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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7031, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([220,213]))
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gp:[g,chi] = znchar(Mod(1083, 7031))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7031.1083");
| Modulus: | \(7031\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7031\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(264\) |
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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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\(\chi_{7031}(24,\cdot)\)
\(\chi_{7031}(56,\cdot)\)
\(\chi_{7031}(103,\cdot)\)
\(\chi_{7031}(135,\cdot)\)
\(\chi_{7031}(261,\cdot)\)
\(\chi_{7031}(293,\cdot)\)
\(\chi_{7031}(419,\cdot)\)
\(\chi_{7031}(451,\cdot)\)
\(\chi_{7031}(577,\cdot)\)
\(\chi_{7031}(609,\cdot)\)
\(\chi_{7031}(656,\cdot)\)
\(\chi_{7031}(688,\cdot)\)
\(\chi_{7031}(735,\cdot)\)
\(\chi_{7031}(814,\cdot)\)
\(\chi_{7031}(893,\cdot)\)
\(\chi_{7031}(925,\cdot)\)
\(\chi_{7031}(972,\cdot)\)
\(\chi_{7031}(1083,\cdot)\)
\(\chi_{7031}(1130,\cdot)\)
\(\chi_{7031}(1320,\cdot)\)
\(\chi_{7031}(1478,\cdot)\)
\(\chi_{7031}(1715,\cdot)\)
\(\chi_{7031}(1794,\cdot)\)
\(\chi_{7031}(1841,\cdot)\)
\(\chi_{7031}(1920,\cdot)\)
\(\chi_{7031}(1952,\cdot)\)
\(\chi_{7031}(1999,\cdot)\)
\(\chi_{7031}(2078,\cdot)\)
\(\chi_{7031}(2110,\cdot)\)
\(\chi_{7031}(2268,\cdot)\)
...
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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 };
\((1425,5610)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{71}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 7031 }(1083, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{233}{264}\right)\) | \(e\left(\frac{137}{264}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{29}{66}\right)\) |
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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)
