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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,35,21,26]))
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pari:[g,chi] = znchar(Mod(4559,8820))
| Modulus: | \(8820\) | |
| Conductor: | \(8820\) |
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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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\(\chi_{8820}(779,\cdot)\)
\(\chi_{8820}(1019,\cdot)\)
\(\chi_{8820}(2279,\cdot)\)
\(\chi_{8820}(3299,\cdot)\)
\(\chi_{8820}(3539,\cdot)\)
\(\chi_{8820}(4559,\cdot)\)
\(\chi_{8820}(4799,\cdot)\)
\(\chi_{8820}(5819,\cdot)\)
\(\chi_{8820}(6059,\cdot)\)
\(\chi_{8820}(7079,\cdot)\)
\(\chi_{8820}(8339,\cdot)\)
\(\chi_{8820}(8579,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((-1,e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(4559, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(-1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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sage:chi.jacobi_sum(n)
