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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,21,21,5]))
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pari:[g,chi] = znchar(Mod(2909,5160))
| Modulus: | \(5160\) | |
| Conductor: | \(5160\) |
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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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\(\chi_{5160}(29,\cdot)\)
\(\chi_{5160}(149,\cdot)\)
\(\chi_{5160}(749,\cdot)\)
\(\chi_{5160}(1109,\cdot)\)
\(\chi_{5160}(2069,\cdot)\)
\(\chi_{5160}(2309,\cdot)\)
\(\chi_{5160}(2549,\cdot)\)
\(\chi_{5160}(2669,\cdot)\)
\(\chi_{5160}(2909,\cdot)\)
\(\chi_{5160}(3029,\cdot)\)
\(\chi_{5160}(3509,\cdot)\)
\(\chi_{5160}(3989,\cdot)\)
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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 ]
\((3871,2581,1721,3097,4561)\) → \((1,-1,-1,-1,e\left(\frac{5}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(2909, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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sage:chi.jacobi_sum(n)
