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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(223767, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([115,69,120]))
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gp:[g,chi] = znchar(Mod(46022, 223767))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("223767.46022");
| Modulus: | \(223767\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9729\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{9729}(7106,\cdot)\) |
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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: | 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);
|
\(\chi_{223767}(3173,\cdot)\)
\(\chi_{223767}(6347,\cdot)\)
\(\chi_{223767}(11108,\cdot)\)
\(\chi_{223767}(30152,\cdot)\)
\(\chi_{223767}(31739,\cdot)\)
\(\chi_{223767}(36500,\cdot)\)
\(\chi_{223767}(39674,\cdot)\)
\(\chi_{223767}(41261,\cdot)\)
\(\chi_{223767}(46022,\cdot)\)
\(\chi_{223767}(49196,\cdot)\)
\(\chi_{223767}(55544,\cdot)\)
\(\chi_{223767}(60305,\cdot)\)
\(\chi_{223767}(65066,\cdot)\)
\(\chi_{223767}(69827,\cdot)\)
\(\chi_{223767}(77762,\cdot)\)
\(\chi_{223767}(84110,\cdot)\)
\(\chi_{223767}(93632,\cdot)\)
\(\chi_{223767}(103154,\cdot)\)
\(\chi_{223767}(106328,\cdot)\)
\(\chi_{223767}(107915,\cdot)\)
\(\chi_{223767}(111089,\cdot)\)
\(\chi_{223767}(112676,\cdot)\)
\(\chi_{223767}(115850,\cdot)\)
\(\chi_{223767}(120611,\cdot)\)
\(\chi_{223767}(126959,\cdot)\)
\(\chi_{223767}(130133,\cdot)\)
\(\chi_{223767}(134894,\cdot)\)
\(\chi_{223767}(139655,\cdot)\)
\(\chi_{223767}(141242,\cdot)\)
\(\chi_{223767}(144416,\cdot)\)
...
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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 };
\((49727,215308,33328)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{20}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 223767 }(46022, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) |
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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)
