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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7120, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([44,66,22,27]))
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gp:[g,chi] = znchar(Mod(1587, 7120))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7120.1587");
| Modulus: | \(7120\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7120\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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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\(\chi_{7120}(43,\cdot)\)
\(\chi_{7120}(147,\cdot)\)
\(\chi_{7120}(363,\cdot)\)
\(\chi_{7120}(387,\cdot)\)
\(\chi_{7120}(547,\cdot)\)
\(\chi_{7120}(923,\cdot)\)
\(\chi_{7120}(1003,\cdot)\)
\(\chi_{7120}(1163,\cdot)\)
\(\chi_{7120}(1187,\cdot)\)
\(\chi_{7120}(1243,\cdot)\)
\(\chi_{7120}(1587,\cdot)\)
\(\chi_{7120}(1643,\cdot)\)
\(\chi_{7120}(1803,\cdot)\)
\(\chi_{7120}(1883,\cdot)\)
\(\chi_{7120}(1907,\cdot)\)
\(\chi_{7120}(1987,\cdot)\)
\(\chi_{7120}(2867,\cdot)\)
\(\chi_{7120}(2923,\cdot)\)
\(\chi_{7120}(3003,\cdot)\)
\(\chi_{7120}(3163,\cdot)\)
\(\chi_{7120}(3267,\cdot)\)
\(\chi_{7120}(3347,\cdot)\)
\(\chi_{7120}(3563,\cdot)\)
\(\chi_{7120}(3587,\cdot)\)
\(\chi_{7120}(3643,\cdot)\)
\(\chi_{7120}(3803,\cdot)\)
\(\chi_{7120}(3883,\cdot)\)
\(\chi_{7120}(4067,\cdot)\)
\(\chi_{7120}(4307,\cdot)\)
\(\chi_{7120}(4387,\cdot)\)
...
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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 };
\((2671,1781,5697,5521)\) → \((-1,-i,i,e\left(\frac{27}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 7120 }(1587, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{37}{88}\right)\) |
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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)
