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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,45,104]))
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gp:[g,chi] = znchar(Mod(11, 1728))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.11");
| Modulus: | \(1728\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1728\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{1728}(11,\cdot)\)
\(\chi_{1728}(59,\cdot)\)
\(\chi_{1728}(83,\cdot)\)
\(\chi_{1728}(131,\cdot)\)
\(\chi_{1728}(155,\cdot)\)
\(\chi_{1728}(203,\cdot)\)
\(\chi_{1728}(227,\cdot)\)
\(\chi_{1728}(275,\cdot)\)
\(\chi_{1728}(299,\cdot)\)
\(\chi_{1728}(347,\cdot)\)
\(\chi_{1728}(371,\cdot)\)
\(\chi_{1728}(419,\cdot)\)
\(\chi_{1728}(443,\cdot)\)
\(\chi_{1728}(491,\cdot)\)
\(\chi_{1728}(515,\cdot)\)
\(\chi_{1728}(563,\cdot)\)
\(\chi_{1728}(587,\cdot)\)
\(\chi_{1728}(635,\cdot)\)
\(\chi_{1728}(659,\cdot)\)
\(\chi_{1728}(707,\cdot)\)
\(\chi_{1728}(731,\cdot)\)
\(\chi_{1728}(779,\cdot)\)
\(\chi_{1728}(803,\cdot)\)
\(\chi_{1728}(851,\cdot)\)
\(\chi_{1728}(875,\cdot)\)
\(\chi_{1728}(923,\cdot)\)
\(\chi_{1728}(947,\cdot)\)
\(\chi_{1728}(995,\cdot)\)
\(\chi_{1728}(1019,\cdot)\)
\(\chi_{1728}(1067,\cdot)\)
...
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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: |
$\Q(\zeta_{144})$ |
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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 144 polynomial (not computed) |
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sage:chi.fixed_field()
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\((703,325,1217)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1728 }(11, a) \) |
\(1\) | \(1\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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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)
