Properties

Label 76.7.c.b.37.7
Level $76$
Weight $7$
Character 76.37
Analytic conductor $17.484$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.7
Root \(40.3415i\) of defining polynomial
Character \(\chi\) \(=\) 76.37
Dual form 76.7.c.b.37.2

$q$-expansion

\(f(q)\) \(=\) \(q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} +O(q^{10})\) \(q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} -1591.27 q^{11} +143.277i q^{13} -7535.62i q^{15} +4771.21 q^{17} +(-1010.48 - 6784.16i) q^{19} +8160.80i q^{21} +7737.11 q^{23} +19267.6 q^{25} -6835.36i q^{27} +8820.72i q^{29} -51550.3i q^{31} -64194.3i q^{33} -37787.4 q^{35} -94640.6i q^{37} -5779.99 q^{39} +78747.2i q^{41} -129047. q^{43} +167824. q^{45} -97220.5 q^{47} -76726.6 q^{49} +192478. i q^{51} -46143.8i q^{53} +297243. q^{55} +(273683. - 40764.1i) q^{57} +61729.4i q^{59} -85634.0 q^{61} -181747. q^{63} -26763.5i q^{65} -49044.4i q^{67} +312127. i q^{69} +408517. i q^{71} -130029. q^{73} +777286. i q^{75} -321903. q^{77} -106942. i q^{79} -379212. q^{81} -12258.3 q^{83} -891242. q^{85} -355841. q^{87} -561920. i q^{89} +28983.8i q^{91} +2.07962e6 q^{93} +(188753. + 1.26725e6i) q^{95} -851962. i q^{97} +1.42966e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} + 362q^{7} - 4348q^{9} + O(q^{10}) \) \( 8q + 2q^{5} + 362q^{7} - 4348q^{9} + 902q^{11} + 1550q^{17} + 6232q^{19} - 18820q^{23} - 12158q^{25} - 101762q^{35} + 167028q^{39} - 335042q^{43} - 57230q^{45} - 570394q^{47} + 448182q^{49} + 1089198q^{55} + 341316q^{57} - 632014q^{61} + 328174q^{63} - 852938q^{73} + 1850530q^{77} - 1819456q^{81} + 441200q^{83} - 1828374q^{85} + 1483380q^{87} + 2131176q^{93} + 627950q^{95} - 865394q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 40.3415i 1.49413i 0.664751 + 0.747065i \(0.268538\pi\)
−0.664751 + 0.747065i \(0.731462\pi\)
\(4\) 0 0
\(5\) −186.796 −1.49437 −0.747183 0.664618i \(-0.768594\pi\)
−0.747183 + 0.664618i \(0.768594\pi\)
\(6\) 0 0
\(7\) 202.293 0.589775 0.294888 0.955532i \(-0.404718\pi\)
0.294888 + 0.955532i \(0.404718\pi\)
\(8\) 0 0
\(9\) −898.437 −1.23242
\(10\) 0 0
\(11\) −1591.27 −1.19555 −0.597773 0.801666i \(-0.703947\pi\)
−0.597773 + 0.801666i \(0.703947\pi\)
\(12\) 0 0
\(13\) 143.277i 0.0652146i 0.999468 + 0.0326073i \(0.0103811\pi\)
−0.999468 + 0.0326073i \(0.989619\pi\)
\(14\) 0 0
\(15\) 7535.62i 2.23278i
\(16\) 0 0
\(17\) 4771.21 0.971141 0.485570 0.874198i \(-0.338612\pi\)
0.485570 + 0.874198i \(0.338612\pi\)
\(18\) 0 0
\(19\) −1010.48 6784.16i −0.147321 0.989089i
\(20\) 0 0
\(21\) 8160.80i 0.881201i
\(22\) 0 0
\(23\) 7737.11 0.635910 0.317955 0.948106i \(-0.397004\pi\)
0.317955 + 0.948106i \(0.397004\pi\)
\(24\) 0 0
\(25\) 19267.6 1.23313
\(26\) 0 0
\(27\) 6835.36i 0.347272i
\(28\) 0 0
\(29\) 8820.72i 0.361668i 0.983514 + 0.180834i \(0.0578796\pi\)
−0.983514 + 0.180834i \(0.942120\pi\)
\(30\) 0 0
\(31\) 51550.3i 1.73040i −0.501428 0.865199i \(-0.667192\pi\)
0.501428 0.865199i \(-0.332808\pi\)
\(32\) 0 0
\(33\) 64194.3i 1.78630i
\(34\) 0 0
\(35\) −37787.4 −0.881340
\(36\) 0 0
\(37\) 94640.6i 1.86841i −0.356735 0.934206i \(-0.616110\pi\)
0.356735 0.934206i \(-0.383890\pi\)
\(38\) 0 0
\(39\) −5779.99 −0.0974392
\(40\) 0 0
\(41\) 78747.2i 1.14257i 0.820751 + 0.571286i \(0.193555\pi\)
−0.820751 + 0.571286i \(0.806445\pi\)
\(42\) 0 0
\(43\) −129047. −1.62308 −0.811542 0.584294i \(-0.801371\pi\)
−0.811542 + 0.584294i \(0.801371\pi\)
\(44\) 0 0
\(45\) 167824. 1.84169
\(46\) 0 0
\(47\) −97220.5 −0.936406 −0.468203 0.883621i \(-0.655098\pi\)
−0.468203 + 0.883621i \(0.655098\pi\)
\(48\) 0 0
\(49\) −76726.6 −0.652165
\(50\) 0 0
\(51\) 192478.i 1.45101i
\(52\) 0 0
\(53\) 46143.8i 0.309946i −0.987919 0.154973i \(-0.950471\pi\)
0.987919 0.154973i \(-0.0495290\pi\)
\(54\) 0 0
\(55\) 297243. 1.78658
\(56\) 0 0
\(57\) 273683. 40764.1i 1.47783 0.220117i
\(58\) 0 0
\(59\) 61729.4i 0.300563i 0.988643 + 0.150282i \(0.0480181\pi\)
−0.988643 + 0.150282i \(0.951982\pi\)
\(60\) 0 0
\(61\) −85634.0 −0.377274 −0.188637 0.982047i \(-0.560407\pi\)
−0.188637 + 0.982047i \(0.560407\pi\)
\(62\) 0 0
\(63\) −181747. −0.726853
\(64\) 0 0
\(65\) 26763.5i 0.0974545i
\(66\) 0 0
\(67\) 49044.4i 0.163067i −0.996671 0.0815334i \(-0.974018\pi\)
0.996671 0.0815334i \(-0.0259817\pi\)
\(68\) 0 0
\(69\) 312127.i 0.950132i
\(70\) 0 0
\(71\) 408517.i 1.14139i 0.821161 + 0.570696i \(0.193327\pi\)
−0.821161 + 0.570696i \(0.806673\pi\)
\(72\) 0 0
\(73\) −130029. −0.334250 −0.167125 0.985936i \(-0.553448\pi\)
−0.167125 + 0.985936i \(0.553448\pi\)
\(74\) 0 0
\(75\) 777286.i 1.84246i
\(76\) 0 0
\(77\) −321903. −0.705103
\(78\) 0 0
\(79\) 106942.i 0.216904i −0.994102 0.108452i \(-0.965411\pi\)
0.994102 0.108452i \(-0.0345893\pi\)
\(80\) 0 0
\(81\) −379212. −0.713554
\(82\) 0 0
\(83\) −12258.3 −0.0214386 −0.0107193 0.999943i \(-0.503412\pi\)
−0.0107193 + 0.999943i \(0.503412\pi\)
\(84\) 0 0
\(85\) −891242. −1.45124
\(86\) 0 0
\(87\) −355841. −0.540379
\(88\) 0 0
\(89\) 561920.i 0.797085i −0.917150 0.398542i \(-0.869516\pi\)
0.917150 0.398542i \(-0.130484\pi\)
\(90\) 0 0
\(91\) 28983.8i 0.0384620i
\(92\) 0 0
\(93\) 2.07962e6 2.58544
\(94\) 0 0
\(95\) 188753. + 1.26725e6i 0.220152 + 1.47806i
\(96\) 0 0
\(97\) 851962.i 0.933480i −0.884395 0.466740i \(-0.845428\pi\)
0.884395 0.466740i \(-0.154572\pi\)
\(98\) 0 0
\(99\) 1.42966e6 1.47342
\(100\) 0 0
\(101\) −1.73735e6 −1.68625 −0.843127 0.537714i \(-0.819288\pi\)
−0.843127 + 0.537714i \(0.819288\pi\)
\(102\) 0 0
\(103\) 1.34950e6i 1.23498i −0.786577 0.617492i \(-0.788149\pi\)
0.786577 0.617492i \(-0.211851\pi\)
\(104\) 0 0
\(105\) 1.52440e6i 1.31684i
\(106\) 0 0
\(107\) 1.32161e6i 1.07883i −0.842042 0.539413i \(-0.818646\pi\)
0.842042 0.539413i \(-0.181354\pi\)
\(108\) 0 0
\(109\) 1.99638e6i 1.54157i 0.637093 + 0.770787i \(0.280137\pi\)
−0.637093 + 0.770787i \(0.719863\pi\)
\(110\) 0 0
\(111\) 3.81795e6 2.79165
\(112\) 0 0
\(113\) 1.42114e6i 0.984920i −0.870335 0.492460i \(-0.836098\pi\)
0.870335 0.492460i \(-0.163902\pi\)
\(114\) 0 0
\(115\) −1.44526e6 −0.950282
\(116\) 0 0
\(117\) 128725.i 0.0803721i
\(118\) 0 0
\(119\) 965182. 0.572754
\(120\) 0 0
\(121\) 760581. 0.429328
\(122\) 0 0
\(123\) −3.17678e6 −1.70715
\(124\) 0 0
\(125\) −680431. −0.348381
\(126\) 0 0
\(127\) 3.62081e6i 1.76764i 0.467823 + 0.883822i \(0.345039\pi\)
−0.467823 + 0.883822i \(0.654961\pi\)
\(128\) 0 0
\(129\) 5.20593e6i 2.42510i
\(130\) 0 0
\(131\) −4.12259e6 −1.83382 −0.916909 0.399096i \(-0.869324\pi\)
−0.916909 + 0.399096i \(0.869324\pi\)
\(132\) 0 0
\(133\) −204412. 1.37239e6i −0.0868864 0.583340i
\(134\) 0 0
\(135\) 1.27682e6i 0.518952i
\(136\) 0 0
\(137\) 3.08355e6 1.19920 0.599598 0.800302i \(-0.295327\pi\)
0.599598 + 0.800302i \(0.295327\pi\)
\(138\) 0 0
\(139\) −1.59964e6 −0.595631 −0.297816 0.954623i \(-0.596258\pi\)
−0.297816 + 0.954623i \(0.596258\pi\)
\(140\) 0 0
\(141\) 3.92202e6i 1.39911i
\(142\) 0 0
\(143\) 227992.i 0.0779671i
\(144\) 0 0
\(145\) 1.64767e6i 0.540464i
\(146\) 0 0
\(147\) 3.09527e6i 0.974420i
\(148\) 0 0
\(149\) 3.38080e6 1.02202 0.511011 0.859574i \(-0.329271\pi\)
0.511011 + 0.859574i \(0.329271\pi\)
\(150\) 0 0
\(151\) 3.08302e6i 0.895458i 0.894169 + 0.447729i \(0.147767\pi\)
−0.894169 + 0.447729i \(0.852233\pi\)
\(152\) 0 0
\(153\) −4.28664e6 −1.19686
\(154\) 0 0
\(155\) 9.62938e6i 2.58585i
\(156\) 0 0
\(157\) 3.24712e6 0.839072 0.419536 0.907739i \(-0.362193\pi\)
0.419536 + 0.907739i \(0.362193\pi\)
\(158\) 0 0
\(159\) 1.86151e6 0.463099
\(160\) 0 0
\(161\) 1.56516e6 0.375044
\(162\) 0 0
\(163\) −2.32461e6 −0.536769 −0.268385 0.963312i \(-0.586490\pi\)
−0.268385 + 0.963312i \(0.586490\pi\)
\(164\) 0 0
\(165\) 1.19912e7i 2.66939i
\(166\) 0 0
\(167\) 274073.i 0.0588460i −0.999567 0.0294230i \(-0.990633\pi\)
0.999567 0.0294230i \(-0.00936698\pi\)
\(168\) 0 0
\(169\) 4.80628e6 0.995747
\(170\) 0 0
\(171\) 907850. + 6.09514e6i 0.181562 + 1.21898i
\(172\) 0 0
\(173\) 2.32755e6i 0.449532i 0.974413 + 0.224766i \(0.0721617\pi\)
−0.974413 + 0.224766i \(0.927838\pi\)
\(174\) 0 0
\(175\) 3.89771e6 0.727269
\(176\) 0 0
\(177\) −2.49026e6 −0.449081
\(178\) 0 0
\(179\) 6.83222e6i 1.19125i −0.803263 0.595625i \(-0.796905\pi\)
0.803263 0.595625i \(-0.203095\pi\)
\(180\) 0 0
\(181\) 9.40294e6i 1.58573i −0.609400 0.792863i \(-0.708590\pi\)
0.609400 0.792863i \(-0.291410\pi\)
\(182\) 0 0
\(183\) 3.45460e6i 0.563696i
\(184\) 0 0
\(185\) 1.76785e7i 2.79209i
\(186\) 0 0
\(187\) −7.59229e6 −1.16104
\(188\) 0 0
\(189\) 1.38275e6i 0.204813i
\(190\) 0 0
\(191\) −1.12644e7 −1.61662 −0.808309 0.588759i \(-0.799617\pi\)
−0.808309 + 0.588759i \(0.799617\pi\)
\(192\) 0 0
\(193\) 1.26456e6i 0.175901i 0.996125 + 0.0879504i \(0.0280317\pi\)
−0.996125 + 0.0879504i \(0.971968\pi\)
\(194\) 0 0
\(195\) 1.07968e6 0.145610
\(196\) 0 0
\(197\) −5.92683e6 −0.775218 −0.387609 0.921824i \(-0.626699\pi\)
−0.387609 + 0.921824i \(0.626699\pi\)
\(198\) 0 0
\(199\) 5.49979e6 0.697889 0.348945 0.937143i \(-0.386540\pi\)
0.348945 + 0.937143i \(0.386540\pi\)
\(200\) 0 0
\(201\) 1.97853e6 0.243643
\(202\) 0 0
\(203\) 1.78437e6i 0.213303i
\(204\) 0 0
\(205\) 1.47096e7i 1.70742i
\(206\) 0 0
\(207\) −6.95131e6 −0.783711
\(208\) 0 0
\(209\) 1.60794e6 + 1.07954e7i 0.176129 + 1.18250i
\(210\) 0 0
\(211\) 1.20580e7i 1.28359i −0.766876 0.641795i \(-0.778190\pi\)
0.766876 0.641795i \(-0.221810\pi\)
\(212\) 0 0
\(213\) −1.64802e7 −1.70539
\(214\) 0 0
\(215\) 2.41053e7 2.42548
\(216\) 0 0
\(217\) 1.04283e7i 1.02055i
\(218\) 0 0
\(219\) 5.24556e6i 0.499412i
\(220\) 0 0
\(221\) 683603.i 0.0633326i
\(222\) 0 0
\(223\) 1.50611e7i 1.35813i 0.734077 + 0.679066i \(0.237615\pi\)
−0.734077 + 0.679066i \(0.762385\pi\)
\(224\) 0 0
\(225\) −1.73108e7 −1.51974
\(226\) 0 0
\(227\) 1.46485e7i 1.25232i 0.779694 + 0.626161i \(0.215375\pi\)
−0.779694 + 0.626161i \(0.784625\pi\)
\(228\) 0 0
\(229\) 1.69472e7 1.41121 0.705603 0.708607i \(-0.250676\pi\)
0.705603 + 0.708607i \(0.250676\pi\)
\(230\) 0 0
\(231\) 1.29860e7i 1.05352i
\(232\) 0 0
\(233\) −3.81937e6 −0.301942 −0.150971 0.988538i \(-0.548240\pi\)
−0.150971 + 0.988538i \(0.548240\pi\)
\(234\) 0 0
\(235\) 1.81604e7 1.39933
\(236\) 0 0
\(237\) 4.31420e6 0.324082
\(238\) 0 0
\(239\) 2.51010e7 1.83864 0.919322 0.393507i \(-0.128738\pi\)
0.919322 + 0.393507i \(0.128738\pi\)
\(240\) 0 0
\(241\) 7.55161e6i 0.539496i −0.962931 0.269748i \(-0.913060\pi\)
0.962931 0.269748i \(-0.0869404\pi\)
\(242\) 0 0
\(243\) 2.02810e7i 1.41342i
\(244\) 0 0
\(245\) 1.43322e7 0.974574
\(246\) 0 0
\(247\) 972011. 144778.i 0.0645031 0.00960750i
\(248\) 0 0
\(249\) 494519.i 0.0320321i
\(250\) 0 0
\(251\) −2.68658e7 −1.69894 −0.849471 0.527635i \(-0.823079\pi\)
−0.849471 + 0.527635i \(0.823079\pi\)
\(252\) 0 0
\(253\) −1.23118e7 −0.760259
\(254\) 0 0
\(255\) 3.59541e7i 2.16834i
\(256\) 0 0
\(257\) 1.72065e7i 1.01366i −0.862046 0.506831i \(-0.830817\pi\)
0.862046 0.506831i \(-0.169183\pi\)
\(258\) 0 0
\(259\) 1.91451e7i 1.10194i
\(260\) 0 0
\(261\) 7.92486e6i 0.445728i
\(262\) 0 0
\(263\) −7.35861e6 −0.404510 −0.202255 0.979333i \(-0.564827\pi\)
−0.202255 + 0.979333i \(0.564827\pi\)
\(264\) 0 0
\(265\) 8.61947e6i 0.463173i
\(266\) 0 0
\(267\) 2.26687e7 1.19095
\(268\) 0 0
\(269\) 2.68576e7i 1.37978i 0.723913 + 0.689891i \(0.242342\pi\)
−0.723913 + 0.689891i \(0.757658\pi\)
\(270\) 0 0
\(271\) −1.84648e6 −0.0927762 −0.0463881 0.998923i \(-0.514771\pi\)
−0.0463881 + 0.998923i \(0.514771\pi\)
\(272\) 0 0
\(273\) −1.16925e6 −0.0574672
\(274\) 0 0
\(275\) −3.06600e7 −1.47426
\(276\) 0 0
\(277\) 1.00493e7 0.472819 0.236410 0.971653i \(-0.424029\pi\)
0.236410 + 0.971653i \(0.424029\pi\)
\(278\) 0 0
\(279\) 4.63147e7i 2.13259i
\(280\) 0 0
\(281\) 1.79351e7i 0.808321i 0.914688 + 0.404160i \(0.132436\pi\)
−0.914688 + 0.404160i \(0.867564\pi\)
\(282\) 0 0
\(283\) −3.11035e7 −1.37230 −0.686151 0.727460i \(-0.740701\pi\)
−0.686151 + 0.727460i \(0.740701\pi\)
\(284\) 0 0
\(285\) −5.11229e7 + 7.61457e6i −2.20841 + 0.328936i
\(286\) 0 0
\(287\) 1.59300e7i 0.673860i
\(288\) 0 0
\(289\) −1.37309e6 −0.0568860
\(290\) 0 0
\(291\) 3.43694e7 1.39474
\(292\) 0 0
\(293\) 2.91396e7i 1.15846i 0.815165 + 0.579229i \(0.196646\pi\)
−0.815165 + 0.579229i \(0.803354\pi\)
\(294\) 0 0
\(295\) 1.15308e7i 0.449152i
\(296\) 0 0
\(297\) 1.08769e7i 0.415180i
\(298\) 0 0
\(299\) 1.10855e6i 0.0414706i
\(300\) 0 0
\(301\) −2.61052e7 −0.957254
\(302\) 0 0
\(303\) 7.00873e7i 2.51948i
\(304\) 0 0
\(305\) 1.59961e7 0.563785
\(306\) 0 0
\(307\) 4.02183e7i 1.38998i 0.719020 + 0.694989i \(0.244591\pi\)
−0.719020 + 0.694989i \(0.755409\pi\)
\(308\) 0 0
\(309\) 5.44409e7 1.84523
\(310\) 0 0
\(311\) −1.73795e7 −0.577771 −0.288885 0.957364i \(-0.593285\pi\)
−0.288885 + 0.957364i \(0.593285\pi\)
\(312\) 0 0
\(313\) 3.25313e7 1.06089 0.530443 0.847721i \(-0.322026\pi\)
0.530443 + 0.847721i \(0.322026\pi\)
\(314\) 0 0
\(315\) 3.39497e7 1.08618
\(316\) 0 0
\(317\) 2.86504e7i 0.899400i 0.893180 + 0.449700i \(0.148469\pi\)
−0.893180 + 0.449700i \(0.851531\pi\)
\(318\) 0 0
\(319\) 1.40361e7i 0.432390i
\(320\) 0 0
\(321\) 5.33156e7 1.61190
\(322\) 0 0
\(323\) −4.82120e6 3.23687e7i −0.143070 0.960544i
\(324\) 0 0
\(325\) 2.76060e6i 0.0804181i
\(326\) 0 0
\(327\) −8.05371e7 −2.30331
\(328\) 0 0
\(329\) −1.96670e7 −0.552269
\(330\) 0 0
\(331\) 4.74259e7i 1.30777i 0.756594 + 0.653885i \(0.226862\pi\)
−0.756594 + 0.653885i \(0.773138\pi\)
\(332\) 0 0
\(333\) 8.50287e7i 2.30268i
\(334\) 0 0
\(335\) 9.16129e6i 0.243681i
\(336\) 0 0
\(337\) 2.20894e7i 0.577157i −0.957456 0.288579i \(-0.906817\pi\)
0.957456 0.288579i \(-0.0931827\pi\)
\(338\) 0 0
\(339\) 5.73309e7 1.47160
\(340\) 0 0
\(341\) 8.20305e7i 2.06877i
\(342\) 0 0
\(343\) −3.93208e7 −0.974406
\(344\) 0 0
\(345\) 5.83039e7i 1.41984i
\(346\) 0 0
\(347\) 2.73262e7 0.654019 0.327009 0.945021i \(-0.393959\pi\)
0.327009 + 0.945021i \(0.393959\pi\)
\(348\) 0 0
\(349\) 4.12946e7 0.971442 0.485721 0.874114i \(-0.338557\pi\)
0.485721 + 0.874114i \(0.338557\pi\)
\(350\) 0 0
\(351\) 979348. 0.0226473
\(352\) 0 0
\(353\) −5.44872e7 −1.23871 −0.619356 0.785110i \(-0.712606\pi\)
−0.619356 + 0.785110i \(0.712606\pi\)
\(354\) 0 0
\(355\) 7.63092e7i 1.70566i
\(356\) 0 0
\(357\) 3.89369e7i 0.855770i
\(358\) 0 0
\(359\) −1.80050e7 −0.389144 −0.194572 0.980888i \(-0.562332\pi\)
−0.194572 + 0.980888i \(0.562332\pi\)
\(360\) 0 0
\(361\) −4.50038e7 + 1.37105e7i −0.956593 + 0.291428i
\(362\) 0 0
\(363\) 3.06830e7i 0.641472i
\(364\) 0 0
\(365\) 2.42888e7 0.499491
\(366\) 0 0
\(367\) −8.33600e7 −1.68640 −0.843198 0.537603i \(-0.819330\pi\)
−0.843198 + 0.537603i \(0.819330\pi\)
\(368\) 0 0
\(369\) 7.07494e7i 1.40813i
\(370\) 0 0
\(371\) 9.33456e6i 0.182798i
\(372\) 0 0
\(373\) 2.46066e7i 0.474159i −0.971490 0.237080i \(-0.923810\pi\)
0.971490 0.237080i \(-0.0761902\pi\)
\(374\) 0 0
\(375\) 2.74496e7i 0.520526i
\(376\) 0 0
\(377\) −1.26380e6 −0.0235860
\(378\) 0 0
\(379\) 1.80987e7i 0.332452i −0.986088 0.166226i \(-0.946842\pi\)
0.986088 0.166226i \(-0.0531581\pi\)
\(380\) 0 0
\(381\) −1.46069e8 −2.64109
\(382\) 0 0
\(383\) 4.68217e7i 0.833395i −0.909045 0.416697i \(-0.863188\pi\)
0.909045 0.416697i \(-0.136812\pi\)
\(384\) 0 0
\(385\) 6.01300e7 1.05368
\(386\) 0 0
\(387\) 1.15940e8 2.00033
\(388\) 0 0
\(389\) −2.49752e7 −0.424288 −0.212144 0.977238i \(-0.568045\pi\)
−0.212144 + 0.977238i \(0.568045\pi\)
\(390\) 0 0
\(391\) 3.69154e7 0.617558
\(392\) 0 0
\(393\) 1.66312e8i 2.73996i
\(394\) 0 0
\(395\) 1.99763e7i 0.324134i
\(396\) 0 0
\(397\) 1.03759e6 0.0165827 0.00829133 0.999966i \(-0.497361\pi\)
0.00829133 + 0.999966i \(0.497361\pi\)
\(398\) 0 0
\(399\) 5.53642e7 8.24630e6i 0.871586 0.129820i
\(400\) 0 0
\(401\) 1.02883e8i 1.59555i −0.602953 0.797777i \(-0.706009\pi\)
0.602953 0.797777i \(-0.293991\pi\)
\(402\) 0 0
\(403\) 7.38595e6 0.112847
\(404\) 0 0
\(405\) 7.08352e7 1.06631
\(406\) 0 0
\(407\) 1.50599e8i 2.23377i
\(408\) 0 0
\(409\) 7.78751e7i 1.13823i −0.822259 0.569113i \(-0.807287\pi\)
0.822259 0.569113i \(-0.192713\pi\)
\(410\) 0 0
\(411\) 1.24395e8i 1.79175i
\(412\) 0 0
\(413\) 1.24874e7i 0.177265i
\(414\) 0 0
\(415\) 2.28980e6 0.0320371
\(416\) 0 0
\(417\) 6.45318e7i 0.889950i
\(418\) 0 0
\(419\) −1.20628e8 −1.63986 −0.819932 0.572461i \(-0.805989\pi\)
−0.819932 + 0.572461i \(0.805989\pi\)
\(420\) 0 0
\(421\) 9.94059e7i 1.33219i −0.745867 0.666095i \(-0.767965\pi\)
0.745867 0.666095i \(-0.232035\pi\)
\(422\) 0 0
\(423\) 8.73465e7 1.15405
\(424\) 0 0
\(425\) 9.19301e7 1.19754
\(426\) 0 0
\(427\) −1.73231e7 −0.222507
\(428\) 0 0
\(429\) 9.19753e6 0.116493
\(430\) 0 0
\(431\) 3.81811e7i 0.476888i −0.971156 0.238444i \(-0.923363\pi\)
0.971156 0.238444i \(-0.0766373\pi\)
\(432\) 0 0
\(433\) 1.37273e8i 1.69091i 0.534043 + 0.845457i \(0.320672\pi\)
−0.534043 + 0.845457i \(0.679328\pi\)
\(434\) 0 0
\(435\) 6.64696e7 0.807524
\(436\) 0 0
\(437\) −7.81817e6 5.24898e7i −0.0936830 0.628971i
\(438\) 0 0
\(439\) 9.17437e6i 0.108438i 0.998529 + 0.0542192i \(0.0172670\pi\)
−0.998529 + 0.0542192i \(0.982733\pi\)
\(440\) 0 0
\(441\) 6.89341e7 0.803745
\(442\) 0 0
\(443\) 1.79271e6 0.0206204 0.0103102 0.999947i \(-0.496718\pi\)
0.0103102 + 0.999947i \(0.496718\pi\)
\(444\) 0 0
\(445\) 1.04964e8i 1.19114i
\(446\) 0 0
\(447\) 1.36387e8i 1.52703i
\(448\) 0 0
\(449\) 6.43497e6i 0.0710898i 0.999368 + 0.0355449i \(0.0113167\pi\)
−0.999368 + 0.0355449i \(0.988683\pi\)
\(450\) 0 0
\(451\) 1.25308e8i 1.36600i
\(452\) 0 0
\(453\) −1.24374e8 −1.33793
\(454\) 0 0
\(455\) 5.41406e6i 0.0574763i
\(456\) 0 0
\(457\) −1.28044e8 −1.34156 −0.670779 0.741658i \(-0.734040\pi\)
−0.670779 + 0.741658i \(0.734040\pi\)
\(458\) 0 0
\(459\) 3.26130e7i 0.337250i
\(460\) 0 0
\(461\) 3.10758e7 0.317190 0.158595 0.987344i \(-0.449304\pi\)
0.158595 + 0.987344i \(0.449304\pi\)
\(462\) 0 0
\(463\) 8.91490e7 0.898201 0.449100 0.893481i \(-0.351745\pi\)
0.449100 + 0.893481i \(0.351745\pi\)
\(464\) 0 0
\(465\) −3.88464e8 −3.86359
\(466\) 0 0
\(467\) −2.26320e7 −0.222214 −0.111107 0.993808i \(-0.535440\pi\)
−0.111107 + 0.993808i \(0.535440\pi\)
\(468\) 0 0
\(469\) 9.92134e6i 0.0961727i
\(470\) 0 0
\(471\) 1.30994e8i 1.25368i
\(472\) 0 0
\(473\) 2.05348e8 1.94047
\(474\) 0 0
\(475\) −1.94695e7 1.30715e8i −0.181666 1.21967i
\(476\) 0 0
\(477\) 4.14573e7i 0.381985i
\(478\) 0 0
\(479\) −1.34746e8 −1.22605 −0.613026 0.790062i \(-0.710048\pi\)
−0.613026 + 0.790062i \(0.710048\pi\)
\(480\) 0 0
\(481\) 1.35598e7 0.121848
\(482\) 0 0
\(483\) 6.31410e7i 0.560364i
\(484\) 0 0
\(485\) 1.59143e8i 1.39496i
\(486\) 0 0
\(487\) 4.94767e7i 0.428365i −0.976794 0.214182i \(-0.931291\pi\)
0.976794 0.214182i \(-0.0687087\pi\)
\(488\) 0 0
\(489\) 9.37783e7i 0.802003i
\(490\) 0 0
\(491\) 1.86984e8 1.57965 0.789825 0.613333i \(-0.210172\pi\)
0.789825 + 0.613333i \(0.210172\pi\)
\(492\) 0 0
\(493\) 4.20855e7i 0.351230i
\(494\) 0 0
\(495\) −2.67054e8 −2.20183
\(496\) 0 0
\(497\) 8.26400e7i 0.673165i
\(498\) 0 0
\(499\) −6.68560e6 −0.0538070 −0.0269035 0.999638i \(-0.508565\pi\)
−0.0269035 + 0.999638i \(0.508565\pi\)
\(500\) 0 0
\(501\) 1.10565e7 0.0879236
\(502\) 0 0
\(503\) −1.07046e8 −0.841139 −0.420570 0.907260i \(-0.638170\pi\)
−0.420570 + 0.907260i \(0.638170\pi\)
\(504\) 0 0
\(505\) 3.24529e8 2.51988
\(506\) 0 0
\(507\) 1.93893e8i 1.48778i
\(508\) 0 0
\(509\) 8.78261e7i 0.665994i −0.942928 0.332997i \(-0.891940\pi\)
0.942928 0.332997i \(-0.108060\pi\)
\(510\) 0 0
\(511\) −2.63039e7 −0.197132
\(512\) 0 0
\(513\) −4.63722e7 + 6.90697e6i −0.343483 + 0.0511606i
\(514\) 0 0
\(515\) 2.52081e8i 1.84552i
\(516\) 0 0
\(517\) 1.54704e8 1.11952
\(518\) 0 0
\(519\) −9.38968e7 −0.671659
\(520\) 0 0
\(521\) 5.29466e7i 0.374391i 0.982323 + 0.187195i \(0.0599397\pi\)
−0.982323 + 0.187195i \(0.940060\pi\)
\(522\) 0 0
\(523\) 2.04408e8i 1.42887i 0.699701 + 0.714436i \(0.253317\pi\)
−0.699701 + 0.714436i \(0.746683\pi\)
\(524\) 0 0
\(525\) 1.57239e8i 1.08663i
\(526\) 0 0
\(527\) 2.45958e8i 1.68046i
\(528\) 0 0
\(529\) −8.81730e7 −0.595619
\(530\) 0 0
\(531\) 5.54600e7i 0.370422i
\(532\) 0 0
\(533\) −1.12826e7 −0.0745124
\(534\) 0 0
\(535\) 2.46871e8i 1.61216i
\(536\) 0 0
\(537\) 2.75622e8 1.77988
\(538\) 0 0
\(539\) 1.22093e8 0.779693
\(540\) 0 0
\(541\) 1.07904e7 0.0681469 0.0340735 0.999419i \(-0.489152\pi\)
0.0340735 + 0.999419i \(0.489152\pi\)
\(542\) 0 0
\(543\) 3.79329e8 2.36928
\(544\) 0 0
\(545\) 3.72916e8i 2.30368i
\(546\) 0 0
\(547\) 5.89052e7i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(548\) 0 0
\(549\) 7.69368e7 0.464961
\(550\) 0 0
\(551\) 5.98411e7 8.91313e6i 0.357722 0.0532814i
\(552\) 0 0
\(553\) 2.16336e7i 0.127924i
\(554\) 0 0
\(555\) −7.13176e8 −4.17175
\(556\) 0 0
\(557\) 4.50709e7 0.260814 0.130407 0.991461i \(-0.458372\pi\)
0.130407 + 0.991461i \(0.458372\pi\)
\(558\) 0 0
\(559\) 1.84893e7i 0.105849i
\(560\) 0 0
\(561\) 3.06285e8i 1.73475i
\(562\) 0 0
\(563\) 2.83179e8i 1.58685i −0.608670 0.793424i \(-0.708297\pi\)
0.608670 0.793424i \(-0.291703\pi\)
\(564\) 0 0
\(565\) 2.65463e8i 1.47183i
\(566\) 0 0
\(567\) −7.67119e7 −0.420837
\(568\) 0 0
\(569\) 2.97743e8i 1.61624i −0.589021 0.808118i \(-0.700486\pi\)
0.589021 0.808118i \(-0.299514\pi\)
\(570\) 0 0
\(571\) 4.51806e7 0.242685 0.121343 0.992611i \(-0.461280\pi\)
0.121343 + 0.992611i \(0.461280\pi\)
\(572\) 0 0
\(573\) 4.54422e8i 2.41544i
\(574\) 0 0
\(575\) 1.49076e8 0.784159
\(576\) 0 0
\(577\) −1.15048e8 −0.598896 −0.299448 0.954113i \(-0.596802\pi\)
−0.299448 + 0.954113i \(0.596802\pi\)
\(578\) 0 0
\(579\) −5.10143e7 −0.262819
\(580\) 0 0
\(581\) −2.47977e6 −0.0126440
\(582\) 0 0
\(583\) 7.34273e7i 0.370554i
\(584\) 0 0
\(585\) 2.40453e7i 0.120105i
\(586\) 0 0
\(587\) 1.59622e8 0.789183 0.394592 0.918857i \(-0.370886\pi\)
0.394592 + 0.918857i \(0.370886\pi\)
\(588\) 0 0
\(589\) −3.49726e8 + 5.20904e7i −1.71152 + 0.254925i
\(590\) 0 0
\(591\) 2.39097e8i 1.15828i
\(592\) 0 0
\(593\) −6.60997e7 −0.316983 −0.158491 0.987360i \(-0.550663\pi\)
−0.158491 + 0.987360i \(0.550663\pi\)
\(594\) 0 0
\(595\) −1.80292e8 −0.855905
\(596\) 0 0
\(597\) 2.21870e8i 1.04274i
\(598\) 0 0
\(599\) 1.30989e8i 0.609475i −0.952436 0.304738i \(-0.901431\pi\)
0.952436 0.304738i \(-0.0985688\pi\)
\(600\) 0 0
\(601\) 2.96588e8i 1.36625i 0.730303 + 0.683124i \(0.239379\pi\)
−0.730303 + 0.683124i \(0.760621\pi\)
\(602\) 0 0
\(603\) 4.40634e7i 0.200967i
\(604\) 0 0
\(605\) −1.42073e8 −0.641573
\(606\) 0 0
\(607\) 7.29083e7i 0.325995i 0.986626 + 0.162997i \(0.0521162\pi\)
−0.986626 + 0.162997i \(0.947884\pi\)
\(608\) 0 0
\(609\) −7.19841e7 −0.318702
\(610\) 0 0
\(611\) 1.39294e7i 0.0610674i
\(612\) 0 0
\(613\) 3.92971e8 1.70600 0.853000 0.521910i \(-0.174780\pi\)
0.853000 + 0.521910i \(0.174780\pi\)
\(614\) 0 0
\(615\) 5.93409e8 2.55111
\(616\) 0 0
\(617\) −2.27576e8 −0.968884 −0.484442 0.874823i \(-0.660977\pi\)
−0.484442 + 0.874823i \(0.660977\pi\)
\(618\) 0 0
\(619\) −1.93252e8 −0.814804 −0.407402 0.913249i \(-0.633565\pi\)
−0.407402 + 0.913249i \(0.633565\pi\)
\(620\) 0 0
\(621\) 5.28860e7i 0.220834i
\(622\) 0 0
\(623\) 1.13672e8i 0.470101i
\(624\) 0 0
\(625\) −1.73955e8 −0.712521
\(626\) 0 0
\(627\) −4.35504e8 + 6.48668e7i −1.76681 + 0.263160i
\(628\) 0 0
\(629\) 4.51551e8i 1.81449i
\(630\) 0 0
\(631\) 1.93900e7 0.0771773 0.0385887 0.999255i \(-0.487714\pi\)
0.0385887 + 0.999255i \(0.487714\pi\)
\(632\) 0 0
\(633\) 4.86436e8 1.91785
\(634\) 0 0
\(635\) 6.76353e8i 2.64151i
\(636\) 0 0
\(637\) 1.09931e7i 0.0425307i
\(638\) 0 0
\(639\) 3.67027e8i 1.40668i
\(640\) 0 0
\(641\) 3.28211e8i 1.24617i 0.782152 + 0.623087i \(0.214122\pi\)
−0.782152 + 0.623087i \(0.785878\pi\)
\(642\) 0 0
\(643\) 2.37495e7 0.0893349 0.0446675 0.999002i \(-0.485777\pi\)
0.0446675 + 0.999002i \(0.485777\pi\)
\(644\) 0 0
\(645\) 9.72446e8i 3.62398i
\(646\) 0 0
\(647\) 1.17583e7 0.0434143 0.0217072 0.999764i \(-0.493090\pi\)
0.0217072 + 0.999764i \(0.493090\pi\)
\(648\) 0 0
\(649\) 9.82282e7i 0.359337i
\(650\) 0 0
\(651\) 4.20692e8 1.52483
\(652\) 0 0
\(653\) 1.27271e8 0.457079 0.228539 0.973535i \(-0.426605\pi\)
0.228539 + 0.973535i \(0.426605\pi\)
\(654\) 0 0
\(655\) 7.70082e8 2.74040
\(656\) 0 0
\(657\) 1.16823e8 0.411937
\(658\) 0 0
\(659\) 2.32452e8i 0.812227i 0.913823 + 0.406114i \(0.133116\pi\)
−0.913823 + 0.406114i \(0.866884\pi\)
\(660\) 0 0
\(661\) 3.14463e8i 1.08884i −0.838812 0.544421i \(-0.816750\pi\)
0.838812 0.544421i \(-0.183250\pi\)
\(662\) 0 0
\(663\) −2.75776e7 −0.0946271
\(664\) 0 0
\(665\) 3.81833e7 + 2.56356e8i 0.129840 + 0.871723i
\(666\) 0 0
\(667\) 6.82469e7i 0.229988i
\(668\) 0 0
\(669\) −6.07587e8 −2.02923
\(670\) 0 0
\(671\) 1.36267e8 0.451048
\(672\) 0 0
\(673\) 2.76451e8i 0.906929i −0.891274 0.453465i \(-0.850188\pi\)
0.891274 0.453465i \(-0.149812\pi\)
\(674\) 0 0
\(675\) 1.31701e8i 0.428232i
\(676\) 0 0
\(677\) 4.08200e8i 1.31555i 0.753215 + 0.657775i \(0.228502\pi\)
−0.753215 + 0.657775i \(0.771498\pi\)
\(678\) 0 0
\(679\) 1.72346e8i 0.550543i
\(680\) 0 0
\(681\) −5.90943e8 −1.87113
\(682\) 0 0
\(683\) 3.61070e8i 1.13326i −0.823973 0.566630i \(-0.808247\pi\)
0.823973 0.566630i \(-0.191753\pi\)
\(684\) 0 0
\(685\) −5.75995e8 −1.79204
\(686\) 0 0
\(687\) 6.83674e8i 2.10853i
\(688\) 0 0
\(689\) 6.61133e6 0.0202130
\(690\) 0 0
\(691\) 3.47750e8 1.05398 0.526991 0.849871i \(-0.323320\pi\)
0.526991 + 0.849871i \(0.323320\pi\)
\(692\) 0 0
\(693\) 2.89209e8 0.868986
\(694\) 0 0
\(695\) 2.98806e8 0.890091
\(696\) 0 0
\(697\) 3.75720e8i 1.10960i
\(698\) 0 0
\(699\) 1.54079e8i 0.451141i
\(700\) 0 0
\(701\) −8.95408e7 −0.259936 −0.129968 0.991518i \(-0.541487\pi\)
−0.129968 + 0.991518i \(0.541487\pi\)
\(702\) 0 0
\(703\) −6.42057e8 + 9.56321e7i −1.84802 + 0.275257i
\(704\) 0 0
\(705\) 7.32617e8i 2.09079i
\(706\) 0 0
\(707\) −3.51453e8 −0.994511
\(708\) 0 0
\(709\) 1.93298e8 0.542362 0.271181 0.962528i \(-0.412586\pi\)
0.271181 + 0.962528i \(0.412586\pi\)
\(710\) 0 0
\(711\) 9.60807e7i 0.267317i
\(712\) 0 0
\(713\) 3.98851e8i 1.10038i
\(714\) 0 0
\(715\) 4.25879e7i 0.116511i
\(716\) 0 0
\(717\) 1.01261e9i 2.74717i
\(718\) 0 0
\(719\) −3.52698e8 −0.948891 −0.474445 0.880285i \(-0.657351\pi\)
−0.474445 + 0.880285i \(0.657351\pi\)
\(720\) 0 0
\(721\) 2.72994e8i 0.728363i
\(722\) 0 0
\(723\) 3.04643e8 0.806077
\(724\) 0 0
\(725\) 1.69954e8i 0.445983i
\(726\) 0 0
\(727\) −2.02827e8 −0.527865 −0.263932 0.964541i \(-0.585020\pi\)
−0.263932 + 0.964541i \(0.585020\pi\)
\(728\) 0 0
\(729\) 5.41719e8 1.39827
\(730\) 0 0
\(731\) −6.15709e8 −1.57624
\(732\) 0 0
\(733\) −4.60785e8 −1.17000 −0.585001 0.811033i \(-0.698906\pi\)
−0.585001 + 0.811033i \(0.698906\pi\)
\(734\) 0 0
\(735\) 5.78183e8i 1.45614i
\(736\) 0 0
\(737\) 7.80430e7i 0.194954i
\(738\) 0 0
\(739\) −2.61415e8 −0.647734 −0.323867 0.946103i \(-0.604983\pi\)
−0.323867 + 0.946103i \(0.604983\pi\)
\(740\) 0 0
\(741\) 5.84055e6 + 3.92124e7i 0.0143549 + 0.0963760i
\(742\) 0 0
\(743\) 3.25053e8i 0.792480i 0.918147 + 0.396240i \(0.129685\pi\)
−0.918147 + 0.396240i \(0.870315\pi\)
\(744\) 0 0
\(745\) −6.31519e8 −1.52728
\(746\) 0 0
\(747\) 1.10133e7 0.0264215
\(748\) 0 0
\(749\) 2.67352e8i 0.636264i
\(750\) 0 0
\(751\) 2.90899e8i 0.686787i 0.939192 + 0.343393i \(0.111576\pi\)
−0.939192 + 0.343393i \(0.888424\pi\)
\(752\) 0 0
\(753\) 1.08381e9i 2.53844i
\(754\) 0 0
\(755\) 5.75895e8i 1.33814i
\(756\) 0 0
\(757\) 7.60667e8 1.75350 0.876752 0.480943i \(-0.159706\pi\)
0.876752 + 0.480943i \(0.159706\pi\)
\(758\) 0 0
\(759\) 4.96678e8i 1.13593i
\(760\) 0 0
\(761\) −3.20532e8 −0.727306 −0.363653 0.931534i \(-0.618471\pi\)
−0.363653 + 0.931534i \(0.618471\pi\)
\(762\) 0 0
\(763\) 4.03854e8i 0.909182i
\(764\) 0 0
\(765\) 8.00726e8 1.78854
\(766\) 0 0
\(767\) −8.84438e6 −0.0196011
\(768\) 0 0
\(769\) 4.20733e8 0.925183 0.462592 0.886571i \(-0.346920\pi\)
0.462592 + 0.886571i \(0.346920\pi\)
\(770\) 0 0
\(771\) 6.94136e8 1.51454
\(772\) 0 0
\(773\) 6.21759e8i 1.34612i 0.739588 + 0.673060i \(0.235020\pi\)
−0.739588 + 0.673060i \(0.764980\pi\)
\(774\) 0 0
\(775\) 9.93253e8i 2.13381i
\(776\) 0 0
\(777\) 7.72343e8 1.64645
\(778\) 0 0
\(779\) 5.34233e8 7.95722e7i 1.13010 0.168325i
\(780\) 0 0
\(781\) 6.50061e8i 1.36459i
\(782\) 0 0
\(783\) 6.02928e7 0.125597
\(784\) 0 0
\(785\) −6.06548e8 −1.25388
\(786\) 0 0
\(787\) 1.97860e7i 0.0405913i 0.999794 + 0.0202956i \(0.00646075\pi\)
−0.999794 + 0.0202956i \(0.993539\pi\)
\(788\) 0 0
\(789\) 2.96858e8i 0.604390i
\(790\) 0 0
\(791\) 2.87486e8i 0.580881i
\(792\) 0 0
\(793\) 1.22693e7i 0.0246038i
\(794\) 0 0
\(795\) −3.47722e8 −0.692040
\(796\) 0