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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(277984, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,0,24,81,22]))
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gp:[g,chi] = znchar(Mod(31, 277984))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277984.31");
| Modulus: | \(277984\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(34748\) |
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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: | no, induced from \(\chi_{34748}(31,\cdot)\) |
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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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\(\chi_{277984}(31,\cdot)\)
\(\chi_{277984}(6079,\cdot)\)
\(\chi_{277984}(8223,\cdot)\)
\(\chi_{277984}(13471,\cdot)\)
\(\chi_{277984}(18527,\cdot)\)
\(\chi_{277984}(28703,\cdot)\)
\(\chi_{277984}(35775,\cdot)\)
\(\chi_{277984}(43039,\cdot)\)
\(\chi_{277984}(45727,\cdot)\)
\(\chi_{277984}(66239,\cdot)\)
\(\chi_{277984}(70047,\cdot)\)
\(\chi_{277984}(79775,\cdot)\)
\(\chi_{277984}(84255,\cdot)\)
\(\chi_{277984}(95007,\cdot)\)
\(\chi_{277984}(99167,\cdot)\)
\(\chi_{277984}(101631,\cdot)\)
\(\chi_{277984}(102975,\cdot)\)
\(\chi_{277984}(105983,\cdot)\)
\(\chi_{277984}(118751,\cdot)\)
\(\chi_{277984}(127935,\cdot)\)
\(\chi_{277984}(133311,\cdot)\)
\(\chi_{277984}(133439,\cdot)\)
\(\chi_{277984}(143167,\cdot)\)
\(\chi_{277984}(144639,\cdot)\)
\(\chi_{277984}(150239,\cdot)\)
\(\chi_{277984}(156063,\cdot)\)
\(\chi_{277984}(169599,\cdot)\)
\(\chi_{277984}(180703,\cdot)\)
\(\chi_{277984}(183391,\cdot)\)
\(\chi_{277984}(184511,\cdot)\)
...
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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()
|
\((17375,243237,79425,261633,186593)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right),e\left(\frac{11}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 277984 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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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)
