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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2940, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,21,37]))
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pari:[g,chi] = znchar(Mod(1739,2940))
| Modulus: | \(2940\) | |
| Conductor: | \(2940\) |
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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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\(\chi_{2940}(59,\cdot)\)
\(\chi_{2940}(299,\cdot)\)
\(\chi_{2940}(479,\cdot)\)
\(\chi_{2940}(719,\cdot)\)
\(\chi_{2940}(899,\cdot)\)
\(\chi_{2940}(1139,\cdot)\)
\(\chi_{2940}(1319,\cdot)\)
\(\chi_{2940}(1559,\cdot)\)
\(\chi_{2940}(1739,\cdot)\)
\(\chi_{2940}(2159,\cdot)\)
\(\chi_{2940}(2399,\cdot)\)
\(\chi_{2940}(2819,\cdot)\)
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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 ]
\((1471,1961,1177,1081)\) → \((-1,-1,-1,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2940 }(1739, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |
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sage:chi.jacobi_sum(n)
