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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2560, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([64,29,64]))
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gp:[g,chi] = znchar(Mod(939, 2560))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2560.939");
| Modulus: | \(2560\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2560\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(128\) |
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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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\(\chi_{2560}(19,\cdot)\)
\(\chi_{2560}(59,\cdot)\)
\(\chi_{2560}(99,\cdot)\)
\(\chi_{2560}(139,\cdot)\)
\(\chi_{2560}(179,\cdot)\)
\(\chi_{2560}(219,\cdot)\)
\(\chi_{2560}(259,\cdot)\)
\(\chi_{2560}(299,\cdot)\)
\(\chi_{2560}(339,\cdot)\)
\(\chi_{2560}(379,\cdot)\)
\(\chi_{2560}(419,\cdot)\)
\(\chi_{2560}(459,\cdot)\)
\(\chi_{2560}(499,\cdot)\)
\(\chi_{2560}(539,\cdot)\)
\(\chi_{2560}(579,\cdot)\)
\(\chi_{2560}(619,\cdot)\)
\(\chi_{2560}(659,\cdot)\)
\(\chi_{2560}(699,\cdot)\)
\(\chi_{2560}(739,\cdot)\)
\(\chi_{2560}(779,\cdot)\)
\(\chi_{2560}(819,\cdot)\)
\(\chi_{2560}(859,\cdot)\)
\(\chi_{2560}(899,\cdot)\)
\(\chi_{2560}(939,\cdot)\)
\(\chi_{2560}(979,\cdot)\)
\(\chi_{2560}(1019,\cdot)\)
\(\chi_{2560}(1059,\cdot)\)
\(\chi_{2560}(1099,\cdot)\)
\(\chi_{2560}(1139,\cdot)\)
\(\chi_{2560}(1179,\cdot)\)
...
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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 };
\((511,1541,1537)\) → \((-1,e\left(\frac{29}{128}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2560 }(939, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{83}{128}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{101}{128}\right)\) |
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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)
