A396259 Decimal expansion of the smallest positive root to 64*x^6 - 672*x^4 + 1764*x^2 - 1183 = 0.

1, 0, 2, 2, 6, 1, 8, 7, 9, 1, 8, 7, 1, 7, 9, 4, 1, 3, 0, 8, 7, 4, 5, 2, 5, 7, 0, 3, 2, 0, 2, 5, 4, 3, 0, 3, 7, 8, 4, 1, 8, 7, 3, 9, 6, 4, 5, 8, 4, 8, 3, 7, 4, 9, 6, 7, 5, 7, 7, 3, 7, 6, 4, 4, 5, 3, 4, 7, 4, 0, 7, 8, 3, 2, 8, 6, 6, 6, 7, 5, 1, 1, 2, 2, 2, 4, 0, 7, 5, 7

Offset

1,3

Comments

Imaginary part of the Gauss sum tau(chi) = Sum_{a=0..6} chi(a)*exp(2*Pi*i/7), where chi is the Dirichlet character modulo 7 such that chi(3) = exp(2*Pi*i/6). Note that tau(chi) is a root to x^12 + 497*x^6 + 117649 = 0.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for algebraic numbers, degree 6.

Example

1.02261879187179413087...

Mathematica

First[RealDigits[Root[64*#^6 - 672*#^4 + 1764*#^2 - 1183 &, 4], 10, 100]] (* Paolo Xausa, May 21 2026 *)

Prog

(PARI) solve(x=1.0, 1.1, 64*x^6 - 672*x^4 + 1764*x^2 - 1183)

Crossrefs

See A396258 for table of Gauss sums of nontrivial Dirichlet characters modulo 7.

Sequence in context: A155818 A010245 A154196 * A248617 A225975 A016529

Adjacent sequences: A396256 A396257 A396258 * A396260 A396261 A396262

Keyword

nonn,cons,easy

Author

Jianing Song, May 20 2026

Status

approved