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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,22,3]))
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pari:[g,chi] = znchar(Mod(2144,2415))
| Modulus: | \(2415\) | |
| Conductor: | \(2415\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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)
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\(\chi_{2415}(44,\cdot)\)
\(\chi_{2415}(74,\cdot)\)
\(\chi_{2415}(149,\cdot)\)
\(\chi_{2415}(359,\cdot)\)
\(\chi_{2415}(389,\cdot)\)
\(\chi_{2415}(494,\cdot)\)
\(\chi_{2415}(569,\cdot)\)
\(\chi_{2415}(674,\cdot)\)
\(\chi_{2415}(704,\cdot)\)
\(\chi_{2415}(779,\cdot)\)
\(\chi_{2415}(884,\cdot)\)
\(\chi_{2415}(914,\cdot)\)
\(\chi_{2415}(1019,\cdot)\)
\(\chi_{2415}(1124,\cdot)\)
\(\chi_{2415}(1229,\cdot)\)
\(\chi_{2415}(1514,\cdot)\)
\(\chi_{2415}(1859,\cdot)\)
\(\chi_{2415}(2039,\cdot)\)
\(\chi_{2415}(2144,\cdot)\)
\(\chi_{2415}(2384,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(2144, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(1\) | \(e\left(\frac{59}{66}\right)\) |
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sage:chi.jacobi_sum(n)
