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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([11,194]))
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gp:[g,chi] = znchar(Mod(1052, 3025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3025.1052");
| Modulus: | \(3025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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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\(\chi_{3025}(28,\cdot)\)
\(\chi_{3025}(63,\cdot)\)
\(\chi_{3025}(72,\cdot)\)
\(\chi_{3025}(73,\cdot)\)
\(\chi_{3025}(162,\cdot)\)
\(\chi_{3025}(167,\cdot)\)
\(\chi_{3025}(227,\cdot)\)
\(\chi_{3025}(303,\cdot)\)
\(\chi_{3025}(338,\cdot)\)
\(\chi_{3025}(347,\cdot)\)
\(\chi_{3025}(348,\cdot)\)
\(\chi_{3025}(437,\cdot)\)
\(\chi_{3025}(442,\cdot)\)
\(\chi_{3025}(502,\cdot)\)
\(\chi_{3025}(508,\cdot)\)
\(\chi_{3025}(613,\cdot)\)
\(\chi_{3025}(622,\cdot)\)
\(\chi_{3025}(623,\cdot)\)
\(\chi_{3025}(712,\cdot)\)
\(\chi_{3025}(777,\cdot)\)
\(\chi_{3025}(783,\cdot)\)
\(\chi_{3025}(853,\cdot)\)
\(\chi_{3025}(888,\cdot)\)
\(\chi_{3025}(897,\cdot)\)
\(\chi_{3025}(898,\cdot)\)
\(\chi_{3025}(987,\cdot)\)
\(\chi_{3025}(992,\cdot)\)
\(\chi_{3025}(1052,\cdot)\)
\(\chi_{3025}(1058,\cdot)\)
\(\chi_{3025}(1128,\cdot)\)
...
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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: |
$\Q(\zeta_{220})$ |
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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 220 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((727,2301)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{97}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(1052, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{39}{110}\right)\) |
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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)
