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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([133,110]))
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gp:[g,chi] = znchar(Mod(13, 1225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1225.13");
| Modulus: | \(1225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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);
|
| 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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\(\chi_{1225}(13,\cdot)\)
\(\chi_{1225}(27,\cdot)\)
\(\chi_{1225}(62,\cdot)\)
\(\chi_{1225}(83,\cdot)\)
\(\chi_{1225}(153,\cdot)\)
\(\chi_{1225}(167,\cdot)\)
\(\chi_{1225}(188,\cdot)\)
\(\chi_{1225}(202,\cdot)\)
\(\chi_{1225}(223,\cdot)\)
\(\chi_{1225}(237,\cdot)\)
\(\chi_{1225}(258,\cdot)\)
\(\chi_{1225}(272,\cdot)\)
\(\chi_{1225}(328,\cdot)\)
\(\chi_{1225}(363,\cdot)\)
\(\chi_{1225}(377,\cdot)\)
\(\chi_{1225}(398,\cdot)\)
\(\chi_{1225}(412,\cdot)\)
\(\chi_{1225}(433,\cdot)\)
\(\chi_{1225}(447,\cdot)\)
\(\chi_{1225}(503,\cdot)\)
\(\chi_{1225}(517,\cdot)\)
\(\chi_{1225}(552,\cdot)\)
\(\chi_{1225}(573,\cdot)\)
\(\chi_{1225}(608,\cdot)\)
\(\chi_{1225}(622,\cdot)\)
\(\chi_{1225}(678,\cdot)\)
\(\chi_{1225}(692,\cdot)\)
\(\chi_{1225}(713,\cdot)\)
\(\chi_{1225}(727,\cdot)\)
\(\chi_{1225}(748,\cdot)\)
...
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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: |
$\Q(\zeta_{140})$ |
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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 140 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1177,101)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) |
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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)
