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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([11,58]))
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gp:[g,chi] = znchar(Mod(1777, 3025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3025.1777");
| 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);
|
| 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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\(\chi_{3025}(13,\cdot)\)
\(\chi_{3025}(117,\cdot)\)
\(\chi_{3025}(127,\cdot)\)
\(\chi_{3025}(183,\cdot)\)
\(\chi_{3025}(228,\cdot)\)
\(\chi_{3025}(248,\cdot)\)
\(\chi_{3025}(272,\cdot)\)
\(\chi_{3025}(288,\cdot)\)
\(\chi_{3025}(387,\cdot)\)
\(\chi_{3025}(392,\cdot)\)
\(\chi_{3025}(402,\cdot)\)
\(\chi_{3025}(458,\cdot)\)
\(\chi_{3025}(503,\cdot)\)
\(\chi_{3025}(523,\cdot)\)
\(\chi_{3025}(547,\cdot)\)
\(\chi_{3025}(563,\cdot)\)
\(\chi_{3025}(662,\cdot)\)
\(\chi_{3025}(667,\cdot)\)
\(\chi_{3025}(677,\cdot)\)
\(\chi_{3025}(733,\cdot)\)
\(\chi_{3025}(778,\cdot)\)
\(\chi_{3025}(798,\cdot)\)
\(\chi_{3025}(822,\cdot)\)
\(\chi_{3025}(937,\cdot)\)
\(\chi_{3025}(942,\cdot)\)
\(\chi_{3025}(952,\cdot)\)
\(\chi_{3025}(1053,\cdot)\)
\(\chi_{3025}(1073,\cdot)\)
\(\chi_{3025}(1097,\cdot)\)
\(\chi_{3025}(1113,\cdot)\)
...
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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));
|
| Fixed field: |
Number field defined by a degree 220 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((727,2301)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{29}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(1777, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{9}{22}\right)\) |
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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)
