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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7680, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([0,93,64,96]))
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gp:[g,chi] = znchar(Mod(3413, 7680))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7680.3413");
| Modulus: | \(7680\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7680\) |
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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: | 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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\(\chi_{7680}(53,\cdot)\)
\(\chi_{7680}(77,\cdot)\)
\(\chi_{7680}(293,\cdot)\)
\(\chi_{7680}(317,\cdot)\)
\(\chi_{7680}(533,\cdot)\)
\(\chi_{7680}(557,\cdot)\)
\(\chi_{7680}(773,\cdot)\)
\(\chi_{7680}(797,\cdot)\)
\(\chi_{7680}(1013,\cdot)\)
\(\chi_{7680}(1037,\cdot)\)
\(\chi_{7680}(1253,\cdot)\)
\(\chi_{7680}(1277,\cdot)\)
\(\chi_{7680}(1493,\cdot)\)
\(\chi_{7680}(1517,\cdot)\)
\(\chi_{7680}(1733,\cdot)\)
\(\chi_{7680}(1757,\cdot)\)
\(\chi_{7680}(1973,\cdot)\)
\(\chi_{7680}(1997,\cdot)\)
\(\chi_{7680}(2213,\cdot)\)
\(\chi_{7680}(2237,\cdot)\)
\(\chi_{7680}(2453,\cdot)\)
\(\chi_{7680}(2477,\cdot)\)
\(\chi_{7680}(2693,\cdot)\)
\(\chi_{7680}(2717,\cdot)\)
\(\chi_{7680}(2933,\cdot)\)
\(\chi_{7680}(2957,\cdot)\)
\(\chi_{7680}(3173,\cdot)\)
\(\chi_{7680}(3197,\cdot)\)
\(\chi_{7680}(3413,\cdot)\)
\(\chi_{7680}(3437,\cdot)\)
...
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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 };
\((511,6661,2561,1537)\) → \((1,e\left(\frac{93}{128}\right),-1,-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7680 }(3413, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{3}{64}\right)\) |
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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)
