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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27783, base_ring=CyclotomicField(2646))
M = H._module
chi = DirichletCharacter(H, M([245,432]))
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gp:[g,chi] = znchar(Mod(680, 27783))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27783.680");
| Modulus: | \(27783\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(27783\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2646\) |
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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);
|
\(\chi_{27783}(29,\cdot)\)
\(\chi_{27783}(92,\cdot)\)
\(\chi_{27783}(113,\cdot)\)
\(\chi_{27783}(155,\cdot)\)
\(\chi_{27783}(176,\cdot)\)
\(\chi_{27783}(218,\cdot)\)
\(\chi_{27783}(239,\cdot)\)
\(\chi_{27783}(281,\cdot)\)
\(\chi_{27783}(302,\cdot)\)
\(\chi_{27783}(365,\cdot)\)
\(\chi_{27783}(407,\cdot)\)
\(\chi_{27783}(428,\cdot)\)
\(\chi_{27783}(470,\cdot)\)
\(\chi_{27783}(533,\cdot)\)
\(\chi_{27783}(554,\cdot)\)
\(\chi_{27783}(596,\cdot)\)
\(\chi_{27783}(617,\cdot)\)
\(\chi_{27783}(659,\cdot)\)
\(\chi_{27783}(680,\cdot)\)
\(\chi_{27783}(722,\cdot)\)
\(\chi_{27783}(743,\cdot)\)
\(\chi_{27783}(806,\cdot)\)
\(\chi_{27783}(848,\cdot)\)
\(\chi_{27783}(869,\cdot)\)
\(\chi_{27783}(911,\cdot)\)
\(\chi_{27783}(974,\cdot)\)
\(\chi_{27783}(995,\cdot)\)
\(\chi_{27783}(1037,\cdot)\)
\(\chi_{27783}(1058,\cdot)\)
\(\chi_{27783}(1100,\cdot)\)
...
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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 };
\((21953,11665)\) → \((e\left(\frac{5}{54}\right),e\left(\frac{8}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 27783 }(680, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2027}{2646}\right)\) | \(e\left(\frac{704}{1323}\right)\) | \(e\left(\frac{2287}{2646}\right)\) | \(e\left(\frac{263}{882}\right)\) | \(e\left(\frac{278}{441}\right)\) | \(e\left(\frac{431}{2646}\right)\) | \(e\left(\frac{359}{1323}\right)\) | \(e\left(\frac{85}{1323}\right)\) | \(e\left(\frac{121}{882}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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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)
