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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256025, base_ring=CyclotomicField(1260))
M = H._module
chi = DirichletCharacter(H, M([819,600,252,490]))
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gp:[g,chi] = znchar(Mod(1192, 256025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256025.1192");
| Modulus: | \(256025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(256025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1260\) |
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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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\(\chi_{256025}(1192,\cdot)\)
\(\chi_{256025}(1362,\cdot)\)
\(\chi_{256025}(1808,\cdot)\)
\(\chi_{256025}(2522,\cdot)\)
\(\chi_{256025}(3327,\cdot)\)
\(\chi_{256025}(3392,\cdot)\)
\(\chi_{256025}(4288,\cdot)\)
\(\chi_{256025}(4953,\cdot)\)
\(\chi_{256025}(5373,\cdot)\)
\(\chi_{256025}(6862,\cdot)\)
\(\chi_{256025}(7177,\cdot)\)
\(\chi_{256025}(7583,\cdot)\)
\(\chi_{256025}(9788,\cdot)\)
\(\chi_{256025}(10453,\cdot)\)
\(\chi_{256025}(11433,\cdot)\)
\(\chi_{256025}(12422,\cdot)\)
\(\chi_{256025}(12637,\cdot)\)
\(\chi_{256025}(13633,\cdot)\)
\(\chi_{256025}(16228,\cdot)\)
\(\chi_{256025}(16487,\cdot)\)
\(\chi_{256025}(17077,\cdot)\)
\(\chi_{256025}(19413,\cdot)\)
\(\chi_{256025}(20078,\cdot)\)
\(\chi_{256025}(21613,\cdot)\)
\(\chi_{256025}(22278,\cdot)\)
\(\chi_{256025}(22642,\cdot)\)
\(\chi_{256025}(23972,\cdot)\)
\(\chi_{256025}(26387,\cdot)\)
\(\chi_{256025}(28142,\cdot)\)
\(\chi_{256025}(28198,\cdot)\)
...
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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 };
\((112652,198551,232751,67376)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{10}{21}\right),e\left(\frac{1}{5}\right),e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 256025 }(1192, a) \) |
\(1\) | \(1\) | \(e\left(\frac{781}{1260}\right)\) | \(e\left(\frac{859}{1260}\right)\) | \(e\left(\frac{151}{630}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{229}{630}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{263}{1260}\right)\) | \(e\left(\frac{151}{315}\right)\) | \(e\left(\frac{11}{252}\right)\) |
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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)
