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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2825, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([28,95]))
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gp:[g,chi] = znchar(Mod(1116, 2825))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2825.1116");
| Modulus: | \(2825\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2825\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{2825}(56,\cdot)\)
\(\chi_{2825}(81,\cdot)\)
\(\chi_{2825}(111,\cdot)\)
\(\chi_{2825}(121,\cdot)\)
\(\chi_{2825}(166,\cdot)\)
\(\chi_{2825}(286,\cdot)\)
\(\chi_{2825}(331,\cdot)\)
\(\chi_{2825}(341,\cdot)\)
\(\chi_{2825}(371,\cdot)\)
\(\chi_{2825}(396,\cdot)\)
\(\chi_{2825}(466,\cdot)\)
\(\chi_{2825}(621,\cdot)\)
\(\chi_{2825}(646,\cdot)\)
\(\chi_{2825}(686,\cdot)\)
\(\chi_{2825}(731,\cdot)\)
\(\chi_{2825}(896,\cdot)\)
\(\chi_{2825}(906,\cdot)\)
\(\chi_{2825}(936,\cdot)\)
\(\chi_{2825}(961,\cdot)\)
\(\chi_{2825}(1031,\cdot)\)
\(\chi_{2825}(1116,\cdot)\)
\(\chi_{2825}(1186,\cdot)\)
\(\chi_{2825}(1211,\cdot)\)
\(\chi_{2825}(1241,\cdot)\)
\(\chi_{2825}(1296,\cdot)\)
\(\chi_{2825}(1416,\cdot)\)
\(\chi_{2825}(1461,\cdot)\)
\(\chi_{2825}(1471,\cdot)\)
\(\chi_{2825}(1596,\cdot)\)
\(\chi_{2825}(1681,\cdot)\)
...
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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 };
\((227,2376)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{19}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2825 }(1116, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{51}{70}\right)\) |
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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)
