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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11552, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([114,57,100]))
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gp:[g,chi] = znchar(Mod(7, 11552))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11552.7");
| Modulus: | \(11552\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5776\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(228\) |
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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: | no, induced from \(\chi_{5776}(1451,\cdot)\) |
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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: | no |
| 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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\(\chi_{11552}(7,\cdot)\)
\(\chi_{11552}(87,\cdot)\)
\(\chi_{11552}(311,\cdot)\)
\(\chi_{11552}(391,\cdot)\)
\(\chi_{11552}(615,\cdot)\)
\(\chi_{11552}(695,\cdot)\)
\(\chi_{11552}(919,\cdot)\)
\(\chi_{11552}(999,\cdot)\)
\(\chi_{11552}(1223,\cdot)\)
\(\chi_{11552}(1303,\cdot)\)
\(\chi_{11552}(1527,\cdot)\)
\(\chi_{11552}(1607,\cdot)\)
\(\chi_{11552}(1831,\cdot)\)
\(\chi_{11552}(1911,\cdot)\)
\(\chi_{11552}(2135,\cdot)\)
\(\chi_{11552}(2215,\cdot)\)
\(\chi_{11552}(2439,\cdot)\)
\(\chi_{11552}(2519,\cdot)\)
\(\chi_{11552}(2743,\cdot)\)
\(\chi_{11552}(2823,\cdot)\)
\(\chi_{11552}(3047,\cdot)\)
\(\chi_{11552}(3127,\cdot)\)
\(\chi_{11552}(3351,\cdot)\)
\(\chi_{11552}(3431,\cdot)\)
\(\chi_{11552}(3655,\cdot)\)
\(\chi_{11552}(3735,\cdot)\)
\(\chi_{11552}(3959,\cdot)\)
\(\chi_{11552}(4343,\cdot)\)
\(\chi_{11552}(4567,\cdot)\)
\(\chi_{11552}(4647,\cdot)\)
...
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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 };
\((5055,1445,2529)\) → \((-1,i,e\left(\frac{25}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 11552 }(7, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{228}\right)\) | \(e\left(\frac{1}{228}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{11}{228}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{1}{228}\right)\) | \(e\left(\frac{2}{57}\right)\) |
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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)
