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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65325, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([330,198,385,60]))
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gp:[g,chi] = znchar(Mod(11789, 65325))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65325.11789");
| Modulus: | \(65325\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(65325\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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);
|
\(\chi_{65325}(59,\cdot)\)
\(\chi_{65325}(89,\cdot)\)
\(\chi_{65325}(344,\cdot)\)
\(\chi_{65325}(509,\cdot)\)
\(\chi_{65325}(734,\cdot)\)
\(\chi_{65325}(1064,\cdot)\)
\(\chi_{65325}(1094,\cdot)\)
\(\chi_{65325}(1514,\cdot)\)
\(\chi_{65325}(2069,\cdot)\)
\(\chi_{65325}(4379,\cdot)\)
\(\chi_{65325}(5384,\cdot)\)
\(\chi_{65325}(5519,\cdot)\)
\(\chi_{65325}(6389,\cdot)\)
\(\chi_{65325}(6494,\cdot)\)
\(\chi_{65325}(7529,\cdot)\)
\(\chi_{65325}(8054,\cdot)\)
\(\chi_{65325}(8504,\cdot)\)
\(\chi_{65325}(8534,\cdot)\)
\(\chi_{65325}(9059,\cdot)\)
\(\chi_{65325}(9509,\cdot)\)
\(\chi_{65325}(10064,\cdot)\)
\(\chi_{65325}(10394,\cdot)\)
\(\chi_{65325}(10784,\cdot)\)
\(\chi_{65325}(11069,\cdot)\)
\(\chi_{65325}(11204,\cdot)\)
\(\chi_{65325}(11564,\cdot)\)
\(\chi_{65325}(11789,\cdot)\)
\(\chi_{65325}(12209,\cdot)\)
\(\chi_{65325}(12404,\cdot)\)
\(\chi_{65325}(12569,\cdot)\)
...
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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 };
\((21776,5227,10051,28276)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 65325 }(11789, a) \) |
\(1\) | \(1\) | \(e\left(\frac{313}{660}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{493}{660}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{149}{660}\right)\) | \(e\left(\frac{73}{330}\right)\) |
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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)
