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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(264275, base_ring=CyclotomicField(930))
M = H._module
chi = DirichletCharacter(H, M([837,465,212]))
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gp:[g,chi] = znchar(Mod(27444, 264275))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("264275.27444");
| Modulus: | \(264275\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(264275\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(930\) |
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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);
|
\(\chi_{264275}(1154,\cdot)\)
\(\chi_{264275}(1869,\cdot)\)
\(\chi_{264275}(3739,\cdot)\)
\(\chi_{264275}(4234,\cdot)\)
\(\chi_{264275}(6489,\cdot)\)
\(\chi_{264275}(6654,\cdot)\)
\(\chi_{264275}(7809,\cdot)\)
\(\chi_{264275}(7919,\cdot)\)
\(\chi_{264275}(9679,\cdot)\)
\(\chi_{264275}(10394,\cdot)\)
\(\chi_{264275}(12759,\cdot)\)
\(\chi_{264275}(15014,\cdot)\)
\(\chi_{264275}(15179,\cdot)\)
\(\chi_{264275}(16334,\cdot)\)
\(\chi_{264275}(16444,\cdot)\)
\(\chi_{264275}(18204,\cdot)\)
\(\chi_{264275}(18919,\cdot)\)
\(\chi_{264275}(20789,\cdot)\)
\(\chi_{264275}(21284,\cdot)\)
\(\chi_{264275}(23539,\cdot)\)
\(\chi_{264275}(23704,\cdot)\)
\(\chi_{264275}(24859,\cdot)\)
\(\chi_{264275}(24969,\cdot)\)
\(\chi_{264275}(26729,\cdot)\)
\(\chi_{264275}(27444,\cdot)\)
\(\chi_{264275}(29314,\cdot)\)
\(\chi_{264275}(29809,\cdot)\)
\(\chi_{264275}(32064,\cdot)\)
\(\chi_{264275}(32229,\cdot)\)
\(\chi_{264275}(33384,\cdot)\)
...
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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 };
\((63427,24026,89376)\) → \((e\left(\frac{9}{10}\right),-1,e\left(\frac{106}{465}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 264275 }(27444, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{20}{31}\right)\) | \(e\left(\frac{491}{930}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{161}{930}\right)\) | \(e\left(\frac{463}{465}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{26}{465}\right)\) | \(e\left(\frac{761}{930}\right)\) | \(e\left(\frac{1}{93}\right)\) | \(e\left(\frac{298}{465}\right)\) |
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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)
