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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1900, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([90,99,20]))
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gp:[g,chi] = znchar(Mod(23, 1900))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1900.23");
| Modulus: | \(1900\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1900\) |
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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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\(\chi_{1900}(23,\cdot)\)
\(\chi_{1900}(47,\cdot)\)
\(\chi_{1900}(63,\cdot)\)
\(\chi_{1900}(123,\cdot)\)
\(\chi_{1900}(187,\cdot)\)
\(\chi_{1900}(263,\cdot)\)
\(\chi_{1900}(283,\cdot)\)
\(\chi_{1900}(327,\cdot)\)
\(\chi_{1900}(347,\cdot)\)
\(\chi_{1900}(367,\cdot)\)
\(\chi_{1900}(403,\cdot)\)
\(\chi_{1900}(423,\cdot)\)
\(\chi_{1900}(427,\cdot)\)
\(\chi_{1900}(503,\cdot)\)
\(\chi_{1900}(567,\cdot)\)
\(\chi_{1900}(587,\cdot)\)
\(\chi_{1900}(663,\cdot)\)
\(\chi_{1900}(727,\cdot)\)
\(\chi_{1900}(747,\cdot)\)
\(\chi_{1900}(783,\cdot)\)
\(\chi_{1900}(803,\cdot)\)
\(\chi_{1900}(823,\cdot)\)
\(\chi_{1900}(883,\cdot)\)
\(\chi_{1900}(947,\cdot)\)
\(\chi_{1900}(967,\cdot)\)
\(\chi_{1900}(1023,\cdot)\)
\(\chi_{1900}(1087,\cdot)\)
\(\chi_{1900}(1127,\cdot)\)
\(\chi_{1900}(1163,\cdot)\)
\(\chi_{1900}(1183,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{180})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 180 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((951,77,401)\) → \((-1,e\left(\frac{11}{20}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1900 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
