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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7025, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([49,34]))
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gp:[g,chi] = znchar(Mod(4053, 7025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7025.4053");
| Modulus: | \(7025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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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\(\chi_{7025}(2,\cdot)\)
\(\chi_{7025}(8,\cdot)\)
\(\chi_{7025}(498,\cdot)\)
\(\chi_{7025}(512,\cdot)\)
\(\chi_{7025}(758,\cdot)\)
\(\chi_{7025}(1013,\cdot)\)
\(\chi_{7025}(1167,\cdot)\)
\(\chi_{7025}(1523,\cdot)\)
\(\chi_{7025}(1623,\cdot)\)
\(\chi_{7025}(1992,\cdot)\)
\(\chi_{7025}(2048,\cdot)\)
\(\chi_{7025}(2137,\cdot)\)
\(\chi_{7025}(2513,\cdot)\)
\(\chi_{7025}(2558,\cdot)\)
\(\chi_{7025}(3027,\cdot)\)
\(\chi_{7025}(3442,\cdot)\)
\(\chi_{7025}(3513,\cdot)\)
\(\chi_{7025}(3637,\cdot)\)
\(\chi_{7025}(3772,\cdot)\)
\(\chi_{7025}(3942,\cdot)\)
\(\chi_{7025}(4052,\cdot)\)
\(\chi_{7025}(4053,\cdot)\)
\(\chi_{7025}(4152,\cdot)\)
\(\chi_{7025}(4258,\cdot)\)
\(\chi_{7025}(4373,\cdot)\)
\(\chi_{7025}(4438,\cdot)\)
\(\chi_{7025}(4498,\cdot)\)
\(\chi_{7025}(4577,\cdot)\)
\(\chi_{7025}(4622,\cdot)\)
\(\chi_{7025}(4637,\cdot)\)
...
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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 };
\((5902,2251)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{17}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 7025 }(4053, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{117}{140}\right)\) |
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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)
