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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13000, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([50,50,64,25]))
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gp:[g,chi] = znchar(Mod(2491, 13000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13000.2491");
| Modulus: | \(13000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13000\) |
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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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\(\chi_{13000}(291,\cdot)\)
\(\chi_{13000}(411,\cdot)\)
\(\chi_{13000}(811,\cdot)\)
\(\chi_{13000}(931,\cdot)\)
\(\chi_{13000}(1331,\cdot)\)
\(\chi_{13000}(1971,\cdot)\)
\(\chi_{13000}(2371,\cdot)\)
\(\chi_{13000}(2491,\cdot)\)
\(\chi_{13000}(2891,\cdot)\)
\(\chi_{13000}(3011,\cdot)\)
\(\chi_{13000}(3411,\cdot)\)
\(\chi_{13000}(3531,\cdot)\)
\(\chi_{13000}(3931,\cdot)\)
\(\chi_{13000}(4571,\cdot)\)
\(\chi_{13000}(4971,\cdot)\)
\(\chi_{13000}(5091,\cdot)\)
\(\chi_{13000}(5491,\cdot)\)
\(\chi_{13000}(5611,\cdot)\)
\(\chi_{13000}(6011,\cdot)\)
\(\chi_{13000}(6131,\cdot)\)
\(\chi_{13000}(6531,\cdot)\)
\(\chi_{13000}(7171,\cdot)\)
\(\chi_{13000}(7571,\cdot)\)
\(\chi_{13000}(7691,\cdot)\)
\(\chi_{13000}(8091,\cdot)\)
\(\chi_{13000}(8211,\cdot)\)
\(\chi_{13000}(8611,\cdot)\)
\(\chi_{13000}(8731,\cdot)\)
\(\chi_{13000}(9131,\cdot)\)
\(\chi_{13000}(9771,\cdot)\)
...
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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 };
\((9751,6501,12377,12001)\) → \((-1,-1,e\left(\frac{16}{25}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 13000 }(2491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) |
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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)
