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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2277, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([220,66,150]))
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gp:[g,chi] = znchar(Mod(1159, 2277))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2277.1159");
| Modulus: | \(2277\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2277\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(165\) |
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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);
|
\(\chi_{2277}(4,\cdot)\)
\(\chi_{2277}(16,\cdot)\)
\(\chi_{2277}(25,\cdot)\)
\(\chi_{2277}(31,\cdot)\)
\(\chi_{2277}(49,\cdot)\)
\(\chi_{2277}(58,\cdot)\)
\(\chi_{2277}(124,\cdot)\)
\(\chi_{2277}(169,\cdot)\)
\(\chi_{2277}(196,\cdot)\)
\(\chi_{2277}(202,\cdot)\)
\(\chi_{2277}(223,\cdot)\)
\(\chi_{2277}(256,\cdot)\)
\(\chi_{2277}(301,\cdot)\)
\(\chi_{2277}(328,\cdot)\)
\(\chi_{2277}(394,\cdot)\)
\(\chi_{2277}(400,\cdot)\)
\(\chi_{2277}(427,\cdot)\)
\(\chi_{2277}(445,\cdot)\)
\(\chi_{2277}(466,\cdot)\)
\(\chi_{2277}(499,\cdot)\)
\(\chi_{2277}(565,\cdot)\)
\(\chi_{2277}(610,\cdot)\)
\(\chi_{2277}(625,\cdot)\)
\(\chi_{2277}(652,\cdot)\)
\(\chi_{2277}(742,\cdot)\)
\(\chi_{2277}(763,\cdot)\)
\(\chi_{2277}(790,\cdot)\)
\(\chi_{2277}(808,\cdot)\)
\(\chi_{2277}(817,\cdot)\)
\(\chi_{2277}(823,\cdot)\)
...
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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 };
\((254,1036,1684)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{5}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 2277 }(1159, a) \) |
\(1\) | \(1\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) |
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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)
