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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4205, base_ring=CyclotomicField(406))
M = H._module
chi = DirichletCharacter(H, M([203,46]))
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gp:[g,chi] = znchar(Mod(2344, 4205))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4205.2344");
| Modulus: | \(4205\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4205\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(406\) |
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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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\(\chi_{4205}(24,\cdot)\)
\(\chi_{4205}(49,\cdot)\)
\(\chi_{4205}(54,\cdot)\)
\(\chi_{4205}(74,\cdot)\)
\(\chi_{4205}(94,\cdot)\)
\(\chi_{4205}(139,\cdot)\)
\(\chi_{4205}(169,\cdot)\)
\(\chi_{4205}(194,\cdot)\)
\(\chi_{4205}(199,\cdot)\)
\(\chi_{4205}(219,\cdot)\)
\(\chi_{4205}(239,\cdot)\)
\(\chi_{4205}(284,\cdot)\)
\(\chi_{4205}(314,\cdot)\)
\(\chi_{4205}(339,\cdot)\)
\(\chi_{4205}(344,\cdot)\)
\(\chi_{4205}(364,\cdot)\)
\(\chi_{4205}(384,\cdot)\)
\(\chi_{4205}(429,\cdot)\)
\(\chi_{4205}(459,\cdot)\)
\(\chi_{4205}(484,\cdot)\)
\(\chi_{4205}(489,\cdot)\)
\(\chi_{4205}(509,\cdot)\)
\(\chi_{4205}(529,\cdot)\)
\(\chi_{4205}(604,\cdot)\)
\(\chi_{4205}(629,\cdot)\)
\(\chi_{4205}(634,\cdot)\)
\(\chi_{4205}(654,\cdot)\)
\(\chi_{4205}(674,\cdot)\)
\(\chi_{4205}(719,\cdot)\)
\(\chi_{4205}(749,\cdot)\)
...
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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 };
\((842,3366)\) → \((-1,e\left(\frac{23}{203}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 4205 }(2344, a) \) |
\(1\) | \(1\) | \(e\left(\frac{249}{406}\right)\) | \(e\left(\frac{139}{406}\right)\) | \(e\left(\frac{46}{203}\right)\) | \(e\left(\frac{194}{203}\right)\) | \(e\left(\frac{41}{406}\right)\) | \(e\left(\frac{341}{406}\right)\) | \(e\left(\frac{139}{203}\right)\) | \(e\left(\frac{36}{203}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{205}{406}\right)\) |
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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)
