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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([77,117]))
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gp:[g,chi] = znchar(Mod(1049, 1323))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.1049");
| 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: | 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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\(\chi_{1323}(20,\cdot)\)
\(\chi_{1323}(41,\cdot)\)
\(\chi_{1323}(83,\cdot)\)
\(\chi_{1323}(104,\cdot)\)
\(\chi_{1323}(167,\cdot)\)
\(\chi_{1323}(209,\cdot)\)
\(\chi_{1323}(230,\cdot)\)
\(\chi_{1323}(272,\cdot)\)
\(\chi_{1323}(335,\cdot)\)
\(\chi_{1323}(356,\cdot)\)
\(\chi_{1323}(398,\cdot)\)
\(\chi_{1323}(419,\cdot)\)
\(\chi_{1323}(461,\cdot)\)
\(\chi_{1323}(482,\cdot)\)
\(\chi_{1323}(524,\cdot)\)
\(\chi_{1323}(545,\cdot)\)
\(\chi_{1323}(608,\cdot)\)
\(\chi_{1323}(650,\cdot)\)
\(\chi_{1323}(671,\cdot)\)
\(\chi_{1323}(713,\cdot)\)
\(\chi_{1323}(776,\cdot)\)
\(\chi_{1323}(797,\cdot)\)
\(\chi_{1323}(839,\cdot)\)
\(\chi_{1323}(860,\cdot)\)
\(\chi_{1323}(902,\cdot)\)
\(\chi_{1323}(923,\cdot)\)
\(\chi_{1323}(965,\cdot)\)
\(\chi_{1323}(986,\cdot)\)
\(\chi_{1323}(1049,\cdot)\)
\(\chi_{1323}(1091,\cdot)\)
...
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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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\((785,1081)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(1049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
