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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2675, base_ring=CyclotomicField(530))
M = H._module
chi = DirichletCharacter(H, M([424,10]))
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gp:[g,chi] = znchar(Mod(111, 2675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2675.111");
| Modulus: | \(2675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(265\) |
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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);
|
\(\chi_{2675}(11,\cdot)\)
\(\chi_{2675}(16,\cdot)\)
\(\chi_{2675}(36,\cdot)\)
\(\chi_{2675}(41,\cdot)\)
\(\chi_{2675}(56,\cdot)\)
\(\chi_{2675}(61,\cdot)\)
\(\chi_{2675}(81,\cdot)\)
\(\chi_{2675}(86,\cdot)\)
\(\chi_{2675}(111,\cdot)\)
\(\chi_{2675}(116,\cdot)\)
\(\chi_{2675}(121,\cdot)\)
\(\chi_{2675}(136,\cdot)\)
\(\chi_{2675}(141,\cdot)\)
\(\chi_{2675}(146,\cdot)\)
\(\chi_{2675}(156,\cdot)\)
\(\chi_{2675}(171,\cdot)\)
\(\chi_{2675}(186,\cdot)\)
\(\chi_{2675}(196,\cdot)\)
\(\chi_{2675}(206,\cdot)\)
\(\chi_{2675}(241,\cdot)\)
\(\chi_{2675}(256,\cdot)\)
\(\chi_{2675}(261,\cdot)\)
\(\chi_{2675}(266,\cdot)\)
\(\chi_{2675}(271,\cdot)\)
\(\chi_{2675}(306,\cdot)\)
\(\chi_{2675}(316,\cdot)\)
\(\chi_{2675}(331,\cdot)\)
\(\chi_{2675}(346,\cdot)\)
\(\chi_{2675}(356,\cdot)\)
\(\chi_{2675}(361,\cdot)\)
...
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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 };
\((1927,751)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{1}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2675 }(111, a) \) |
\(1\) | \(1\) | \(e\left(\frac{217}{265}\right)\) | \(e\left(\frac{244}{265}\right)\) | \(e\left(\frac{169}{265}\right)\) | \(e\left(\frac{196}{265}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{121}{265}\right)\) | \(e\left(\frac{223}{265}\right)\) | \(e\left(\frac{57}{265}\right)\) | \(e\left(\frac{148}{265}\right)\) | \(e\left(\frac{123}{265}\right)\) |
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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)
