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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79632, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,117,0,52,50]))
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gp:[g,chi] = znchar(Mod(35821, 79632))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("79632.35821");
| Modulus: | \(79632\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8848\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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 \(\chi_{8848}(429,\cdot)\) |
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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);
|
\(\chi_{79632}(37,\cdot)\)
\(\chi_{79632}(613,\cdot)\)
\(\chi_{79632}(3061,\cdot)\)
\(\chi_{79632}(6157,\cdot)\)
\(\chi_{79632}(7669,\cdot)\)
\(\chi_{79632}(8101,\cdot)\)
\(\chi_{79632}(9613,\cdot)\)
\(\chi_{79632}(9685,\cdot)\)
\(\chi_{79632}(10621,\cdot)\)
\(\chi_{79632}(12205,\cdot)\)
\(\chi_{79632}(17173,\cdot)\)
\(\chi_{79632}(17749,\cdot)\)
\(\chi_{79632}(20701,\cdot)\)
\(\chi_{79632}(20773,\cdot)\)
\(\chi_{79632}(21709,\cdot)\)
\(\chi_{79632}(22285,\cdot)\)
\(\chi_{79632}(24301,\cdot)\)
\(\chi_{79632}(25309,\cdot)\)
\(\chi_{79632}(25741,\cdot)\)
\(\chi_{79632}(27253,\cdot)\)
\(\chi_{79632}(27325,\cdot)\)
\(\chi_{79632}(27757,\cdot)\)
\(\chi_{79632}(35821,\cdot)\)
\(\chi_{79632}(37909,\cdot)\)
\(\chi_{79632}(39853,\cdot)\)
\(\chi_{79632}(40429,\cdot)\)
\(\chi_{79632}(42877,\cdot)\)
\(\chi_{79632}(45973,\cdot)\)
\(\chi_{79632}(47485,\cdot)\)
\(\chi_{79632}(47917,\cdot)\)
...
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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 };
\((69679,19909,8849,22753,29233)\) → \((1,-i,1,e\left(\frac{1}{3}\right),e\left(\frac{25}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 79632 }(35821, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(-1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{79}{156}\right)\) |
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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)
