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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54691, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([100,30,113]))
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gp:[g,chi] = znchar(Mod(30623, 54691))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54691.30623");
| Modulus: | \(54691\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(54691\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{54691}(1230,\cdot)\)
\(\chi_{54691}(3076,\cdot)\)
\(\chi_{54691}(9238,\cdot)\)
\(\chi_{54691}(10496,\cdot)\)
\(\chi_{54691}(10678,\cdot)\)
\(\chi_{54691}(13135,\cdot)\)
\(\chi_{54691}(22615,\cdot)\)
\(\chi_{54691}(23210,\cdot)\)
\(\chi_{54691}(23460,\cdot)\)
\(\chi_{54691}(24097,\cdot)\)
\(\chi_{54691}(24757,\cdot)\)
\(\chi_{54691}(25329,\cdot)\)
\(\chi_{54691}(25826,\cdot)\)
\(\chi_{54691}(27786,\cdot)\)
\(\chi_{54691}(27968,\cdot)\)
\(\chi_{54691}(29466,\cdot)\)
\(\chi_{54691}(29944,\cdot)\)
\(\chi_{54691}(30623,\cdot)\)
\(\chi_{54691}(31195,\cdot)\)
\(\chi_{54691}(31832,\cdot)\)
\(\chi_{54691}(34081,\cdot)\)
\(\chi_{54691}(34494,\cdot)\)
\(\chi_{54691}(35222,\cdot)\)
\(\chi_{54691}(38540,\cdot)\)
\(\chi_{54691}(39772,\cdot)\)
\(\chi_{54691}(42892,\cdot)\)
\(\chi_{54691}(44959,\cdot)\)
\(\chi_{54691}(46506,\cdot)\)
\(\chi_{54691}(48004,\cdot)\)
\(\chi_{54691}(50263,\cdot)\)
...
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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 };
\((15627,21036,45683)\) → \((e\left(\frac{5}{6}\right),i,e\left(\frac{113}{120}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 54691 }(30623, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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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)
