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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6035, base_ring=CyclotomicField(560))
M = H._module
chi = DirichletCharacter(H, M([280,385,296]))
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gp:[g,chi] = znchar(Mod(874, 6035))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6035.874");
| Modulus: | \(6035\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6035\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(560\) |
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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);
|
\(\chi_{6035}(44,\cdot)\)
\(\chi_{6035}(99,\cdot)\)
\(\chi_{6035}(124,\cdot)\)
\(\chi_{6035}(139,\cdot)\)
\(\chi_{6035}(164,\cdot)\)
\(\chi_{6035}(184,\cdot)\)
\(\chi_{6035}(194,\cdot)\)
\(\chi_{6035}(209,\cdot)\)
\(\chi_{6035}(224,\cdot)\)
\(\chi_{6035}(244,\cdot)\)
\(\chi_{6035}(269,\cdot)\)
\(\chi_{6035}(414,\cdot)\)
\(\chi_{6035}(439,\cdot)\)
\(\chi_{6035}(454,\cdot)\)
\(\chi_{6035}(479,\cdot)\)
\(\chi_{6035}(504,\cdot)\)
\(\chi_{6035}(539,\cdot)\)
\(\chi_{6035}(549,\cdot)\)
\(\chi_{6035}(564,\cdot)\)
\(\chi_{6035}(589,\cdot)\)
\(\chi_{6035}(624,\cdot)\)
\(\chi_{6035}(674,\cdot)\)
\(\chi_{6035}(694,\cdot)\)
\(\chi_{6035}(704,\cdot)\)
\(\chi_{6035}(754,\cdot)\)
\(\chi_{6035}(779,\cdot)\)
\(\chi_{6035}(794,\cdot)\)
\(\chi_{6035}(809,\cdot)\)
\(\chi_{6035}(844,\cdot)\)
\(\chi_{6035}(874,\cdot)\)
...
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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 };
\((3622,5681,3486)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{37}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 6035 }(874, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{280}\right)\) | \(e\left(\frac{521}{560}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{127}{560}\right)\) | \(e\left(\frac{331}{560}\right)\) | \(e\left(\frac{249}{280}\right)\) | \(e\left(\frac{241}{280}\right)\) | \(e\left(\frac{111}{560}\right)\) | \(e\left(\frac{293}{560}\right)\) | \(e\left(\frac{121}{140}\right)\) |
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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)
