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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5082, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,55,36]))
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gp:[g,chi] = znchar(Mod(97, 5082))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5082.97");
| Modulus: | \(5082\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(847\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{847}(97,\cdot)\) |
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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: | 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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\(\chi_{5082}(97,\cdot)\)
\(\chi_{5082}(181,\cdot)\)
\(\chi_{5082}(223,\cdot)\)
\(\chi_{5082}(433,\cdot)\)
\(\chi_{5082}(559,\cdot)\)
\(\chi_{5082}(643,\cdot)\)
\(\chi_{5082}(685,\cdot)\)
\(\chi_{5082}(895,\cdot)\)
\(\chi_{5082}(1021,\cdot)\)
\(\chi_{5082}(1105,\cdot)\)
\(\chi_{5082}(1147,\cdot)\)
\(\chi_{5082}(1357,\cdot)\)
\(\chi_{5082}(1483,\cdot)\)
\(\chi_{5082}(1567,\cdot)\)
\(\chi_{5082}(1609,\cdot)\)
\(\chi_{5082}(1819,\cdot)\)
\(\chi_{5082}(2029,\cdot)\)
\(\chi_{5082}(2071,\cdot)\)
\(\chi_{5082}(2281,\cdot)\)
\(\chi_{5082}(2407,\cdot)\)
\(\chi_{5082}(2491,\cdot)\)
\(\chi_{5082}(2533,\cdot)\)
\(\chi_{5082}(2869,\cdot)\)
\(\chi_{5082}(2953,\cdot)\)
\(\chi_{5082}(2995,\cdot)\)
\(\chi_{5082}(3205,\cdot)\)
\(\chi_{5082}(3331,\cdot)\)
\(\chi_{5082}(3457,\cdot)\)
\(\chi_{5082}(3667,\cdot)\)
\(\chi_{5082}(3793,\cdot)\)
...
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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: |
$\Q(\zeta_{55})$ |
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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 110 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3389,4357,2059)\) → \((1,-1,e\left(\frac{18}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5082 }(97, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) |
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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)
