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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676000, base_ring=CyclotomicField(7800))
M = H._module
chi = DirichletCharacter(H, M([3900,975,6006,800]))
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gp:[g,chi] = znchar(Mod(1147, 676000))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("676000.1147");
| Modulus: | \(676000\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(676000\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(7800\) |
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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: | yes |
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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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\(\chi_{676000}(3,\cdot)\)
\(\chi_{676000}(347,\cdot)\)
\(\chi_{676000}(1147,\cdot)\)
\(\chi_{676000}(1283,\cdot)\)
\(\chi_{676000}(1387,\cdot)\)
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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: |
$\Q(\zeta_{7800})$ |
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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 7800 polynomial (not computed) |
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sage:chi.fixed_field()
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\((126751,422501,389377,12001)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{77}{100}\right),e\left(\frac{4}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 676000 }(1147, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7667}{7800}\right)\) | \(e\left(\frac{34}{195}\right)\) | \(e\left(\frac{3767}{3900}\right)\) | \(e\left(\frac{1631}{7800}\right)\) | \(e\left(\frac{2669}{3900}\right)\) | \(e\left(\frac{541}{600}\right)\) | \(e\left(\frac{409}{2600}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{2467}{2600}\right)\) | \(e\left(\frac{1697}{7800}\right)\) |
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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)
