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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,16,21]))
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pari:[g,chi] = znchar(Mod(1299,2548))
| Modulus: | \(2548\) | |
| Conductor: | \(2548\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2548}(51,\cdot)\)
\(\chi_{2548}(207,\cdot)\)
\(\chi_{2548}(415,\cdot)\)
\(\chi_{2548}(571,\cdot)\)
\(\chi_{2548}(779,\cdot)\)
\(\chi_{2548}(935,\cdot)\)
\(\chi_{2548}(1143,\cdot)\)
\(\chi_{2548}(1299,\cdot)\)
\(\chi_{2548}(1507,\cdot)\)
\(\chi_{2548}(1663,\cdot)\)
\(\chi_{2548}(1871,\cdot)\)
\(\chi_{2548}(2391,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((1275,885,197)\) → \((-1,e\left(\frac{8}{21}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2548 }(1299, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) |
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sage:chi.jacobi_sum(n)
