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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,42,53]))
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gp:[g,chi] = znchar(Mod(233, 2667))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2667.233");
| Modulus: | \(2667\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2667\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{2667}(86,\cdot)\)
\(\chi_{2667}(170,\cdot)\)
\(\chi_{2667}(212,\cdot)\)
\(\chi_{2667}(233,\cdot)\)
\(\chi_{2667}(368,\cdot)\)
\(\chi_{2667}(464,\cdot)\)
\(\chi_{2667}(473,\cdot)\)
\(\chi_{2667}(515,\cdot)\)
\(\chi_{2667}(620,\cdot)\)
\(\chi_{2667}(641,\cdot)\)
\(\chi_{2667}(674,\cdot)\)
\(\chi_{2667}(683,\cdot)\)
\(\chi_{2667}(935,\cdot)\)
\(\chi_{2667}(947,\cdot)\)
\(\chi_{2667}(956,\cdot)\)
\(\chi_{2667}(1019,\cdot)\)
\(\chi_{2667}(1094,\cdot)\)
\(\chi_{2667}(1166,\cdot)\)
\(\chi_{2667}(1208,\cdot)\)
\(\chi_{2667}(1325,\cdot)\)
\(\chi_{2667}(1388,\cdot)\)
\(\chi_{2667}(1577,\cdot)\)
\(\chi_{2667}(1640,\cdot)\)
\(\chi_{2667}(1871,\cdot)\)
\(\chi_{2667}(1934,\cdot)\)
\(\chi_{2667}(2006,\cdot)\)
\(\chi_{2667}(2123,\cdot)\)
\(\chi_{2667}(2216,\cdot)\)
\(\chi_{2667}(2300,\cdot)\)
\(\chi_{2667}(2342,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
| Field of values: |
$\Q(\zeta_{63})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 126 polynomial (not computed) |
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sage:chi.fixed_field()
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\((890,1144,2416)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{53}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(233, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(1\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
