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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65869, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([295,232]))
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gp:[g,chi] = znchar(Mod(15696, 65869))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65869.15696");
| Modulus: | \(65869\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(65869\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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);
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\(\chi_{65869}(76,\cdot)\)
\(\chi_{65869}(1007,\cdot)\)
\(\chi_{65869}(1405,\cdot)\)
\(\chi_{65869}(1850,\cdot)\)
\(\chi_{65869}(5321,\cdot)\)
\(\chi_{65869}(5365,\cdot)\)
\(\chi_{65869}(5567,\cdot)\)
\(\chi_{65869}(5788,\cdot)\)
\(\chi_{65869}(5997,\cdot)\)
\(\chi_{65869}(6129,\cdot)\)
\(\chi_{65869}(8844,\cdot)\)
\(\chi_{65869}(9209,\cdot)\)
\(\chi_{65869}(9462,\cdot)\)
\(\chi_{65869}(9778,\cdot)\)
\(\chi_{65869}(10250,\cdot)\)
\(\chi_{65869}(10365,\cdot)\)
\(\chi_{65869}(10741,\cdot)\)
\(\chi_{65869}(10822,\cdot)\)
\(\chi_{65869}(10920,\cdot)\)
\(\chi_{65869}(12007,\cdot)\)
\(\chi_{65869}(12819,\cdot)\)
\(\chi_{65869}(12952,\cdot)\)
\(\chi_{65869}(14594,\cdot)\)
\(\chi_{65869}(15610,\cdot)\)
\(\chi_{65869}(15696,\cdot)\)
\(\chi_{65869}(15975,\cdot)\)
\(\chi_{65869}(18210,\cdot)\)
\(\chi_{65869}(20580,\cdot)\)
\(\chi_{65869}(20755,\cdot)\)
\(\chi_{65869}(22681,\cdot)\)
...
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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 };
\((64877,996)\) → \((e\left(\frac{59}{66}\right),e\left(\frac{116}{165}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 65869 }(15696, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) |
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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)
