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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,77,39]))
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gp:[g,chi] = znchar(Mod(6584, 9075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9075.6584");
| Modulus: | \(9075\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9075\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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);
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| 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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\(\chi_{9075}(29,\cdot)\)
\(\chi_{9075}(464,\cdot)\)
\(\chi_{9075}(569,\cdot)\)
\(\chi_{9075}(809,\cdot)\)
\(\chi_{9075}(854,\cdot)\)
\(\chi_{9075}(1289,\cdot)\)
\(\chi_{9075}(1394,\cdot)\)
\(\chi_{9075}(1634,\cdot)\)
\(\chi_{9075}(1679,\cdot)\)
\(\chi_{9075}(2114,\cdot)\)
\(\chi_{9075}(2219,\cdot)\)
\(\chi_{9075}(2459,\cdot)\)
\(\chi_{9075}(2504,\cdot)\)
\(\chi_{9075}(2939,\cdot)\)
\(\chi_{9075}(3044,\cdot)\)
\(\chi_{9075}(3284,\cdot)\)
\(\chi_{9075}(3329,\cdot)\)
\(\chi_{9075}(3764,\cdot)\)
\(\chi_{9075}(4109,\cdot)\)
\(\chi_{9075}(4694,\cdot)\)
\(\chi_{9075}(4979,\cdot)\)
\(\chi_{9075}(5414,\cdot)\)
\(\chi_{9075}(5519,\cdot)\)
\(\chi_{9075}(5759,\cdot)\)
\(\chi_{9075}(5804,\cdot)\)
\(\chi_{9075}(6239,\cdot)\)
\(\chi_{9075}(6344,\cdot)\)
\(\chi_{9075}(6584,\cdot)\)
\(\chi_{9075}(6629,\cdot)\)
\(\chi_{9075}(7064,\cdot)\)
...
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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: |
$\Q(\zeta_{55})$ |
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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 110 polynomial (not computed) |
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sage:chi.fixed_field()
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\((3026,727,5326)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{39}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(6584, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) |
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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)
