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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2112, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,45,40,32]))
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gp:[g,chi] = znchar(Mod(1061, 2112))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2112.1061");
| Modulus: | \(2112\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2112\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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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\(\chi_{2112}(5,\cdot)\)
\(\chi_{2112}(53,\cdot)\)
\(\chi_{2112}(125,\cdot)\)
\(\chi_{2112}(245,\cdot)\)
\(\chi_{2112}(269,\cdot)\)
\(\chi_{2112}(317,\cdot)\)
\(\chi_{2112}(389,\cdot)\)
\(\chi_{2112}(509,\cdot)\)
\(\chi_{2112}(533,\cdot)\)
\(\chi_{2112}(581,\cdot)\)
\(\chi_{2112}(653,\cdot)\)
\(\chi_{2112}(773,\cdot)\)
\(\chi_{2112}(797,\cdot)\)
\(\chi_{2112}(845,\cdot)\)
\(\chi_{2112}(917,\cdot)\)
\(\chi_{2112}(1037,\cdot)\)
\(\chi_{2112}(1061,\cdot)\)
\(\chi_{2112}(1109,\cdot)\)
\(\chi_{2112}(1181,\cdot)\)
\(\chi_{2112}(1301,\cdot)\)
\(\chi_{2112}(1325,\cdot)\)
\(\chi_{2112}(1373,\cdot)\)
\(\chi_{2112}(1445,\cdot)\)
\(\chi_{2112}(1565,\cdot)\)
\(\chi_{2112}(1589,\cdot)\)
\(\chi_{2112}(1637,\cdot)\)
\(\chi_{2112}(1709,\cdot)\)
\(\chi_{2112}(1829,\cdot)\)
\(\chi_{2112}(1853,\cdot)\)
\(\chi_{2112}(1901,\cdot)\)
...
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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 };
\((2047,133,1409,1729)\) → \((1,e\left(\frac{9}{16}\right),-1,e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2112 }(1061, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{80}\right)\) |
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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)
