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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1992, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([41,0,0,58]))
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gp:[g,chi] = znchar(Mod(151, 1992))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1992.151");
| Modulus: | \(1992\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(332\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(82\) |
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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: | no, induced from \(\chi_{332}(151,\cdot)\) |
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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: | no |
| 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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\(\chi_{1992}(7,\cdot)\)
\(\chi_{1992}(31,\cdot)\)
\(\chi_{1992}(127,\cdot)\)
\(\chi_{1992}(151,\cdot)\)
\(\chi_{1992}(175,\cdot)\)
\(\chi_{1992}(199,\cdot)\)
\(\chi_{1992}(247,\cdot)\)
\(\chi_{1992}(319,\cdot)\)
\(\chi_{1992}(343,\cdot)\)
\(\chi_{1992}(391,\cdot)\)
\(\chi_{1992}(463,\cdot)\)
\(\chi_{1992}(535,\cdot)\)
\(\chi_{1992}(559,\cdot)\)
\(\chi_{1992}(607,\cdot)\)
\(\chi_{1992}(727,\cdot)\)
\(\chi_{1992}(751,\cdot)\)
\(\chi_{1992}(775,\cdot)\)
\(\chi_{1992}(847,\cdot)\)
\(\chi_{1992}(871,\cdot)\)
\(\chi_{1992}(895,\cdot)\)
\(\chi_{1992}(943,\cdot)\)
\(\chi_{1992}(991,\cdot)\)
\(\chi_{1992}(1183,\cdot)\)
\(\chi_{1992}(1231,\cdot)\)
\(\chi_{1992}(1255,\cdot)\)
\(\chi_{1992}(1351,\cdot)\)
\(\chi_{1992}(1423,\cdot)\)
\(\chi_{1992}(1447,\cdot)\)
\(\chi_{1992}(1519,\cdot)\)
\(\chi_{1992}(1543,\cdot)\)
...
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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 };
\((1495,997,665,1081)\) → \((-1,1,1,e\left(\frac{29}{41}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1992 }(151, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{13}{82}\right)\) | \(e\left(\frac{39}{82}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{77}{82}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{31}{82}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
