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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,99,102]))
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gp:[g,chi] = znchar(Mod(1762, 9075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9075.1762");
| Modulus: | \(9075\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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 \(\chi_{3025}(1762,\cdot)\) |
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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);
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\(\chi_{9075}(13,\cdot)\)
\(\chi_{9075}(127,\cdot)\)
\(\chi_{9075}(523,\cdot)\)
\(\chi_{9075}(547,\cdot)\)
\(\chi_{9075}(667,\cdot)\)
\(\chi_{9075}(733,\cdot)\)
\(\chi_{9075}(778,\cdot)\)
\(\chi_{9075}(937,\cdot)\)
\(\chi_{9075}(952,\cdot)\)
\(\chi_{9075}(1348,\cdot)\)
\(\chi_{9075}(1372,\cdot)\)
\(\chi_{9075}(1558,\cdot)\)
\(\chi_{9075}(1603,\cdot)\)
\(\chi_{9075}(1663,\cdot)\)
\(\chi_{9075}(1762,\cdot)\)
\(\chi_{9075}(1777,\cdot)\)
\(\chi_{9075}(2173,\cdot)\)
\(\chi_{9075}(2197,\cdot)\)
\(\chi_{9075}(2317,\cdot)\)
\(\chi_{9075}(2383,\cdot)\)
\(\chi_{9075}(2428,\cdot)\)
\(\chi_{9075}(2488,\cdot)\)
\(\chi_{9075}(2587,\cdot)\)
\(\chi_{9075}(2602,\cdot)\)
\(\chi_{9075}(3142,\cdot)\)
\(\chi_{9075}(3208,\cdot)\)
\(\chi_{9075}(3253,\cdot)\)
\(\chi_{9075}(3313,\cdot)\)
\(\chi_{9075}(3412,\cdot)\)
\(\chi_{9075}(3427,\cdot)\)
...
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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: |
$\Q(\zeta_{220})$ |
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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 220 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3026,727,5326)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{51}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(1762, a) \) |
\(1\) | \(1\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{89}{220}\right)\) |
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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)
