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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87120, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([0,0,220,165,192]))
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gp:[g,chi] = znchar(Mod(4657, 87120))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87120.4657");
| Modulus: | \(87120\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5445\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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 \(\chi_{5445}(4657,\cdot)\) |
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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);
|
\(\chi_{87120}(97,\cdot)\)
\(\chi_{87120}(817,\cdot)\)
\(\chi_{87120}(1633,\cdot)\)
\(\chi_{87120}(3073,\cdot)\)
\(\chi_{87120}(3217,\cdot)\)
\(\chi_{87120}(3793,\cdot)\)
\(\chi_{87120}(4273,\cdot)\)
\(\chi_{87120}(4513,\cdot)\)
\(\chi_{87120}(4657,\cdot)\)
\(\chi_{87120}(5377,\cdot)\)
\(\chi_{87120}(5713,\cdot)\)
\(\chi_{87120}(5857,\cdot)\)
\(\chi_{87120}(6097,\cdot)\)
\(\chi_{87120}(6433,\cdot)\)
\(\chi_{87120}(7153,\cdot)\)
\(\chi_{87120}(7297,\cdot)\)
\(\chi_{87120}(8017,\cdot)\)
\(\chi_{87120}(8737,\cdot)\)
\(\chi_{87120}(9553,\cdot)\)
\(\chi_{87120}(10993,\cdot)\)
\(\chi_{87120}(11137,\cdot)\)
\(\chi_{87120}(11713,\cdot)\)
\(\chi_{87120}(12193,\cdot)\)
\(\chi_{87120}(12433,\cdot)\)
\(\chi_{87120}(12577,\cdot)\)
\(\chi_{87120}(13297,\cdot)\)
\(\chi_{87120}(13777,\cdot)\)
\(\chi_{87120}(14017,\cdot)\)
\(\chi_{87120}(14353,\cdot)\)
\(\chi_{87120}(15073,\cdot)\)
...
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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 };
\((32671,21781,19361,69697,14401)\) → \((1,1,e\left(\frac{1}{3}\right),i,e\left(\frac{16}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 87120 }(4657, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{409}{660}\right)\) | \(e\left(\frac{527}{660}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{47}{132}\right)\) |
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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)
