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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8100, base_ring=CyclotomicField(270))
M = H._module
chi = DirichletCharacter(H, M([0,245,108]))
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gp:[g,chi] = znchar(Mod(281, 8100))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8100.281");
| 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: | \(2025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(270\) |
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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 \(\chi_{2025}(281,\cdot)\) |
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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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\(\chi_{8100}(41,\cdot)\)
\(\chi_{8100}(221,\cdot)\)
\(\chi_{8100}(281,\cdot)\)
\(\chi_{8100}(461,\cdot)\)
\(\chi_{8100}(581,\cdot)\)
\(\chi_{8100}(641,\cdot)\)
\(\chi_{8100}(761,\cdot)\)
\(\chi_{8100}(821,\cdot)\)
\(\chi_{8100}(941,\cdot)\)
\(\chi_{8100}(1121,\cdot)\)
\(\chi_{8100}(1181,\cdot)\)
\(\chi_{8100}(1361,\cdot)\)
\(\chi_{8100}(1481,\cdot)\)
\(\chi_{8100}(1541,\cdot)\)
\(\chi_{8100}(1661,\cdot)\)
\(\chi_{8100}(1721,\cdot)\)
\(\chi_{8100}(1841,\cdot)\)
\(\chi_{8100}(2021,\cdot)\)
\(\chi_{8100}(2081,\cdot)\)
\(\chi_{8100}(2261,\cdot)\)
\(\chi_{8100}(2381,\cdot)\)
\(\chi_{8100}(2441,\cdot)\)
\(\chi_{8100}(2561,\cdot)\)
\(\chi_{8100}(2621,\cdot)\)
\(\chi_{8100}(2741,\cdot)\)
\(\chi_{8100}(2921,\cdot)\)
\(\chi_{8100}(2981,\cdot)\)
\(\chi_{8100}(3161,\cdot)\)
\(\chi_{8100}(3281,\cdot)\)
\(\chi_{8100}(3341,\cdot)\)
...
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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: |
$\Q(\zeta_{135})$ |
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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 270 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((4051,6401,7777)\) → \((1,e\left(\frac{49}{54}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8100 }(281, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{53}{270}\right)\) | \(e\left(\frac{116}{135}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{103}{270}\right)\) | \(e\left(\frac{101}{270}\right)\) | \(e\left(\frac{47}{135}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{187}{270}\right)\) |
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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)
