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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([741,160]))
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pari:[g,chi] = znchar(Mod(113,4225))
| Modulus: | \(4225\) | |
| Conductor: | \(4225\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(780\) |
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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)
|
\(\chi_{4225}(3,\cdot)\)
\(\chi_{4225}(42,\cdot)\)
\(\chi_{4225}(48,\cdot)\)
\(\chi_{4225}(87,\cdot)\)
\(\chi_{4225}(113,\cdot)\)
\(\chi_{4225}(133,\cdot)\)
\(\chi_{4225}(152,\cdot)\)
\(\chi_{4225}(172,\cdot)\)
\(\chi_{4225}(178,\cdot)\)
\(\chi_{4225}(198,\cdot)\)
\(\chi_{4225}(217,\cdot)\)
\(\chi_{4225}(237,\cdot)\)
\(\chi_{4225}(263,\cdot)\)
\(\chi_{4225}(302,\cdot)\)
\(\chi_{4225}(308,\cdot)\)
\(\chi_{4225}(328,\cdot)\)
\(\chi_{4225}(347,\cdot)\)
\(\chi_{4225}(367,\cdot)\)
\(\chi_{4225}(373,\cdot)\)
\(\chi_{4225}(412,\cdot)\)
\(\chi_{4225}(438,\cdot)\)
\(\chi_{4225}(458,\cdot)\)
\(\chi_{4225}(477,\cdot)\)
\(\chi_{4225}(497,\cdot)\)
\(\chi_{4225}(503,\cdot)\)
\(\chi_{4225}(523,\cdot)\)
\(\chi_{4225}(542,\cdot)\)
\(\chi_{4225}(562,\cdot)\)
\(\chi_{4225}(588,\cdot)\)
\(\chi_{4225}(627,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{8}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(113, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{121}{780}\right)\) | \(e\left(\frac{67}{780}\right)\) | \(e\left(\frac{121}{390}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{121}{260}\right)\) | \(e\left(\frac{67}{390}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{103}{260}\right)\) | \(e\left(\frac{111}{130}\right)\) |
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sage:chi.jacobi_sum(n)
