sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(29)
sage: chi = H[8]
pari: [g,chi] = znchar(Mod(8,29))
Basic properties
sage: chi.conductor()
pari: znconreyconductor(g,chi)
| ||
Conductor | = | 29 |
sage: chi.multiplicative_order()
pari: charorder(g,chi)
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Order | = | 28 |
Real | = | No |
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
| ||
Primitive | = | Yes |
sage: chi.is_odd()
pari: zncharisodd(g,chi)
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Parity | = | Odd |
Orbit label | = | 29.f |
Orbit index | = | 6 |
Galois orbit
sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(\chi_{29}(2,\cdot)\) \(\chi_{29}(3,\cdot)\) \(\chi_{29}(8,\cdot)\) \(\chi_{29}(10,\cdot)\) \(\chi_{29}(11,\cdot)\) \(\chi_{29}(14,\cdot)\) \(\chi_{29}(15,\cdot)\) \(\chi_{29}(18,\cdot)\) \(\chi_{29}(19,\cdot)\) \(\chi_{29}(21,\cdot)\) \(\chi_{29}(26,\cdot)\) \(\chi_{29}(27,\cdot)\)
Values on generators
\(2\) → \(e\left(\frac{3}{28}\right)\)
Values
-1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
\(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
Related number fields
Field of values | \(\Q(\zeta_{28})\) |
Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{29}(8,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(8,r) e\left(\frac{2r}{29}\right) = -5.3824881894+-0.1697671685i \)
Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
\( \displaystyle J(\chi_{29}(8,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(8,r) \chi_{29}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{29}(8,·))
= \sum_{r \in \Z/29\Z}
\chi_{29}(8,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right)
= -2.3804169867+6.8028409057i \)