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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([110,121,40]))
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gp:[g,chi] = znchar(Mod(1223, 2300))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2300.1223");
| Modulus: | \(2300\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2300\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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;
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| 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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\(\chi_{2300}(3,\cdot)\)
\(\chi_{2300}(27,\cdot)\)
\(\chi_{2300}(87,\cdot)\)
\(\chi_{2300}(123,\cdot)\)
\(\chi_{2300}(127,\cdot)\)
\(\chi_{2300}(147,\cdot)\)
\(\chi_{2300}(163,\cdot)\)
\(\chi_{2300}(167,\cdot)\)
\(\chi_{2300}(187,\cdot)\)
\(\chi_{2300}(223,\cdot)\)
\(\chi_{2300}(303,\cdot)\)
\(\chi_{2300}(347,\cdot)\)
\(\chi_{2300}(363,\cdot)\)
\(\chi_{2300}(403,\cdot)\)
\(\chi_{2300}(423,\cdot)\)
\(\chi_{2300}(427,\cdot)\)
\(\chi_{2300}(463,\cdot)\)
\(\chi_{2300}(487,\cdot)\)
\(\chi_{2300}(547,\cdot)\)
\(\chi_{2300}(583,\cdot)\)
\(\chi_{2300}(587,\cdot)\)
\(\chi_{2300}(623,\cdot)\)
\(\chi_{2300}(627,\cdot)\)
\(\chi_{2300}(647,\cdot)\)
\(\chi_{2300}(683,\cdot)\)
\(\chi_{2300}(703,\cdot)\)
\(\chi_{2300}(763,\cdot)\)
\(\chi_{2300}(767,\cdot)\)
\(\chi_{2300}(823,\cdot)\)
\(\chi_{2300}(863,\cdot)\)
...
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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: |
$\Q(\zeta_{220})$ |
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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 220 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1151,277,1201)\) → \((-1,e\left(\frac{11}{20}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 2300 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) |
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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)
