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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5803, base_ring=CyclotomicField(828))
M = H._module
chi = DirichletCharacter(H, M([414,539]))
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gp:[g,chi] = znchar(Mod(475, 5803))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5803.475");
| Modulus: | \(5803\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5803\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(828\) |
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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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\(\chi_{5803}(6,\cdot)\)
\(\chi_{5803}(90,\cdot)\)
\(\chi_{5803}(104,\cdot)\)
\(\chi_{5803}(118,\cdot)\)
\(\chi_{5803}(160,\cdot)\)
\(\chi_{5803}(181,\cdot)\)
\(\chi_{5803}(188,\cdot)\)
\(\chi_{5803}(202,\cdot)\)
\(\chi_{5803}(209,\cdot)\)
\(\chi_{5803}(230,\cdot)\)
\(\chi_{5803}(251,\cdot)\)
\(\chi_{5803}(265,\cdot)\)
\(\chi_{5803}(279,\cdot)\)
\(\chi_{5803}(286,\cdot)\)
\(\chi_{5803}(293,\cdot)\)
\(\chi_{5803}(314,\cdot)\)
\(\chi_{5803}(321,\cdot)\)
\(\chi_{5803}(328,\cdot)\)
\(\chi_{5803}(349,\cdot)\)
\(\chi_{5803}(356,\cdot)\)
\(\chi_{5803}(377,\cdot)\)
\(\chi_{5803}(384,\cdot)\)
\(\chi_{5803}(412,\cdot)\)
\(\chi_{5803}(419,\cdot)\)
\(\chi_{5803}(426,\cdot)\)
\(\chi_{5803}(440,\cdot)\)
\(\chi_{5803}(447,\cdot)\)
\(\chi_{5803}(475,\cdot)\)
\(\chi_{5803}(496,\cdot)\)
\(\chi_{5803}(517,\cdot)\)
...
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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 };
\((2488,1660)\) → \((-1,e\left(\frac{539}{828}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5803 }(475, a) \) |
\(1\) | \(1\) | \(e\left(\frac{539}{828}\right)\) | \(e\left(\frac{109}{414}\right)\) | \(e\left(\frac{125}{414}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{757}{828}\right)\) | \(e\left(\frac{263}{276}\right)\) | \(e\left(\frac{109}{207}\right)\) | \(e\left(\frac{11}{276}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
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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)
