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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([33,33,11,42]))
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pari:[g,chi] = znchar(Mod(59,2415))
| 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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\(\chi_{2415}(59,\cdot)\)
\(\chi_{2415}(164,\cdot)\)
\(\chi_{2415}(269,\cdot)\)
\(\chi_{2415}(374,\cdot)\)
\(\chi_{2415}(404,\cdot)\)
\(\chi_{2415}(509,\cdot)\)
\(\chi_{2415}(584,\cdot)\)
\(\chi_{2415}(614,\cdot)\)
\(\chi_{2415}(719,\cdot)\)
\(\chi_{2415}(794,\cdot)\)
\(\chi_{2415}(899,\cdot)\)
\(\chi_{2415}(929,\cdot)\)
\(\chi_{2415}(1139,\cdot)\)
\(\chi_{2415}(1214,\cdot)\)
\(\chi_{2415}(1244,\cdot)\)
\(\chi_{2415}(1319,\cdot)\)
\(\chi_{2415}(1559,\cdot)\)
\(\chi_{2415}(1664,\cdot)\)
\(\chi_{2415}(1844,\cdot)\)
\(\chi_{2415}(2189,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(59, a) \) |
\(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(-1\) | \(e\left(\frac{17}{33}\right)\) |
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sage:chi.jacobi_sum(n)
