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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,42,55]))
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pari:[g,chi] = znchar(Mod(59,1300))
| Modulus: | \(1300\) | |
| Conductor: | \(1300\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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
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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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\(\chi_{1300}(19,\cdot)\)
\(\chi_{1300}(59,\cdot)\)
\(\chi_{1300}(119,\cdot)\)
\(\chi_{1300}(219,\cdot)\)
\(\chi_{1300}(279,\cdot)\)
\(\chi_{1300}(319,\cdot)\)
\(\chi_{1300}(379,\cdot)\)
\(\chi_{1300}(479,\cdot)\)
\(\chi_{1300}(539,\cdot)\)
\(\chi_{1300}(579,\cdot)\)
\(\chi_{1300}(639,\cdot)\)
\(\chi_{1300}(739,\cdot)\)
\(\chi_{1300}(839,\cdot)\)
\(\chi_{1300}(1059,\cdot)\)
\(\chi_{1300}(1159,\cdot)\)
\(\chi_{1300}(1259,\cdot)\)
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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 ]
\((651,677,301)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1300 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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sage:chi.jacobi_sum(n)
