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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4740, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,78,117,16]))
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gp:[g,chi] = znchar(Mod(83, 4740))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4740.83");
| Modulus: | \(4740\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4740\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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;
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| 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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\(\chi_{4740}(83,\cdot)\)
\(\chi_{4740}(167,\cdot)\)
\(\chi_{4740}(203,\cdot)\)
\(\chi_{4740}(263,\cdot)\)
\(\chi_{4740}(287,\cdot)\)
\(\chi_{4740}(347,\cdot)\)
\(\chi_{4740}(467,\cdot)\)
\(\chi_{4740}(683,\cdot)\)
\(\chi_{4740}(743,\cdot)\)
\(\chi_{4740}(803,\cdot)\)
\(\chi_{4740}(863,\cdot)\)
\(\chi_{4740}(1043,\cdot)\)
\(\chi_{4740}(1067,\cdot)\)
\(\chi_{4740}(1103,\cdot)\)
\(\chi_{4740}(1187,\cdot)\)
\(\chi_{4740}(1283,\cdot)\)
\(\chi_{4740}(1427,\cdot)\)
\(\chi_{4740}(1703,\cdot)\)
\(\chi_{4740}(1763,\cdot)\)
\(\chi_{4740}(1787,\cdot)\)
\(\chi_{4740}(1907,\cdot)\)
\(\chi_{4740}(2063,\cdot)\)
\(\chi_{4740}(2183,\cdot)\)
\(\chi_{4740}(2243,\cdot)\)
\(\chi_{4740}(2327,\cdot)\)
\(\chi_{4740}(2363,\cdot)\)
\(\chi_{4740}(2627,\cdot)\)
\(\chi_{4740}(2807,\cdot)\)
\(\chi_{4740}(2927,\cdot)\)
\(\chi_{4740}(2963,\cdot)\)
...
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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: |
$\Q(\zeta_{156})$ |
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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 156 polynomial (not computed) |
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sage:chi.fixed_field()
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\((2371,3161,1897,1741)\) → \((-1,-1,-i,e\left(\frac{4}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4740 }(83, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{5}{26}\right)\) |
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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)
