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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,80,117]))
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gp:[g,chi] = znchar(Mod(7, 1476))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1476.7");
| Modulus: | \(1476\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1476\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{1476}(7,\cdot)\)
\(\chi_{1476}(67,\cdot)\)
\(\chi_{1476}(151,\cdot)\)
\(\chi_{1476}(175,\cdot)\)
\(\chi_{1476}(211,\cdot)\)
\(\chi_{1476}(259,\cdot)\)
\(\chi_{1476}(391,\cdot)\)
\(\chi_{1476}(403,\cdot)\)
\(\chi_{1476}(427,\cdot)\)
\(\chi_{1476}(439,\cdot)\)
\(\chi_{1476}(463,\cdot)\)
\(\chi_{1476}(475,\cdot)\)
\(\chi_{1476}(499,\cdot)\)
\(\chi_{1476}(511,\cdot)\)
\(\chi_{1476}(643,\cdot)\)
\(\chi_{1476}(691,\cdot)\)
\(\chi_{1476}(727,\cdot)\)
\(\chi_{1476}(751,\cdot)\)
\(\chi_{1476}(835,\cdot)\)
\(\chi_{1476}(895,\cdot)\)
\(\chi_{1476}(931,\cdot)\)
\(\chi_{1476}(967,\cdot)\)
\(\chi_{1476}(1003,\cdot)\)
\(\chi_{1476}(1051,\cdot)\)
\(\chi_{1476}(1159,\cdot)\)
\(\chi_{1476}(1183,\cdot)\)
\(\chi_{1476}(1195,\cdot)\)
\(\chi_{1476}(1219,\cdot)\)
\(\chi_{1476}(1327,\cdot)\)
\(\chi_{1476}(1375,\cdot)\)
...
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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: |
$\Q(\zeta_{120})$ |
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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 120 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((739,821,1441)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{39}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1476 }(7, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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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)
