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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32704, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,9,24,28]))
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gp:[g,chi] = znchar(Mod(955, 32704))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32704.955");
| Modulus: | \(32704\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(32704\) |
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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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\(\chi_{32704}(523,\cdot)\)
\(\chi_{32704}(955,\cdot)\)
\(\chi_{32704}(1235,\cdot)\)
\(\chi_{32704}(2371,\cdot)\)
\(\chi_{32704}(2651,\cdot)\)
\(\chi_{32704}(3043,\cdot)\)
\(\chi_{32704}(3323,\cdot)\)
\(\chi_{32704}(3923,\cdot)\)
\(\chi_{32704}(4259,\cdot)\)
\(\chi_{32704}(5171,\cdot)\)
\(\chi_{32704}(6107,\cdot)\)
\(\chi_{32704}(6443,\cdot)\)
\(\chi_{32704}(8699,\cdot)\)
\(\chi_{32704}(9131,\cdot)\)
\(\chi_{32704}(9411,\cdot)\)
\(\chi_{32704}(10547,\cdot)\)
\(\chi_{32704}(10827,\cdot)\)
\(\chi_{32704}(11219,\cdot)\)
\(\chi_{32704}(11499,\cdot)\)
\(\chi_{32704}(12099,\cdot)\)
\(\chi_{32704}(12435,\cdot)\)
\(\chi_{32704}(13347,\cdot)\)
\(\chi_{32704}(14283,\cdot)\)
\(\chi_{32704}(14619,\cdot)\)
\(\chi_{32704}(16875,\cdot)\)
\(\chi_{32704}(17307,\cdot)\)
\(\chi_{32704}(17587,\cdot)\)
\(\chi_{32704}(18723,\cdot)\)
\(\chi_{32704}(19003,\cdot)\)
\(\chi_{32704}(19395,\cdot)\)
...
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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 };
\((1023,30661,14017,6721)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{7}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 32704 }(955, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
