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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3395, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([72,48,77]))
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gp:[g,chi] = znchar(Mod(993, 3395))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3395.993");
| Modulus: | \(3395\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3395\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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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\(\chi_{3395}(13,\cdot)\)
\(\chi_{3395}(83,\cdot)\)
\(\chi_{3395}(118,\cdot)\)
\(\chi_{3395}(153,\cdot)\)
\(\chi_{3395}(223,\cdot)\)
\(\chi_{3395}(447,\cdot)\)
\(\chi_{3395}(468,\cdot)\)
\(\chi_{3395}(622,\cdot)\)
\(\chi_{3395}(993,\cdot)\)
\(\chi_{3395}(1007,\cdot)\)
\(\chi_{3395}(1077,\cdot)\)
\(\chi_{3395}(1238,\cdot)\)
\(\chi_{3395}(1287,\cdot)\)
\(\chi_{3395}(1462,\cdot)\)
\(\chi_{3395}(1567,\cdot)\)
\(\chi_{3395}(1707,\cdot)\)
\(\chi_{3395}(1763,\cdot)\)
\(\chi_{3395}(1882,\cdot)\)
\(\chi_{3395}(2008,\cdot)\)
\(\chi_{3395}(2022,\cdot)\)
\(\chi_{3395}(2078,\cdot)\)
\(\chi_{3395}(2113,\cdot)\)
\(\chi_{3395}(2127,\cdot)\)
\(\chi_{3395}(2148,\cdot)\)
\(\chi_{3395}(2218,\cdot)\)
\(\chi_{3395}(2302,\cdot)\)
\(\chi_{3395}(2323,\cdot)\)
\(\chi_{3395}(2512,\cdot)\)
\(\chi_{3395}(2582,\cdot)\)
\(\chi_{3395}(2967,\cdot)\)
...
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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 };
\((2717,486,1751)\) → \((-i,-1,e\left(\frac{77}{96}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 3395 }(993, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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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)
