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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2646, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([7,66]))
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gp:[g,chi] = znchar(Mod(1271, 2646))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2646.1271");
| Modulus: | \(2646\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| 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;
|
| Primitive: | no, induced from \(\chi_{1323}(1271,\cdot)\) |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| 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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\(\chi_{2646}(11,\cdot)\)
\(\chi_{2646}(23,\cdot)\)
\(\chi_{2646}(137,\cdot)\)
\(\chi_{2646}(149,\cdot)\)
\(\chi_{2646}(389,\cdot)\)
\(\chi_{2646}(401,\cdot)\)
\(\chi_{2646}(515,\cdot)\)
\(\chi_{2646}(527,\cdot)\)
\(\chi_{2646}(641,\cdot)\)
\(\chi_{2646}(653,\cdot)\)
\(\chi_{2646}(767,\cdot)\)
\(\chi_{2646}(779,\cdot)\)
\(\chi_{2646}(893,\cdot)\)
\(\chi_{2646}(905,\cdot)\)
\(\chi_{2646}(1019,\cdot)\)
\(\chi_{2646}(1031,\cdot)\)
\(\chi_{2646}(1271,\cdot)\)
\(\chi_{2646}(1283,\cdot)\)
\(\chi_{2646}(1397,\cdot)\)
\(\chi_{2646}(1409,\cdot)\)
\(\chi_{2646}(1523,\cdot)\)
\(\chi_{2646}(1535,\cdot)\)
\(\chi_{2646}(1649,\cdot)\)
\(\chi_{2646}(1661,\cdot)\)
\(\chi_{2646}(1775,\cdot)\)
\(\chi_{2646}(1787,\cdot)\)
\(\chi_{2646}(1901,\cdot)\)
\(\chi_{2646}(1913,\cdot)\)
\(\chi_{2646}(2153,\cdot)\)
\(\chi_{2646}(2165,\cdot)\)
...
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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()
|
\((785,1081)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{11}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2646 }(1271, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(1\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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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)
