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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,2]))
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pari:[g,chi] = znchar(Mod(1364,2667))
| Modulus: | \(2667\) | |
| Conductor: | \(2667\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2667}(188,\cdot)\)
\(\chi_{2667}(419,\cdot)\)
\(\chi_{2667}(503,\cdot)\)
\(\chi_{2667}(608,\cdot)\)
\(\chi_{2667}(965,\cdot)\)
\(\chi_{2667}(1133,\cdot)\)
\(\chi_{2667}(1343,\cdot)\)
\(\chi_{2667}(1364,\cdot)\)
\(\chi_{2667}(1574,\cdot)\)
\(\chi_{2667}(1952,\cdot)\)
\(\chi_{2667}(2057,\cdot)\)
\(\chi_{2667}(2246,\cdot)\)
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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 ]
\((890,1144,2416)\) → \((-1,-1,e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(1364, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
