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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20335, base_ring=CyclotomicField(1722))
M = H._module
chi = DirichletCharacter(H, M([861,1394,1407]))
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gp:[g,chi] = znchar(Mod(39, 20335))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20335.39");
| Modulus: | \(20335\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20335\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1722\) |
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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: | odd |
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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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\(\chi_{20335}(39,\cdot)\)
\(\chi_{20335}(74,\cdot)\)
\(\chi_{20335}(149,\cdot)\)
\(\chi_{20335}(179,\cdot)\)
\(\chi_{20335}(184,\cdot)\)
\(\chi_{20335}(219,\cdot)\)
\(\chi_{20335}(254,\cdot)\)
\(\chi_{20335}(284,\cdot)\)
\(\chi_{20335}(354,\cdot)\)
\(\chi_{20335}(389,\cdot)\)
\(\chi_{20335}(394,\cdot)\)
\(\chi_{20335}(429,\cdot)\)
\(\chi_{20335}(494,\cdot)\)
\(\chi_{20335}(564,\cdot)\)
\(\chi_{20335}(599,\cdot)\)
\(\chi_{20335}(634,\cdot)\)
\(\chi_{20335}(639,\cdot)\)
\(\chi_{20335}(669,\cdot)\)
\(\chi_{20335}(709,\cdot)\)
\(\chi_{20335}(744,\cdot)\)
\(\chi_{20335}(779,\cdot)\)
\(\chi_{20335}(809,\cdot)\)
\(\chi_{20335}(844,\cdot)\)
\(\chi_{20335}(849,\cdot)\)
\(\chi_{20335}(884,\cdot)\)
\(\chi_{20335}(919,\cdot)\)
\(\chi_{20335}(984,\cdot)\)
\(\chi_{20335}(989,\cdot)\)
\(\chi_{20335}(1054,\cdot)\)
\(\chi_{20335}(1094,\cdot)\)
...
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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 };
\((12202,6226,15191)\) → \((-1,e\left(\frac{17}{21}\right),e\left(\frac{67}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 20335 }(39, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{314}{861}\right)\) | \(e\left(\frac{239}{1722}\right)\) | \(e\left(\frac{628}{861}\right)\) | \(e\left(\frac{289}{574}\right)\) | \(e\left(\frac{27}{287}\right)\) | \(e\left(\frac{239}{861}\right)\) | \(e\left(\frac{853}{861}\right)\) | \(e\left(\frac{1495}{1722}\right)\) | \(e\left(\frac{37}{287}\right)\) | \(e\left(\frac{395}{861}\right)\) |
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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)
