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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3175, base_ring=CyclotomicField(630))
M = H._module
chi = DirichletCharacter(H, M([504,550]))
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gp:[g,chi] = znchar(Mod(161, 3175))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3175.161");
| Modulus: | \(3175\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3175\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(315\) |
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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);
|
\(\chi_{3175}(11,\cdot)\)
\(\chi_{3175}(21,\cdot)\)
\(\chi_{3175}(31,\cdot)\)
\(\chi_{3175}(36,\cdot)\)
\(\chi_{3175}(41,\cdot)\)
\(\chi_{3175}(71,\cdot)\)
\(\chi_{3175}(81,\cdot)\)
\(\chi_{3175}(121,\cdot)\)
\(\chi_{3175}(136,\cdot)\)
\(\chi_{3175}(161,\cdot)\)
\(\chi_{3175}(171,\cdot)\)
\(\chi_{3175}(196,\cdot)\)
\(\chi_{3175}(206,\cdot)\)
\(\chi_{3175}(211,\cdot)\)
\(\chi_{3175}(231,\cdot)\)
\(\chi_{3175}(271,\cdot)\)
\(\chi_{3175}(296,\cdot)\)
\(\chi_{3175}(316,\cdot)\)
\(\chi_{3175}(336,\cdot)\)
\(\chi_{3175}(396,\cdot)\)
\(\chi_{3175}(411,\cdot)\)
\(\chi_{3175}(416,\cdot)\)
\(\chi_{3175}(441,\cdot)\)
\(\chi_{3175}(496,\cdot)\)
\(\chi_{3175}(521,\cdot)\)
\(\chi_{3175}(596,\cdot)\)
\(\chi_{3175}(606,\cdot)\)
\(\chi_{3175}(621,\cdot)\)
\(\chi_{3175}(646,\cdot)\)
\(\chi_{3175}(656,\cdot)\)
...
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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 };
\((1652,3051)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{55}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3175 }(161, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{149}{315}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{41}{315}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{298}{315}\right)\) | \(e\left(\frac{52}{315}\right)\) | \(e\left(\frac{248}{315}\right)\) | \(e\left(\frac{83}{315}\right)\) |
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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)
