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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31775, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([6,56,87]))
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gp:[g,chi] = znchar(Mod(227, 31775))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31775.227");
| Modulus: | \(31775\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(31775\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{31775}(227,\cdot)\)
\(\chi_{31775}(503,\cdot)\)
\(\chi_{31775}(1073,\cdot)\)
\(\chi_{31775}(1247,\cdot)\)
\(\chi_{31775}(2117,\cdot)\)
\(\chi_{31775}(2308,\cdot)\)
\(\chi_{31775}(6958,\cdot)\)
\(\chi_{31775}(7613,\cdot)\)
\(\chi_{31775}(8752,\cdot)\)
\(\chi_{31775}(11428,\cdot)\)
\(\chi_{31775}(12192,\cdot)\)
\(\chi_{31775}(12348,\cdot)\)
\(\chi_{31775}(12738,\cdot)\)
\(\chi_{31775}(14497,\cdot)\)
\(\chi_{31775}(15633,\cdot)\)
\(\chi_{31775}(18463,\cdot)\)
\(\chi_{31775}(19673,\cdot)\)
\(\chi_{31775}(20137,\cdot)\)
\(\chi_{31775}(20283,\cdot)\)
\(\chi_{31775}(20567,\cdot)\)
\(\chi_{31775}(20653,\cdot)\)
\(\chi_{31775}(20727,\cdot)\)
\(\chi_{31775}(22462,\cdot)\)
\(\chi_{31775}(23053,\cdot)\)
\(\chi_{31775}(23588,\cdot)\)
\(\chi_{31775}(23722,\cdot)\)
\(\chi_{31775}(23797,\cdot)\)
\(\chi_{31775}(25262,\cdot)\)
\(\chi_{31775}(27587,\cdot)\)
\(\chi_{31775}(29252,\cdot)\)
...
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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 };
\((7627,1026,6976)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{7}{15}\right),e\left(\frac{29}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 31775 }(227, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{67}{120}\right)\) |
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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)
