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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39326, base_ring=CyclotomicField(4134))
M = H._module
chi = DirichletCharacter(H, M([1378,1206]))
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gp:[g,chi] = znchar(Mod(331, 39326))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39326.331");
| Modulus: | \(39326\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19663\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2067\) |
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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 \(\chi_{19663}(331,\cdot)\) |
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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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\(\chi_{39326}(81,\cdot)\)
\(\chi_{39326}(95,\cdot)\)
\(\chi_{39326}(121,\cdot)\)
\(\chi_{39326}(205,\cdot)\)
\(\chi_{39326}(261,\cdot)\)
\(\chi_{39326}(275,\cdot)\)
\(\chi_{39326}(289,\cdot)\)
\(\chi_{39326}(331,\cdot)\)
\(\chi_{39326}(333,\cdot)\)
\(\chi_{39326}(387,\cdot)\)
\(\chi_{39326}(415,\cdot)\)
\(\chi_{39326}(417,\cdot)\)
\(\chi_{39326}(471,\cdot)\)
\(\chi_{39326}(473,\cdot)\)
\(\chi_{39326}(487,\cdot)\)
\(\chi_{39326}(501,\cdot)\)
\(\chi_{39326}(513,\cdot)\)
\(\chi_{39326}(543,\cdot)\)
\(\chi_{39326}(599,\cdot)\)
\(\chi_{39326}(611,\cdot)\)
\(\chi_{39326}(625,\cdot)\)
\(\chi_{39326}(627,\cdot)\)
\(\chi_{39326}(683,\cdot)\)
\(\chi_{39326}(725,\cdot)\)
\(\chi_{39326}(823,\cdot)\)
\(\chi_{39326}(837,\cdot)\)
\(\chi_{39326}(863,\cdot)\)
\(\chi_{39326}(947,\cdot)\)
\(\chi_{39326}(1003,\cdot)\)
\(\chi_{39326}(1017,\cdot)\)
...
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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: |
$\Q(\zeta_{2067})$ |
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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 2067 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((22473,8429)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{201}{689}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 39326 }(331, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1697}{2067}\right)\) | \(e\left(\frac{976}{2067}\right)\) | \(e\left(\frac{1327}{2067}\right)\) | \(e\left(\frac{407}{2067}\right)\) | \(e\left(\frac{456}{689}\right)\) | \(e\left(\frac{202}{689}\right)\) | \(e\left(\frac{1571}{2067}\right)\) | \(e\left(\frac{1966}{2067}\right)\) | \(e\left(\frac{142}{159}\right)\) | \(e\left(\frac{1952}{2067}\right)\) |
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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)
