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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104000, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,5,16,40]))
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gp:[g,chi] = znchar(Mod(75451, 104000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104000.75451");
| Modulus: | \(104000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20800\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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 \(\chi_{20800}(8891,\cdot)\) |
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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: | no |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{104000}(51,\cdot)\)
\(\chi_{104000}(2651,\cdot)\)
\(\chi_{104000}(7851,\cdot)\)
\(\chi_{104000}(10451,\cdot)\)
\(\chi_{104000}(13051,\cdot)\)
\(\chi_{104000}(15651,\cdot)\)
\(\chi_{104000}(20851,\cdot)\)
\(\chi_{104000}(23451,\cdot)\)
\(\chi_{104000}(26051,\cdot)\)
\(\chi_{104000}(28651,\cdot)\)
\(\chi_{104000}(33851,\cdot)\)
\(\chi_{104000}(36451,\cdot)\)
\(\chi_{104000}(39051,\cdot)\)
\(\chi_{104000}(41651,\cdot)\)
\(\chi_{104000}(46851,\cdot)\)
\(\chi_{104000}(49451,\cdot)\)
\(\chi_{104000}(52051,\cdot)\)
\(\chi_{104000}(54651,\cdot)\)
\(\chi_{104000}(59851,\cdot)\)
\(\chi_{104000}(62451,\cdot)\)
\(\chi_{104000}(65051,\cdot)\)
\(\chi_{104000}(67651,\cdot)\)
\(\chi_{104000}(72851,\cdot)\)
\(\chi_{104000}(75451,\cdot)\)
\(\chi_{104000}(78051,\cdot)\)
\(\chi_{104000}(80651,\cdot)\)
\(\chi_{104000}(85851,\cdot)\)
\(\chi_{104000}(88451,\cdot)\)
\(\chi_{104000}(91051,\cdot)\)
\(\chi_{104000}(93651,\cdot)\)
...
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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 };
\((74751,58501,77377,64001)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 104000 }(75451, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) |
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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)
