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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24648, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,78,78,39,140]))
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pari:[g,chi] = znchar(Mod(8315,24648))
| Modulus: | \(24648\) | |
| Conductor: | \(24648\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{24648}(83,\cdot)\)
\(\chi_{24648}(203,\cdot)\)
\(\chi_{24648}(1763,\cdot)\)
\(\chi_{24648}(2579,\cdot)\)
\(\chi_{24648}(2699,\cdot)\)
\(\chi_{24648}(3011,\cdot)\)
\(\chi_{24648}(3323,\cdot)\)
\(\chi_{24648}(3947,\cdot)\)
\(\chi_{24648}(4139,\cdot)\)
\(\chi_{24648}(4259,\cdot)\)
\(\chi_{24648}(5075,\cdot)\)
\(\chi_{24648}(5699,\cdot)\)
\(\chi_{24648}(6443,\cdot)\)
\(\chi_{24648}(6755,\cdot)\)
\(\chi_{24648}(7067,\cdot)\)
\(\chi_{24648}(7379,\cdot)\)
\(\chi_{24648}(8315,\cdot)\)
\(\chi_{24648}(8627,\cdot)\)
\(\chi_{24648}(8819,\cdot)\)
\(\chi_{24648}(9443,\cdot)\)
\(\chi_{24648}(9563,\cdot)\)
\(\chi_{24648}(10691,\cdot)\)
\(\chi_{24648}(12059,\cdot)\)
\(\chi_{24648}(12563,\cdot)\)
\(\chi_{24648}(13187,\cdot)\)
\(\chi_{24648}(13619,\cdot)\)
\(\chi_{24648}(14555,\cdot)\)
\(\chi_{24648}(15059,\cdot)\)
\(\chi_{24648}(15179,\cdot)\)
\(\chi_{24648}(15371,\cdot)\)
...
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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 ]
\((18487,12325,16433,11377,12169)\) → \((-1,-1,-1,i,e\left(\frac{35}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(8315, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{55}{78}\right)\) |
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sage:chi.jacobi_sum(n)
