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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6500, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([0,279,125]))
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gp:[g,chi] = znchar(Mod(1917, 6500))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6500.1917");
| Modulus: | \(6500\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1625\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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 \(\chi_{1625}(292,\cdot)\) |
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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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\(\chi_{6500}(33,\cdot)\)
\(\chi_{6500}(97,\cdot)\)
\(\chi_{6500}(197,\cdot)\)
\(\chi_{6500}(453,\cdot)\)
\(\chi_{6500}(553,\cdot)\)
\(\chi_{6500}(617,\cdot)\)
\(\chi_{6500}(713,\cdot)\)
\(\chi_{6500}(717,\cdot)\)
\(\chi_{6500}(813,\cdot)\)
\(\chi_{6500}(877,\cdot)\)
\(\chi_{6500}(973,\cdot)\)
\(\chi_{6500}(977,\cdot)\)
\(\chi_{6500}(1073,\cdot)\)
\(\chi_{6500}(1137,\cdot)\)
\(\chi_{6500}(1233,\cdot)\)
\(\chi_{6500}(1237,\cdot)\)
\(\chi_{6500}(1333,\cdot)\)
\(\chi_{6500}(1397,\cdot)\)
\(\chi_{6500}(1497,\cdot)\)
\(\chi_{6500}(1753,\cdot)\)
\(\chi_{6500}(1853,\cdot)\)
\(\chi_{6500}(1917,\cdot)\)
\(\chi_{6500}(2013,\cdot)\)
\(\chi_{6500}(2017,\cdot)\)
\(\chi_{6500}(2113,\cdot)\)
\(\chi_{6500}(2177,\cdot)\)
\(\chi_{6500}(2273,\cdot)\)
\(\chi_{6500}(2277,\cdot)\)
\(\chi_{6500}(2373,\cdot)\)
\(\chi_{6500}(2437,\cdot)\)
...
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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: |
$\Q(\zeta_{300})$ |
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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 300 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3251,5877,5501)\) → \((1,e\left(\frac{93}{100}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6500 }(1917, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{300}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{217}{300}\right)\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{49}{150}\right)\) |
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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)
