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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195337, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([157,209]))
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gp:[g,chi] = znchar(Mod(14403, 195337))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195337.14403");
| Modulus: | \(195337\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(195337\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(228\) |
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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);
|
\(\chi_{195337}(98,\cdot)\)
\(\chi_{195337}(7059,\cdot)\)
\(\chi_{195337}(8628,\cdot)\)
\(\chi_{195337}(9481,\cdot)\)
\(\chi_{195337}(11707,\cdot)\)
\(\chi_{195337}(12697,\cdot)\)
\(\chi_{195337}(13550,\cdot)\)
\(\chi_{195337}(14403,\cdot)\)
\(\chi_{195337}(19854,\cdot)\)
\(\chi_{195337}(21943,\cdot)\)
\(\chi_{195337}(22080,\cdot)\)
\(\chi_{195337}(22276,\cdot)\)
\(\chi_{195337}(25492,\cdot)\)
\(\chi_{195337}(26541,\cdot)\)
\(\chi_{195337}(28051,\cdot)\)
\(\chi_{195337}(28904,\cdot)\)
\(\chi_{195337}(31326,\cdot)\)
\(\chi_{195337}(32512,\cdot)\)
\(\chi_{195337}(33885,\cdot)\)
\(\chi_{195337}(35924,\cdot)\)
\(\chi_{195337}(45307,\cdot)\)
\(\chi_{195337}(48003,\cdot)\)
\(\chi_{195337}(48856,\cdot)\)
\(\chi_{195337}(51415,\cdot)\)
\(\chi_{195337}(59092,\cdot)\)
\(\chi_{195337}(59475,\cdot)\)
\(\chi_{195337}(61181,\cdot)\)
\(\chi_{195337}(69328,\cdot)\)
\(\chi_{195337}(76152,\cdot)\)
\(\chi_{195337}(77721,\cdot)\)
...
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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 };
\((24738,34122)\) → \((e\left(\frac{157}{228}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 195337 }(14403, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{139}{228}\right)\) | \(e\left(\frac{145}{228}\right)\) |
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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)
