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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3753, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,8]))
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gp:[g,chi] = znchar(Mod(1646, 3753))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3753.1646");
| Modulus: | \(3753\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(417\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{417}(395,\cdot)\) |
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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: | 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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\(\chi_{3753}(107,\cdot)\)
\(\chi_{3753}(188,\cdot)\)
\(\chi_{3753}(377,\cdot)\)
\(\chi_{3753}(458,\cdot)\)
\(\chi_{3753}(539,\cdot)\)
\(\chi_{3753}(593,\cdot)\)
\(\chi_{3753}(674,\cdot)\)
\(\chi_{3753}(863,\cdot)\)
\(\chi_{3753}(917,\cdot)\)
\(\chi_{3753}(971,\cdot)\)
\(\chi_{3753}(998,\cdot)\)
\(\chi_{3753}(1322,\cdot)\)
\(\chi_{3753}(1403,\cdot)\)
\(\chi_{3753}(1457,\cdot)\)
\(\chi_{3753}(1511,\cdot)\)
\(\chi_{3753}(1538,\cdot)\)
\(\chi_{3753}(1646,\cdot)\)
\(\chi_{3753}(1673,\cdot)\)
\(\chi_{3753}(1754,\cdot)\)
\(\chi_{3753}(1781,\cdot)\)
\(\chi_{3753}(1835,\cdot)\)
\(\chi_{3753}(1943,\cdot)\)
\(\chi_{3753}(1970,\cdot)\)
\(\chi_{3753}(1997,\cdot)\)
\(\chi_{3753}(2024,\cdot)\)
\(\chi_{3753}(2105,\cdot)\)
\(\chi_{3753}(2132,\cdot)\)
\(\chi_{3753}(2240,\cdot)\)
\(\chi_{3753}(2348,\cdot)\)
\(\chi_{3753}(2429,\cdot)\)
...
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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 };
\((974,2782)\) → \((-1,e\left(\frac{4}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3753 }(1646, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{16}{69}\right)\) |
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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)
