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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(4332))
M = H._module
chi = DirichletCharacter(H, M([2166,3249,3610,2144]))
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gp:[g,chi] = znchar(Mod(7763, 987696))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("987696.7763");
| Modulus: | \(987696\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(987696\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(4332\) |
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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);
|
\(\chi_{987696}(83,\cdot)\)
\(\chi_{987696}(923,\cdot)\)
\(\chi_{987696}(1451,\cdot)\)
\(\chi_{987696}(2291,\cdot)\)
\(\chi_{987696}(3659,\cdot)\)
\(\chi_{987696}(4187,\cdot)\)
\(\chi_{987696}(5027,\cdot)\)
\(\chi_{987696}(5555,\cdot)\)
\(\chi_{987696}(6395,\cdot)\)
\(\chi_{987696}(6923,\cdot)\)
\(\chi_{987696}(7763,\cdot)\)
\(\chi_{987696}(8291,\cdot)\)
\(\chi_{987696}(9131,\cdot)\)
\(\chi_{987696}(9659,\cdot)\)
\(\chi_{987696}(10499,\cdot)\)
\(\chi_{987696}(11027,\cdot)\)
\(\chi_{987696}(11867,\cdot)\)
\(\chi_{987696}(12395,\cdot)\)
\(\chi_{987696}(13235,\cdot)\)
\(\chi_{987696}(13763,\cdot)\)
\(\chi_{987696}(14603,\cdot)\)
\(\chi_{987696}(15131,\cdot)\)
\(\chi_{987696}(15971,\cdot)\)
\(\chi_{987696}(16499,\cdot)\)
\(\chi_{987696}(17339,\cdot)\)
\(\chi_{987696}(17867,\cdot)\)
\(\chi_{987696}(18707,\cdot)\)
\(\chi_{987696}(19235,\cdot)\)
\(\chi_{987696}(20075,\cdot)\)
\(\chi_{987696}(20603,\cdot)\)
...
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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 };
\((617311,740773,438977,857377)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{536}{1083}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 987696 }(7763, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1831}{4332}\right)\) | \(e\left(\frac{904}{1083}\right)\) | \(e\left(\frac{169}{4332}\right)\) | \(e\left(\frac{1001}{1444}\right)\) | \(e\left(\frac{193}{2166}\right)\) | \(e\left(\frac{231}{722}\right)\) | \(e\left(\frac{1831}{2166}\right)\) | \(e\left(\frac{3521}{4332}\right)\) | \(e\left(\frac{619}{2166}\right)\) | \(e\left(\frac{1115}{4332}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
