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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2552, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([0,70,28,45]))
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gp:[g,chi] = znchar(Mod(1005, 2552))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2552.1005");
| Modulus: | \(2552\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2552\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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);
|
\(\chi_{2552}(37,\cdot)\)
\(\chi_{2552}(69,\cdot)\)
\(\chi_{2552}(213,\cdot)\)
\(\chi_{2552}(229,\cdot)\)
\(\chi_{2552}(269,\cdot)\)
\(\chi_{2552}(301,\cdot)\)
\(\chi_{2552}(317,\cdot)\)
\(\chi_{2552}(333,\cdot)\)
\(\chi_{2552}(421,\cdot)\)
\(\chi_{2552}(445,\cdot)\)
\(\chi_{2552}(533,\cdot)\)
\(\chi_{2552}(565,\cdot)\)
\(\chi_{2552}(653,\cdot)\)
\(\chi_{2552}(669,\cdot)\)
\(\chi_{2552}(685,\cdot)\)
\(\chi_{2552}(757,\cdot)\)
\(\chi_{2552}(773,\cdot)\)
\(\chi_{2552}(797,\cdot)\)
\(\chi_{2552}(885,\cdot)\)
\(\chi_{2552}(917,\cdot)\)
\(\chi_{2552}(949,\cdot)\)
\(\chi_{2552}(1005,\cdot)\)
\(\chi_{2552}(1149,\cdot)\)
\(\chi_{2552}(1181,\cdot)\)
\(\chi_{2552}(1197,\cdot)\)
\(\chi_{2552}(1237,\cdot)\)
\(\chi_{2552}(1373,\cdot)\)
\(\chi_{2552}(1389,\cdot)\)
\(\chi_{2552}(1413,\cdot)\)
\(\chi_{2552}(1461,\cdot)\)
...
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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 };
\((639,1277,233,89)\) → \((1,-1,e\left(\frac{1}{5}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2552 }(1005, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
