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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2465, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([84,49,72]))
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gp:[g,chi] = znchar(Mod(1898, 2465))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2465.1898");
| Modulus: | \(2465\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2465\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(112\) |
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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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\(\chi_{2465}(22,\cdot)\)
\(\chi_{2465}(122,\cdot)\)
\(\chi_{2465}(167,\cdot)\)
\(\chi_{2465}(207,\cdot)\)
\(\chi_{2465}(267,\cdot)\)
\(\chi_{2465}(283,\cdot)\)
\(\chi_{2465}(352,\cdot)\)
\(\chi_{2465}(448,\cdot)\)
\(\chi_{2465}(618,\cdot)\)
\(\chi_{2465}(673,\cdot)\)
\(\chi_{2465}(702,\cdot)\)
\(\chi_{2465}(738,\cdot)\)
\(\chi_{2465}(758,\cdot)\)
\(\chi_{2465}(787,\cdot)\)
\(\chi_{2465}(788,\cdot)\)
\(\chi_{2465}(847,\cdot)\)
\(\chi_{2465}(908,\cdot)\)
\(\chi_{2465}(932,\cdot)\)
\(\chi_{2465}(963,\cdot)\)
\(\chi_{2465}(1048,\cdot)\)
\(\chi_{2465}(1057,\cdot)\)
\(\chi_{2465}(1078,\cdot)\)
\(\chi_{2465}(1202,\cdot)\)
\(\chi_{2465}(1227,\cdot)\)
\(\chi_{2465}(1298,\cdot)\)
\(\chi_{2465}(1372,\cdot)\)
\(\chi_{2465}(1397,\cdot)\)
\(\chi_{2465}(1542,\cdot)\)
\(\chi_{2465}(1588,\cdot)\)
\(\chi_{2465}(1608,\cdot)\)
...
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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 };
\((987,581,1191)\) → \((-i,e\left(\frac{7}{16}\right),e\left(\frac{9}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2465 }(1898, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{4}{7}\right)\) |
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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)
