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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7050, base_ring=CyclotomicField(92))
M = H._module
chi = DirichletCharacter(H, M([0,23,42]))
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gp:[g,chi] = znchar(Mod(4057, 7050))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7050.4057");
| Modulus: | \(7050\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(235\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(92\) |
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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: | no, induced from \(\chi_{235}(62,\cdot)\) |
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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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\(\chi_{7050}(43,\cdot)\)
\(\chi_{7050}(193,\cdot)\)
\(\chi_{7050}(493,\cdot)\)
\(\chi_{7050}(607,\cdot)\)
\(\chi_{7050}(757,\cdot)\)
\(\chi_{7050}(793,\cdot)\)
\(\chi_{7050}(1057,\cdot)\)
\(\chi_{7050}(1357,\cdot)\)
\(\chi_{7050}(1393,\cdot)\)
\(\chi_{7050}(1543,\cdot)\)
\(\chi_{7050}(1843,\cdot)\)
\(\chi_{7050}(1957,\cdot)\)
\(\chi_{7050}(1993,\cdot)\)
\(\chi_{7050}(2107,\cdot)\)
\(\chi_{7050}(2407,\cdot)\)
\(\chi_{7050}(2557,\cdot)\)
\(\chi_{7050}(2893,\cdot)\)
\(\chi_{7050}(3043,\cdot)\)
\(\chi_{7050}(3193,\cdot)\)
\(\chi_{7050}(3457,\cdot)\)
\(\chi_{7050}(3493,\cdot)\)
\(\chi_{7050}(3607,\cdot)\)
\(\chi_{7050}(3757,\cdot)\)
\(\chi_{7050}(3793,\cdot)\)
\(\chi_{7050}(4057,\cdot)\)
\(\chi_{7050}(4243,\cdot)\)
\(\chi_{7050}(4357,\cdot)\)
\(\chi_{7050}(4393,\cdot)\)
\(\chi_{7050}(4543,\cdot)\)
\(\chi_{7050}(4693,\cdot)\)
...
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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 };
\((2351,5077,5551)\) → \((1,i,e\left(\frac{21}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7050 }(4057, a) \) |
\(1\) | \(1\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) |
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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)
