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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(83200, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,65,56,0]))
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gp:[g,chi] = znchar(Mod(33359, 83200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("83200.33359");
| Modulus: | \(83200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1600\) |
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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 \(\chi_{1600}(1259,\cdot)\) |
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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);
|
\(\chi_{83200}(79,\cdot)\)
\(\chi_{83200}(2159,\cdot)\)
\(\chi_{83200}(4239,\cdot)\)
\(\chi_{83200}(6319,\cdot)\)
\(\chi_{83200}(10479,\cdot)\)
\(\chi_{83200}(12559,\cdot)\)
\(\chi_{83200}(14639,\cdot)\)
\(\chi_{83200}(16719,\cdot)\)
\(\chi_{83200}(20879,\cdot)\)
\(\chi_{83200}(22959,\cdot)\)
\(\chi_{83200}(25039,\cdot)\)
\(\chi_{83200}(27119,\cdot)\)
\(\chi_{83200}(31279,\cdot)\)
\(\chi_{83200}(33359,\cdot)\)
\(\chi_{83200}(35439,\cdot)\)
\(\chi_{83200}(37519,\cdot)\)
\(\chi_{83200}(41679,\cdot)\)
\(\chi_{83200}(43759,\cdot)\)
\(\chi_{83200}(45839,\cdot)\)
\(\chi_{83200}(47919,\cdot)\)
\(\chi_{83200}(52079,\cdot)\)
\(\chi_{83200}(54159,\cdot)\)
\(\chi_{83200}(56239,\cdot)\)
\(\chi_{83200}(58319,\cdot)\)
\(\chi_{83200}(62479,\cdot)\)
\(\chi_{83200}(64559,\cdot)\)
\(\chi_{83200}(66639,\cdot)\)
\(\chi_{83200}(68719,\cdot)\)
\(\chi_{83200}(72879,\cdot)\)
\(\chi_{83200}(74959,\cdot)\)
...
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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 };
\((74751,16901,56577,64001)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{7}{10}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 83200 }(33359, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) |
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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)
