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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8325, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([60,153,85]))
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gp:[g,chi] = znchar(Mod(2497, 8325))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8325.2497");
| Modulus: | \(8325\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8325\) |
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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;
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| 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);
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| 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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\(\chi_{8325}(13,\cdot)\)
\(\chi_{8325}(52,\cdot)\)
\(\chi_{8325}(133,\cdot)\)
\(\chi_{8325}(187,\cdot)\)
\(\chi_{8325}(313,\cdot)\)
\(\chi_{8325}(463,\cdot)\)
\(\chi_{8325}(1327,\cdot)\)
\(\chi_{8325}(1537,\cdot)\)
\(\chi_{8325}(1633,\cdot)\)
\(\chi_{8325}(1663,\cdot)\)
\(\chi_{8325}(1678,\cdot)\)
\(\chi_{8325}(1717,\cdot)\)
\(\chi_{8325}(1798,\cdot)\)
\(\chi_{8325}(1852,\cdot)\)
\(\chi_{8325}(1978,\cdot)\)
\(\chi_{8325}(2128,\cdot)\)
\(\chi_{8325}(2497,\cdot)\)
\(\chi_{8325}(2947,\cdot)\)
\(\chi_{8325}(2992,\cdot)\)
\(\chi_{8325}(3202,\cdot)\)
\(\chi_{8325}(3298,\cdot)\)
\(\chi_{8325}(3328,\cdot)\)
\(\chi_{8325}(3463,\cdot)\)
\(\chi_{8325}(3517,\cdot)\)
\(\chi_{8325}(4162,\cdot)\)
\(\chi_{8325}(4612,\cdot)\)
\(\chi_{8325}(4867,\cdot)\)
\(\chi_{8325}(4963,\cdot)\)
\(\chi_{8325}(5008,\cdot)\)
\(\chi_{8325}(5047,\cdot)\)
...
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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));
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| Fixed field: |
Number field defined by a degree 180 polynomial (not computed) |
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sage:chi.fixed_field()
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\((3701,7327,5626)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right),e\left(\frac{17}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 8325 }(2497, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{149}{180}\right)\) |
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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)
