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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28158, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([114,19,56]))
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gp:[g,chi] = znchar(Mod(4799, 28158))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28158.4799");
| Modulus: | \(28158\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14079\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(228\) |
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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 \(\chi_{14079}(4799,\cdot)\) |
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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: | 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);
|
\(\chi_{28158}(11,\cdot)\)
\(\chi_{28158}(353,\cdot)\)
\(\chi_{28158}(539,\cdot)\)
\(\chi_{28158}(995,\cdot)\)
\(\chi_{28158}(1493,\cdot)\)
\(\chi_{28158}(1835,\cdot)\)
\(\chi_{28158}(2021,\cdot)\)
\(\chi_{28158}(2477,\cdot)\)
\(\chi_{28158}(2975,\cdot)\)
\(\chi_{28158}(3503,\cdot)\)
\(\chi_{28158}(3959,\cdot)\)
\(\chi_{28158}(4457,\cdot)\)
\(\chi_{28158}(4799,\cdot)\)
\(\chi_{28158}(5441,\cdot)\)
\(\chi_{28158}(5939,\cdot)\)
\(\chi_{28158}(6281,\cdot)\)
\(\chi_{28158}(6467,\cdot)\)
\(\chi_{28158}(6923,\cdot)\)
\(\chi_{28158}(7421,\cdot)\)
\(\chi_{28158}(7763,\cdot)\)
\(\chi_{28158}(7949,\cdot)\)
\(\chi_{28158}(8405,\cdot)\)
\(\chi_{28158}(8903,\cdot)\)
\(\chi_{28158}(9245,\cdot)\)
\(\chi_{28158}(9431,\cdot)\)
\(\chi_{28158}(9887,\cdot)\)
\(\chi_{28158}(10385,\cdot)\)
\(\chi_{28158}(10727,\cdot)\)
\(\chi_{28158}(10913,\cdot)\)
\(\chi_{28158}(11369,\cdot)\)
...
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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 };
\((18773,10831,12637)\) → \((-1,e\left(\frac{1}{12}\right),e\left(\frac{14}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 28158 }(4799, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{173}{228}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{145}{228}\right)\) |
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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)
