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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37224, base_ring=CyclotomicField(230))
M = H._module
chi = DirichletCharacter(H, M([0,0,115,161,210]))
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gp:[g,chi] = znchar(Mod(8873, 37224))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37224.8873");
| Modulus: | \(37224\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1551\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(230\) |
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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 \(\chi_{1551}(1118,\cdot)\) |
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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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\(\chi_{37224}(17,\cdot)\)
\(\chi_{37224}(1097,\cdot)\)
\(\chi_{37224}(1601,\cdot)\)
\(\chi_{37224}(1745,\cdot)\)
\(\chi_{37224}(1889,\cdot)\)
\(\chi_{37224}(2609,\cdot)\)
\(\chi_{37224}(2681,\cdot)\)
\(\chi_{37224}(3185,\cdot)\)
\(\chi_{37224}(3401,\cdot)\)
\(\chi_{37224}(3473,\cdot)\)
\(\chi_{37224}(4121,\cdot)\)
\(\chi_{37224}(4913,\cdot)\)
\(\chi_{37224}(4985,\cdot)\)
\(\chi_{37224}(5057,\cdot)\)
\(\chi_{37224}(5705,\cdot)\)
\(\chi_{37224}(5849,\cdot)\)
\(\chi_{37224}(6353,\cdot)\)
\(\chi_{37224}(6569,\cdot)\)
\(\chi_{37224}(6641,\cdot)\)
\(\chi_{37224}(7289,\cdot)\)
\(\chi_{37224}(7433,\cdot)\)
\(\chi_{37224}(8729,\cdot)\)
\(\chi_{37224}(8873,\cdot)\)
\(\chi_{37224}(9521,\cdot)\)
\(\chi_{37224}(9737,\cdot)\)
\(\chi_{37224}(11249,\cdot)\)
\(\chi_{37224}(11897,\cdot)\)
\(\chi_{37224}(12041,\cdot)\)
\(\chi_{37224}(12113,\cdot)\)
\(\chi_{37224}(12185,\cdot)\)
...
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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 };
\((27919,18613,8273,27073,26137)\) → \((1,1,-1,e\left(\frac{7}{10}\right),e\left(\frac{21}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 37224 }(8873, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{230}\right)\) | \(e\left(\frac{27}{230}\right)\) | \(e\left(\frac{171}{230}\right)\) | \(e\left(\frac{47}{115}\right)\) | \(e\left(\frac{43}{230}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{49}{115}\right)\) | \(e\left(\frac{41}{115}\right)\) | \(e\left(\frac{108}{115}\right)\) | \(e\left(\frac{38}{115}\right)\) |
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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)
