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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3636, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,50,21]))
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gp:[g,chi] = znchar(Mod(931, 3636))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3636.931");
| Modulus: | \(3636\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3636\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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: | 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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\(\chi_{3636}(43,\cdot)\)
\(\chi_{3636}(211,\cdot)\)
\(\chi_{3636}(223,\cdot)\)
\(\chi_{3636}(247,\cdot)\)
\(\chi_{3636}(367,\cdot)\)
\(\chi_{3636}(427,\cdot)\)
\(\chi_{3636}(535,\cdot)\)
\(\chi_{3636}(619,\cdot)\)
\(\chi_{3636}(655,\cdot)\)
\(\chi_{3636}(691,\cdot)\)
\(\chi_{3636}(727,\cdot)\)
\(\chi_{3636}(931,\cdot)\)
\(\chi_{3636}(979,\cdot)\)
\(\chi_{3636}(1087,\cdot)\)
\(\chi_{3636}(1255,\cdot)\)
\(\chi_{3636}(1435,\cdot)\)
\(\chi_{3636}(1447,\cdot)\)
\(\chi_{3636}(1519,\cdot)\)
\(\chi_{3636}(1579,\cdot)\)
\(\chi_{3636}(1591,\cdot)\)
\(\chi_{3636}(1663,\cdot)\)
\(\chi_{3636}(1831,\cdot)\)
\(\chi_{3636}(1867,\cdot)\)
\(\chi_{3636}(1903,\cdot)\)
\(\chi_{3636}(1939,\cdot)\)
\(\chi_{3636}(2191,\cdot)\)
\(\chi_{3636}(2203,\cdot)\)
\(\chi_{3636}(2299,\cdot)\)
\(\chi_{3636}(2419,\cdot)\)
\(\chi_{3636}(2635,\cdot)\)
...
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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 };
\((1819,3233,3133)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3636 }(931, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{139}{150}\right)\) |
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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)
