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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12544, base_ring=CyclotomicField(448))
M = H._module
chi = DirichletCharacter(H, M([224,245,128]))
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gp:[g,chi] = znchar(Mod(771, 12544))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12544.771");
| Modulus: | \(12544\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12544\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(448\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{12544}(43,\cdot)\)
\(\chi_{12544}(155,\cdot)\)
\(\chi_{12544}(211,\cdot)\)
\(\chi_{12544}(267,\cdot)\)
\(\chi_{12544}(323,\cdot)\)
\(\chi_{12544}(379,\cdot)\)
\(\chi_{12544}(435,\cdot)\)
\(\chi_{12544}(547,\cdot)\)
\(\chi_{12544}(603,\cdot)\)
\(\chi_{12544}(659,\cdot)\)
\(\chi_{12544}(715,\cdot)\)
\(\chi_{12544}(771,\cdot)\)
\(\chi_{12544}(827,\cdot)\)
\(\chi_{12544}(939,\cdot)\)
\(\chi_{12544}(995,\cdot)\)
\(\chi_{12544}(1051,\cdot)\)
\(\chi_{12544}(1107,\cdot)\)
\(\chi_{12544}(1163,\cdot)\)
\(\chi_{12544}(1219,\cdot)\)
\(\chi_{12544}(1331,\cdot)\)
\(\chi_{12544}(1387,\cdot)\)
\(\chi_{12544}(1443,\cdot)\)
\(\chi_{12544}(1499,\cdot)\)
\(\chi_{12544}(1555,\cdot)\)
\(\chi_{12544}(1611,\cdot)\)
\(\chi_{12544}(1723,\cdot)\)
\(\chi_{12544}(1779,\cdot)\)
\(\chi_{12544}(1835,\cdot)\)
\(\chi_{12544}(1891,\cdot)\)
\(\chi_{12544}(1947,\cdot)\)
...
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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 };
\((4607,3333,4609)\) → \((-1,e\left(\frac{35}{64}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 12544 }(771, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{415}{448}\right)\) | \(e\left(\frac{373}{448}\right)\) | \(e\left(\frac{191}{224}\right)\) | \(e\left(\frac{185}{448}\right)\) | \(e\left(\frac{59}{448}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{224}\right)\) | \(e\left(\frac{149}{224}\right)\) |
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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)
