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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(621, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,27]))
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gp:[g,chi] = znchar(Mod(287, 621))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("621.287");
| Modulus: | \(621\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(207\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(66\) |
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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 \(\chi_{207}(149,\cdot)\) |
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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: | no |
| 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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\(\chi_{621}(17,\cdot)\)
\(\chi_{621}(44,\cdot)\)
\(\chi_{621}(89,\cdot)\)
\(\chi_{621}(125,\cdot)\)
\(\chi_{621}(143,\cdot)\)
\(\chi_{621}(152,\cdot)\)
\(\chi_{621}(224,\cdot)\)
\(\chi_{621}(251,\cdot)\)
\(\chi_{621}(260,\cdot)\)
\(\chi_{621}(287,\cdot)\)
\(\chi_{621}(314,\cdot)\)
\(\chi_{621}(332,\cdot)\)
\(\chi_{621}(341,\cdot)\)
\(\chi_{621}(359,\cdot)\)
\(\chi_{621}(467,\cdot)\)
\(\chi_{621}(494,\cdot)\)
\(\chi_{621}(503,\cdot)\)
\(\chi_{621}(521,\cdot)\)
\(\chi_{621}(548,\cdot)\)
\(\chi_{621}(557,\cdot)\)
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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 };
\((461,28)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 621 }(287, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
