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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18360, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,0,80,108,63]))
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gp:[g,chi] = znchar(Mod(2833, 18360))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18360.2833");
| Modulus: | \(18360\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2295\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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 \(\chi_{2295}(538,\cdot)\) |
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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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\(\chi_{18360}(97,\cdot)\)
\(\chi_{18360}(193,\cdot)\)
\(\chi_{18360}(313,\cdot)\)
\(\chi_{18360}(337,\cdot)\)
\(\chi_{18360}(673,\cdot)\)
\(\chi_{18360}(1057,\cdot)\)
\(\chi_{18360}(2137,\cdot)\)
\(\chi_{18360}(2353,\cdot)\)
\(\chi_{18360}(2713,\cdot)\)
\(\chi_{18360}(2833,\cdot)\)
\(\chi_{18360}(3337,\cdot)\)
\(\chi_{18360}(4273,\cdot)\)
\(\chi_{18360}(4417,\cdot)\)
\(\chi_{18360}(4873,\cdot)\)
\(\chi_{18360}(5137,\cdot)\)
\(\chi_{18360}(5377,\cdot)\)
\(\chi_{18360}(6217,\cdot)\)
\(\chi_{18360}(6313,\cdot)\)
\(\chi_{18360}(6433,\cdot)\)
\(\chi_{18360}(6457,\cdot)\)
\(\chi_{18360}(6793,\cdot)\)
\(\chi_{18360}(7177,\cdot)\)
\(\chi_{18360}(8257,\cdot)\)
\(\chi_{18360}(8473,\cdot)\)
\(\chi_{18360}(8833,\cdot)\)
\(\chi_{18360}(8953,\cdot)\)
\(\chi_{18360}(9457,\cdot)\)
\(\chi_{18360}(10393,\cdot)\)
\(\chi_{18360}(10537,\cdot)\)
\(\chi_{18360}(10993,\cdot)\)
...
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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 };
\((4591,9181,7481,11017,4321)\) → \((1,1,e\left(\frac{5}{9}\right),-i,e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 18360 }(2833, a) \) |
\(1\) | \(1\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{25}{72}\right)\) |
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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)
