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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([19,75]))
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gp:[g,chi] = znchar(Mod(2279, 7942))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7942.2279");
| Modulus: | \(7942\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3971\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(190\) |
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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 \(\chi_{3971}(2279,\cdot)\) |
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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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\(\chi_{7942}(151,\cdot)\)
\(\chi_{7942}(189,\cdot)\)
\(\chi_{7942}(227,\cdot)\)
\(\chi_{7942}(303,\cdot)\)
\(\chi_{7942}(569,\cdot)\)
\(\chi_{7942}(607,\cdot)\)
\(\chi_{7942}(645,\cdot)\)
\(\chi_{7942}(987,\cdot)\)
\(\chi_{7942}(1025,\cdot)\)
\(\chi_{7942}(1063,\cdot)\)
\(\chi_{7942}(1139,\cdot)\)
\(\chi_{7942}(1405,\cdot)\)
\(\chi_{7942}(1481,\cdot)\)
\(\chi_{7942}(1557,\cdot)\)
\(\chi_{7942}(1823,\cdot)\)
\(\chi_{7942}(1861,\cdot)\)
\(\chi_{7942}(1899,\cdot)\)
\(\chi_{7942}(1975,\cdot)\)
\(\chi_{7942}(2241,\cdot)\)
\(\chi_{7942}(2279,\cdot)\)
\(\chi_{7942}(2317,\cdot)\)
\(\chi_{7942}(2393,\cdot)\)
\(\chi_{7942}(2659,\cdot)\)
\(\chi_{7942}(2697,\cdot)\)
\(\chi_{7942}(2735,\cdot)\)
\(\chi_{7942}(2811,\cdot)\)
\(\chi_{7942}(3077,\cdot)\)
\(\chi_{7942}(3115,\cdot)\)
\(\chi_{7942}(3153,\cdot)\)
\(\chi_{7942}(3229,\cdot)\)
...
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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 };
\((5777,6139)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{15}{38}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 7942 }(2279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{190}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{173}{190}\right)\) | \(e\left(\frac{32}{95}\right)\) | \(e\left(\frac{92}{95}\right)\) | \(e\left(\frac{123}{190}\right)\) | \(e\left(\frac{21}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{91}{95}\right)\) |
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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)
