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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10027, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([85,166]))
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gp:[g,chi] = znchar(Mod(3459, 10027))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10027.3459");
| Modulus: | \(10027\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10027\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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);
|
\(\chi_{10027}(1049,\cdot)\)
\(\chi_{10027}(1053,\cdot)\)
\(\chi_{10027}(1189,\cdot)\)
\(\chi_{10027}(1347,\cdot)\)
\(\chi_{10027}(1441,\cdot)\)
\(\chi_{10027}(2129,\cdot)\)
\(\chi_{10027}(2133,\cdot)\)
\(\chi_{10027}(2646,\cdot)\)
\(\chi_{10027}(2901,\cdot)\)
\(\chi_{10027}(3180,\cdot)\)
\(\chi_{10027}(3197,\cdot)\)
\(\chi_{10027}(3276,\cdot)\)
\(\chi_{10027}(3369,\cdot)\)
\(\chi_{10027}(3382,\cdot)\)
\(\chi_{10027}(3459,\cdot)\)
\(\chi_{10027}(3917,\cdot)\)
\(\chi_{10027}(4568,\cdot)\)
\(\chi_{10027}(5051,\cdot)\)
\(\chi_{10027}(5341,\cdot)\)
\(\chi_{10027}(5683,\cdot)\)
\(\chi_{10027}(5715,\cdot)\)
\(\chi_{10027}(5814,\cdot)\)
\(\chi_{10027}(5864,\cdot)\)
\(\chi_{10027}(6192,\cdot)\)
\(\chi_{10027}(6425,\cdot)\)
\(\chi_{10027}(6470,\cdot)\)
\(\chi_{10027}(6551,\cdot)\)
\(\chi_{10027}(6621,\cdot)\)
\(\chi_{10027}(6636,\cdot)\)
\(\chi_{10027}(7198,\cdot)\)
...
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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 };
\((9215,4071)\) → \((e\left(\frac{17}{36}\right),e\left(\frac{83}{90}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 10027 }(3459, a) \) |
\(1\) | \(1\) | \(e\left(\frac{89}{180}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{59}{90}\right)\) |
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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)
