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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1125, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([125,66]))
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gp:[g,chi] = znchar(Mod(41, 1125))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1125.41");
| Modulus: | \(1125\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1125\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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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\(\chi_{1125}(11,\cdot)\)
\(\chi_{1125}(41,\cdot)\)
\(\chi_{1125}(56,\cdot)\)
\(\chi_{1125}(86,\cdot)\)
\(\chi_{1125}(131,\cdot)\)
\(\chi_{1125}(146,\cdot)\)
\(\chi_{1125}(191,\cdot)\)
\(\chi_{1125}(221,\cdot)\)
\(\chi_{1125}(236,\cdot)\)
\(\chi_{1125}(266,\cdot)\)
\(\chi_{1125}(281,\cdot)\)
\(\chi_{1125}(311,\cdot)\)
\(\chi_{1125}(356,\cdot)\)
\(\chi_{1125}(371,\cdot)\)
\(\chi_{1125}(416,\cdot)\)
\(\chi_{1125}(446,\cdot)\)
\(\chi_{1125}(461,\cdot)\)
\(\chi_{1125}(491,\cdot)\)
\(\chi_{1125}(506,\cdot)\)
\(\chi_{1125}(536,\cdot)\)
\(\chi_{1125}(581,\cdot)\)
\(\chi_{1125}(596,\cdot)\)
\(\chi_{1125}(641,\cdot)\)
\(\chi_{1125}(671,\cdot)\)
\(\chi_{1125}(686,\cdot)\)
\(\chi_{1125}(716,\cdot)\)
\(\chi_{1125}(731,\cdot)\)
\(\chi_{1125}(761,\cdot)\)
\(\chi_{1125}(806,\cdot)\)
\(\chi_{1125}(821,\cdot)\)
...
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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 };
\((1001,127)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{25}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1125 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) |
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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)
