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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([312,117,128]))
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gp:[g,chi] = znchar(Mod(95, 4029))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4029.95");
| Modulus: | \(4029\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4029\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(624\) |
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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: | 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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\(\chi_{4029}(5,\cdot)\)
\(\chi_{4029}(11,\cdot)\)
\(\chi_{4029}(20,\cdot)\)
\(\chi_{4029}(44,\cdot)\)
\(\chi_{4029}(92,\cdot)\)
\(\chi_{4029}(95,\cdot)\)
\(\chi_{4029}(167,\cdot)\)
\(\chi_{4029}(194,\cdot)\)
\(\chi_{4029}(209,\cdot)\)
\(\chi_{4029}(248,\cdot)\)
\(\chi_{4029}(269,\cdot)\)
\(\chi_{4029}(320,\cdot)\)
\(\chi_{4029}(329,\cdot)\)
\(\chi_{4029}(335,\cdot)\)
\(\chi_{4029}(347,\cdot)\)
\(\chi_{4029}(431,\cdot)\)
\(\chi_{4029}(437,\cdot)\)
\(\chi_{4029}(479,\cdot)\)
\(\chi_{4029}(500,\cdot)\)
\(\chi_{4029}(524,\cdot)\)
\(\chi_{4029}(566,\cdot)\)
\(\chi_{4029}(572,\cdot)\)
\(\chi_{4029}(584,\cdot)\)
\(\chi_{4029}(602,\cdot)\)
\(\chi_{4029}(626,\cdot)\)
\(\chi_{4029}(641,\cdot)\)
\(\chi_{4029}(668,\cdot)\)
\(\chi_{4029}(674,\cdot)\)
\(\chi_{4029}(677,\cdot)\)
\(\chi_{4029}(683,\cdot)\)
...
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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 };
\((2687,3556,3163)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{8}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4029 }(95, a) \) |
\(1\) | \(1\) | \(e\left(\frac{295}{312}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{97}{624}\right)\) | \(e\left(\frac{583}{624}\right)\) | \(e\left(\frac{87}{104}\right)\) | \(e\left(\frac{21}{208}\right)\) | \(e\left(\frac{475}{624}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{61}{78}\right)\) |
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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)
