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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10780, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,105,130,378]))
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gp:[g,chi] = znchar(Mod(3097, 10780))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10780.3097");
| Modulus: | \(10780\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2695\) |
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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 \(\chi_{2695}(402,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{10780}(17,\cdot)\)
\(\chi_{10780}(73,\cdot)\)
\(\chi_{10780}(173,\cdot)\)
\(\chi_{10780}(437,\cdot)\)
\(\chi_{10780}(453,\cdot)\)
\(\chi_{10780}(633,\cdot)\)
\(\chi_{10780}(677,\cdot)\)
\(\chi_{10780}(733,\cdot)\)
\(\chi_{10780}(997,\cdot)\)
\(\chi_{10780}(1053,\cdot)\)
\(\chi_{10780}(1333,\cdot)\)
\(\chi_{10780}(1377,\cdot)\)
\(\chi_{10780}(1557,\cdot)\)
\(\chi_{10780}(1613,\cdot)\)
\(\chi_{10780}(1657,\cdot)\)
\(\chi_{10780}(1713,\cdot)\)
\(\chi_{10780}(1977,\cdot)\)
\(\chi_{10780}(1993,\cdot)\)
\(\chi_{10780}(2173,\cdot)\)
\(\chi_{10780}(2217,\cdot)\)
\(\chi_{10780}(2257,\cdot)\)
\(\chi_{10780}(2537,\cdot)\)
\(\chi_{10780}(2593,\cdot)\)
\(\chi_{10780}(2637,\cdot)\)
\(\chi_{10780}(2833,\cdot)\)
\(\chi_{10780}(2917,\cdot)\)
\(\chi_{10780}(3097,\cdot)\)
\(\chi_{10780}(3153,\cdot)\)
\(\chi_{10780}(3197,\cdot)\)
\(\chi_{10780}(3517,\cdot)\)
...
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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 };
\((5391,2157,9901,981)\) → \((1,i,e\left(\frac{13}{42}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 10780 }(3097, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{401}{420}\right)\) |
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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)
