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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3042, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
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gp:[g,chi] = znchar(Mod(2107, 3042))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3042.2107");
| Modulus: | \(3042\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(169\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(13\) |
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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 \(\chi_{169}(79,\cdot)\) |
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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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\(\chi_{3042}(235,\cdot)\)
\(\chi_{3042}(469,\cdot)\)
\(\chi_{3042}(703,\cdot)\)
\(\chi_{3042}(937,\cdot)\)
\(\chi_{3042}(1171,\cdot)\)
\(\chi_{3042}(1405,\cdot)\)
\(\chi_{3042}(1639,\cdot)\)
\(\chi_{3042}(1873,\cdot)\)
\(\chi_{3042}(2107,\cdot)\)
\(\chi_{3042}(2341,\cdot)\)
\(\chi_{3042}(2575,\cdot)\)
\(\chi_{3042}(2809,\cdot)\)
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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 };
\((677,847)\) → \((1,e\left(\frac{2}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(2107, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |
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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)
