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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19435, base_ring=CyclotomicField(1716))
M = H._module
chi = DirichletCharacter(H, M([429,1078,156]))
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gp:[g,chi] = znchar(Mod(8857, 19435))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19435.8857");
| Modulus: | \(19435\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19435\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1716\) |
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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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\(\chi_{19435}(62,\cdot)\)
\(\chi_{19435}(82,\cdot)\)
\(\chi_{19435}(108,\cdot)\)
\(\chi_{19435}(127,\cdot)\)
\(\chi_{19435}(173,\cdot)\)
\(\chi_{19435}(238,\cdot)\)
\(\chi_{19435}(257,\cdot)\)
\(\chi_{19435}(303,\cdot)\)
\(\chi_{19435}(348,\cdot)\)
\(\chi_{19435}(407,\cdot)\)
\(\chi_{19435}(472,\cdot)\)
\(\chi_{19435}(478,\cdot)\)
\(\chi_{19435}(537,\cdot)\)
\(\chi_{19435}(602,\cdot)\)
\(\chi_{19435}(647,\cdot)\)
\(\chi_{19435}(673,\cdot)\)
\(\chi_{19435}(693,\cdot)\)
\(\chi_{19435}(738,\cdot)\)
\(\chi_{19435}(777,\cdot)\)
\(\chi_{19435}(933,\cdot)\)
\(\chi_{19435}(972,\cdot)\)
\(\chi_{19435}(998,\cdot)\)
\(\chi_{19435}(1018,\cdot)\)
\(\chi_{19435}(1083,\cdot)\)
\(\chi_{19435}(1122,\cdot)\)
\(\chi_{19435}(1232,\cdot)\)
\(\chi_{19435}(1258,\cdot)\)
\(\chi_{19435}(1278,\cdot)\)
\(\chi_{19435}(1297,\cdot)\)
\(\chi_{19435}(1317,\cdot)\)
...
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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: |
$\Q(\zeta_{1716})$ |
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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 1716 polynomial (not computed) |
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sage:chi.fixed_field()
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\((11662,11156,5916)\) → \((i,e\left(\frac{49}{78}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 19435 }(8857, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{1716}\right)\) | \(e\left(\frac{175}{1716}\right)\) | \(e\left(\frac{103}{858}\right)\) | \(e\left(\frac{139}{858}\right)\) | \(e\left(\frac{335}{1716}\right)\) | \(e\left(\frac{103}{572}\right)\) | \(e\left(\frac{175}{858}\right)\) | \(e\left(\frac{449}{858}\right)\) | \(e\left(\frac{127}{572}\right)\) | \(e\left(\frac{73}{286}\right)\) |
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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)
