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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18360, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,72,80,36,117]))
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gp:[g,chi] = znchar(Mod(3157, 18360))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18360.3157");
| Modulus: | \(18360\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18360\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{18360}(277,\cdot)\)
\(\chi_{18360}(1213,\cdot)\)
\(\chi_{18360}(1357,\cdot)\)
\(\chi_{18360}(1813,\cdot)\)
\(\chi_{18360}(2077,\cdot)\)
\(\chi_{18360}(2317,\cdot)\)
\(\chi_{18360}(3157,\cdot)\)
\(\chi_{18360}(3253,\cdot)\)
\(\chi_{18360}(3373,\cdot)\)
\(\chi_{18360}(3397,\cdot)\)
\(\chi_{18360}(3733,\cdot)\)
\(\chi_{18360}(4117,\cdot)\)
\(\chi_{18360}(5197,\cdot)\)
\(\chi_{18360}(5413,\cdot)\)
\(\chi_{18360}(5773,\cdot)\)
\(\chi_{18360}(5893,\cdot)\)
\(\chi_{18360}(6397,\cdot)\)
\(\chi_{18360}(7333,\cdot)\)
\(\chi_{18360}(7477,\cdot)\)
\(\chi_{18360}(7933,\cdot)\)
\(\chi_{18360}(8197,\cdot)\)
\(\chi_{18360}(8437,\cdot)\)
\(\chi_{18360}(9277,\cdot)\)
\(\chi_{18360}(9373,\cdot)\)
\(\chi_{18360}(9493,\cdot)\)
\(\chi_{18360}(9517,\cdot)\)
\(\chi_{18360}(9853,\cdot)\)
\(\chi_{18360}(10237,\cdot)\)
\(\chi_{18360}(11317,\cdot)\)
\(\chi_{18360}(11533,\cdot)\)
...
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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 };
\((4591,9181,7481,11017,4321)\) → \((1,-1,e\left(\frac{5}{9}\right),i,e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 18360 }(3157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{7}{72}\right)\) |
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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)
