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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, base_ring=CyclotomicField(76))
M = H._module
chi = DirichletCharacter(H, M([38,57,52]))
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gp:[g,chi] = znchar(Mod(1103, 5415))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5415.1103");
| Modulus: | \(5415\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5415\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(76\) |
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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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\(\chi_{5415}(77,\cdot)\)
\(\chi_{5415}(248,\cdot)\)
\(\chi_{5415}(533,\cdot)\)
\(\chi_{5415}(647,\cdot)\)
\(\chi_{5415}(818,\cdot)\)
\(\chi_{5415}(932,\cdot)\)
\(\chi_{5415}(1103,\cdot)\)
\(\chi_{5415}(1217,\cdot)\)
\(\chi_{5415}(1388,\cdot)\)
\(\chi_{5415}(1502,\cdot)\)
\(\chi_{5415}(1673,\cdot)\)
\(\chi_{5415}(1787,\cdot)\)
\(\chi_{5415}(1958,\cdot)\)
\(\chi_{5415}(2072,\cdot)\)
\(\chi_{5415}(2243,\cdot)\)
\(\chi_{5415}(2357,\cdot)\)
\(\chi_{5415}(2642,\cdot)\)
\(\chi_{5415}(2813,\cdot)\)
\(\chi_{5415}(2927,\cdot)\)
\(\chi_{5415}(3098,\cdot)\)
\(\chi_{5415}(3212,\cdot)\)
\(\chi_{5415}(3383,\cdot)\)
\(\chi_{5415}(3497,\cdot)\)
\(\chi_{5415}(3668,\cdot)\)
\(\chi_{5415}(3782,\cdot)\)
\(\chi_{5415}(3953,\cdot)\)
\(\chi_{5415}(4067,\cdot)\)
\(\chi_{5415}(4238,\cdot)\)
\(\chi_{5415}(4352,\cdot)\)
\(\chi_{5415}(4523,\cdot)\)
...
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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 };
\((3611,2167,5056)\) → \((-1,-i,e\left(\frac{13}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 5415 }(1103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{17}{76}\right)\) |
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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)
