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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9373, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([34,34,44]))
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gp:[g,chi] = znchar(Mod(7933, 9373))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9373.7933");
| Modulus: | \(9373\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9373\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(51\) |
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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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\(\chi_{9373}(16,\cdot)\)
\(\chi_{9373}(256,\cdot)\)
\(\chi_{9373}(289,\cdot)\)
\(\chi_{9373}(347,\cdot)\)
\(\chi_{9373}(471,\cdot)\)
\(\chi_{9373}(1166,\cdot)\)
\(\chi_{9373}(1563,\cdot)\)
\(\chi_{9373}(1745,\cdot)\)
\(\chi_{9373}(1803,\cdot)\)
\(\chi_{9373}(2109,\cdot)\)
\(\chi_{9373}(2167,\cdot)\)
\(\chi_{9373}(2291,\cdot)\)
\(\chi_{9373}(2746,\cdot)\)
\(\chi_{9373}(4624,\cdot)\)
\(\chi_{9373}(5079,\cdot)\)
\(\chi_{9373}(5294,\cdot)\)
\(\chi_{9373}(5385,\cdot)\)
\(\chi_{9373}(5625,\cdot)\)
\(\chi_{9373}(5931,\cdot)\)
\(\chi_{9373}(5989,\cdot)\)
\(\chi_{9373}(6262,\cdot)\)
\(\chi_{9373}(6444,\cdot)\)
\(\chi_{9373}(6750,\cdot)\)
\(\chi_{9373}(7023,\cdot)\)
\(\chi_{9373}(7536,\cdot)\)
\(\chi_{9373}(7751,\cdot)\)
\(\chi_{9373}(7933,\cdot)\)
\(\chi_{9373}(8173,\cdot)\)
\(\chi_{9373}(8537,\cdot)\)
\(\chi_{9373}(9174,\cdot)\)
...
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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 };
\((1340,7932,3095)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{22}{51}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 9373 }(7933, a) \) |
\(1\) | \(1\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) |
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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)
