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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41600, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,45,36,60]))
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gp:[g,chi] = znchar(Mod(2137, 41600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41600.2137");
| Modulus: | \(41600\) |
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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 \(\chi_{20800}(11237,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{41600}(73,\cdot)\)
\(\chi_{41600}(2137,\cdot)\)
\(\chi_{41600}(2153,\cdot)\)
\(\chi_{41600}(4217,\cdot)\)
\(\chi_{41600}(4233,\cdot)\)
\(\chi_{41600}(6297,\cdot)\)
\(\chi_{41600}(6313,\cdot)\)
\(\chi_{41600}(8377,\cdot)\)
\(\chi_{41600}(10473,\cdot)\)
\(\chi_{41600}(12537,\cdot)\)
\(\chi_{41600}(12553,\cdot)\)
\(\chi_{41600}(14617,\cdot)\)
\(\chi_{41600}(14633,\cdot)\)
\(\chi_{41600}(16697,\cdot)\)
\(\chi_{41600}(16713,\cdot)\)
\(\chi_{41600}(18777,\cdot)\)
\(\chi_{41600}(20873,\cdot)\)
\(\chi_{41600}(22937,\cdot)\)
\(\chi_{41600}(22953,\cdot)\)
\(\chi_{41600}(25017,\cdot)\)
\(\chi_{41600}(25033,\cdot)\)
\(\chi_{41600}(27097,\cdot)\)
\(\chi_{41600}(27113,\cdot)\)
\(\chi_{41600}(29177,\cdot)\)
\(\chi_{41600}(31273,\cdot)\)
\(\chi_{41600}(33337,\cdot)\)
\(\chi_{41600}(33353,\cdot)\)
\(\chi_{41600}(35417,\cdot)\)
\(\chi_{41600}(35433,\cdot)\)
\(\chi_{41600}(37497,\cdot)\)
...
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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 };
\((33151,16901,14977,22401)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{9}{20}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 41600 }(2137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) |
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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)
