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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([663,355]))
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pari:[g,chi] = znchar(Mod(1047,4225))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{4225}(28,\cdot)\)
\(\chi_{4225}(37,\cdot)\)
\(\chi_{4225}(58,\cdot)\)
\(\chi_{4225}(72,\cdot)\)
\(\chi_{4225}(102,\cdot)\)
\(\chi_{4225}(123,\cdot)\)
\(\chi_{4225}(137,\cdot)\)
\(\chi_{4225}(158,\cdot)\)
\(\chi_{4225}(167,\cdot)\)
\(\chi_{4225}(202,\cdot)\)
\(\chi_{4225}(223,\cdot)\)
\(\chi_{4225}(253,\cdot)\)
\(\chi_{4225}(267,\cdot)\)
\(\chi_{4225}(288,\cdot)\)
\(\chi_{4225}(297,\cdot)\)
\(\chi_{4225}(353,\cdot)\)
\(\chi_{4225}(362,\cdot)\)
\(\chi_{4225}(383,\cdot)\)
\(\chi_{4225}(397,\cdot)\)
\(\chi_{4225}(448,\cdot)\)
\(\chi_{4225}(462,\cdot)\)
\(\chi_{4225}(483,\cdot)\)
\(\chi_{4225}(492,\cdot)\)
\(\chi_{4225}(513,\cdot)\)
\(\chi_{4225}(527,\cdot)\)
\(\chi_{4225}(548,\cdot)\)
\(\chi_{4225}(578,\cdot)\)
\(\chi_{4225}(592,\cdot)\)
\(\chi_{4225}(613,\cdot)\)
\(\chi_{4225}(622,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{71}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(1047, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{301}{780}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{539}{780}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{301}{390}\right)\) | \(e\left(\frac{373}{780}\right)\) | \(e\left(\frac{259}{260}\right)\) | \(e\left(\frac{33}{130}\right)\) |
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sage:chi.jacobi_sum(n)
