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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6500, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([0,39,25]))
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gp:[g,chi] = znchar(Mod(4489, 6500))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6500.4489");
| 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: | \(150\) |
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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 \(\chi_{1625}(1239,\cdot)\) |
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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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\(\chi_{6500}(69,\cdot)\)
\(\chi_{6500}(309,\cdot)\)
\(\chi_{6500}(329,\cdot)\)
\(\chi_{6500}(569,\cdot)\)
\(\chi_{6500}(589,\cdot)\)
\(\chi_{6500}(829,\cdot)\)
\(\chi_{6500}(1089,\cdot)\)
\(\chi_{6500}(1109,\cdot)\)
\(\chi_{6500}(1369,\cdot)\)
\(\chi_{6500}(1609,\cdot)\)
\(\chi_{6500}(1629,\cdot)\)
\(\chi_{6500}(1869,\cdot)\)
\(\chi_{6500}(1889,\cdot)\)
\(\chi_{6500}(2129,\cdot)\)
\(\chi_{6500}(2389,\cdot)\)
\(\chi_{6500}(2409,\cdot)\)
\(\chi_{6500}(2669,\cdot)\)
\(\chi_{6500}(2909,\cdot)\)
\(\chi_{6500}(2929,\cdot)\)
\(\chi_{6500}(3169,\cdot)\)
\(\chi_{6500}(3189,\cdot)\)
\(\chi_{6500}(3429,\cdot)\)
\(\chi_{6500}(3689,\cdot)\)
\(\chi_{6500}(3709,\cdot)\)
\(\chi_{6500}(3969,\cdot)\)
\(\chi_{6500}(4209,\cdot)\)
\(\chi_{6500}(4229,\cdot)\)
\(\chi_{6500}(4469,\cdot)\)
\(\chi_{6500}(4489,\cdot)\)
\(\chi_{6500}(4729,\cdot)\)
...
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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: |
$\Q(\zeta_{75})$ |
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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 150 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3251,5877,5501)\) → \((1,e\left(\frac{13}{50}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6500 }(4489, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{59}{75}\right)\) |
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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)
