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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1327, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([23]))
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gp:[g,chi] = znchar(Mod(39, 1327))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1327.39");
| Modulus: | \(1327\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1327\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(102\) |
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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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\(\chi_{1327}(39,\cdot)\)
\(\chi_{1327}(91,\cdot)\)
\(\chi_{1327}(142,\cdot)\)
\(\chi_{1327}(145,\cdot)\)
\(\chi_{1327}(171,\cdot)\)
\(\chi_{1327}(175,\cdot)\)
\(\chi_{1327}(207,\cdot)\)
\(\chi_{1327}(296,\cdot)\)
\(\chi_{1327}(359,\cdot)\)
\(\chi_{1327}(399,\cdot)\)
\(\chi_{1327}(403,\cdot)\)
\(\chi_{1327}(418,\cdot)\)
\(\chi_{1327}(420,\cdot)\)
\(\chi_{1327}(440,\cdot)\)
\(\chi_{1327}(445,\cdot)\)
\(\chi_{1327}(498,\cdot)\)
\(\chi_{1327}(596,\cdot)\)
\(\chi_{1327}(812,\cdot)\)
\(\chi_{1327}(847,\cdot)\)
\(\chi_{1327}(871,\cdot)\)
\(\chi_{1327}(942,\cdot)\)
\(\chi_{1327}(1008,\cdot)\)
\(\chi_{1327}(1025,\cdot)\)
\(\chi_{1327}(1056,\cdot)\)
\(\chi_{1327}(1068,\cdot)\)
\(\chi_{1327}(1127,\cdot)\)
\(\chi_{1327}(1133,\cdot)\)
\(\chi_{1327}(1137,\cdot)\)
\(\chi_{1327}(1165,\cdot)\)
\(\chi_{1327}(1223,\cdot)\)
...
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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 };
\(3\) → \(e\left(\frac{23}{102}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1327 }(39, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{4}{51}\right)\) |
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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)
