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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20664, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([60,60,80,40,3]))
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gp:[g,chi] = znchar(Mod(15955, 20664))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20664.15955");
| Modulus: | \(20664\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20664\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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);
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\(\chi_{20664}(67,\cdot)\)
\(\chi_{20664}(835,\cdot)\)
\(\chi_{20664}(2851,\cdot)\)
\(\chi_{20664}(3355,\cdot)\)
\(\chi_{20664}(3595,\cdot)\)
\(\chi_{20664}(4363,\cdot)\)
\(\chi_{20664}(4603,\cdot)\)
\(\chi_{20664}(4867,\cdot)\)
\(\chi_{20664}(5611,\cdot)\)
\(\chi_{20664}(5875,\cdot)\)
\(\chi_{20664}(6115,\cdot)\)
\(\chi_{20664}(6379,\cdot)\)
\(\chi_{20664}(7123,\cdot)\)
\(\chi_{20664}(7387,\cdot)\)
\(\chi_{20664}(7891,\cdot)\)
\(\chi_{20664}(8131,\cdot)\)
\(\chi_{20664}(9907,\cdot)\)
\(\chi_{20664}(11659,\cdot)\)
\(\chi_{20664}(13435,\cdot)\)
\(\chi_{20664}(13675,\cdot)\)
\(\chi_{20664}(14179,\cdot)\)
\(\chi_{20664}(14443,\cdot)\)
\(\chi_{20664}(15187,\cdot)\)
\(\chi_{20664}(15451,\cdot)\)
\(\chi_{20664}(15691,\cdot)\)
\(\chi_{20664}(15955,\cdot)\)
\(\chi_{20664}(16699,\cdot)\)
\(\chi_{20664}(16963,\cdot)\)
\(\chi_{20664}(17203,\cdot)\)
\(\chi_{20664}(17971,\cdot)\)
...
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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 };
\((5167,10333,2297,17713,19153)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 20664 }(15955, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
