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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2401, base_ring=CyclotomicField(2058))
M = H._module
chi = DirichletCharacter(H, M([1822]))
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gp:[g,chi] = znchar(Mod(25, 2401))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2401.25");
| Modulus: | \(2401\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2401\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1029\) |
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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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\(\chi_{2401}(2,\cdot)\)
\(\chi_{2401}(4,\cdot)\)
\(\chi_{2401}(9,\cdot)\)
\(\chi_{2401}(11,\cdot)\)
\(\chi_{2401}(16,\cdot)\)
\(\chi_{2401}(23,\cdot)\)
\(\chi_{2401}(25,\cdot)\)
\(\chi_{2401}(32,\cdot)\)
\(\chi_{2401}(37,\cdot)\)
\(\chi_{2401}(39,\cdot)\)
\(\chi_{2401}(44,\cdot)\)
\(\chi_{2401}(46,\cdot)\)
\(\chi_{2401}(51,\cdot)\)
\(\chi_{2401}(53,\cdot)\)
\(\chi_{2401}(58,\cdot)\)
\(\chi_{2401}(60,\cdot)\)
\(\chi_{2401}(65,\cdot)\)
\(\chi_{2401}(72,\cdot)\)
\(\chi_{2401}(74,\cdot)\)
\(\chi_{2401}(81,\cdot)\)
\(\chi_{2401}(86,\cdot)\)
\(\chi_{2401}(88,\cdot)\)
\(\chi_{2401}(93,\cdot)\)
\(\chi_{2401}(95,\cdot)\)
\(\chi_{2401}(100,\cdot)\)
\(\chi_{2401}(102,\cdot)\)
\(\chi_{2401}(107,\cdot)\)
\(\chi_{2401}(109,\cdot)\)
\(\chi_{2401}(114,\cdot)\)
\(\chi_{2401}(121,\cdot)\)
...
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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: |
$\Q(\zeta_{1029})$ |
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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 1029 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{911}{1029}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2401 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{628}{1029}\right)\) | \(e\left(\frac{911}{1029}\right)\) | \(e\left(\frac{227}{1029}\right)\) | \(e\left(\frac{547}{1029}\right)\) | \(e\left(\frac{170}{343}\right)\) | \(e\left(\frac{285}{343}\right)\) | \(e\left(\frac{793}{1029}\right)\) | \(e\left(\frac{146}{1029}\right)\) | \(e\left(\frac{593}{1029}\right)\) | \(e\left(\frac{109}{1029}\right)\) |
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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)
