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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7667, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([72,75,74]))
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gp:[g,chi] = znchar(Mod(3746, 7667))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7667.3746");
| Modulus: | \(7667\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7667\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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: | yes |
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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: | odd |
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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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\(\chi_{7667}(294,\cdot)\)
\(\chi_{7667}(589,\cdot)\)
\(\chi_{7667}(673,\cdot)\)
\(\chi_{7667}(1899,\cdot)\)
\(\chi_{7667}(1916,\cdot)\)
\(\chi_{7667}(2043,\cdot)\)
\(\chi_{7667}(2318,\cdot)\)
\(\chi_{7667}(2349,\cdot)\)
\(\chi_{7667}(2404,\cdot)\)
\(\chi_{7667}(2570,\cdot)\)
\(\chi_{7667}(2999,\cdot)\)
\(\chi_{7667}(3252,\cdot)\)
\(\chi_{7667}(3269,\cdot)\)
\(\chi_{7667}(3632,\cdot)\)
\(\chi_{7667}(3746,\cdot)\)
\(\chi_{7667}(4176,\cdot)\)
\(\chi_{7667}(4298,\cdot)\)
\(\chi_{7667}(4529,\cdot)\)
\(\chi_{7667}(4604,\cdot)\)
\(\chi_{7667}(4644,\cdot)\)
\(\chi_{7667}(4825,\cdot)\)
\(\chi_{7667}(5110,\cdot)\)
\(\chi_{7667}(5605,\cdot)\)
\(\chi_{7667}(5634,\cdot)\)
\(\chi_{7667}(5705,\cdot)\)
\(\chi_{7667}(6608,\cdot)\)
\(\chi_{7667}(6789,\cdot)\)
\(\chi_{7667}(6828,\cdot)\)
\(\chi_{7667}(6899,\cdot)\)
\(\chi_{7667}(6958,\cdot)\)
...
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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 };
\((2092,1805,375)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{15}{16}\right),e\left(\frac{37}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 7667 }(3746, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) |
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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)
