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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4291, base_ring=CyclotomicField(306))
M = H._module
chi = DirichletCharacter(H, M([153,67]))
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gp:[g,chi] = znchar(Mod(1651, 4291))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4291.1651");
| Modulus: | \(4291\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4291\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(306\) |
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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);
|
\(\chi_{4291}(48,\cdot)\)
\(\chi_{4291}(55,\cdot)\)
\(\chi_{4291}(76,\cdot)\)
\(\chi_{4291}(97,\cdot)\)
\(\chi_{4291}(244,\cdot)\)
\(\chi_{4291}(349,\cdot)\)
\(\chi_{4291}(356,\cdot)\)
\(\chi_{4291}(447,\cdot)\)
\(\chi_{4291}(475,\cdot)\)
\(\chi_{4291}(510,\cdot)\)
\(\chi_{4291}(524,\cdot)\)
\(\chi_{4291}(538,\cdot)\)
\(\chi_{4291}(601,\cdot)\)
\(\chi_{4291}(664,\cdot)\)
\(\chi_{4291}(734,\cdot)\)
\(\chi_{4291}(804,\cdot)\)
\(\chi_{4291}(867,\cdot)\)
\(\chi_{4291}(874,\cdot)\)
\(\chi_{4291}(937,\cdot)\)
\(\chi_{4291}(958,\cdot)\)
\(\chi_{4291}(1056,\cdot)\)
\(\chi_{4291}(1077,\cdot)\)
\(\chi_{4291}(1210,\cdot)\)
\(\chi_{4291}(1252,\cdot)\)
\(\chi_{4291}(1266,\cdot)\)
\(\chi_{4291}(1476,\cdot)\)
\(\chi_{4291}(1525,\cdot)\)
\(\chi_{4291}(1567,\cdot)\)
\(\chi_{4291}(1637,\cdot)\)
\(\chi_{4291}(1651,\cdot)\)
...
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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 };
\((2453,3067)\) → \((-1,e\left(\frac{67}{306}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 4291 }(1651, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{306}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{67}{153}\right)\) | \(e\left(\frac{130}{153}\right)\) | \(e\left(\frac{104}{153}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{47}{306}\right)\) | \(e\left(\frac{275}{306}\right)\) |
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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)
