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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4228, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,0,119]))
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gp:[g,chi] = znchar(Mod(967, 4228))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4228.967");
| Modulus: | \(4228\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(604\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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 \(\chi_{604}(363,\cdot)\) |
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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);
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\(\chi_{4228}(15,\cdot)\)
\(\chi_{4228}(71,\cdot)\)
\(\chi_{4228}(379,\cdot)\)
\(\chi_{4228}(435,\cdot)\)
\(\chi_{4228}(715,\cdot)\)
\(\chi_{4228}(967,\cdot)\)
\(\chi_{4228}(995,\cdot)\)
\(\chi_{4228}(1023,\cdot)\)
\(\chi_{4228}(1163,\cdot)\)
\(\chi_{4228}(1191,\cdot)\)
\(\chi_{4228}(1415,\cdot)\)
\(\chi_{4228}(1471,\cdot)\)
\(\chi_{4228}(1499,\cdot)\)
\(\chi_{4228}(1639,\cdot)\)
\(\chi_{4228}(1667,\cdot)\)
\(\chi_{4228}(1807,\cdot)\)
\(\chi_{4228}(1863,\cdot)\)
\(\chi_{4228}(1975,\cdot)\)
\(\chi_{4228}(2059,\cdot)\)
\(\chi_{4228}(2255,\cdot)\)
\(\chi_{4228}(2367,\cdot)\)
\(\chi_{4228}(2395,\cdot)\)
\(\chi_{4228}(2423,\cdot)\)
\(\chi_{4228}(2451,\cdot)\)
\(\chi_{4228}(2479,\cdot)\)
\(\chi_{4228}(2619,\cdot)\)
\(\chi_{4228}(2675,\cdot)\)
\(\chi_{4228}(2731,\cdot)\)
\(\chi_{4228}(2899,\cdot)\)
\(\chi_{4228}(2983,\cdot)\)
...
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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 };
\((2115,605,3781)\) → \((-1,1,e\left(\frac{119}{150}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 4228 }(967, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{75}\right)\) |
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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)
