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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([1607]))
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gp:[g,chi] = znchar(Mod(110, 2401))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2401.110");
| 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: | \(2058\) |
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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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\(\chi_{2401}(3,\cdot)\)
\(\chi_{2401}(5,\cdot)\)
\(\chi_{2401}(10,\cdot)\)
\(\chi_{2401}(12,\cdot)\)
\(\chi_{2401}(17,\cdot)\)
\(\chi_{2401}(24,\cdot)\)
\(\chi_{2401}(26,\cdot)\)
\(\chi_{2401}(33,\cdot)\)
\(\chi_{2401}(38,\cdot)\)
\(\chi_{2401}(40,\cdot)\)
\(\chi_{2401}(45,\cdot)\)
\(\chi_{2401}(47,\cdot)\)
\(\chi_{2401}(52,\cdot)\)
\(\chi_{2401}(54,\cdot)\)
\(\chi_{2401}(59,\cdot)\)
\(\chi_{2401}(61,\cdot)\)
\(\chi_{2401}(66,\cdot)\)
\(\chi_{2401}(73,\cdot)\)
\(\chi_{2401}(75,\cdot)\)
\(\chi_{2401}(82,\cdot)\)
\(\chi_{2401}(87,\cdot)\)
\(\chi_{2401}(89,\cdot)\)
\(\chi_{2401}(94,\cdot)\)
\(\chi_{2401}(96,\cdot)\)
\(\chi_{2401}(101,\cdot)\)
\(\chi_{2401}(103,\cdot)\)
\(\chi_{2401}(108,\cdot)\)
\(\chi_{2401}(110,\cdot)\)
\(\chi_{2401}(115,\cdot)\)
\(\chi_{2401}(122,\cdot)\)
...
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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 2058 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{1607}{2058}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2401 }(110, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{206}{1029}\right)\) | \(e\left(\frac{1607}{2058}\right)\) | \(e\left(\frac{412}{1029}\right)\) | \(e\left(\frac{739}{2058}\right)\) | \(e\left(\frac{673}{686}\right)\) | \(e\left(\frac{206}{343}\right)\) | \(e\left(\frac{578}{1029}\right)\) | \(e\left(\frac{1151}{2058}\right)\) | \(e\left(\frac{283}{1029}\right)\) | \(e\left(\frac{373}{2058}\right)\) |
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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)
