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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(440))
M = H._module
chi = DirichletCharacter(H, M([424,385]))
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gp:[g,chi] = znchar(Mod(53, 2057))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2057.53");
| Modulus: | \(2057\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2057\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(440\) |
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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: | yes |
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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);
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\(\chi_{2057}(15,\cdot)\)
\(\chi_{2057}(25,\cdot)\)
\(\chi_{2057}(26,\cdot)\)
\(\chi_{2057}(36,\cdot)\)
\(\chi_{2057}(42,\cdot)\)
\(\chi_{2057}(49,\cdot)\)
\(\chi_{2057}(53,\cdot)\)
\(\chi_{2057}(59,\cdot)\)
\(\chi_{2057}(60,\cdot)\)
\(\chi_{2057}(70,\cdot)\)
\(\chi_{2057}(93,\cdot)\)
\(\chi_{2057}(104,\cdot)\)
\(\chi_{2057}(168,\cdot)\)
\(\chi_{2057}(179,\cdot)\)
\(\chi_{2057}(185,\cdot)\)
\(\chi_{2057}(196,\cdot)\)
\(\chi_{2057}(212,\cdot)\)
\(\chi_{2057}(213,\cdot)\)
\(\chi_{2057}(223,\cdot)\)
\(\chi_{2057}(229,\cdot)\)
\(\chi_{2057}(236,\cdot)\)
\(\chi_{2057}(240,\cdot)\)
\(\chi_{2057}(246,\cdot)\)
\(\chi_{2057}(247,\cdot)\)
\(\chi_{2057}(257,\cdot)\)
\(\chi_{2057}(280,\cdot)\)
\(\chi_{2057}(291,\cdot)\)
\(\chi_{2057}(355,\cdot)\)
\(\chi_{2057}(383,\cdot)\)
\(\chi_{2057}(389,\cdot)\)
...
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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: |
$\Q(\zeta_{440})$ |
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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 440 polynomial (not computed) |
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sage:chi.fixed_field()
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\((970,122)\) → \((e\left(\frac{53}{55}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{301}{440}\right)\) | \(e\left(\frac{391}{440}\right)\) | \(e\left(\frac{163}{440}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) |
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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)
