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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91091, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([520,546,465]))
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gp:[g,chi] = znchar(Mod(18, 91091))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91091.18");
| Modulus: | \(91091\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13013\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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 \(\chi_{13013}(18,\cdot)\) |
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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);
|
\(\chi_{91091}(18,\cdot)\)
\(\chi_{91091}(226,\cdot)\)
\(\chi_{91091}(655,\cdot)\)
\(\chi_{91091}(2725,\cdot)\)
\(\chi_{91091}(3154,\cdot)\)
\(\chi_{91091}(3203,\cdot)\)
\(\chi_{91091}(3362,\cdot)\)
\(\chi_{91091}(3791,\cdot)\)
\(\chi_{91091}(3999,\cdot)\)
\(\chi_{91091}(4428,\cdot)\)
\(\chi_{91091}(5959,\cdot)\)
\(\chi_{91091}(6388,\cdot)\)
\(\chi_{91091}(6547,\cdot)\)
\(\chi_{91091}(6596,\cdot)\)
\(\chi_{91091}(6976,\cdot)\)
\(\chi_{91091}(7025,\cdot)\)
\(\chi_{91091}(7233,\cdot)\)
\(\chi_{91091}(7662,\cdot)\)
\(\chi_{91091}(9781,\cdot)\)
\(\chi_{91091}(10161,\cdot)\)
\(\chi_{91091}(10369,\cdot)\)
\(\chi_{91091}(10798,\cdot)\)
\(\chi_{91091}(11006,\cdot)\)
\(\chi_{91091}(11435,\cdot)\)
\(\chi_{91091}(12966,\cdot)\)
\(\chi_{91091}(13395,\cdot)\)
\(\chi_{91091}(13554,\cdot)\)
\(\chi_{91091}(13603,\cdot)\)
\(\chi_{91091}(13983,\cdot)\)
\(\chi_{91091}(14032,\cdot)\)
...
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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: |
$\Q(\zeta_{780})$ |
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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 780 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((59489,41406,19944)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{31}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 91091 }(18, a) \) |
\(1\) | \(1\) | \(e\left(\frac{491}{780}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{101}{390}\right)\) | \(e\left(\frac{389}{780}\right)\) | \(e\left(\frac{213}{260}\right)\) | \(e\left(\frac{231}{260}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{179}{260}\right)\) |
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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)
