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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17061, base_ring=CyclotomicField(2530))
M = H._module
chi = DirichletCharacter(H, M([1265,46,660]))
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gp:[g,chi] = znchar(Mod(488, 17061))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17061.488");
| Modulus: | \(17061\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(17061\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2530\) |
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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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\(\chi_{17061}(14,\cdot)\)
\(\chi_{17061}(53,\cdot)\)
\(\chi_{17061}(59,\cdot)\)
\(\chi_{17061}(71,\cdot)\)
\(\chi_{17061}(119,\cdot)\)
\(\chi_{17061}(158,\cdot)\)
\(\chi_{17061}(191,\cdot)\)
\(\chi_{17061}(212,\cdot)\)
\(\chi_{17061}(224,\cdot)\)
\(\chi_{17061}(284,\cdot)\)
\(\chi_{17061}(290,\cdot)\)
\(\chi_{17061}(335,\cdot)\)
\(\chi_{17061}(350,\cdot)\)
\(\chi_{17061}(356,\cdot)\)
\(\chi_{17061}(383,\cdot)\)
\(\chi_{17061}(401,\cdot)\)
\(\chi_{17061}(410,\cdot)\)
\(\chi_{17061}(455,\cdot)\)
\(\chi_{17061}(476,\cdot)\)
\(\chi_{17061}(482,\cdot)\)
\(\chi_{17061}(488,\cdot)\)
\(\chi_{17061}(521,\cdot)\)
\(\chi_{17061}(533,\cdot)\)
\(\chi_{17061}(542,\cdot)\)
\(\chi_{17061}(554,\cdot)\)
\(\chi_{17061}(566,\cdot)\)
\(\chi_{17061}(581,\cdot)\)
\(\chi_{17061}(620,\cdot)\)
\(\chi_{17061}(647,\cdot)\)
\(\chi_{17061}(653,\cdot)\)
...
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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 };
\((11375,16216,9076)\) → \((-1,e\left(\frac{1}{55}\right),e\left(\frac{6}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 17061 }(488, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{541}{2530}\right)\) | \(e\left(\frac{541}{1265}\right)\) | \(e\left(\frac{269}{2530}\right)\) | \(e\left(\frac{601}{1265}\right)\) | \(e\left(\frac{1623}{2530}\right)\) | \(e\left(\frac{81}{253}\right)\) | \(e\left(\frac{893}{1265}\right)\) | \(e\left(\frac{1743}{2530}\right)\) | \(e\left(\frac{1082}{1265}\right)\) | \(e\left(\frac{1429}{2530}\right)\) |
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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)
