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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([110,165,152]))
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pari:[g,chi] = znchar(Mod(383,1815))
| Modulus: | \(1815\) | |
| Conductor: | \(1815\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(220\) |
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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)
|
\(\chi_{1815}(38,\cdot)\)
\(\chi_{1815}(47,\cdot)\)
\(\chi_{1815}(53,\cdot)\)
\(\chi_{1815}(92,\cdot)\)
\(\chi_{1815}(113,\cdot)\)
\(\chi_{1815}(137,\cdot)\)
\(\chi_{1815}(152,\cdot)\)
\(\chi_{1815}(158,\cdot)\)
\(\chi_{1815}(203,\cdot)\)
\(\chi_{1815}(212,\cdot)\)
\(\chi_{1815}(218,\cdot)\)
\(\chi_{1815}(257,\cdot)\)
\(\chi_{1815}(278,\cdot)\)
\(\chi_{1815}(302,\cdot)\)
\(\chi_{1815}(317,\cdot)\)
\(\chi_{1815}(368,\cdot)\)
\(\chi_{1815}(377,\cdot)\)
\(\chi_{1815}(383,\cdot)\)
\(\chi_{1815}(422,\cdot)\)
\(\chi_{1815}(443,\cdot)\)
\(\chi_{1815}(467,\cdot)\)
\(\chi_{1815}(482,\cdot)\)
\(\chi_{1815}(488,\cdot)\)
\(\chi_{1815}(533,\cdot)\)
\(\chi_{1815}(542,\cdot)\)
\(\chi_{1815}(548,\cdot)\)
\(\chi_{1815}(587,\cdot)\)
\(\chi_{1815}(647,\cdot)\)
\(\chi_{1815}(653,\cdot)\)
\(\chi_{1815}(698,\cdot)\)
...
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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 ]
\((1211,727,1696)\) → \((-1,-i,e\left(\frac{38}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 1815 }(383, a) \) |
\(1\) | \(1\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{5}{44}\right)\) |
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sage:chi.jacobi_sum(n)
