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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8779, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([96]))
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gp:[g,chi] = znchar(Mod(1560, 8779))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8779.1560");
| Modulus: | \(8779\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8779\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(77\) |
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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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\(\chi_{8779}(161,\cdot)\)
\(\chi_{8779}(253,\cdot)\)
\(\chi_{8779}(347,\cdot)\)
\(\chi_{8779}(585,\cdot)\)
\(\chi_{8779}(686,\cdot)\)
\(\chi_{8779}(777,\cdot)\)
\(\chi_{8779}(1009,\cdot)\)
\(\chi_{8779}(1078,\cdot)\)
\(\chi_{8779}(1246,\cdot)\)
\(\chi_{8779}(1560,\cdot)\)
\(\chi_{8779}(1591,\cdot)\)
\(\chi_{8779}(1648,\cdot)\)
\(\chi_{8779}(1694,\cdot)\)
\(\chi_{8779}(1817,\cdot)\)
\(\chi_{8779}(1919,\cdot)\)
\(\chi_{8779}(1958,\cdot)\)
\(\chi_{8779}(2072,\cdot)\)
\(\chi_{8779}(2191,\cdot)\)
\(\chi_{8779}(2295,\cdot)\)
\(\chi_{8779}(2452,\cdot)\)
\(\chi_{8779}(2556,\cdot)\)
\(\chi_{8779}(2599,\cdot)\)
\(\chi_{8779}(2662,\cdot)\)
\(\chi_{8779}(2830,\cdot)\)
\(\chi_{8779}(2929,\cdot)\)
\(\chi_{8779}(3186,\cdot)\)
\(\chi_{8779}(3193,\cdot)\)
\(\chi_{8779}(3256,\cdot)\)
\(\chi_{8779}(3387,\cdot)\)
\(\chi_{8779}(3443,\cdot)\)
...
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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 };
\(11\) → \(e\left(\frac{48}{77}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8779 }(1560, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{48}{77}\right)\) |
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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)
