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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40200, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([0,330,0,561,530]))
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gp:[g,chi] = znchar(Mod(4597, 40200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40200.4597");
| Modulus: | \(40200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13400\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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: | no, induced from \(\chi_{13400}(4597,\cdot)\) |
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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);
|
\(\chi_{40200}(13,\cdot)\)
\(\chi_{40200}(517,\cdot)\)
\(\chi_{40200}(637,\cdot)\)
\(\chi_{40200}(733,\cdot)\)
\(\chi_{40200}(1213,\cdot)\)
\(\chi_{40200}(1237,\cdot)\)
\(\chi_{40200}(1453,\cdot)\)
\(\chi_{40200}(1573,\cdot)\)
\(\chi_{40200}(1837,\cdot)\)
\(\chi_{40200}(1933,\cdot)\)
\(\chi_{40200}(2797,\cdot)\)
\(\chi_{40200}(3133,\cdot)\)
\(\chi_{40200}(3277,\cdot)\)
\(\chi_{40200}(3733,\cdot)\)
\(\chi_{40200}(3853,\cdot)\)
\(\chi_{40200}(3973,\cdot)\)
\(\chi_{40200}(3997,\cdot)\)
\(\chi_{40200}(4453,\cdot)\)
\(\chi_{40200}(4597,\cdot)\)
\(\chi_{40200}(4837,\cdot)\)
\(\chi_{40200}(5053,\cdot)\)
\(\chi_{40200}(5773,\cdot)\)
\(\chi_{40200}(6013,\cdot)\)
\(\chi_{40200}(6037,\cdot)\)
\(\chi_{40200}(6277,\cdot)\)
\(\chi_{40200}(6397,\cdot)\)
\(\chi_{40200}(6517,\cdot)\)
\(\chi_{40200}(7213,\cdot)\)
\(\chi_{40200}(7573,\cdot)\)
\(\chi_{40200}(7717,\cdot)\)
...
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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 };
\((30151,20101,26801,35377,13201)\) → \((1,-1,1,e\left(\frac{17}{20}\right),e\left(\frac{53}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 40200 }(4597, a) \) |
\(1\) | \(1\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{599}{660}\right)\) | \(e\left(\frac{293}{660}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{551}{660}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{317}{330}\right)\) |
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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)
