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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8954, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([276,220]))
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gp:[g,chi] = znchar(Mod(1379, 8954))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8954.1379");
| Modulus: | \(8954\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4477\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(165\) |
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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 \(\chi_{4477}(1379,\cdot)\) |
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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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\(\chi_{8954}(47,\cdot)\)
\(\chi_{8954}(137,\cdot)\)
\(\chi_{8954}(433,\cdot)\)
\(\chi_{8954}(581,\cdot)\)
\(\chi_{8954}(713,\cdot)\)
\(\chi_{8954}(861,\cdot)\)
\(\chi_{8954}(951,\cdot)\)
\(\chi_{8954}(1083,\cdot)\)
\(\chi_{8954}(1247,\cdot)\)
\(\chi_{8954}(1379,\cdot)\)
\(\chi_{8954}(1395,\cdot)\)
\(\chi_{8954}(1527,\cdot)\)
\(\chi_{8954}(1543,\cdot)\)
\(\chi_{8954}(1675,\cdot)\)
\(\chi_{8954}(1765,\cdot)\)
\(\chi_{8954}(1897,\cdot)\)
\(\chi_{8954}(2061,\cdot)\)
\(\chi_{8954}(2193,\cdot)\)
\(\chi_{8954}(2209,\cdot)\)
\(\chi_{8954}(2341,\cdot)\)
\(\chi_{8954}(2357,\cdot)\)
\(\chi_{8954}(2489,\cdot)\)
\(\chi_{8954}(2579,\cdot)\)
\(\chi_{8954}(2711,\cdot)\)
\(\chi_{8954}(2875,\cdot)\)
\(\chi_{8954}(3007,\cdot)\)
\(\chi_{8954}(3023,\cdot)\)
\(\chi_{8954}(3171,\cdot)\)
\(\chi_{8954}(3303,\cdot)\)
\(\chi_{8954}(3393,\cdot)\)
...
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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 };
\((1333,3147)\) → \((e\left(\frac{46}{55}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8954 }(1379, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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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)
