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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1625, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([69,25]))
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gp:[g,chi] = znchar(Mod(1087, 1625))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1625.1087");
| Modulus: | \(1625\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1625\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(100\) |
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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);
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\(\chi_{1625}(47,\cdot)\)
\(\chi_{1625}(83,\cdot)\)
\(\chi_{1625}(112,\cdot)\)
\(\chi_{1625}(148,\cdot)\)
\(\chi_{1625}(177,\cdot)\)
\(\chi_{1625}(213,\cdot)\)
\(\chi_{1625}(242,\cdot)\)
\(\chi_{1625}(278,\cdot)\)
\(\chi_{1625}(372,\cdot)\)
\(\chi_{1625}(408,\cdot)\)
\(\chi_{1625}(437,\cdot)\)
\(\chi_{1625}(473,\cdot)\)
\(\chi_{1625}(502,\cdot)\)
\(\chi_{1625}(538,\cdot)\)
\(\chi_{1625}(567,\cdot)\)
\(\chi_{1625}(603,\cdot)\)
\(\chi_{1625}(697,\cdot)\)
\(\chi_{1625}(733,\cdot)\)
\(\chi_{1625}(762,\cdot)\)
\(\chi_{1625}(798,\cdot)\)
\(\chi_{1625}(827,\cdot)\)
\(\chi_{1625}(863,\cdot)\)
\(\chi_{1625}(892,\cdot)\)
\(\chi_{1625}(928,\cdot)\)
\(\chi_{1625}(1022,\cdot)\)
\(\chi_{1625}(1058,\cdot)\)
\(\chi_{1625}(1087,\cdot)\)
\(\chi_{1625}(1123,\cdot)\)
\(\chi_{1625}(1152,\cdot)\)
\(\chi_{1625}(1188,\cdot)\)
...
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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 };
\((1002,626)\) → \((e\left(\frac{69}{100}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 1625 }(1087, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{17}{50}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
