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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125341, base_ring=CyclotomicField(400))
M = H._module
chi = DirichletCharacter(H, M([375,50,252]))
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gp:[g,chi] = znchar(Mod(24027, 125341))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125341.24027");
| Modulus: | \(125341\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(125341\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(400\) |
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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);
|
\(\chi_{125341}(1219,\cdot)\)
\(\chi_{125341}(1523,\cdot)\)
\(\chi_{125341}(4005,\cdot)\)
\(\chi_{125341}(5088,\cdot)\)
\(\chi_{125341}(6195,\cdot)\)
\(\chi_{125341}(6329,\cdot)\)
\(\chi_{125341}(6446,\cdot)\)
\(\chi_{125341}(7436,\cdot)\)
\(\chi_{125341}(7687,\cdot)\)
\(\chi_{125341}(10449,\cdot)\)
\(\chi_{125341}(12067,\cdot)\)
\(\chi_{125341}(13892,\cdot)\)
\(\chi_{125341}(16415,\cdot)\)
\(\chi_{125341}(16654,\cdot)\)
\(\chi_{125341}(17822,\cdot)\)
\(\chi_{125341}(17895,\cdot)\)
\(\chi_{125341}(19834,\cdot)\)
\(\chi_{125341}(19980,\cdot)\)
\(\chi_{125341}(21075,\cdot)\)
\(\chi_{125341}(21221,\cdot)\)
\(\chi_{125341}(24027,\cdot)\)
\(\chi_{125341}(24477,\cdot)\)
\(\chi_{125341}(25268,\cdot)\)
\(\chi_{125341}(26051,\cdot)\)
\(\chi_{125341}(26959,\cdot)\)
\(\chi_{125341}(27426,\cdot)\)
\(\chi_{125341}(28533,\cdot)\)
\(\chi_{125341}(28825,\cdot)\)
\(\chi_{125341}(29441,\cdot)\)
\(\chi_{125341}(31015,\cdot)\)
...
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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 };
\((22120,46360,8688)\) → \((e\left(\frac{15}{16}\right),e\left(\frac{1}{8}\right),e\left(\frac{63}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 125341 }(24027, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{151}{200}\right)\) | \(e\left(\frac{63}{400}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{373}{400}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{43}{400}\right)\) | \(e\left(\frac{53}{200}\right)\) | \(e\left(\frac{63}{200}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{251}{400}\right)\) |
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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)
