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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8100, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([0,80,63]))
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gp:[g,chi] = znchar(Mod(253, 8100))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8100.253");
| Modulus: | \(8100\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{675}(553,\cdot)\) |
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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: | odd |
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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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\(\chi_{8100}(37,\cdot)\)
\(\chi_{8100}(73,\cdot)\)
\(\chi_{8100}(253,\cdot)\)
\(\chi_{8100}(397,\cdot)\)
\(\chi_{8100}(577,\cdot)\)
\(\chi_{8100}(613,\cdot)\)
\(\chi_{8100}(937,\cdot)\)
\(\chi_{8100}(1117,\cdot)\)
\(\chi_{8100}(1153,\cdot)\)
\(\chi_{8100}(1333,\cdot)\)
\(\chi_{8100}(1477,\cdot)\)
\(\chi_{8100}(1873,\cdot)\)
\(\chi_{8100}(2017,\cdot)\)
\(\chi_{8100}(2197,\cdot)\)
\(\chi_{8100}(2233,\cdot)\)
\(\chi_{8100}(2413,\cdot)\)
\(\chi_{8100}(2737,\cdot)\)
\(\chi_{8100}(2773,\cdot)\)
\(\chi_{8100}(2953,\cdot)\)
\(\chi_{8100}(3097,\cdot)\)
\(\chi_{8100}(3277,\cdot)\)
\(\chi_{8100}(3313,\cdot)\)
\(\chi_{8100}(3637,\cdot)\)
\(\chi_{8100}(3817,\cdot)\)
\(\chi_{8100}(3853,\cdot)\)
\(\chi_{8100}(4033,\cdot)\)
\(\chi_{8100}(4177,\cdot)\)
\(\chi_{8100}(4573,\cdot)\)
\(\chi_{8100}(4717,\cdot)\)
\(\chi_{8100}(4897,\cdot)\)
...
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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: |
$\Q(\zeta_{180})$ |
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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 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((4051,6401,7777)\) → \((1,e\left(\frac{4}{9}\right),e\left(\frac{7}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8100 }(253, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{133}{180}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{43}{45}\right)\) |
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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)
