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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3485, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([20,35,22]))
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gp:[g,chi] = znchar(Mod(1422, 3485))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3485.1422");
| Modulus: | \(3485\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3485\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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: | odd |
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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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\(\chi_{3485}(7,\cdot)\)
\(\chi_{3485}(112,\cdot)\)
\(\chi_{3485}(177,\cdot)\)
\(\chi_{3485}(343,\cdot)\)
\(\chi_{3485}(397,\cdot)\)
\(\chi_{3485}(473,\cdot)\)
\(\chi_{3485}(498,\cdot)\)
\(\chi_{3485}(567,\cdot)\)
\(\chi_{3485}(598,\cdot)\)
\(\chi_{3485}(622,\cdot)\)
\(\chi_{3485}(792,\cdot)\)
\(\chi_{3485}(1077,\cdot)\)
\(\chi_{3485}(1083,\cdot)\)
\(\chi_{3485}(1338,\cdot)\)
\(\chi_{3485}(1422,\cdot)\)
\(\chi_{3485}(1592,\cdot)\)
\(\chi_{3485}(1703,\cdot)\)
\(\chi_{3485}(1792,\cdot)\)
\(\chi_{3485}(1933,\cdot)\)
\(\chi_{3485}(2003,\cdot)\)
\(\chi_{3485}(2028,\cdot)\)
\(\chi_{3485}(2102,\cdot)\)
\(\chi_{3485}(2267,\cdot)\)
\(\chi_{3485}(2407,\cdot)\)
\(\chi_{3485}(2598,\cdot)\)
\(\chi_{3485}(2853,\cdot)\)
\(\chi_{3485}(2982,\cdot)\)
\(\chi_{3485}(3133,\cdot)\)
\(\chi_{3485}(3233,\cdot)\)
\(\chi_{3485}(3292,\cdot)\)
...
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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 };
\((2092,411,2466)\) → \((i,e\left(\frac{7}{16}\right),e\left(\frac{11}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3485 }(1422, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) |
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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)
