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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6500, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,216,175]))
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gp:[g,chi] = znchar(Mod(1571, 6500))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6500.1571");
| Modulus: | \(6500\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6500\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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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\(\chi_{6500}(11,\cdot)\)
\(\chi_{6500}(71,\cdot)\)
\(\chi_{6500}(111,\cdot)\)
\(\chi_{6500}(171,\cdot)\)
\(\chi_{6500}(271,\cdot)\)
\(\chi_{6500}(331,\cdot)\)
\(\chi_{6500}(371,\cdot)\)
\(\chi_{6500}(431,\cdot)\)
\(\chi_{6500}(531,\cdot)\)
\(\chi_{6500}(591,\cdot)\)
\(\chi_{6500}(631,\cdot)\)
\(\chi_{6500}(691,\cdot)\)
\(\chi_{6500}(791,\cdot)\)
\(\chi_{6500}(891,\cdot)\)
\(\chi_{6500}(1111,\cdot)\)
\(\chi_{6500}(1211,\cdot)\)
\(\chi_{6500}(1311,\cdot)\)
\(\chi_{6500}(1371,\cdot)\)
\(\chi_{6500}(1411,\cdot)\)
\(\chi_{6500}(1471,\cdot)\)
\(\chi_{6500}(1571,\cdot)\)
\(\chi_{6500}(1631,\cdot)\)
\(\chi_{6500}(1671,\cdot)\)
\(\chi_{6500}(1731,\cdot)\)
\(\chi_{6500}(1831,\cdot)\)
\(\chi_{6500}(1891,\cdot)\)
\(\chi_{6500}(1931,\cdot)\)
\(\chi_{6500}(1991,\cdot)\)
\(\chi_{6500}(2091,\cdot)\)
\(\chi_{6500}(2191,\cdot)\)
...
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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: |
$\Q(\zeta_{300})$ |
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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 300 polynomial (not computed) |
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sage:chi.fixed_field()
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\((3251,5877,5501)\) → \((-1,e\left(\frac{18}{25}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6500 }(1571, a) \) |
\(1\) | \(1\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{73}{75}\right)\) |
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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)
