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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([150,291,125]))
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gp:[g,chi] = znchar(Mod(2047, 6500))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6500.2047");
| 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;
|
| 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: | 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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\(\chi_{6500}(63,\cdot)\)
\(\chi_{6500}(67,\cdot)\)
\(\chi_{6500}(163,\cdot)\)
\(\chi_{6500}(227,\cdot)\)
\(\chi_{6500}(323,\cdot)\)
\(\chi_{6500}(327,\cdot)\)
\(\chi_{6500}(423,\cdot)\)
\(\chi_{6500}(487,\cdot)\)
\(\chi_{6500}(583,\cdot)\)
\(\chi_{6500}(587,\cdot)\)
\(\chi_{6500}(683,\cdot)\)
\(\chi_{6500}(747,\cdot)\)
\(\chi_{6500}(847,\cdot)\)
\(\chi_{6500}(1103,\cdot)\)
\(\chi_{6500}(1203,\cdot)\)
\(\chi_{6500}(1267,\cdot)\)
\(\chi_{6500}(1363,\cdot)\)
\(\chi_{6500}(1367,\cdot)\)
\(\chi_{6500}(1463,\cdot)\)
\(\chi_{6500}(1527,\cdot)\)
\(\chi_{6500}(1623,\cdot)\)
\(\chi_{6500}(1627,\cdot)\)
\(\chi_{6500}(1723,\cdot)\)
\(\chi_{6500}(1787,\cdot)\)
\(\chi_{6500}(1883,\cdot)\)
\(\chi_{6500}(1887,\cdot)\)
\(\chi_{6500}(1983,\cdot)\)
\(\chi_{6500}(2047,\cdot)\)
\(\chi_{6500}(2147,\cdot)\)
\(\chi_{6500}(2403,\cdot)\)
...
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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));
|
| Fixed field: |
Number field defined by a degree 300 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3251,5877,5501)\) → \((-1,e\left(\frac{97}{100}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6500 }(2047, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{287}{300}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{41}{300}\right)\) | \(e\left(\frac{193}{300}\right)\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{121}{150}\right)\) |
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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)
