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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1265, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,66,75]))
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gp:[g,chi] = znchar(Mod(801, 1265))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1265.801");
| Modulus: | \(1265\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(253\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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: | no, induced from \(\chi_{253}(42,\cdot)\) |
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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);
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\(\chi_{1265}(86,\cdot)\)
\(\chi_{1265}(126,\cdot)\)
\(\chi_{1265}(136,\cdot)\)
\(\chi_{1265}(181,\cdot)\)
\(\chi_{1265}(191,\cdot)\)
\(\chi_{1265}(201,\cdot)\)
\(\chi_{1265}(251,\cdot)\)
\(\chi_{1265}(291,\cdot)\)
\(\chi_{1265}(306,\cdot)\)
\(\chi_{1265}(356,\cdot)\)
\(\chi_{1265}(366,\cdot)\)
\(\chi_{1265}(401,\cdot)\)
\(\chi_{1265}(411,\cdot)\)
\(\chi_{1265}(421,\cdot)\)
\(\chi_{1265}(456,\cdot)\)
\(\chi_{1265}(471,\cdot)\)
\(\chi_{1265}(511,\cdot)\)
\(\chi_{1265}(521,\cdot)\)
\(\chi_{1265}(526,\cdot)\)
\(\chi_{1265}(566,\cdot)\)
\(\chi_{1265}(586,\cdot)\)
\(\chi_{1265}(631,\cdot)\)
\(\chi_{1265}(636,\cdot)\)
\(\chi_{1265}(641,\cdot)\)
\(\chi_{1265}(686,\cdot)\)
\(\chi_{1265}(741,\cdot)\)
\(\chi_{1265}(746,\cdot)\)
\(\chi_{1265}(751,\cdot)\)
\(\chi_{1265}(796,\cdot)\)
\(\chi_{1265}(801,\cdot)\)
...
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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 };
\((507,1036,166)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1265 }(801, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) |
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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)
