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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4169, base_ring=CyclotomicField(630))
M = H._module
chi = DirichletCharacter(H, M([63,380]))
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gp:[g,chi] = znchar(Mod(167, 4169))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4169.167");
| Modulus: | \(4169\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4169\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(630\) |
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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);
|
\(\chi_{4169}(6,\cdot)\)
\(\chi_{4169}(30,\cdot)\)
\(\chi_{4169}(41,\cdot)\)
\(\chi_{4169}(83,\cdot)\)
\(\chi_{4169}(139,\cdot)\)
\(\chi_{4169}(150,\cdot)\)
\(\chi_{4169}(167,\cdot)\)
\(\chi_{4169}(205,\cdot)\)
\(\chi_{4169}(222,\cdot)\)
\(\chi_{4169}(244,\cdot)\)
\(\chi_{4169}(316,\cdot)\)
\(\chi_{4169}(371,\cdot)\)
\(\chi_{4169}(393,\cdot)\)
\(\chi_{4169}(402,\cdot)\)
\(\chi_{4169}(409,\cdot)\)
\(\chi_{4169}(415,\cdot)\)
\(\chi_{4169}(420,\cdot)\)
\(\chi_{4169}(446,\cdot)\)
\(\chi_{4169}(546,\cdot)\)
\(\chi_{4169}(558,\cdot)\)
\(\chi_{4169}(601,\cdot)\)
\(\chi_{4169}(611,\cdot)\)
\(\chi_{4169}(623,\cdot)\)
\(\chi_{4169}(646,\cdot)\)
\(\chi_{4169}(695,\cdot)\)
\(\chi_{4169}(699,\cdot)\)
\(\chi_{4169}(710,\cdot)\)
\(\chi_{4169}(750,\cdot)\)
\(\chi_{4169}(772,\cdot)\)
\(\chi_{4169}(788,\cdot)\)
...
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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 };
\((3412,760)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{38}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 4169 }(167, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{443}{630}\right)\) | \(e\left(\frac{137}{315}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{121}{630}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{274}{315}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{53}{63}\right)\) |
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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)
