\)
\(\chi_{2825}(286,\cdot)\)
\(\chi_{2825}(331,\cdot)\)
\(\chi_{2825}(341,\cdot)\)
\(\chi_{2825}(371,\cdot)\)
\(\chi_{2825}(396,\cdot)\)
\(\chi_{2825}(466,\cdot)\)
\(\chi_{2825}(621,\cdot)\)
\(\chi_{2825}(646,\cdot)\)
\(\chi_{2825}(686,\cdot)\)
\(\chi_{2825}(731,\cdot)\)
\(\chi_{2825}(896,\cdot)\)
\(\chi_{2825}(906,\cdot)\)
\(\chi_{2825}(936,\cdot)\)
\(\chi_{2825}(961,\cdot)\)
\(\chi_{2825}(1031,\cdot)\)
\(\chi_{2825}(1116,\cdot)\)
\(\chi_{2825}(1186,\cdot)\)
\(\chi_{2825}(1211,\cdot)\)
\(\chi_{2825}(1241,\cdot)\)
\(\chi_{2825}(1296,\cdot)\)
\(\chi_{2825}(1416,\cdot)\)
\(\chi_{2825}(1461,\cdot)\)
\(\chi_{2825}(1471,\cdot)\)
\(\chi_{2825}(1596,\cdot)\)
\(\chi_{2825}(1681,\cdot)\)
...
![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 };
\((227,2376)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{5}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2825 }(1031, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
