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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192185, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([204,136,123,136]))
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gp:[g,chi] = znchar(Mod(14884, 192185))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192185.14884");
| Modulus: | \(192185\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(192185\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(408\) |
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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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\(\chi_{192185}(144,\cdot)\)
\(\chi_{192185}(3579,\cdot)\)
\(\chi_{192185}(5574,\cdot)\)
\(\chi_{192185}(6129,\cdot)\)
\(\chi_{192185}(6904,\cdot)\)
\(\chi_{192185}(8124,\cdot)\)
\(\chi_{192185}(8899,\cdot)\)
\(\chi_{192185}(9454,\cdot)\)
\(\chi_{192185}(11449,\cdot)\)
\(\chi_{192185}(14884,\cdot)\)
\(\chi_{192185}(16879,\cdot)\)
\(\chi_{192185}(17434,\cdot)\)
\(\chi_{192185}(18209,\cdot)\)
\(\chi_{192185}(19429,\cdot)\)
\(\chi_{192185}(20204,\cdot)\)
\(\chi_{192185}(20759,\cdot)\)
\(\chi_{192185}(22754,\cdot)\)
\(\chi_{192185}(28184,\cdot)\)
\(\chi_{192185}(28739,\cdot)\)
\(\chi_{192185}(29514,\cdot)\)
\(\chi_{192185}(30734,\cdot)\)
\(\chi_{192185}(31509,\cdot)\)
\(\chi_{192185}(32064,\cdot)\)
\(\chi_{192185}(34059,\cdot)\)
\(\chi_{192185}(37494,\cdot)\)
\(\chi_{192185}(39489,\cdot)\)
\(\chi_{192185}(40044,\cdot)\)
\(\chi_{192185}(40819,\cdot)\)
\(\chi_{192185}(42814,\cdot)\)
\(\chi_{192185}(43369,\cdot)\)
...
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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 };
\((115312,54911,68496,111266)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{41}{136}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 192185 }(14884, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{2}{17}\right)\) |
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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)
