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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6675, base_ring=CyclotomicField(440))
M = H._module
chi = DirichletCharacter(H, M([220,44,335]))
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gp:[g,chi] = znchar(Mod(254, 6675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6675.254");
| Modulus: | \(6675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(440\) |
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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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\(\chi_{6675}(14,\cdot)\)
\(\chi_{6675}(29,\cdot)\)
\(\chi_{6675}(59,\cdot)\)
\(\chi_{6675}(104,\cdot)\)
\(\chi_{6675}(119,\cdot)\)
\(\chi_{6675}(164,\cdot)\)
\(\chi_{6675}(209,\cdot)\)
\(\chi_{6675}(239,\cdot)\)
\(\chi_{6675}(254,\cdot)\)
\(\chi_{6675}(329,\cdot)\)
\(\chi_{6675}(359,\cdot)\)
\(\chi_{6675}(389,\cdot)\)
\(\chi_{6675}(404,\cdot)\)
\(\chi_{6675}(419,\cdot)\)
\(\chi_{6675}(464,\cdot)\)
\(\chi_{6675}(569,\cdot)\)
\(\chi_{6675}(629,\cdot)\)
\(\chi_{6675}(689,\cdot)\)
\(\chi_{6675}(719,\cdot)\)
\(\chi_{6675}(794,\cdot)\)
\(\chi_{6675}(839,\cdot)\)
\(\chi_{6675}(884,\cdot)\)
\(\chi_{6675}(914,\cdot)\)
\(\chi_{6675}(944,\cdot)\)
\(\chi_{6675}(1094,\cdot)\)
\(\chi_{6675}(1109,\cdot)\)
\(\chi_{6675}(1154,\cdot)\)
\(\chi_{6675}(1184,\cdot)\)
\(\chi_{6675}(1259,\cdot)\)
\(\chi_{6675}(1289,\cdot)\)
...
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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 };
\((4451,802,2851)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{67}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6675 }(254, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{181}{440}\right)\) | \(e\left(\frac{419}{440}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{197}{440}\right)\) |
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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)
