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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37905, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([0,0,114,88]))
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gp:[g,chi] = znchar(Mod(1696, 37905))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37905.1696");
| Modulus: | \(37905\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2527\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(171\) |
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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 \(\chi_{2527}(1696,\cdot)\) |
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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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\(\chi_{37905}(16,\cdot)\)
\(\chi_{37905}(226,\cdot)\)
\(\chi_{37905}(256,\cdot)\)
\(\chi_{37905}(1201,\cdot)\)
\(\chi_{37905}(1621,\cdot)\)
\(\chi_{37905}(1696,\cdot)\)
\(\chi_{37905}(2011,\cdot)\)
\(\chi_{37905}(2221,\cdot)\)
\(\chi_{37905}(2251,\cdot)\)
\(\chi_{37905}(3196,\cdot)\)
\(\chi_{37905}(3616,\cdot)\)
\(\chi_{37905}(3691,\cdot)\)
\(\chi_{37905}(4006,\cdot)\)
\(\chi_{37905}(4246,\cdot)\)
\(\chi_{37905}(5191,\cdot)\)
\(\chi_{37905}(5611,\cdot)\)
\(\chi_{37905}(5686,\cdot)\)
\(\chi_{37905}(6001,\cdot)\)
\(\chi_{37905}(6211,\cdot)\)
\(\chi_{37905}(6241,\cdot)\)
\(\chi_{37905}(7186,\cdot)\)
\(\chi_{37905}(7606,\cdot)\)
\(\chi_{37905}(7681,\cdot)\)
\(\chi_{37905}(8206,\cdot)\)
\(\chi_{37905}(8236,\cdot)\)
\(\chi_{37905}(9181,\cdot)\)
\(\chi_{37905}(9601,\cdot)\)
\(\chi_{37905}(9676,\cdot)\)
\(\chi_{37905}(9991,\cdot)\)
\(\chi_{37905}(10201,\cdot)\)
...
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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 };
\((25271,7582,21661,32131)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{44}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 37905 }(1696, a) \) |
\(1\) | \(1\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{11}{19}\right)\) |
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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)
