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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1664, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,39,88]))
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gp:[g,chi] = znchar(Mod(1515, 1664))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1664.1515");
| Modulus: | \(1664\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1664\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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);
|
\(\chi_{1664}(11,\cdot)\)
\(\chi_{1664}(19,\cdot)\)
\(\chi_{1664}(59,\cdot)\)
\(\chi_{1664}(67,\cdot)\)
\(\chi_{1664}(219,\cdot)\)
\(\chi_{1664}(227,\cdot)\)
\(\chi_{1664}(267,\cdot)\)
\(\chi_{1664}(275,\cdot)\)
\(\chi_{1664}(427,\cdot)\)
\(\chi_{1664}(435,\cdot)\)
\(\chi_{1664}(475,\cdot)\)
\(\chi_{1664}(483,\cdot)\)
\(\chi_{1664}(635,\cdot)\)
\(\chi_{1664}(643,\cdot)\)
\(\chi_{1664}(683,\cdot)\)
\(\chi_{1664}(691,\cdot)\)
\(\chi_{1664}(843,\cdot)\)
\(\chi_{1664}(851,\cdot)\)
\(\chi_{1664}(891,\cdot)\)
\(\chi_{1664}(899,\cdot)\)
\(\chi_{1664}(1051,\cdot)\)
\(\chi_{1664}(1059,\cdot)\)
\(\chi_{1664}(1099,\cdot)\)
\(\chi_{1664}(1107,\cdot)\)
\(\chi_{1664}(1259,\cdot)\)
\(\chi_{1664}(1267,\cdot)\)
\(\chi_{1664}(1307,\cdot)\)
\(\chi_{1664}(1315,\cdot)\)
\(\chi_{1664}(1467,\cdot)\)
\(\chi_{1664}(1475,\cdot)\)
...
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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 };
\((1535,261,769)\) → \((-1,e\left(\frac{13}{32}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1664 }(1515, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{17}{48}\right)\) |
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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)
