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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(836352, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
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gp:[g,chi] = znchar(Mod(781057, 836352))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("836352.781057");
| Modulus: | \(836352\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(121\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{121}(2,\cdot)\) |
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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: | odd |
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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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\(\chi_{836352}(20737,\cdot)\)
\(\chi_{836352}(27649,\cdot)\)
\(\chi_{836352}(48385,\cdot)\)
\(\chi_{836352}(69121,\cdot)\)
\(\chi_{836352}(96769,\cdot)\)
\(\chi_{836352}(103681,\cdot)\)
\(\chi_{836352}(124417,\cdot)\)
\(\chi_{836352}(145153,\cdot)\)
\(\chi_{836352}(172801,\cdot)\)
\(\chi_{836352}(179713,\cdot)\)
\(\chi_{836352}(200449,\cdot)\)
\(\chi_{836352}(248833,\cdot)\)
\(\chi_{836352}(255745,\cdot)\)
\(\chi_{836352}(276481,\cdot)\)
\(\chi_{836352}(297217,\cdot)\)
\(\chi_{836352}(324865,\cdot)\)
\(\chi_{836352}(331777,\cdot)\)
\(\chi_{836352}(373249,\cdot)\)
\(\chi_{836352}(400897,\cdot)\)
\(\chi_{836352}(407809,\cdot)\)
\(\chi_{836352}(428545,\cdot)\)
\(\chi_{836352}(449281,\cdot)\)
\(\chi_{836352}(476929,\cdot)\)
\(\chi_{836352}(483841,\cdot)\)
\(\chi_{836352}(504577,\cdot)\)
\(\chi_{836352}(525313,\cdot)\)
\(\chi_{836352}(559873,\cdot)\)
\(\chi_{836352}(580609,\cdot)\)
\(\chi_{836352}(601345,\cdot)\)
\(\chi_{836352}(628993,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{55})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 110 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((137215,561925,123905,781057)\) → \((1,1,1,e\left(\frac{1}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 836352 }(781057, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
