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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49619, base_ring=CyclotomicField(812))
M = H._module
chi = DirichletCharacter(H, M([389,308]))
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gp:[g,chi] = znchar(Mod(997, 49619))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49619.997");
| Modulus: | \(49619\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(49619\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(812\) |
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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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\(\chi_{49619}(79,\cdot)\)
\(\chi_{49619}(163,\cdot)\)
\(\chi_{49619}(166,\cdot)\)
\(\chi_{49619}(287,\cdot)\)
\(\chi_{49619}(694,\cdot)\)
\(\chi_{49619}(897,\cdot)\)
\(\chi_{49619}(931,\cdot)\)
\(\chi_{49619}(936,\cdot)\)
\(\chi_{49619}(971,\cdot)\)
\(\chi_{49619}(997,\cdot)\)
\(\chi_{49619}(1377,\cdot)\)
\(\chi_{49619}(1556,\cdot)\)
\(\chi_{49619}(1613,\cdot)\)
\(\chi_{49619}(1664,\cdot)\)
\(\chi_{49619}(2172,\cdot)\)
\(\chi_{49619}(2346,\cdot)\)
\(\chi_{49619}(2700,\cdot)\)
\(\chi_{49619}(2860,\cdot)\)
\(\chi_{49619}(2972,\cdot)\)
\(\chi_{49619}(2998,\cdot)\)
\(\chi_{49619}(3662,\cdot)\)
\(\chi_{49619}(3781,\cdot)\)
\(\chi_{49619}(3942,\cdot)\)
\(\chi_{49619}(3965,\cdot)\)
\(\chi_{49619}(4120,\cdot)\)
\(\chi_{49619}(4155,\cdot)\)
\(\chi_{49619}(4294,\cdot)\)
\(\chi_{49619}(4352,\cdot)\)
\(\chi_{49619}(4506,\cdot)\)
\(\chi_{49619}(4600,\cdot)\)
...
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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 };
\((46257,3365)\) → \((e\left(\frac{389}{812}\right),e\left(\frac{11}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 49619 }(997, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{697}{812}\right)\) | \(e\left(\frac{181}{812}\right)\) | \(e\left(\frac{291}{406}\right)\) | \(e\left(\frac{387}{406}\right)\) | \(e\left(\frac{33}{406}\right)\) | \(e\left(\frac{194}{203}\right)\) | \(e\left(\frac{467}{812}\right)\) | \(e\left(\frac{181}{406}\right)\) | \(e\left(\frac{659}{812}\right)\) | \(e\left(\frac{233}{812}\right)\) |
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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)
