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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8245, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([24,24,59]))
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gp:[g,chi] = znchar(Mod(2257, 8245))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8245.2257");
| Modulus: | \(8245\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8245\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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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\(\chi_{8245}(208,\cdot)\)
\(\chi_{8245}(642,\cdot)\)
\(\chi_{8245}(1143,\cdot)\)
\(\chi_{8245}(1492,\cdot)\)
\(\chi_{8245}(2078,\cdot)\)
\(\chi_{8245}(2163,\cdot)\)
\(\chi_{8245}(2172,\cdot)\)
\(\chi_{8245}(2248,\cdot)\)
\(\chi_{8245}(2257,\cdot)\)
\(\chi_{8245}(2333,\cdot)\)
\(\chi_{8245}(2512,\cdot)\)
\(\chi_{8245}(2852,\cdot)\)
\(\chi_{8245}(3022,\cdot)\)
\(\chi_{8245}(3532,\cdot)\)
\(\chi_{8245}(3778,\cdot)\)
\(\chi_{8245}(3863,\cdot)\)
\(\chi_{8245}(3948,\cdot)\)
\(\chi_{8245}(4033,\cdot)\)
\(\chi_{8245}(4552,\cdot)\)
\(\chi_{8245}(4968,\cdot)\)
\(\chi_{8245}(5827,\cdot)\)
\(\chi_{8245}(5903,\cdot)\)
\(\chi_{8245}(6328,\cdot)\)
\(\chi_{8245}(6583,\cdot)\)
\(\chi_{8245}(6847,\cdot)\)
\(\chi_{8245}(7357,\cdot)\)
\(\chi_{8245}(7527,\cdot)\)
\(\chi_{8245}(7773,\cdot)\)
\(\chi_{8245}(7867,\cdot)\)
\(\chi_{8245}(8028,\cdot)\)
...
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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 };
\((6597,1941,2721)\) → \((i,i,e\left(\frac{59}{96}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 8245 }(2257, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{96}\right)\) |
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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)
