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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34400, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,105,42,20]))
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gp:[g,chi] = znchar(Mod(10329, 34400))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("34400.10329");
| Modulus: | \(34400\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(17200\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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: | no, induced from \(\chi_{17200}(14629,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{34400}(9,\cdot)\)
\(\chi_{34400}(169,\cdot)\)
\(\chi_{34400}(569,\cdot)\)
\(\chi_{34400}(969,\cdot)\)
\(\chi_{34400}(1529,\cdot)\)
\(\chi_{34400}(2009,\cdot)\)
\(\chi_{34400}(2089,\cdot)\)
\(\chi_{34400}(2489,\cdot)\)
\(\chi_{34400}(2809,\cdot)\)
\(\chi_{34400}(3369,\cdot)\)
\(\chi_{34400}(3609,\cdot)\)
\(\chi_{34400}(4009,\cdot)\)
\(\chi_{34400}(4409,\cdot)\)
\(\chi_{34400}(4489,\cdot)\)
\(\chi_{34400}(4969,\cdot)\)
\(\chi_{34400}(5529,\cdot)\)
\(\chi_{34400}(5689,\cdot)\)
\(\chi_{34400}(5929,\cdot)\)
\(\chi_{34400}(6809,\cdot)\)
\(\chi_{34400}(6889,\cdot)\)
\(\chi_{34400}(7929,\cdot)\)
\(\chi_{34400}(8409,\cdot)\)
\(\chi_{34400}(8889,\cdot)\)
\(\chi_{34400}(8969,\cdot)\)
\(\chi_{34400}(9129,\cdot)\)
\(\chi_{34400}(9369,\cdot)\)
\(\chi_{34400}(9689,\cdot)\)
\(\chi_{34400}(10329,\cdot)\)
\(\chi_{34400}(10489,\cdot)\)
\(\chi_{34400}(10889,\cdot)\)
...
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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 };
\((10751,12901,1377,8001)\) → \((1,i,e\left(\frac{1}{10}\right),e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 34400 }(10329, a) \) |
\(1\) | \(1\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{69}{140}\right)\) |
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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)
