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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,14,33]))
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pari:[g,chi] = znchar(Mod(17,2001))
| Modulus: | \(2001\) | |
| Conductor: | \(2001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
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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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\(\chi_{2001}(17,\cdot)\)
\(\chi_{2001}(191,\cdot)\)
\(\chi_{2001}(365,\cdot)\)
\(\chi_{2001}(389,\cdot)\)
\(\chi_{2001}(452,\cdot)\)
\(\chi_{2001}(539,\cdot)\)
\(\chi_{2001}(563,\cdot)\)
\(\chi_{2001}(626,\cdot)\)
\(\chi_{2001}(824,\cdot)\)
\(\chi_{2001}(911,\cdot)\)
\(\chi_{2001}(1148,\cdot)\)
\(\chi_{2001}(1259,\cdot)\)
\(\chi_{2001}(1322,\cdot)\)
\(\chi_{2001}(1433,\cdot)\)
\(\chi_{2001}(1583,\cdot)\)
\(\chi_{2001}(1607,\cdot)\)
\(\chi_{2001}(1670,\cdot)\)
\(\chi_{2001}(1694,\cdot)\)
\(\chi_{2001}(1781,\cdot)\)
\(\chi_{2001}(1868,\cdot)\)
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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 ]
\((668,1132,553)\) → \((-1,e\left(\frac{7}{22}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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sage:chi.jacobi_sum(n)
