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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6825, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,9,30,5]))
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pari:[g,chi] = znchar(Mod(1133,6825))
| Modulus: | \(6825\) | |
| Conductor: | \(6825\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{6825}(167,\cdot)\)
\(\chi_{6825}(188,\cdot)\)
\(\chi_{6825}(1112,\cdot)\)
\(\chi_{6825}(1133,\cdot)\)
\(\chi_{6825}(1553,\cdot)\)
\(\chi_{6825}(2477,\cdot)\)
\(\chi_{6825}(2498,\cdot)\)
\(\chi_{6825}(2897,\cdot)\)
\(\chi_{6825}(3842,\cdot)\)
\(\chi_{6825}(3863,\cdot)\)
\(\chi_{6825}(4262,\cdot)\)
\(\chi_{6825}(4283,\cdot)\)
\(\chi_{6825}(5228,\cdot)\)
\(\chi_{6825}(5627,\cdot)\)
\(\chi_{6825}(5648,\cdot)\)
\(\chi_{6825}(6572,\cdot)\)
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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 ]
\((2276,3277,976,4201)\) → \((-1,e\left(\frac{3}{20}\right),-1,e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 6825 }(1133, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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sage:chi.jacobi_sum(n)
