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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2301, base_ring=CyclotomicField(348))
M = H._module
chi = DirichletCharacter(H, M([174,203,342]))
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gp:[g,chi] = znchar(Mod(89, 2301))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2301.89");
| Modulus: | \(2301\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2301\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(348\) |
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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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\(\chi_{2301}(2,\cdot)\)
\(\chi_{2301}(11,\cdot)\)
\(\chi_{2301}(32,\cdot)\)
\(\chi_{2301}(50,\cdot)\)
\(\chi_{2301}(89,\cdot)\)
\(\chi_{2301}(98,\cdot)\)
\(\chi_{2301}(128,\cdot)\)
\(\chi_{2301}(149,\cdot)\)
\(\chi_{2301}(158,\cdot)\)
\(\chi_{2301}(188,\cdot)\)
\(\chi_{2301}(215,\cdot)\)
\(\chi_{2301}(227,\cdot)\)
\(\chi_{2301}(254,\cdot)\)
\(\chi_{2301}(266,\cdot)\)
\(\chi_{2301}(275,\cdot)\)
\(\chi_{2301}(305,\cdot)\)
\(\chi_{2301}(332,\cdot)\)
\(\chi_{2301}(362,\cdot)\)
\(\chi_{2301}(392,\cdot)\)
\(\chi_{2301}(401,\cdot)\)
\(\chi_{2301}(410,\cdot)\)
\(\chi_{2301}(431,\cdot)\)
\(\chi_{2301}(509,\cdot)\)
\(\chi_{2301}(527,\cdot)\)
\(\chi_{2301}(539,\cdot)\)
\(\chi_{2301}(578,\cdot)\)
\(\chi_{2301}(587,\cdot)\)
\(\chi_{2301}(596,\cdot)\)
\(\chi_{2301}(644,\cdot)\)
\(\chi_{2301}(683,\cdot)\)
...
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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: |
$\Q(\zeta_{348})$ |
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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 348 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((1535,886,2185)\) → \((-1,e\left(\frac{7}{12}\right),e\left(\frac{57}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2301 }(89, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{348}\right)\) | \(e\left(\frac{23}{174}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{37}{348}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{53}{348}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{23}{87}\right)\) | \(e\left(\frac{85}{87}\right)\) |
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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)
