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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([7,66]))
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gp:[g,chi] = znchar(Mod(1271, 1323))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.1271");
| Modulus: | \(1323\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
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| Conductor: | \(1323\) |
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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;
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| 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: | odd |
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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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\(\chi_{1323}(11,\cdot)\)
\(\chi_{1323}(23,\cdot)\)
\(\chi_{1323}(74,\cdot)\)
\(\chi_{1323}(86,\cdot)\)
\(\chi_{1323}(137,\cdot)\)
\(\chi_{1323}(149,\cdot)\)
\(\chi_{1323}(200,\cdot)\)
\(\chi_{1323}(212,\cdot)\)
\(\chi_{1323}(326,\cdot)\)
\(\chi_{1323}(338,\cdot)\)
\(\chi_{1323}(389,\cdot)\)
\(\chi_{1323}(401,\cdot)\)
\(\chi_{1323}(452,\cdot)\)
\(\chi_{1323}(464,\cdot)\)
\(\chi_{1323}(515,\cdot)\)
\(\chi_{1323}(527,\cdot)\)
\(\chi_{1323}(578,\cdot)\)
\(\chi_{1323}(590,\cdot)\)
\(\chi_{1323}(641,\cdot)\)
\(\chi_{1323}(653,\cdot)\)
\(\chi_{1323}(767,\cdot)\)
\(\chi_{1323}(779,\cdot)\)
\(\chi_{1323}(830,\cdot)\)
\(\chi_{1323}(842,\cdot)\)
\(\chi_{1323}(893,\cdot)\)
\(\chi_{1323}(905,\cdot)\)
\(\chi_{1323}(956,\cdot)\)
\(\chi_{1323}(968,\cdot)\)
\(\chi_{1323}(1019,\cdot)\)
\(\chi_{1323}(1031,\cdot)\)
...
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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));
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| Fixed field: |
Number field defined by a degree 126 polynomial (not computed) |
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sage:chi.fixed_field()
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\((785,1081)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{11}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(1271, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(1\) |
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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)
