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sage:H = DirichletGroup(1080000)
chi = H[1000001]
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pari:[g,chi] = znchar(Mod(1000001,1080000))
| Modulus: | \(1080000\) | |
| Conductor: | \(27\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(18\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | no |
| Primitive: | no, induced from \(\chi_{27}(2,\cdot)\) |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
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| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\((978751,202501,1000001,29377)\) → \((1,1,e\left(\frac{1}{18}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1080000 }(1000001, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
