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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,6,10]))
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pari:[g,chi] = znchar(Mod(341,2600))
| Modulus: | \(2600\) | |
| Conductor: | \(2600\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(30\) |
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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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\(\chi_{2600}(61,\cdot)\)
\(\chi_{2600}(341,\cdot)\)
\(\chi_{2600}(581,\cdot)\)
\(\chi_{2600}(861,\cdot)\)
\(\chi_{2600}(1381,\cdot)\)
\(\chi_{2600}(1621,\cdot)\)
\(\chi_{2600}(2141,\cdot)\)
\(\chi_{2600}(2421,\cdot)\)
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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 ]
\((1951,1301,1977,1601)\) → \((1,-1,e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2600 }(341, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) |
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sage:chi.jacobi_sum(n)
