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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7957, base_ring=CyclotomicField(216))
M = H._module
chi = DirichletCharacter(H, M([9,154]))
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gp:[g,chi] = znchar(Mod(3191, 7957))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7957.3191");
| Modulus: | \(7957\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7957\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(216\) |
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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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\(\chi_{7957}(212,\cdot)\)
\(\chi_{7957}(309,\cdot)\)
\(\chi_{7957}(563,\cdot)\)
\(\chi_{7957}(614,\cdot)\)
\(\chi_{7957}(713,\cdot)\)
\(\chi_{7957}(773,\cdot)\)
\(\chi_{7957}(970,\cdot)\)
\(\chi_{7957}(1258,\cdot)\)
\(\chi_{7957}(1284,\cdot)\)
\(\chi_{7957}(1321,\cdot)\)
\(\chi_{7957}(1370,\cdot)\)
\(\chi_{7957}(1649,\cdot)\)
\(\chi_{7957}(1795,\cdot)\)
\(\chi_{7957}(1915,\cdot)\)
\(\chi_{7957}(1992,\cdot)\)
\(\chi_{7957}(2014,\cdot)\)
\(\chi_{7957}(2027,\cdot)\)
\(\chi_{7957}(2169,\cdot)\)
\(\chi_{7957}(2233,\cdot)\)
\(\chi_{7957}(2329,\cdot)\)
\(\chi_{7957}(2465,\cdot)\)
\(\chi_{7957}(2572,\cdot)\)
\(\chi_{7957}(2731,\cdot)\)
\(\chi_{7957}(2781,\cdot)\)
\(\chi_{7957}(2795,\cdot)\)
\(\chi_{7957}(3000,\cdot)\)
\(\chi_{7957}(3010,\cdot)\)
\(\chi_{7957}(3191,\cdot)\)
\(\chi_{7957}(3219,\cdot)\)
\(\chi_{7957}(3365,\cdot)\)
...
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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 };
\((7086,1096)\) → \((e\left(\frac{1}{24}\right),e\left(\frac{77}{108}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 7957 }(3191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{49}{216}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{193}{216}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{43}{216}\right)\) | \(e\left(\frac{101}{216}\right)\) |
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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)
