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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3700, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([90,27,130]))
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gp:[g,chi] = znchar(Mod(1483, 3700))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3700.1483");
| Modulus: | \(3700\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3700\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{3700}(3,\cdot)\)
\(\chi_{3700}(67,\cdot)\)
\(\chi_{3700}(247,\cdot)\)
\(\chi_{3700}(263,\cdot)\)
\(\chi_{3700}(287,\cdot)\)
\(\chi_{3700}(363,\cdot)\)
\(\chi_{3700}(447,\cdot)\)
\(\chi_{3700}(583,\cdot)\)
\(\chi_{3700}(687,\cdot)\)
\(\chi_{3700}(983,\cdot)\)
\(\chi_{3700}(987,\cdot)\)
\(\chi_{3700}(1003,\cdot)\)
\(\chi_{3700}(1027,\cdot)\)
\(\chi_{3700}(1103,\cdot)\)
\(\chi_{3700}(1187,\cdot)\)
\(\chi_{3700}(1283,\cdot)\)
\(\chi_{3700}(1323,\cdot)\)
\(\chi_{3700}(1427,\cdot)\)
\(\chi_{3700}(1447,\cdot)\)
\(\chi_{3700}(1483,\cdot)\)
\(\chi_{3700}(1547,\cdot)\)
\(\chi_{3700}(1723,\cdot)\)
\(\chi_{3700}(1727,\cdot)\)
\(\chi_{3700}(1767,\cdot)\)
\(\chi_{3700}(1927,\cdot)\)
\(\chi_{3700}(2023,\cdot)\)
\(\chi_{3700}(2063,\cdot)\)
\(\chi_{3700}(2167,\cdot)\)
\(\chi_{3700}(2187,\cdot)\)
\(\chi_{3700}(2223,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1851,1777,1001)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3700 }(1483, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{180}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) |
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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)
