Properties

Label 304.8.a.d.1.1
Level $304$
Weight $8$
Character 304.1
Self dual yes
Analytic conductor $94.965$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(94.9650477472\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(13.0797\) of defining polynomial
Character \(\chi\) \(=\) 304.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} +O(q^{10})\) \(q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} -252.042 q^{11} -1065.19 q^{13} +1012.17 q^{15} -18769.7 q^{17} +6859.00 q^{19} -2483.74 q^{21} -11559.9 q^{23} +19513.8 q^{25} +14134.4 q^{27} -46290.0 q^{29} -46848.2 q^{31} +816.422 q^{33} -239593. q^{35} -182916. q^{37} +3450.42 q^{39} +819661. q^{41} -477471. q^{43} +680098. q^{45} -992580. q^{47} -235611. q^{49} +60799.7 q^{51} -852516. q^{53} +78756.0 q^{55} -22217.9 q^{57} +1.92404e6 q^{59} +209564. q^{61} -1.66887e6 q^{63} +332843. q^{65} +2.32447e6 q^{67} +37445.1 q^{69} +5.37237e6 q^{71} -3.71614e6 q^{73} -63209.9 q^{75} -193257. q^{77} +1.30851e6 q^{79} +4.71424e6 q^{81} +6.51683e6 q^{83} +5.86502e6 q^{85} +149944. q^{87} +3.99586e6 q^{89} -816756. q^{91} +151753. q^{93} -2.14325e6 q^{95} -6.90391e6 q^{97} +548570. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + O(q^{10}) \) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + 3295q^{11} - 13427q^{13} + 34782q^{15} - 32256q^{17} + 13718q^{19} + 103797q^{21} + 82525q^{23} + 159919q^{25} + 75141q^{27} - 12749q^{29} - 258944q^{31} + 257052q^{33} + 448167q^{35} - 149260q^{37} - 889557q^{39} + 339130q^{41} + 83869q^{43} + 2097243q^{45} - 1471025q^{47} + 1105372q^{49} - 913437q^{51} - 945643q^{53} + 1736899q^{55} + 473271q^{57} + 969009q^{59} - 1506755q^{61} + 2791179q^{63} - 5445956q^{65} + 1848219q^{67} + 6834063q^{69} + 3417184q^{71} - 2499822q^{73} + 10079553q^{75} + 5025267q^{77} - 2636926q^{79} + 2491398q^{81} + 10059354q^{83} - 439425q^{85} + 2572923q^{87} - 3506160q^{89} - 19003851q^{91} - 15169884q^{93} + 1063145q^{95} + 5893526q^{97} + 11301453q^{99} + O(q^{100}) \)

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
\(3\) −3.23924 −0.0692657 −0.0346329 0.999400i \(-0.511026\pi\)
−0.0346329 + 0.999400i \(0.511026\pi\)
\(4\) 0 0
\(5\) −312.472 −1.11793 −0.558967 0.829190i \(-0.688802\pi\)
−0.558967 + 0.829190i \(0.688802\pi\)
\(6\) 0 0
\(7\) 766.767 0.844929 0.422465 0.906379i \(-0.361165\pi\)
0.422465 + 0.906379i \(0.361165\pi\)
\(8\) 0 0
\(9\) −2176.51 −0.995202
\(10\) 0 0
\(11\) −252.042 −0.0570950 −0.0285475 0.999592i \(-0.509088\pi\)
−0.0285475 + 0.999592i \(0.509088\pi\)
\(12\) 0 0
\(13\) −1065.19 −0.134471 −0.0672353 0.997737i \(-0.521418\pi\)
−0.0672353 + 0.997737i \(0.521418\pi\)
\(14\) 0 0
\(15\) 1012.17 0.0774345
\(16\) 0 0
\(17\) −18769.7 −0.926589 −0.463295 0.886204i \(-0.653333\pi\)
−0.463295 + 0.886204i \(0.653333\pi\)
\(18\) 0 0
\(19\) 6859.00 0.229416
\(20\) 0 0
\(21\) −2483.74 −0.0585246
\(22\) 0 0
\(23\) −11559.9 −0.198109 −0.0990546 0.995082i \(-0.531582\pi\)
−0.0990546 + 0.995082i \(0.531582\pi\)
\(24\) 0 0
\(25\) 19513.8 0.249777
\(26\) 0 0
\(27\) 14134.4 0.138199
\(28\) 0 0
\(29\) −46290.0 −0.352448 −0.176224 0.984350i \(-0.556388\pi\)
−0.176224 + 0.984350i \(0.556388\pi\)
\(30\) 0 0
\(31\) −46848.2 −0.282441 −0.141220 0.989978i \(-0.545103\pi\)
−0.141220 + 0.989978i \(0.545103\pi\)
\(32\) 0 0
\(33\) 816.422 0.00395472
\(34\) 0 0
\(35\) −239593. −0.944575
\(36\) 0 0
\(37\) −182916. −0.593672 −0.296836 0.954928i \(-0.595932\pi\)
−0.296836 + 0.954928i \(0.595932\pi\)
\(38\) 0 0
\(39\) 3450.42 0.00931420
\(40\) 0 0
\(41\) 819661. 1.85734 0.928669 0.370910i \(-0.120954\pi\)
0.928669 + 0.370910i \(0.120954\pi\)
\(42\) 0 0
\(43\) −477471. −0.915814 −0.457907 0.889000i \(-0.651401\pi\)
−0.457907 + 0.889000i \(0.651401\pi\)
\(44\) 0 0
\(45\) 680098. 1.11257
\(46\) 0 0
\(47\) −992580. −1.39451 −0.697257 0.716821i \(-0.745596\pi\)
−0.697257 + 0.716821i \(0.745596\pi\)
\(48\) 0 0
\(49\) −235611. −0.286095
\(50\) 0 0
\(51\) 60799.7 0.0641809
\(52\) 0 0
\(53\) −852516. −0.786569 −0.393285 0.919417i \(-0.628661\pi\)
−0.393285 + 0.919417i \(0.628661\pi\)
\(54\) 0 0
\(55\) 78756.0 0.0638284
\(56\) 0 0
\(57\) −22217.9 −0.0158906
\(58\) 0 0
\(59\) 1.92404e6 1.21964 0.609821 0.792539i \(-0.291241\pi\)
0.609821 + 0.792539i \(0.291241\pi\)
\(60\) 0 0
\(61\) 209564. 0.118212 0.0591062 0.998252i \(-0.481175\pi\)
0.0591062 + 0.998252i \(0.481175\pi\)
\(62\) 0 0
\(63\) −1.66887e6 −0.840875
\(64\) 0 0
\(65\) 332843. 0.150329
\(66\) 0 0
\(67\) 2.32447e6 0.944198 0.472099 0.881546i \(-0.343497\pi\)
0.472099 + 0.881546i \(0.343497\pi\)
\(68\) 0 0
\(69\) 37445.1 0.0137222
\(70\) 0 0
\(71\) 5.37237e6 1.78140 0.890700 0.454591i \(-0.150215\pi\)
0.890700 + 0.454591i \(0.150215\pi\)
\(72\) 0 0
\(73\) −3.71614e6 −1.11805 −0.559027 0.829150i \(-0.688825\pi\)
−0.559027 + 0.829150i \(0.688825\pi\)
\(74\) 0 0
\(75\) −63209.9 −0.0173010
\(76\) 0 0
\(77\) −193257. −0.0482412
\(78\) 0 0
\(79\) 1.30851e6 0.298596 0.149298 0.988792i \(-0.452299\pi\)
0.149298 + 0.988792i \(0.452299\pi\)
\(80\) 0 0
\(81\) 4.71424e6 0.985630
\(82\) 0 0
\(83\) 6.51683e6 1.25102 0.625508 0.780218i \(-0.284892\pi\)
0.625508 + 0.780218i \(0.284892\pi\)
\(84\) 0 0
\(85\) 5.86502e6 1.03587
\(86\) 0 0
\(87\) 149944. 0.0244125
\(88\) 0 0
\(89\) 3.99586e6 0.600822 0.300411 0.953810i \(-0.402876\pi\)
0.300411 + 0.953810i \(0.402876\pi\)
\(90\) 0 0
\(91\) −816756. −0.113618
\(92\) 0 0
\(93\) 151753. 0.0195635
\(94\) 0 0
\(95\) −2.14325e6 −0.256472
\(96\) 0 0
\(97\) −6.90391e6 −0.768058 −0.384029 0.923321i \(-0.625464\pi\)
−0.384029 + 0.923321i \(0.625464\pi\)
\(98\) 0 0
\(99\) 548570. 0.0568210
\(100\) 0 0
\(101\) 2.81610e6 0.271971 0.135986 0.990711i \(-0.456580\pi\)
0.135986 + 0.990711i \(0.456580\pi\)
\(102\) 0 0
\(103\) −7.86273e6 −0.708995 −0.354497 0.935057i \(-0.615348\pi\)
−0.354497 + 0.935057i \(0.615348\pi\)
\(104\) 0 0
\(105\) 776100. 0.0654267
\(106\) 0 0
\(107\) −611626. −0.0482662 −0.0241331 0.999709i \(-0.507683\pi\)
−0.0241331 + 0.999709i \(0.507683\pi\)
\(108\) 0 0
\(109\) 5.23210e6 0.386976 0.193488 0.981103i \(-0.438020\pi\)
0.193488 + 0.981103i \(0.438020\pi\)
\(110\) 0 0
\(111\) 592510. 0.0411211
\(112\) 0 0
\(113\) −3.58527e6 −0.233747 −0.116874 0.993147i \(-0.537287\pi\)
−0.116874 + 0.993147i \(0.537287\pi\)
\(114\) 0 0
\(115\) 3.61213e6 0.221473
\(116\) 0 0
\(117\) 2.31840e6 0.133825
\(118\) 0 0
\(119\) −1.43920e7 −0.782902
\(120\) 0 0
\(121\) −1.94236e7 −0.996740
\(122\) 0 0
\(123\) −2.65508e6 −0.128650
\(124\) 0 0
\(125\) 1.83144e7 0.838700
\(126\) 0 0
\(127\) 2.99465e7 1.29728 0.648639 0.761096i \(-0.275339\pi\)
0.648639 + 0.761096i \(0.275339\pi\)
\(128\) 0 0
\(129\) 1.54664e6 0.0634345
\(130\) 0 0
\(131\) 8.11292e6 0.315303 0.157651 0.987495i \(-0.449608\pi\)
0.157651 + 0.987495i \(0.449608\pi\)
\(132\) 0 0
\(133\) 5.25926e6 0.193840
\(134\) 0 0
\(135\) −4.41662e6 −0.154498
\(136\) 0 0
\(137\) 1.44157e7 0.478977 0.239488 0.970899i \(-0.423020\pi\)
0.239488 + 0.970899i \(0.423020\pi\)
\(138\) 0 0
\(139\) 3.77446e7 1.19207 0.596036 0.802957i \(-0.296741\pi\)
0.596036 + 0.802957i \(0.296741\pi\)
\(140\) 0 0
\(141\) 3.21520e6 0.0965920
\(142\) 0 0
\(143\) 268473. 0.00767759
\(144\) 0 0
\(145\) 1.44643e7 0.394013
\(146\) 0 0
\(147\) 763200. 0.0198165
\(148\) 0 0
\(149\) 7.30679e7 1.80957 0.904784 0.425872i \(-0.140033\pi\)
0.904784 + 0.425872i \(0.140033\pi\)
\(150\) 0 0
\(151\) 4.14056e7 0.978678 0.489339 0.872094i \(-0.337238\pi\)
0.489339 + 0.872094i \(0.337238\pi\)
\(152\) 0 0
\(153\) 4.08525e7 0.922144
\(154\) 0 0
\(155\) 1.46388e7 0.315750
\(156\) 0 0
\(157\) 2.93945e7 0.606203 0.303101 0.952958i \(-0.401978\pi\)
0.303101 + 0.952958i \(0.401978\pi\)
\(158\) 0 0
\(159\) 2.76150e6 0.0544823
\(160\) 0 0
\(161\) −8.86371e6 −0.167388
\(162\) 0 0
\(163\) −9.03924e6 −0.163484 −0.0817420 0.996654i \(-0.526048\pi\)
−0.0817420 + 0.996654i \(0.526048\pi\)
\(164\) 0 0
\(165\) −255109. −0.00442112
\(166\) 0 0
\(167\) 8.16728e7 1.35697 0.678484 0.734615i \(-0.262637\pi\)
0.678484 + 0.734615i \(0.262637\pi\)
\(168\) 0 0
\(169\) −6.16139e7 −0.981918
\(170\) 0 0
\(171\) −1.49287e7 −0.228315
\(172\) 0 0
\(173\) 1.33191e8 1.95575 0.977874 0.209197i \(-0.0670849\pi\)
0.977874 + 0.209197i \(0.0670849\pi\)
\(174\) 0 0
\(175\) 1.49626e7 0.211044
\(176\) 0 0
\(177\) −6.23243e6 −0.0844794
\(178\) 0 0
\(179\) 4.90910e7 0.639759 0.319879 0.947458i \(-0.396358\pi\)
0.319879 + 0.947458i \(0.396358\pi\)
\(180\) 0 0
\(181\) 8.11979e7 1.01782 0.508908 0.860821i \(-0.330049\pi\)
0.508908 + 0.860821i \(0.330049\pi\)
\(182\) 0 0
\(183\) −678828. −0.00818806
\(184\) 0 0
\(185\) 5.71563e7 0.663686
\(186\) 0 0
\(187\) 4.73076e6 0.0529036
\(188\) 0 0
\(189\) 1.08378e7 0.116768
\(190\) 0 0
\(191\) −7.40713e7 −0.769190 −0.384595 0.923086i \(-0.625659\pi\)
−0.384595 + 0.923086i \(0.625659\pi\)
\(192\) 0 0
\(193\) −9.70125e7 −0.971353 −0.485676 0.874139i \(-0.661427\pi\)
−0.485676 + 0.874139i \(0.661427\pi\)
\(194\) 0 0
\(195\) −1.07816e6 −0.0104127
\(196\) 0 0
\(197\) −4.34770e7 −0.405161 −0.202581 0.979266i \(-0.564933\pi\)
−0.202581 + 0.979266i \(0.564933\pi\)
\(198\) 0 0
\(199\) −3.89651e7 −0.350502 −0.175251 0.984524i \(-0.556074\pi\)
−0.175251 + 0.984524i \(0.556074\pi\)
\(200\) 0 0
\(201\) −7.52952e6 −0.0654005
\(202\) 0 0
\(203\) −3.54937e7 −0.297793
\(204\) 0 0
\(205\) −2.56121e8 −2.07638
\(206\) 0 0
\(207\) 2.51601e7 0.197159
\(208\) 0 0
\(209\) −1.72875e6 −0.0130985
\(210\) 0 0
\(211\) 3.85264e7 0.282339 0.141169 0.989985i \(-0.454914\pi\)
0.141169 + 0.989985i \(0.454914\pi\)
\(212\) 0 0
\(213\) −1.74024e7 −0.123390
\(214\) 0 0
\(215\) 1.49196e8 1.02382
\(216\) 0 0
\(217\) −3.59217e7 −0.238642
\(218\) 0 0
\(219\) 1.20375e7 0.0774428
\(220\) 0 0
\(221\) 1.99934e7 0.124599
\(222\) 0 0
\(223\) 2.03401e8 1.22825 0.614124 0.789209i \(-0.289509\pi\)
0.614124 + 0.789209i \(0.289509\pi\)
\(224\) 0 0
\(225\) −4.24720e7 −0.248579
\(226\) 0 0
\(227\) −2.24944e8 −1.27639 −0.638195 0.769875i \(-0.720319\pi\)
−0.638195 + 0.769875i \(0.720319\pi\)
\(228\) 0 0
\(229\) −2.38882e8 −1.31450 −0.657248 0.753674i \(-0.728280\pi\)
−0.657248 + 0.753674i \(0.728280\pi\)
\(230\) 0 0
\(231\) 626006. 0.00334146
\(232\) 0 0
\(233\) 3.23019e8 1.67295 0.836475 0.548005i \(-0.184613\pi\)
0.836475 + 0.548005i \(0.184613\pi\)
\(234\) 0 0
\(235\) 3.10153e8 1.55897
\(236\) 0 0
\(237\) −4.23859e6 −0.0206825
\(238\) 0 0
\(239\) 5.18160e7 0.245511 0.122756 0.992437i \(-0.460827\pi\)
0.122756 + 0.992437i \(0.460827\pi\)
\(240\) 0 0
\(241\) 1.15926e8 0.533484 0.266742 0.963768i \(-0.414053\pi\)
0.266742 + 0.963768i \(0.414053\pi\)
\(242\) 0 0
\(243\) −4.61825e7 −0.206469
\(244\) 0 0
\(245\) 7.36219e7 0.319835
\(246\) 0 0
\(247\) −7.30617e6 −0.0308497
\(248\) 0 0
\(249\) −2.11096e7 −0.0866525
\(250\) 0 0
\(251\) 4.03761e7 0.161163 0.0805817 0.996748i \(-0.474322\pi\)
0.0805817 + 0.996748i \(0.474322\pi\)
\(252\) 0 0
\(253\) 2.91356e6 0.0113110
\(254\) 0 0
\(255\) −1.89982e7 −0.0717500
\(256\) 0 0
\(257\) 2.67661e8 0.983602 0.491801 0.870708i \(-0.336339\pi\)
0.491801 + 0.870708i \(0.336339\pi\)
\(258\) 0 0
\(259\) −1.40254e8 −0.501611
\(260\) 0 0
\(261\) 1.00751e8 0.350757
\(262\) 0 0
\(263\) −5.12778e8 −1.73814 −0.869068 0.494692i \(-0.835281\pi\)
−0.869068 + 0.494692i \(0.835281\pi\)
\(264\) 0 0
\(265\) 2.66387e8 0.879333
\(266\) 0 0
\(267\) −1.29435e7 −0.0416163
\(268\) 0 0
\(269\) 5.75792e8 1.80357 0.901784 0.432187i \(-0.142258\pi\)
0.901784 + 0.432187i \(0.142258\pi\)
\(270\) 0 0
\(271\) −5.24839e8 −1.60189 −0.800946 0.598737i \(-0.795670\pi\)
−0.800946 + 0.598737i \(0.795670\pi\)
\(272\) 0 0
\(273\) 2.64567e6 0.00786984
\(274\) 0 0
\(275\) −4.91829e6 −0.0142610
\(276\) 0 0
\(277\) −4.64620e8 −1.31347 −0.656733 0.754123i \(-0.728062\pi\)
−0.656733 + 0.754123i \(0.728062\pi\)
\(278\) 0 0
\(279\) 1.01966e8 0.281086
\(280\) 0 0
\(281\) −1.75429e8 −0.471660 −0.235830 0.971794i \(-0.575781\pi\)
−0.235830 + 0.971794i \(0.575781\pi\)
\(282\) 0 0
\(283\) −1.44948e8 −0.380154 −0.190077 0.981769i \(-0.560874\pi\)
−0.190077 + 0.981769i \(0.560874\pi\)
\(284\) 0 0
\(285\) 6.94248e6 0.0177647
\(286\) 0 0
\(287\) 6.28489e8 1.56932
\(288\) 0 0
\(289\) −5.80353e7 −0.141433
\(290\) 0 0
\(291\) 2.23634e7 0.0532001
\(292\) 0 0
\(293\) −4.24522e8 −0.985970 −0.492985 0.870038i \(-0.664094\pi\)
−0.492985 + 0.870038i \(0.664094\pi\)
\(294\) 0 0
\(295\) −6.01210e8 −1.36348
\(296\) 0 0
\(297\) −3.56247e6 −0.00789047
\(298\) 0 0
\(299\) 1.23135e7 0.0266399
\(300\) 0 0
\(301\) −3.66109e8 −0.773798
\(302\) 0 0
\(303\) −9.12200e6 −0.0188383
\(304\) 0 0
\(305\) −6.54830e7 −0.132154
\(306\) 0 0
\(307\) −4.27528e8 −0.843297 −0.421649 0.906759i \(-0.638548\pi\)
−0.421649 + 0.906759i \(0.638548\pi\)
\(308\) 0 0
\(309\) 2.54692e7 0.0491090
\(310\) 0 0
\(311\) 1.70275e8 0.320989 0.160495 0.987037i \(-0.448691\pi\)
0.160495 + 0.987037i \(0.448691\pi\)
\(312\) 0 0
\(313\) −4.02143e8 −0.741269 −0.370634 0.928779i \(-0.620860\pi\)
−0.370634 + 0.928779i \(0.620860\pi\)
\(314\) 0 0
\(315\) 5.21477e8 0.940043
\(316\) 0 0
\(317\) −1.90630e8 −0.336112 −0.168056 0.985777i \(-0.553749\pi\)
−0.168056 + 0.985777i \(0.553749\pi\)
\(318\) 0 0
\(319\) 1.16670e7 0.0201230
\(320\) 0 0
\(321\) 1.98120e6 0.00334319
\(322\) 0 0
\(323\) −1.28742e8 −0.212574
\(324\) 0 0
\(325\) −2.07860e7 −0.0335876
\(326\) 0 0
\(327\) −1.69480e7 −0.0268041
\(328\) 0 0
\(329\) −7.61077e8 −1.17827
\(330\) 0 0
\(331\) 4.12008e8 0.624464 0.312232 0.950006i \(-0.398923\pi\)
0.312232 + 0.950006i \(0.398923\pi\)
\(332\) 0 0
\(333\) 3.98119e8 0.590824
\(334\) 0 0
\(335\) −7.26333e8 −1.05555
\(336\) 0 0
\(337\) −6.99272e8 −0.995271 −0.497636 0.867386i \(-0.665798\pi\)
−0.497636 + 0.867386i \(0.665798\pi\)
\(338\) 0 0
\(339\) 1.16135e7 0.0161907
\(340\) 0 0
\(341\) 1.18077e7 0.0161259
\(342\) 0 0
\(343\) −8.12125e8 −1.08666
\(344\) 0 0
\(345\) −1.17005e7 −0.0153405
\(346\) 0 0
\(347\) 9.29885e8 1.19475 0.597374 0.801963i \(-0.296211\pi\)
0.597374 + 0.801963i \(0.296211\pi\)
\(348\) 0 0
\(349\) −4.43776e8 −0.558823 −0.279412 0.960171i \(-0.590139\pi\)
−0.279412 + 0.960171i \(0.590139\pi\)
\(350\) 0 0
\(351\) −1.50559e7 −0.0185837
\(352\) 0 0
\(353\) −1.49388e9 −1.80761 −0.903804 0.427946i \(-0.859237\pi\)
−0.903804 + 0.427946i \(0.859237\pi\)
\(354\) 0 0
\(355\) −1.67872e9 −1.99149
\(356\) 0 0
\(357\) 4.66192e7 0.0542283
\(358\) 0 0
\(359\) 1.77937e8 0.202972 0.101486 0.994837i \(-0.467640\pi\)
0.101486 + 0.994837i \(0.467640\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 6.29178e7 0.0690399
\(364\) 0 0
\(365\) 1.16119e9 1.24991
\(366\) 0 0
\(367\) 1.37515e9 1.45218 0.726088 0.687601i \(-0.241336\pi\)
0.726088 + 0.687601i \(0.241336\pi\)
\(368\) 0 0
\(369\) −1.78400e9 −1.84843
\(370\) 0 0
\(371\) −6.53681e8 −0.664595
\(372\) 0 0
\(373\) 1.55845e9 1.55493 0.777466 0.628925i \(-0.216505\pi\)
0.777466 + 0.628925i \(0.216505\pi\)
\(374\) 0 0
\(375\) −5.93245e7 −0.0580932
\(376\) 0 0
\(377\) 4.93079e7 0.0473938
\(378\) 0 0
\(379\) −8.84549e8 −0.834613 −0.417306 0.908766i \(-0.637026\pi\)
−0.417306 + 0.908766i \(0.637026\pi\)
\(380\) 0 0
\(381\) −9.70038e7 −0.0898569
\(382\) 0 0
\(383\) −1.44573e9 −1.31490 −0.657449 0.753499i \(-0.728365\pi\)
−0.657449 + 0.753499i \(0.728365\pi\)
\(384\) 0 0
\(385\) 6.03875e7 0.0539305
\(386\) 0 0
\(387\) 1.03922e9 0.911420
\(388\) 0 0
\(389\) −1.02164e9 −0.879986 −0.439993 0.898001i \(-0.645019\pi\)
−0.439993 + 0.898001i \(0.645019\pi\)
\(390\) 0 0
\(391\) 2.16975e8 0.183566
\(392\) 0 0
\(393\) −2.62797e7 −0.0218397
\(394\) 0 0
\(395\) −4.08874e8 −0.333811
\(396\) 0 0
\(397\) 1.80050e8 0.144419 0.0722097 0.997389i \(-0.476995\pi\)
0.0722097 + 0.997389i \(0.476995\pi\)
\(398\) 0 0
\(399\) −1.70360e7 −0.0134265
\(400\) 0 0
\(401\) 5.06670e8 0.392392 0.196196 0.980565i \(-0.437141\pi\)
0.196196 + 0.980565i \(0.437141\pi\)
\(402\) 0 0
\(403\) 4.99025e7 0.0379799
\(404\) 0 0
\(405\) −1.47307e9 −1.10187
\(406\) 0 0
\(407\) 4.61026e7 0.0338957
\(408\) 0 0
\(409\) 1.14898e9 0.830388 0.415194 0.909733i \(-0.363714\pi\)
0.415194 + 0.909733i \(0.363714\pi\)
\(410\) 0 0
\(411\) −4.66959e7 −0.0331767
\(412\) 0 0
\(413\) 1.47529e9 1.03051
\(414\) 0 0
\(415\) −2.03633e9 −1.39855
\(416\) 0 0
\(417\) −1.22264e8 −0.0825698
\(418\) 0 0
\(419\) −2.02289e9 −1.34346 −0.671729 0.740797i \(-0.734448\pi\)
−0.671729 + 0.740797i \(0.734448\pi\)
\(420\) 0 0
\(421\) 2.55503e9 1.66882 0.834408 0.551147i \(-0.185810\pi\)
0.834408 + 0.551147i \(0.185810\pi\)
\(422\) 0 0
\(423\) 2.16036e9 1.38782
\(424\) 0 0
\(425\) −3.66269e8 −0.231441
\(426\) 0 0
\(427\) 1.60687e8 0.0998811
\(428\) 0 0
\(429\) −869648. −0.000531794 0
\(430\) 0 0
\(431\) −1.47316e9 −0.886295 −0.443147 0.896449i \(-0.646138\pi\)
−0.443147 + 0.896449i \(0.646138\pi\)
\(432\) 0 0
\(433\) −9.33243e8 −0.552443 −0.276221 0.961094i \(-0.589082\pi\)
−0.276221 + 0.961094i \(0.589082\pi\)
\(434\) 0 0
\(435\) −4.68534e7 −0.0272916
\(436\) 0 0
\(437\) −7.92890e7 −0.0454494
\(438\) 0 0
\(439\) 1.94411e9 1.09672 0.548358 0.836244i \(-0.315253\pi\)
0.548358 + 0.836244i \(0.315253\pi\)
\(440\) 0 0
\(441\) 5.12809e8 0.284722
\(442\) 0 0
\(443\) 2.35592e9 1.28750 0.643751 0.765235i \(-0.277377\pi\)
0.643751 + 0.765235i \(0.277377\pi\)
\(444\) 0 0
\(445\) −1.24860e9 −0.671679
\(446\) 0 0
\(447\) −2.36684e8 −0.125341
\(448\) 0 0
\(449\) −1.83374e9 −0.956042 −0.478021 0.878349i \(-0.658646\pi\)
−0.478021 + 0.878349i \(0.658646\pi\)
\(450\) 0 0
\(451\) −2.06589e8 −0.106045
\(452\) 0 0
\(453\) −1.34123e8 −0.0677889
\(454\) 0 0
\(455\) 2.55213e8 0.127018
\(456\) 0 0
\(457\) −3.76683e9 −1.84616 −0.923079 0.384611i \(-0.874336\pi\)
−0.923079 + 0.384611i \(0.874336\pi\)
\(458\) 0 0
\(459\) −2.65300e8 −0.128054
\(460\) 0 0
\(461\) 3.30560e9 1.57144 0.785720 0.618583i \(-0.212293\pi\)
0.785720 + 0.618583i \(0.212293\pi\)
\(462\) 0 0
\(463\) 3.66326e9 1.71528 0.857638 0.514253i \(-0.171931\pi\)
0.857638 + 0.514253i \(0.171931\pi\)
\(464\) 0 0
\(465\) −4.74184e7 −0.0218707
\(466\) 0 0
\(467\) 2.22674e9 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(468\) 0 0
\(469\) 1.78233e9 0.797780
\(470\) 0 0
\(471\) −9.52159e7 −0.0419891
\(472\) 0 0
\(473\) 1.20342e8 0.0522883
\(474\) 0 0
\(475\) 1.33845e8 0.0573028
\(476\) 0 0
\(477\) 1.85551e9 0.782796
\(478\) 0 0
\(479\) −1.43508e9 −0.596625 −0.298313 0.954468i \(-0.596424\pi\)
−0.298313 + 0.954468i \(0.596424\pi\)
\(480\) 0 0
\(481\) 1.94842e8 0.0798314
\(482\) 0 0
\(483\) 2.87117e7 0.0115943
\(484\) 0 0
\(485\) 2.15728e9 0.858638
\(486\) 0 0
\(487\) 2.11642e9 0.830330 0.415165 0.909746i \(-0.363724\pi\)
0.415165 + 0.909746i \(0.363724\pi\)
\(488\) 0 0
\(489\) 2.92802e7 0.0113238
\(490\) 0 0
\(491\) −4.07105e9 −1.55211 −0.776053 0.630668i \(-0.782781\pi\)
−0.776053 + 0.630668i \(0.782781\pi\)
\(492\) 0 0
\(493\) 8.68852e8 0.326574
\(494\) 0 0
\(495\) −1.71413e8 −0.0635222
\(496\) 0 0
\(497\) 4.11935e9 1.50516
\(498\) 0 0
\(499\) −3.93039e9 −1.41607 −0.708034 0.706178i \(-0.750418\pi\)
−0.708034 + 0.706178i \(0.750418\pi\)
\(500\) 0 0
\(501\) −2.64558e8 −0.0939914
\(502\) 0 0
\(503\) 1.70030e9 0.595714 0.297857 0.954610i \(-0.403728\pi\)
0.297857 + 0.954610i \(0.403728\pi\)
\(504\) 0 0
\(505\) −8.79952e8 −0.304046
\(506\) 0 0
\(507\) 1.99582e8 0.0680132
\(508\) 0 0
\(509\) −2.62955e8 −0.0883830 −0.0441915 0.999023i \(-0.514071\pi\)
−0.0441915 + 0.999023i \(0.514071\pi\)
\(510\) 0 0
\(511\) −2.84942e9 −0.944676
\(512\) 0 0
\(513\) 9.69481e7 0.0317051
\(514\) 0 0
\(515\) 2.45688e9 0.792609
\(516\) 0 0
\(517\) 2.50171e8 0.0796197
\(518\) 0 0
\(519\) −4.31436e8 −0.135466
\(520\) 0 0
\(521\) −1.86990e9 −0.579277 −0.289639 0.957136i \(-0.593535\pi\)
−0.289639 + 0.957136i \(0.593535\pi\)
\(522\) 0 0
\(523\) 5.49460e9 1.67950 0.839750 0.542972i \(-0.182701\pi\)
0.839750 + 0.542972i \(0.182701\pi\)
\(524\) 0 0
\(525\) −4.84673e7 −0.0146181
\(526\) 0 0
\(527\) 8.79329e8 0.261706
\(528\) 0 0
\(529\) −3.27120e9 −0.960753
\(530\) 0 0
\(531\) −4.18769e9 −1.21379
\(532\) 0 0
\(533\) −8.73098e8 −0.249757
\(534\) 0 0
\(535\) 1.91116e8 0.0539584
\(536\) 0 0
\(537\) −1.59017e8 −0.0443133
\(538\) 0 0
\(539\) 5.93838e7 0.0163346
\(540\) 0 0
\(541\) 6.97892e9 1.89495 0.947475 0.319830i \(-0.103626\pi\)
0.947475 + 0.319830i \(0.103626\pi\)
\(542\) 0 0
\(543\) −2.63019e8 −0.0704998
\(544\) 0 0
\(545\) −1.63489e9 −0.432613
\(546\) 0 0
\(547\) 5.87859e9 1.53574 0.767870 0.640606i \(-0.221317\pi\)
0.767870 + 0.640606i \(0.221317\pi\)
\(548\) 0 0
\(549\) −4.56118e8 −0.117645
\(550\) 0 0
\(551\) −3.17503e8 −0.0808570
\(552\) 0 0
\(553\) 1.00333e9 0.252292
\(554\) 0 0
\(555\) −1.85143e8 −0.0459707
\(556\) 0 0
\(557\) −2.03155e9 −0.498119 −0.249060 0.968488i \(-0.580122\pi\)
−0.249060 + 0.968488i \(0.580122\pi\)
\(558\) 0 0
\(559\) 5.08599e8 0.123150
\(560\) 0 0
\(561\) −1.53240e7 −0.00366440
\(562\) 0 0
\(563\) −1.68161e9 −0.397142 −0.198571 0.980087i \(-0.563630\pi\)
−0.198571 + 0.980087i \(0.563630\pi\)
\(564\) 0 0
\(565\) 1.12030e9 0.261314
\(566\) 0 0
\(567\) 3.61472e9 0.832787
\(568\) 0 0
\(569\) −4.57746e9 −1.04167 −0.520837 0.853656i \(-0.674380\pi\)
−0.520837 + 0.853656i \(0.674380\pi\)
\(570\) 0 0
\(571\) 5.79221e9 1.30202 0.651010 0.759069i \(-0.274345\pi\)
0.651010 + 0.759069i \(0.274345\pi\)
\(572\) 0 0
\(573\) 2.39935e8 0.0532785
\(574\) 0 0
\(575\) −2.25577e8 −0.0494831
\(576\) 0 0
\(577\) −3.74692e9 −0.812005 −0.406003 0.913872i \(-0.633078\pi\)
−0.406003 + 0.913872i \(0.633078\pi\)
\(578\) 0 0
\(579\) 3.14246e8 0.0672814
\(580\) 0 0
\(581\) 4.99689e9 1.05702
\(582\) 0 0
\(583\) 2.14869e8 0.0449091
\(584\) 0 0
\(585\) −7.24436e8 −0.149608
\(586\) 0 0
\(587\) 5.69193e9 1.16152 0.580760 0.814075i \(-0.302756\pi\)
0.580760 + 0.814075i \(0.302756\pi\)
\(588\) 0 0
\(589\) −3.21332e8 −0.0647963
\(590\) 0 0
\(591\) 1.40832e8 0.0280638
\(592\) 0 0
\(593\) 1.45596e9 0.286720 0.143360 0.989671i \(-0.454209\pi\)
0.143360 + 0.989671i \(0.454209\pi\)
\(594\) 0 0
\(595\) 4.49711e9 0.875233
\(596\) 0 0
\(597\) 1.26217e8 0.0242778
\(598\) 0 0
\(599\) 6.21735e9 1.18198 0.590992 0.806677i \(-0.298736\pi\)
0.590992 + 0.806677i \(0.298736\pi\)
\(600\) 0 0
\(601\) 9.98776e9 1.87675 0.938377 0.345613i \(-0.112329\pi\)
0.938377 + 0.345613i \(0.112329\pi\)
\(602\) 0 0
\(603\) −5.05924e9 −0.939668
\(604\) 0 0
\(605\) 6.06935e9 1.11429
\(606\) 0 0
\(607\) −3.17827e9 −0.576807 −0.288404 0.957509i \(-0.593124\pi\)
−0.288404 + 0.957509i \(0.593124\pi\)
\(608\) 0 0
\(609\) 1.14972e8 0.0206269
\(610\) 0 0
\(611\) 1.05729e9 0.187521
\(612\) 0 0
\(613\) −6.34969e8 −0.111337 −0.0556687 0.998449i \(-0.517729\pi\)
−0.0556687 + 0.998449i \(0.517729\pi\)
\(614\) 0 0
\(615\) 8.29637e8 0.143822
\(616\) 0 0
\(617\) −4.48835e7 −0.00769287 −0.00384643 0.999993i \(-0.501224\pi\)
−0.00384643 + 0.999993i \(0.501224\pi\)
\(618\) 0 0
\(619\) 9.79132e8 0.165930 0.0829648 0.996552i \(-0.473561\pi\)
0.0829648 + 0.996552i \(0.473561\pi\)
\(620\) 0 0
\(621\) −1.63392e8 −0.0273785
\(622\) 0 0
\(623\) 3.06390e9 0.507652
\(624\) 0 0
\(625\) −7.24724e9 −1.18739
\(626\) 0 0
\(627\) 5.59984e6 0.000907276 0
\(628\) 0 0
\(629\) 3.43330e9 0.550090
\(630\) 0 0
\(631\) 2.42563e9 0.384345 0.192172 0.981361i \(-0.438447\pi\)
0.192172 + 0.981361i \(0.438447\pi\)
\(632\) 0 0
\(633\) −1.24796e8 −0.0195564
\(634\) 0 0
\(635\) −9.35744e9 −1.45027
\(636\) 0 0
\(637\) 2.50972e8 0.0384713
\(638\) 0 0
\(639\) −1.16930e10 −1.77285
\(640\) 0 0
\(641\) 9.23132e8 0.138440 0.0692199 0.997601i \(-0.477949\pi\)
0.0692199 + 0.997601i \(0.477949\pi\)
\(642\) 0 0
\(643\) −6.86334e9 −1.01812 −0.509058 0.860732i \(-0.670006\pi\)
−0.509058 + 0.860732i \(0.670006\pi\)
\(644\) 0 0
\(645\) −4.83282e8 −0.0709156
\(646\) 0 0
\(647\) −5.02399e8 −0.0729262 −0.0364631 0.999335i \(-0.511609\pi\)
−0.0364631 + 0.999335i \(0.511609\pi\)
\(648\) 0 0
\(649\) −4.84939e8 −0.0696355
\(650\) 0 0
\(651\) 1.16359e8 0.0165297
\(652\) 0 0
\(653\) 4.23523e9 0.595225 0.297612 0.954687i \(-0.403810\pi\)
0.297612 + 0.954687i \(0.403810\pi\)
\(654\) 0 0
\(655\) −2.53506e9 −0.352488
\(656\) 0 0
\(657\) 8.08822e9 1.11269
\(658\) 0 0
\(659\) 1.34554e10 1.83146 0.915731 0.401791i \(-0.131612\pi\)
0.915731 + 0.401791i \(0.131612\pi\)
\(660\) 0 0
\(661\) 1.23518e10 1.66351 0.831753 0.555146i \(-0.187338\pi\)
0.831753 + 0.555146i \(0.187338\pi\)
\(662\) 0 0
\(663\) −6.47634e7 −0.00863043
\(664\) 0 0
\(665\) −1.64337e9 −0.216700
\(666\) 0 0
\(667\) 5.35106e8 0.0698231
\(668\) 0 0
\(669\) −6.58864e8 −0.0850755
\(670\) 0 0
\(671\) −5.28189e7 −0.00674933
\(672\) 0 0
\(673\) −1.52638e9 −0.193024 −0.0965119 0.995332i \(-0.530769\pi\)
−0.0965119 + 0.995332i \(0.530769\pi\)
\(674\) 0 0
\(675\) 2.75817e8 0.0345189
\(676\) 0 0
\(677\) −1.55415e9 −0.192500 −0.0962502 0.995357i \(-0.530685\pi\)
−0.0962502 + 0.995357i \(0.530685\pi\)
\(678\) 0 0
\(679\) −5.29369e9 −0.648954
\(680\) 0 0
\(681\) 7.28646e8 0.0884100
\(682\) 0 0
\(683\) 3.21824e9 0.386497 0.193249 0.981150i \(-0.438098\pi\)
0.193249 + 0.981150i \(0.438098\pi\)
\(684\) 0 0
\(685\) −4.50451e9 −0.535465
\(686\) 0 0
\(687\) 7.73796e8 0.0910495
\(688\) 0 0
\(689\) 9.08095e8 0.105770
\(690\) 0 0
\(691\) −9.15999e9 −1.05614 −0.528071 0.849200i \(-0.677084\pi\)
−0.528071 + 0.849200i \(0.677084\pi\)
\(692\) 0 0
\(693\) 4.20626e8 0.0480098
\(694\) 0 0
\(695\) −1.17941e10 −1.33266
\(696\) 0 0
\(697\) −1.53848e10 −1.72099
\(698\) 0 0
\(699\) −1.04634e9 −0.115878
\(700\) 0 0
\(701\) −7.97717e9 −0.874653 −0.437327 0.899303i \(-0.644075\pi\)
−0.437327 + 0.899303i \(0.644075\pi\)
\(702\) 0 0
\(703\) −1.25462e9 −0.136198
\(704\) 0 0
\(705\) −1.00466e9 −0.107984
\(706\) 0 0
\(707\) 2.15929e9 0.229796
\(708\) 0 0
\(709\) 7.97010e9 0.839851 0.419925 0.907559i \(-0.362056\pi\)
0.419925 + 0.907559i \(0.362056\pi\)
\(710\) 0 0
\(711\) −2.84799e9 −0.297163
\(712\) 0 0
\(713\) 5.41559e8 0.0559541
\(714\) 0 0
\(715\) −8.38904e7 −0.00858304
\(716\) 0 0
\(717\) −1.67844e8 −0.0170055
\(718\) 0 0
\(719\) −1.00263e10 −1.00598 −0.502991 0.864292i \(-0.667767\pi\)
−0.502991 + 0.864292i \(0.667767\pi\)
\(720\) 0 0
\(721\) −6.02888e9 −0.599050
\(722\) 0 0
\(723\) −3.75512e8 −0.0369522
\(724\) 0 0
\(725\) −9.03295e8 −0.0880333
\(726\) 0 0
\(727\) 1.65401e10 1.59650 0.798248 0.602329i \(-0.205760\pi\)
0.798248 + 0.602329i \(0.205760\pi\)
\(728\) 0 0
\(729\) −1.01604e10 −0.971329
\(730\) 0 0
\(731\) 8.96200e9 0.848583
\(732\) 0 0
\(733\) 9.81994e9 0.920969 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(734\) 0 0
\(735\) −2.38479e8 −0.0221536
\(736\) 0 0
\(737\) −5.85864e8 −0.0539089
\(738\) 0 0
\(739\) −5.41304e9 −0.493385 −0.246692 0.969094i \(-0.579344\pi\)
−0.246692 + 0.969094i \(0.579344\pi\)
\(740\) 0 0
\(741\) 2.36664e7 0.00213682
\(742\) 0 0
\(743\) 8.26105e9 0.738881 0.369440 0.929254i \(-0.379549\pi\)
0.369440 + 0.929254i \(0.379549\pi\)
\(744\) 0 0
\(745\) −2.28317e10 −2.02298
\(746\) 0 0
\(747\) −1.41839e10 −1.24501
\(748\) 0 0
\(749\) −4.68975e8 −0.0407815
\(750\) 0 0
\(751\) 1.44332e10 1.24343 0.621715 0.783243i \(-0.286436\pi\)
0.621715 + 0.783243i \(0.286436\pi\)
\(752\) 0 0
\(753\) −1.30788e8 −0.0111631
\(754\) 0 0
\(755\) −1.29381e10 −1.09410
\(756\) 0 0
\(757\) −8.31294e9 −0.696496 −0.348248 0.937402i \(-0.613223\pi\)
−0.348248 + 0.937402i \(0.613223\pi\)
\(758\) 0 0
\(759\) −9.43772e6 −0.000783467 0
\(760\) 0 0
\(761\) −1.96636e10 −1.61739 −0.808697 0.588226i \(-0.799827\pi\)
−0.808697 + 0.588226i \(0.799827\pi\)
\(762\) 0 0
\(763\) 4.01180e9 0.326967
\(764\) 0 0
\(765\) −1.27653e10 −1.03090
\(766\) 0 0
\(767\) −2.04948e9 −0.164006
\(768\) 0 0
\(769\) 1.27102e10 1.00788 0.503942 0.863738i \(-0.331883\pi\)
0.503942 + 0.863738i \(0.331883\pi\)
\(770\) 0 0
\(771\) −8.67018e8 −0.0681299
\(772\) 0 0
\(773\) −2.87614e9 −0.223966 −0.111983 0.993710i \(-0.535720\pi\)
−0.111983 + 0.993710i \(0.535720\pi\)
\(774\) 0 0
\(775\) −9.14188e8 −0.0705472
\(776\) 0 0
\(777\) 4.54317e8 0.0347444
\(778\) 0 0
\(779\) 5.62206e9 0.426102
\(780\) 0 0
\(781\) −1.35406e9 −0.101709
\(782\) 0 0
\(783\) −6.54283e8 −0.0487079
\(784\) 0 0
\(785\) −9.18498e9 −0.677695
\(786\) 0 0
\(787\) −5.61813e9 −0.410847 −0.205424 0.978673i \(-0.565857\pi\)
−0.205424 + 0.978673i \(0.565857\pi\)
\(788\) 0 0
\(789\) 1.66101e9 0.120393
\(790\) 0 0
\(791\) −2.74906e9 −0.197500
\(792\) 0 0
\(793\) −2.23227e8 −0.0158961
\(794\) 0 0
\(795\) −8.62892e8 −0.0609076
\(796\) 0 0
\(797\) −4.98926e9 −0.349085 −0.174543 0.984650i \(-0.555845\pi\)
−0.174543 + 0.984650i \(0.555845\pi\)
\(798\) 0 0
\(799\) 1.86305e10 1.29214
\(800\) 0 0
\(801\) −8.69703e9 −0.597939
\(802\) 0 0
\(803\) 9.36623e8 0.0638352
\(804\) 0 0
\(805\) 2.76966e9 0.187129
\(806\) 0 0
\(807\) −1.86513e9 −0.124925
\(808\) 0 0
\(809\) 2.60176e10 1.72762 0.863809 0.503820i \(-0.168073\pi\)
0.863809 + 0.503820i \(0.168073\pi\)
\(810\) 0 0
\(811\) −1.36033e10 −0.895513 −0.447756 0.894156i \(-0.647777\pi\)
−0.447756 + 0.894156i \(0.647777\pi\)
\(812\) 0 0
\(813\) 1.70008e9 0.110956
\(814\) 0 0
\(815\) 2.82451e9 0.182764
\(816\) 0 0
\(817\) −3.27497e9 −0.210102
\(818\) 0 0
\(819\) 1.77768e9 0.113073
\(820\) 0 0
\(821\) 5.59803e9 0.353048 0.176524 0.984296i \(-0.443515\pi\)
0.176524 + 0.984296i \(0.443515\pi\)
\(822\) 0 0
\(823\) −2.83609e10 −1.77346 −0.886728 0.462292i \(-0.847027\pi\)
−0.886728 + 0.462292i \(0.847027\pi\)
\(824\) 0 0
\(825\) 1.59315e7 0.000987799 0
\(826\) 0 0
\(827\) 1.94087e10 1.19324 0.596619 0.802524i \(-0.296510\pi\)
0.596619 + 0.802524i \(0.296510\pi\)
\(828\) 0 0
\(829\) 1.25701e10 0.766301 0.383150 0.923686i \(-0.374839\pi\)
0.383150 + 0.923686i \(0.374839\pi\)
\(830\) 0 0
\(831\) 1.50501e9 0.0909781
\(832\) 0 0
\(833\) 4.42236e9 0.265092
\(834\) 0 0
\(835\) −2.55205e10 −1.51700
\(836\) 0 0
\(837\) −6.62173e8 −0.0390331
\(838\) 0 0
\(839\) −1.95535e10 −1.14303 −0.571515 0.820592i \(-0.693644\pi\)
−0.571515 + 0.820592i \(0.693644\pi\)
\(840\) 0 0
\(841\) −1.51071e10 −0.875781
\(842\) 0 0
\(843\) 5.68256e8 0.0326699
\(844\) 0 0
\(845\) 1.92526e10 1.09772
\(846\) 0 0
\(847\) −1.48934e10 −0.842175
\(848\) 0 0
\(849\) 4.69521e8 0.0263316
\(850\) 0 0
\(851\) 2.11449e9 0.117612
\(852\) 0 0
\(853\) 1.14196e10 0.629986 0.314993 0.949094i \(-0.397998\pi\)
0.314993 + 0.949094i \(0.397998\pi\)
\(854\) 0 0
\(855\) 4.66479e9 0.255241
\(856\) 0 0
\(857\) −1.94401e10 −1.05503 −0.527516 0.849545i \(-0.676877\pi\)
−0.527516 + 0.849545i \(0.676877\pi\)
\(858\) 0 0
\(859\) −1.07841e10 −0.580509 −0.290254 0.956950i \(-0.593740\pi\)
−0.290254 + 0.956950i \(0.593740\pi\)
\(860\) 0 0
\(861\) −2.03583e9 −0.108700
\(862\) 0 0
\(863\) 8.45236e8 0.0447651 0.0223826 0.999749i \(-0.492875\pi\)
0.0223826 + 0.999749i \(0.492875\pi\)
\(864\) 0 0
\(865\) −4.16184e10 −2.18640
\(866\) 0 0
\(867\) 1.87990e8 0.00979643
\(868\) 0 0
\(869\) −3.29800e8 −0.0170483
\(870\) 0 0
\(871\) −2.47602e9 −0.126967
\(872\) 0 0
\(873\) 1.50264e10 0.764373
\(874\) 0 0
\(875\) 1.40428e10 0.708642
\(876\) 0 0
\(877\) 2.17563e10 1.08915 0.544574 0.838713i \(-0.316691\pi\)
0.544574 + 0.838713i \(0.316691\pi\)
\(878\) 0 0
\(879\) 1.37513e9 0.0682939
\(880\) 0 0
\(881\) 2.18056e10 1.07436 0.537182 0.843466i \(-0.319489\pi\)
0.537182 + 0.843466i \(0.319489\pi\)
\(882\) 0 0
\(883\) −2.25662e9 −0.110305 −0.0551526 0.998478i \(-0.517565\pi\)
−0.0551526 + 0.998478i \(0.517565\pi\)
\(884\) 0 0
\(885\) 1.94746e9 0.0944424
\(886\) 0 0
\(887\) 1.72271e10 0.828857 0.414428 0.910082i \(-0.363982\pi\)
0.414428 + 0.910082i \(0.363982\pi\)
\(888\) 0 0
\(889\) 2.29620e10 1.09611
\(890\) 0 0
\(891\) −1.18818e9 −0.0562745
\(892\) 0 0
\(893\) −6.80810e9 −0.319923
\(894\) 0 0
\(895\) −1.53396e10 −0.715208
\(896\) 0 0
\(897\) −3.98863e7 −0.00184523
\(898\) 0 0
\(899\) 2.16861e9 0.0995455
\(900\) 0 0
\(901\) 1.60015e10 0.728827
\(902\) 0 0
\(903\) 1.18591e9 0.0535976
\(904\) 0 0
\(905\) −2.53721e10 −1.13785
\(906\) 0 0
\(907\) −2.32557e10 −1.03491 −0.517456 0.855710i \(-0.673121\pi\)
−0.517456 + 0.855710i \(0.673121\pi\)
\(908\) 0 0
\(909\) −6.12926e9 −0.270666
\(910\) 0 0
\(911\) 4.22775e10 1.85266 0.926329 0.376715i \(-0.122946\pi\)
0.926329 + 0.376715i \(0.122946\pi\)
\(912\) 0 0
\(913\) −1.64251e9 −0.0714267
\(914\) 0 0
\(915\) 2.12115e8 0.00915372
\(916\) 0 0
\(917\) 6.22072e9 0.266408
\(918\) 0 0
\(919\) −2.74897e10 −1.16833 −0.584164 0.811635i \(-0.698578\pi\)
−0.584164 + 0.811635i \(0.698578\pi\)
\(920\) 0 0
\(921\) 1.38487e9 0.0584116
\(922\) 0 0
\(923\) −5.72261e9 −0.239546
\(924\) 0 0
\(925\) −3.56940e9 −0.148286
\(926\) 0 0
\(927\) 1.71133e10 0.705593
\(928\) 0 0
\(929\) −2.42814e10 −0.993614 −0.496807 0.867861i \(-0.665494\pi\)
−0.496807 + 0.867861i \(0.665494\pi\)
\(930\) 0 0
\(931\) −1.61606e9 −0.0656346
\(932\) 0 0
\(933\) −5.51563e8 −0.0222336
\(934\) 0 0
\(935\) −1.47823e9 −0.0591427
\(936\) 0 0
\(937\) −3.53708e10 −1.40461 −0.702305 0.711876i \(-0.747846\pi\)
−0.702305 + 0.711876i \(0.747846\pi\)
\(938\) 0 0
\(939\) 1.30264e9 0.0513445
\(940\) 0 0
\(941\) −3.60255e10 −1.40944 −0.704719 0.709486i \(-0.748927\pi\)
−0.704719 + 0.709486i \(0.748927\pi\)
\(942\) 0 0
\(943\) −9.47516e9 −0.367956
\(944\) 0 0
\(945\) −3.38652e9 −0.130539
\(946\) 0 0
\(947\) 2.20930e10 0.845336 0.422668 0.906285i \(-0.361094\pi\)
0.422668 + 0.906285i \(0.361094\pi\)
\(948\) 0 0
\(949\) 3.95842e9 0.150345
\(950\) 0 0
\(951\) 6.17496e8 0.0232810
\(952\) 0 0
\(953\) −2.36409e9 −0.0884787 −0.0442393 0.999021i \(-0.514086\pi\)
−0.0442393 + 0.999021i \(0.514086\pi\)
\(954\) 0 0
\(955\) 2.31452e10 0.859903
\(956\) 0 0
\(957\) −3.77922e7 −0.00139383
\(958\) 0 0
\(959\) 1.10535e10 0.404702
\(960\) 0 0
\(961\) −2.53179e10 −0.920227
\(962\) 0 0
\(963\) 1.33121e9 0.0480346
\(964\) 0 0
\(965\) 3.03137e10 1.08591
\(966\) 0 0
\(967\) −2.69098e10 −0.957014 −0.478507 0.878084i \(-0.658822\pi\)
−0.478507 + 0.878084i \(0.658822\pi\)
\(968\) 0 0
\(969\) 4.17025e8 0.0147241
\(970\) 0 0
\(971\) 4.71594e9 0.165311 0.0826553 0.996578i \(-0.473660\pi\)
0.0826553 + 0.996578i \(0.473660\pi\)
\(972\) 0 0
\(973\) 2.89413e10 1.00722
\(974\) 0 0
\(975\) 6.73308e7 0.00232647
\(976\) 0 0
\(977\) 1.79514e10 0.615839 0.307919 0.951412i \(-0.400367\pi\)
0.307919 + 0.951412i \(0.400367\pi\)
\(978\) 0 0
\(979\) −1.00712e9 −0.0343039
\(980\) 0 0
\(981\) −1.13877e10 −0.385119
\(982\) 0 0
\(983\) −2.39450e10 −0.804039 −0.402019 0.915631i \(-0.631692\pi\)
−0.402019 + 0.915631i \(0.631692\pi\)
\(984\) 0 0
\(985\) 1.35854e10 0.452944
\(986\) 0 0
\(987\) 2.46531e9 0.0816134
\(988\) 0 0
\(989\) 5.51949e9 0.181431
\(990\) 0 0
\(991\) −1.07956e10 −0.352362 −0.176181 0.984358i \(-0.556374\pi\)
−0.176181 + 0.984358i \(0.556374\pi\)
\(992\) 0 0
\(993\) −1.33459e9 −0.0432540
\(994\) 0 0
\(995\) 1.21755e10 0.391838
\(996\) 0 0
\(997\) −3.54933e10 −1.13426 −0.567131 0.823627i \(-0.691947\pi\)
−0.567131 + 0.823627i \(0.691947\pi\)
\(998\) 0 0
\(999\) −2.58542e9 −0.0820450
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 304.8.a.d.1.1 2
4.3 odd 2 38.8.a.d.1.2 2
12.11 even 2 342.8.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.2 2 4.3 odd 2
304.8.a.d.1.1 2 1.1 even 1 trivial
342.8.a.g.1.2 2 12.11 even 2