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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5400, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([0,90,50,63]))
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gp:[g,chi] = znchar(Mod(653, 5400))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5400.653");
| Modulus: | \(5400\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5400\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{5400}(77,\cdot)\)
\(\chi_{5400}(173,\cdot)\)
\(\chi_{5400}(317,\cdot)\)
\(\chi_{5400}(437,\cdot)\)
\(\chi_{5400}(533,\cdot)\)
\(\chi_{5400}(653,\cdot)\)
\(\chi_{5400}(677,\cdot)\)
\(\chi_{5400}(797,\cdot)\)
\(\chi_{5400}(1013,\cdot)\)
\(\chi_{5400}(1037,\cdot)\)
\(\chi_{5400}(1253,\cdot)\)
\(\chi_{5400}(1373,\cdot)\)
\(\chi_{5400}(1397,\cdot)\)
\(\chi_{5400}(1517,\cdot)\)
\(\chi_{5400}(1613,\cdot)\)
\(\chi_{5400}(1733,\cdot)\)
\(\chi_{5400}(1877,\cdot)\)
\(\chi_{5400}(1973,\cdot)\)
\(\chi_{5400}(2117,\cdot)\)
\(\chi_{5400}(2237,\cdot)\)
\(\chi_{5400}(2333,\cdot)\)
\(\chi_{5400}(2453,\cdot)\)
\(\chi_{5400}(2477,\cdot)\)
\(\chi_{5400}(2597,\cdot)\)
\(\chi_{5400}(2813,\cdot)\)
\(\chi_{5400}(2837,\cdot)\)
\(\chi_{5400}(3053,\cdot)\)
\(\chi_{5400}(3173,\cdot)\)
\(\chi_{5400}(3197,\cdot)\)
\(\chi_{5400}(3317,\cdot)\)
...
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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 };
\((1351,2701,1001,2377)\) → \((1,-1,e\left(\frac{5}{18}\right),e\left(\frac{7}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5400 }(653, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{163}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{90}\right)\) |
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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)
