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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4545, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([50,0,91]))
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gp:[g,chi] = znchar(Mod(1241, 4545))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4545.1241");
| Modulus: | \(4545\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(303\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(100\) |
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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 \(\chi_{303}(29,\cdot)\) |
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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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\(\chi_{4545}(26,\cdot)\)
\(\chi_{4545}(116,\cdot)\)
\(\chi_{4545}(296,\cdot)\)
\(\chi_{4545}(341,\cdot)\)
\(\chi_{4545}(386,\cdot)\)
\(\chi_{4545}(431,\cdot)\)
\(\chi_{4545}(476,\cdot)\)
\(\chi_{4545}(566,\cdot)\)
\(\chi_{4545}(656,\cdot)\)
\(\chi_{4545}(836,\cdot)\)
\(\chi_{4545}(881,\cdot)\)
\(\chi_{4545}(1061,\cdot)\)
\(\chi_{4545}(1151,\cdot)\)
\(\chi_{4545}(1241,\cdot)\)
\(\chi_{4545}(1286,\cdot)\)
\(\chi_{4545}(1331,\cdot)\)
\(\chi_{4545}(1376,\cdot)\)
\(\chi_{4545}(1421,\cdot)\)
\(\chi_{4545}(1601,\cdot)\)
\(\chi_{4545}(1691,\cdot)\)
\(\chi_{4545}(1826,\cdot)\)
\(\chi_{4545}(1871,\cdot)\)
\(\chi_{4545}(1916,\cdot)\)
\(\chi_{4545}(1961,\cdot)\)
\(\chi_{4545}(2321,\cdot)\)
\(\chi_{4545}(2591,\cdot)\)
\(\chi_{4545}(2681,\cdot)\)
\(\chi_{4545}(2816,\cdot)\)
\(\chi_{4545}(2996,\cdot)\)
\(\chi_{4545}(3041,\cdot)\)
...
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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 };
\((506,3637,406)\) → \((-1,1,e\left(\frac{91}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4545 }(1241, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{25}\right)\) |
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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)
