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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1304, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([81,81,67]))
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gp:[g,chi] = znchar(Mod(619, 1304))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1304.619");
| Modulus: | \(1304\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1304\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(162\) |
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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: | even |
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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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\(\chi_{1304}(3,\cdot)\)
\(\chi_{1304}(11,\cdot)\)
\(\chi_{1304}(19,\cdot)\)
\(\chi_{1304}(67,\cdot)\)
\(\chi_{1304}(75,\cdot)\)
\(\chi_{1304}(107,\cdot)\)
\(\chi_{1304}(139,\cdot)\)
\(\chi_{1304}(147,\cdot)\)
\(\chi_{1304}(195,\cdot)\)
\(\chi_{1304}(235,\cdot)\)
\(\chi_{1304}(243,\cdot)\)
\(\chi_{1304}(275,\cdot)\)
\(\chi_{1304}(283,\cdot)\)
\(\chi_{1304}(291,\cdot)\)
\(\chi_{1304}(355,\cdot)\)
\(\chi_{1304}(371,\cdot)\)
\(\chi_{1304}(427,\cdot)\)
\(\chi_{1304}(435,\cdot)\)
\(\chi_{1304}(443,\cdot)\)
\(\chi_{1304}(475,\cdot)\)
\(\chi_{1304}(491,\cdot)\)
\(\chi_{1304}(507,\cdot)\)
\(\chi_{1304}(531,\cdot)\)
\(\chi_{1304}(539,\cdot)\)
\(\chi_{1304}(555,\cdot)\)
\(\chi_{1304}(571,\cdot)\)
\(\chi_{1304}(595,\cdot)\)
\(\chi_{1304}(603,\cdot)\)
\(\chi_{1304}(611,\cdot)\)
\(\chi_{1304}(619,\cdot)\)
...
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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 };
\((327,653,817)\) → \((-1,-1,e\left(\frac{67}{162}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1304 }(619, a) \) |
\(1\) | \(1\) | \(e\left(\frac{125}{162}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{44}{81}\right)\) | \(e\left(\frac{71}{162}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{77}{162}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{113}{162}\right)\) | \(e\left(\frac{25}{54}\right)\) |
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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)
