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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,68,133]))
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pari:[g,chi] = znchar(Mod(467,2312))
| Modulus: | \(2312\) | |
| Conductor: | \(2312\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(136\) |
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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)
|
\(\chi_{2312}(19,\cdot)\)
\(\chi_{2312}(43,\cdot)\)
\(\chi_{2312}(59,\cdot)\)
\(\chi_{2312}(83,\cdot)\)
\(\chi_{2312}(195,\cdot)\)
\(\chi_{2312}(219,\cdot)\)
\(\chi_{2312}(291,\cdot)\)
\(\chi_{2312}(315,\cdot)\)
\(\chi_{2312}(331,\cdot)\)
\(\chi_{2312}(355,\cdot)\)
\(\chi_{2312}(427,\cdot)\)
\(\chi_{2312}(451,\cdot)\)
\(\chi_{2312}(467,\cdot)\)
\(\chi_{2312}(491,\cdot)\)
\(\chi_{2312}(563,\cdot)\)
\(\chi_{2312}(587,\cdot)\)
\(\chi_{2312}(603,\cdot)\)
\(\chi_{2312}(627,\cdot)\)
\(\chi_{2312}(699,\cdot)\)
\(\chi_{2312}(723,\cdot)\)
\(\chi_{2312}(739,\cdot)\)
\(\chi_{2312}(763,\cdot)\)
\(\chi_{2312}(835,\cdot)\)
\(\chi_{2312}(859,\cdot)\)
\(\chi_{2312}(875,\cdot)\)
\(\chi_{2312}(899,\cdot)\)
\(\chi_{2312}(971,\cdot)\)
\(\chi_{2312}(995,\cdot)\)
\(\chi_{2312}(1011,\cdot)\)
\(\chi_{2312}(1035,\cdot)\)
...
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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 ]
\((1735,1157,1737)\) → \((-1,-1,e\left(\frac{133}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(467, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{79}{136}\right)\) |
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sage:chi.jacobi_sum(n)
