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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42267, base_ring=CyclotomicField(288))
M = H._module
chi = DirichletCharacter(H, M([144,148,111]))
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gp:[g,chi] = znchar(Mod(2477, 42267))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42267.2477");
| Modulus: | \(42267\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(42267\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(288\) |
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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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\(\chi_{42267}(107,\cdot)\)
\(\chi_{42267}(644,\cdot)\)
\(\chi_{42267}(983,\cdot)\)
\(\chi_{42267}(1637,\cdot)\)
\(\chi_{42267}(1820,\cdot)\)
\(\chi_{42267}(2177,\cdot)\)
\(\chi_{42267}(2291,\cdot)\)
\(\chi_{42267}(2414,\cdot)\)
\(\chi_{42267}(2477,\cdot)\)
\(\chi_{42267}(3086,\cdot)\)
\(\chi_{42267}(3566,\cdot)\)
\(\chi_{42267}(3785,\cdot)\)
\(\chi_{42267}(5267,\cdot)\)
\(\chi_{42267}(5504,\cdot)\)
\(\chi_{42267}(5873,\cdot)\)
\(\chi_{42267}(6371,\cdot)\)
\(\chi_{42267}(6464,\cdot)\)
\(\chi_{42267}(6530,\cdot)\)
\(\chi_{42267}(6980,\cdot)\)
\(\chi_{42267}(7238,\cdot)\)
\(\chi_{42267}(7637,\cdot)\)
\(\chi_{42267}(8204,\cdot)\)
\(\chi_{42267}(8567,\cdot)\)
\(\chi_{42267}(8588,\cdot)\)
\(\chi_{42267}(8786,\cdot)\)
\(\chi_{42267}(9065,\cdot)\)
\(\chi_{42267}(10322,\cdot)\)
\(\chi_{42267}(10517,\cdot)\)
\(\chi_{42267}(10559,\cdot)\)
\(\chi_{42267}(10790,\cdot)\)
...
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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 };
\((14090,24898,3286)\) → \((-1,e\left(\frac{37}{72}\right),e\left(\frac{37}{96}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 42267 }(2477, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{115}{288}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{85}{288}\right)\) | \(e\left(\frac{191}{288}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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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)
