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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(400))
M = H._module
chi = DirichletCharacter(H, M([200,225,136,0]))
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gp:[g,chi] = znchar(Mod(12559, 416000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.12559");
| Modulus: | \(416000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(400\) |
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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 \(\chi_{8000}(4059,\cdot)\) |
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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);
|
\(\chi_{416000}(79,\cdot)\)
\(\chi_{416000}(2159,\cdot)\)
\(\chi_{416000}(4239,\cdot)\)
\(\chi_{416000}(6319,\cdot)\)
\(\chi_{416000}(10479,\cdot)\)
\(\chi_{416000}(12559,\cdot)\)
\(\chi_{416000}(14639,\cdot)\)
\(\chi_{416000}(16719,\cdot)\)
\(\chi_{416000}(20879,\cdot)\)
\(\chi_{416000}(22959,\cdot)\)
\(\chi_{416000}(25039,\cdot)\)
\(\chi_{416000}(27119,\cdot)\)
\(\chi_{416000}(31279,\cdot)\)
\(\chi_{416000}(33359,\cdot)\)
\(\chi_{416000}(35439,\cdot)\)
\(\chi_{416000}(37519,\cdot)\)
\(\chi_{416000}(41679,\cdot)\)
\(\chi_{416000}(43759,\cdot)\)
\(\chi_{416000}(45839,\cdot)\)
\(\chi_{416000}(47919,\cdot)\)
\(\chi_{416000}(52079,\cdot)\)
\(\chi_{416000}(54159,\cdot)\)
\(\chi_{416000}(56239,\cdot)\)
\(\chi_{416000}(58319,\cdot)\)
\(\chi_{416000}(62479,\cdot)\)
\(\chi_{416000}(64559,\cdot)\)
\(\chi_{416000}(66639,\cdot)\)
\(\chi_{416000}(68719,\cdot)\)
\(\chi_{416000}(72879,\cdot)\)
\(\chi_{416000}(74959,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{400})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 400 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{17}{50}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(12559, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{227}{400}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{27}{200}\right)\) | \(e\left(\frac{61}{400}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{223}{400}\right)\) | \(e\left(\frac{237}{400}\right)\) | \(e\left(\frac{183}{200}\right)\) | \(e\left(\frac{281}{400}\right)\) | \(e\left(\frac{107}{400}\right)\) |
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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)
