Maass forms near the avoided crossing in Figure 4.4.2 These are odd Maass forms, so they have a Fourier expansion of the form Sum[coeff[[n]] Sqrt[y] BesselK[i R, 2 Pi N y] Sin[2 Pi n x], {n,1,36}] Not all of the given digits of the coefficients are accurate. The first few coefficients are probably accurate to 8 or 9 digits, and the accuracy decreases to one or two digits for the last coefficients. (* The smaller R-value *) R=14.545881953 a = 4.83 b = 0 coeff={1, -2.2280504050072527, 3.0424893777573425, -1.965550491070423, 2.410698653887103, 0.054927671776902484, -2.482095340276804, -1.8306353096256078, -0.8373082360421508, 1.5694725543385522, -0.23901149481317122, 0.3764548204474639, 3.1697080023254394, -0.23868295050282234, -0.20257556554991502, 1.3340190554297708, 2.1726724553256997, -1.4437156060878191, -1.1632790821402514, -0.826072759720858, -1.6185968953516383, 2.4665600720264784, 0.8489529515760925, -3.201024177845283, -2.501769606324363, 0.6438321529174921, -0.38479058036162406, 1.6903502164288742, -0.7338230451940458, 1.1715514695508562, -3.4908919397225637, 1.5282821526327544, -1.0564515292063257, 1.4036183820739636, 0.992985929008604, -2.4602940595941565} (* the larger R-value *) R=14.56188811 a = 4.83 b = 0 coeff={1, 0.4784206004233451, -0.27140580229409267, -0.670154624114623, 0.36203448783770237, -0.050610792318109166, 0.13569798166901714, 1.4902192606634594, -0.5247661829623281, -0.4197044589958622, -0.18152315788532605, -0.5562291125356656, -0.4723138562785235, -0.06459704247193175, 1.8032527762217003, -0.3876466857244512, 0.07961047443763385, -0.30470672921803144, -0.8452890402044172, -0.2987894380077331, -0.38404034628932093, -0.35857562995490105, -0.9963845654504494, -0.15797611266678313, -0.5726491594866806, -1.4535605831183949, -0.1072385885140804, 0.612565775031238, -0.8839910969963721, 0.5246521493866817, -0.4973574251562981, 0.2511162960646506, 0.09126268718308682, -0.5097393091727321, -1.012237721229005, -0.14478958660728206}