Properties

Label 43.10.a.a.1.7
Level 43
Weight 10
Character 43.1
Self dual yes
Analytic conductor 22.147
Analytic rank 1
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(7.62988\) of \(x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.62988 q^{2} +189.920 q^{3} -419.265 q^{4} +636.384 q^{5} -1828.91 q^{6} -3288.51 q^{7} +8967.97 q^{8} +16386.6 q^{9} +O(q^{10})\) \(q-9.62988 q^{2} +189.920 q^{3} -419.265 q^{4} +636.384 q^{5} -1828.91 q^{6} -3288.51 q^{7} +8967.97 q^{8} +16386.6 q^{9} -6128.30 q^{10} -71891.5 q^{11} -79626.9 q^{12} -56499.0 q^{13} +31667.9 q^{14} +120862. q^{15} +128303. q^{16} +227428. q^{17} -157801. q^{18} -258196. q^{19} -266814. q^{20} -624553. q^{21} +692307. q^{22} -205452. q^{23} +1.70320e6 q^{24} -1.54814e6 q^{25} +544079. q^{26} -626056. q^{27} +1.37876e6 q^{28} +568487. q^{29} -1.16389e6 q^{30} -6.46168e6 q^{31} -5.82715e6 q^{32} -1.36536e7 q^{33} -2.19010e6 q^{34} -2.09275e6 q^{35} -6.87033e6 q^{36} -1.41255e7 q^{37} +2.48640e6 q^{38} -1.07303e7 q^{39} +5.70707e6 q^{40} -1.35846e7 q^{41} +6.01437e6 q^{42} -3.41880e6 q^{43} +3.01416e7 q^{44} +1.04282e7 q^{45} +1.97848e6 q^{46} +2.14428e7 q^{47} +2.43674e7 q^{48} -2.95393e7 q^{49} +1.49084e7 q^{50} +4.31931e7 q^{51} +2.36881e7 q^{52} +3.02124e7 q^{53} +6.02884e6 q^{54} -4.57506e7 q^{55} -2.94912e7 q^{56} -4.90366e7 q^{57} -5.47446e6 q^{58} +1.16373e8 q^{59} -5.06733e7 q^{60} -1.09646e8 q^{61} +6.22252e7 q^{62} -5.38874e7 q^{63} -9.57661e6 q^{64} -3.59551e7 q^{65} +1.31483e8 q^{66} +1.24276e8 q^{67} -9.53526e7 q^{68} -3.90194e7 q^{69} +2.01529e7 q^{70} +2.04397e8 q^{71} +1.46954e8 q^{72} +1.16000e8 q^{73} +1.36027e8 q^{74} -2.94023e8 q^{75} +1.08253e8 q^{76} +2.36416e8 q^{77} +1.03331e8 q^{78} +4.47267e8 q^{79} +8.16502e7 q^{80} -4.41438e8 q^{81} +1.30818e8 q^{82} -5.31305e8 q^{83} +2.61853e8 q^{84} +1.44731e8 q^{85} +3.29226e7 q^{86} +1.07967e8 q^{87} -6.44721e8 q^{88} +2.15750e7 q^{89} -1.00422e8 q^{90} +1.85797e8 q^{91} +8.61388e7 q^{92} -1.22720e9 q^{93} -2.06492e8 q^{94} -1.64312e8 q^{95} -1.10669e9 q^{96} +9.39408e8 q^{97} +2.84460e8 q^{98} -1.17806e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.62988 −0.425585 −0.212792 0.977097i \(-0.568256\pi\)
−0.212792 + 0.977097i \(0.568256\pi\)
\(3\) 189.920 1.35371 0.676854 0.736117i \(-0.263343\pi\)
0.676854 + 0.736117i \(0.263343\pi\)
\(4\) −419.265 −0.818878
\(5\) 636.384 0.455359 0.227680 0.973736i \(-0.426886\pi\)
0.227680 + 0.973736i \(0.426886\pi\)
\(6\) −1828.91 −0.576117
\(7\) −3288.51 −0.517675 −0.258838 0.965921i \(-0.583339\pi\)
−0.258838 + 0.965921i \(0.583339\pi\)
\(8\) 8967.97 0.774086
\(9\) 16386.6 0.832525
\(10\) −6128.30 −0.193794
\(11\) −71891.5 −1.48051 −0.740254 0.672327i \(-0.765295\pi\)
−0.740254 + 0.672327i \(0.765295\pi\)
\(12\) −79626.9 −1.10852
\(13\) −56499.0 −0.548651 −0.274325 0.961637i \(-0.588455\pi\)
−0.274325 + 0.961637i \(0.588455\pi\)
\(14\) 31667.9 0.220315
\(15\) 120862. 0.616423
\(16\) 128303. 0.489439
\(17\) 227428. 0.660425 0.330212 0.943907i \(-0.392880\pi\)
0.330212 + 0.943907i \(0.392880\pi\)
\(18\) −157801. −0.354310
\(19\) −258196. −0.454526 −0.227263 0.973833i \(-0.572978\pi\)
−0.227263 + 0.973833i \(0.572978\pi\)
\(20\) −266814. −0.372884
\(21\) −624553. −0.700781
\(22\) 692307. 0.630081
\(23\) −205452. −0.153086 −0.0765429 0.997066i \(-0.524388\pi\)
−0.0765429 + 0.997066i \(0.524388\pi\)
\(24\) 1.70320e6 1.04789
\(25\) −1.54814e6 −0.792648
\(26\) 544079. 0.233497
\(27\) −626056. −0.226713
\(28\) 1.37876e6 0.423913
\(29\) 568487. 0.149255 0.0746276 0.997211i \(-0.476223\pi\)
0.0746276 + 0.997211i \(0.476223\pi\)
\(30\) −1.16389e6 −0.262340
\(31\) −6.46168e6 −1.25666 −0.628330 0.777947i \(-0.716261\pi\)
−0.628330 + 0.777947i \(0.716261\pi\)
\(32\) −5.82715e6 −0.982384
\(33\) −1.36536e7 −2.00417
\(34\) −2.19010e6 −0.281066
\(35\) −2.09275e6 −0.235728
\(36\) −6.87033e6 −0.681736
\(37\) −1.41255e7 −1.23907 −0.619535 0.784969i \(-0.712679\pi\)
−0.619535 + 0.784969i \(0.712679\pi\)
\(38\) 2.48640e6 0.193439
\(39\) −1.07303e7 −0.742713
\(40\) 5.70707e6 0.352487
\(41\) −1.35846e7 −0.750791 −0.375395 0.926865i \(-0.622493\pi\)
−0.375395 + 0.926865i \(0.622493\pi\)
\(42\) 6.01437e6 0.298242
\(43\) −3.41880e6 −0.152499
\(44\) 3.01416e7 1.21235
\(45\) 1.04282e7 0.379098
\(46\) 1.97848e6 0.0651509
\(47\) 2.14428e7 0.640976 0.320488 0.947253i \(-0.396153\pi\)
0.320488 + 0.947253i \(0.396153\pi\)
\(48\) 2.43674e7 0.662557
\(49\) −2.95393e7 −0.732012
\(50\) 1.49084e7 0.337339
\(51\) 4.31931e7 0.894022
\(52\) 2.36881e7 0.449278
\(53\) 3.02124e7 0.525949 0.262975 0.964803i \(-0.415296\pi\)
0.262975 + 0.964803i \(0.415296\pi\)
\(54\) 6.02884e6 0.0964855
\(55\) −4.57506e7 −0.674163
\(56\) −2.94912e7 −0.400725
\(57\) −4.90366e7 −0.615295
\(58\) −5.47446e6 −0.0635207
\(59\) 1.16373e8 1.25031 0.625154 0.780501i \(-0.285036\pi\)
0.625154 + 0.780501i \(0.285036\pi\)
\(60\) −5.06733e7 −0.504775
\(61\) −1.09646e8 −1.01393 −0.506965 0.861967i \(-0.669233\pi\)
−0.506965 + 0.861967i \(0.669233\pi\)
\(62\) 6.22252e7 0.534815
\(63\) −5.38874e7 −0.430977
\(64\) −9.57661e6 −0.0713513
\(65\) −3.59551e7 −0.249833
\(66\) 1.31483e8 0.852946
\(67\) 1.24276e8 0.753444 0.376722 0.926326i \(-0.377051\pi\)
0.376722 + 0.926326i \(0.377051\pi\)
\(68\) −9.53526e7 −0.540807
\(69\) −3.90194e7 −0.207233
\(70\) 2.01529e7 0.100322
\(71\) 2.04397e8 0.954578 0.477289 0.878746i \(-0.341620\pi\)
0.477289 + 0.878746i \(0.341620\pi\)
\(72\) 1.46954e8 0.644446
\(73\) 1.16000e8 0.478086 0.239043 0.971009i \(-0.423166\pi\)
0.239043 + 0.971009i \(0.423166\pi\)
\(74\) 1.36027e8 0.527329
\(75\) −2.94023e8 −1.07301
\(76\) 1.08253e8 0.372201
\(77\) 2.36416e8 0.766422
\(78\) 1.03331e8 0.316087
\(79\) 4.47267e8 1.29195 0.645974 0.763359i \(-0.276451\pi\)
0.645974 + 0.763359i \(0.276451\pi\)
\(80\) 8.16502e7 0.222870
\(81\) −4.41438e8 −1.13943
\(82\) 1.30818e8 0.319525
\(83\) −5.31305e8 −1.22883 −0.614416 0.788982i \(-0.710608\pi\)
−0.614416 + 0.788982i \(0.710608\pi\)
\(84\) 2.61853e8 0.573854
\(85\) 1.44731e8 0.300730
\(86\) 3.29226e7 0.0649010
\(87\) 1.07967e8 0.202048
\(88\) −6.44721e8 −1.14604
\(89\) 2.15750e7 0.0364499 0.0182249 0.999834i \(-0.494198\pi\)
0.0182249 + 0.999834i \(0.494198\pi\)
\(90\) −1.00422e8 −0.161338
\(91\) 1.85797e8 0.284023
\(92\) 8.61388e7 0.125358
\(93\) −1.22720e9 −1.70115
\(94\) −2.06492e8 −0.272790
\(95\) −1.64312e8 −0.206972
\(96\) −1.10669e9 −1.32986
\(97\) 9.39408e8 1.07741 0.538705 0.842494i \(-0.318914\pi\)
0.538705 + 0.842494i \(0.318914\pi\)
\(98\) 2.84460e8 0.311533
\(99\) −1.17806e9 −1.23256
\(100\) 6.49082e8 0.649082
\(101\) 1.98427e8 0.189738 0.0948691 0.995490i \(-0.469757\pi\)
0.0948691 + 0.995490i \(0.469757\pi\)
\(102\) −4.15944e8 −0.380482
\(103\) −3.43220e8 −0.300473 −0.150236 0.988650i \(-0.548003\pi\)
−0.150236 + 0.988650i \(0.548003\pi\)
\(104\) −5.06682e8 −0.424703
\(105\) −3.97455e8 −0.319107
\(106\) −2.90942e8 −0.223836
\(107\) 1.59458e9 1.17603 0.588017 0.808849i \(-0.299909\pi\)
0.588017 + 0.808849i \(0.299909\pi\)
\(108\) 2.62484e8 0.185650
\(109\) −1.36307e8 −0.0924912 −0.0462456 0.998930i \(-0.514726\pi\)
−0.0462456 + 0.998930i \(0.514726\pi\)
\(110\) 4.40573e8 0.286913
\(111\) −2.68271e9 −1.67734
\(112\) −4.21926e8 −0.253370
\(113\) 6.49652e8 0.374824 0.187412 0.982281i \(-0.439990\pi\)
0.187412 + 0.982281i \(0.439990\pi\)
\(114\) 4.72216e8 0.261860
\(115\) −1.30746e8 −0.0697090
\(116\) −2.38347e8 −0.122222
\(117\) −9.25826e8 −0.456765
\(118\) −1.12065e9 −0.532112
\(119\) −7.47897e8 −0.341885
\(120\) 1.08389e9 0.477165
\(121\) 2.81045e9 1.19190
\(122\) 1.05588e9 0.431513
\(123\) −2.57998e9 −1.01635
\(124\) 2.70916e9 1.02905
\(125\) −2.22815e9 −0.816299
\(126\) 5.18929e8 0.183417
\(127\) 8.75307e8 0.298568 0.149284 0.988794i \(-0.452303\pi\)
0.149284 + 0.988794i \(0.452303\pi\)
\(128\) 3.07572e9 1.01275
\(129\) −6.49298e8 −0.206438
\(130\) 3.46243e8 0.106325
\(131\) −5.17356e9 −1.53486 −0.767429 0.641134i \(-0.778464\pi\)
−0.767429 + 0.641134i \(0.778464\pi\)
\(132\) 5.72450e9 1.64117
\(133\) 8.49079e8 0.235297
\(134\) −1.19676e9 −0.320654
\(135\) −3.98412e8 −0.103236
\(136\) 2.03957e9 0.511226
\(137\) 2.77573e9 0.673185 0.336593 0.941650i \(-0.390725\pi\)
0.336593 + 0.941650i \(0.390725\pi\)
\(138\) 3.75752e8 0.0881953
\(139\) −5.81753e9 −1.32182 −0.660910 0.750465i \(-0.729830\pi\)
−0.660910 + 0.750465i \(0.729830\pi\)
\(140\) 8.77419e8 0.193033
\(141\) 4.07242e9 0.867694
\(142\) −1.96832e9 −0.406253
\(143\) 4.06180e9 0.812281
\(144\) 2.10245e9 0.407470
\(145\) 3.61776e8 0.0679648
\(146\) −1.11707e9 −0.203466
\(147\) −5.61011e9 −0.990931
\(148\) 5.92233e9 1.01465
\(149\) −9.17312e9 −1.52468 −0.762340 0.647176i \(-0.775950\pi\)
−0.762340 + 0.647176i \(0.775950\pi\)
\(150\) 2.83140e9 0.456658
\(151\) 1.08986e10 1.70598 0.852990 0.521927i \(-0.174787\pi\)
0.852990 + 0.521927i \(0.174787\pi\)
\(152\) −2.31550e9 −0.351842
\(153\) 3.72676e9 0.549820
\(154\) −2.27665e9 −0.326177
\(155\) −4.11211e9 −0.572232
\(156\) 4.49884e9 0.608191
\(157\) 6.31663e9 0.829730 0.414865 0.909883i \(-0.363829\pi\)
0.414865 + 0.909883i \(0.363829\pi\)
\(158\) −4.30713e9 −0.549833
\(159\) 5.73794e9 0.711981
\(160\) −3.70830e9 −0.447338
\(161\) 6.75629e8 0.0792487
\(162\) 4.25099e9 0.484923
\(163\) 1.41043e10 1.56498 0.782489 0.622665i \(-0.213950\pi\)
0.782489 + 0.622665i \(0.213950\pi\)
\(164\) 5.69555e9 0.614806
\(165\) −8.68895e9 −0.912620
\(166\) 5.11640e9 0.522972
\(167\) −9.73712e9 −0.968739 −0.484369 0.874864i \(-0.660951\pi\)
−0.484369 + 0.874864i \(0.660951\pi\)
\(168\) −5.60097e9 −0.542465
\(169\) −7.41236e9 −0.698983
\(170\) −1.39375e9 −0.127986
\(171\) −4.23095e9 −0.378404
\(172\) 1.43339e9 0.124878
\(173\) −2.77504e9 −0.235539 −0.117769 0.993041i \(-0.537574\pi\)
−0.117769 + 0.993041i \(0.537574\pi\)
\(174\) −1.03971e9 −0.0859885
\(175\) 5.09107e9 0.410334
\(176\) −9.22393e9 −0.724618
\(177\) 2.21015e10 1.69255
\(178\) −2.07765e8 −0.0155125
\(179\) −1.51599e10 −1.10372 −0.551860 0.833937i \(-0.686082\pi\)
−0.551860 + 0.833937i \(0.686082\pi\)
\(180\) −4.37217e9 −0.310435
\(181\) −1.80133e10 −1.24750 −0.623749 0.781624i \(-0.714391\pi\)
−0.623749 + 0.781624i \(0.714391\pi\)
\(182\) −1.78921e9 −0.120876
\(183\) −2.08239e10 −1.37256
\(184\) −1.84249e9 −0.118502
\(185\) −8.98924e9 −0.564222
\(186\) 1.18178e10 0.723984
\(187\) −1.63501e10 −0.977764
\(188\) −8.99024e9 −0.524881
\(189\) 2.05879e9 0.117364
\(190\) 1.58230e9 0.0880843
\(191\) 1.94270e9 0.105623 0.0528113 0.998605i \(-0.483182\pi\)
0.0528113 + 0.998605i \(0.483182\pi\)
\(192\) −1.81879e9 −0.0965888
\(193\) −4.22076e8 −0.0218969 −0.0109485 0.999940i \(-0.503485\pi\)
−0.0109485 + 0.999940i \(0.503485\pi\)
\(194\) −9.04638e9 −0.458529
\(195\) −6.82858e9 −0.338201
\(196\) 1.23848e10 0.599429
\(197\) 2.39596e10 1.13339 0.566697 0.823926i \(-0.308221\pi\)
0.566697 + 0.823926i \(0.308221\pi\)
\(198\) 1.13445e10 0.524558
\(199\) −3.27995e10 −1.48261 −0.741307 0.671166i \(-0.765794\pi\)
−0.741307 + 0.671166i \(0.765794\pi\)
\(200\) −1.38837e10 −0.613578
\(201\) 2.36025e10 1.01994
\(202\) −1.91083e9 −0.0807496
\(203\) −1.86947e9 −0.0772658
\(204\) −1.81094e10 −0.732095
\(205\) −8.64501e9 −0.341880
\(206\) 3.30516e9 0.127876
\(207\) −3.36665e9 −0.127448
\(208\) −7.24902e9 −0.268531
\(209\) 1.85621e10 0.672929
\(210\) 3.82745e9 0.135807
\(211\) −7.88494e9 −0.273859 −0.136930 0.990581i \(-0.543723\pi\)
−0.136930 + 0.990581i \(0.543723\pi\)
\(212\) −1.26670e10 −0.430688
\(213\) 3.88190e10 1.29222
\(214\) −1.53556e10 −0.500502
\(215\) −2.17567e9 −0.0694416
\(216\) −5.61445e9 −0.175495
\(217\) 2.12493e10 0.650542
\(218\) 1.31262e9 0.0393628
\(219\) 2.20308e10 0.647189
\(220\) 1.91817e10 0.552057
\(221\) −1.28494e10 −0.362342
\(222\) 2.58342e10 0.713850
\(223\) 1.50654e10 0.407952 0.203976 0.978976i \(-0.434614\pi\)
0.203976 + 0.978976i \(0.434614\pi\)
\(224\) 1.91626e10 0.508556
\(225\) −2.53687e10 −0.659899
\(226\) −6.25607e9 −0.159519
\(227\) 2.82057e10 0.705051 0.352525 0.935802i \(-0.385323\pi\)
0.352525 + 0.935802i \(0.385323\pi\)
\(228\) 2.05593e10 0.503851
\(229\) 2.85908e10 0.687016 0.343508 0.939150i \(-0.388385\pi\)
0.343508 + 0.939150i \(0.388385\pi\)
\(230\) 1.25907e9 0.0296671
\(231\) 4.49001e10 1.03751
\(232\) 5.09818e9 0.115536
\(233\) 3.60569e10 0.801469 0.400735 0.916194i \(-0.368755\pi\)
0.400735 + 0.916194i \(0.368755\pi\)
\(234\) 8.91559e9 0.194392
\(235\) 1.36459e10 0.291874
\(236\) −4.87910e10 −1.02385
\(237\) 8.49449e10 1.74892
\(238\) 7.20216e9 0.145501
\(239\) −6.82558e10 −1.35316 −0.676580 0.736369i \(-0.736539\pi\)
−0.676580 + 0.736369i \(0.736539\pi\)
\(240\) 1.55070e10 0.301701
\(241\) −4.14356e10 −0.791219 −0.395610 0.918419i \(-0.629467\pi\)
−0.395610 + 0.918419i \(0.629467\pi\)
\(242\) −2.70643e10 −0.507256
\(243\) −7.15151e10 −1.31574
\(244\) 4.59707e10 0.830285
\(245\) −1.87984e10 −0.333329
\(246\) 2.48449e10 0.432543
\(247\) 1.45878e10 0.249376
\(248\) −5.79482e10 −0.972764
\(249\) −1.00905e11 −1.66348
\(250\) 2.14568e10 0.347404
\(251\) −4.84151e10 −0.769927 −0.384963 0.922932i \(-0.625786\pi\)
−0.384963 + 0.922932i \(0.625786\pi\)
\(252\) 2.25931e10 0.352918
\(253\) 1.47702e10 0.226645
\(254\) −8.42910e9 −0.127066
\(255\) 2.74874e10 0.407101
\(256\) −2.47156e10 −0.359659
\(257\) 1.70355e10 0.243588 0.121794 0.992555i \(-0.461135\pi\)
0.121794 + 0.992555i \(0.461135\pi\)
\(258\) 6.25267e9 0.0878570
\(259\) 4.64518e10 0.641436
\(260\) 1.50747e10 0.204583
\(261\) 9.31556e9 0.124259
\(262\) 4.98207e10 0.653212
\(263\) 5.93948e10 0.765504 0.382752 0.923851i \(-0.374976\pi\)
0.382752 + 0.923851i \(0.374976\pi\)
\(264\) −1.22445e11 −1.55140
\(265\) 1.92267e10 0.239496
\(266\) −8.17653e9 −0.100139
\(267\) 4.09753e9 0.0493425
\(268\) −5.21047e10 −0.616978
\(269\) −1.07712e11 −1.25423 −0.627117 0.778925i \(-0.715765\pi\)
−0.627117 + 0.778925i \(0.715765\pi\)
\(270\) 3.83666e9 0.0439356
\(271\) 1.11447e11 1.25518 0.627592 0.778543i \(-0.284041\pi\)
0.627592 + 0.778543i \(0.284041\pi\)
\(272\) 2.91797e10 0.323237
\(273\) 3.52866e10 0.384484
\(274\) −2.67299e10 −0.286497
\(275\) 1.11298e11 1.17352
\(276\) 1.63595e10 0.169699
\(277\) 1.47167e10 0.150194 0.0750968 0.997176i \(-0.476073\pi\)
0.0750968 + 0.997176i \(0.476073\pi\)
\(278\) 5.60222e10 0.562546
\(279\) −1.05885e11 −1.04620
\(280\) −1.87677e10 −0.182474
\(281\) 1.28452e11 1.22903 0.614513 0.788906i \(-0.289352\pi\)
0.614513 + 0.788906i \(0.289352\pi\)
\(282\) −3.92169e10 −0.369277
\(283\) −1.55862e10 −0.144445 −0.0722224 0.997389i \(-0.523009\pi\)
−0.0722224 + 0.997389i \(0.523009\pi\)
\(284\) −8.56965e10 −0.781682
\(285\) −3.12061e10 −0.280180
\(286\) −3.91147e10 −0.345694
\(287\) 4.46730e10 0.388666
\(288\) −9.54870e10 −0.817859
\(289\) −6.68645e10 −0.563839
\(290\) −3.48386e9 −0.0289248
\(291\) 1.78412e11 1.45850
\(292\) −4.86349e10 −0.391494
\(293\) −1.78122e11 −1.41193 −0.705965 0.708247i \(-0.749486\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(294\) 5.40247e10 0.421725
\(295\) 7.40577e10 0.569339
\(296\) −1.26677e11 −0.959148
\(297\) 4.50081e10 0.335650
\(298\) 8.83360e10 0.648881
\(299\) 1.16078e10 0.0839906
\(300\) 1.23274e11 0.878667
\(301\) 1.12427e10 0.0789447
\(302\) −1.04952e11 −0.726039
\(303\) 3.76853e10 0.256850
\(304\) −3.31274e10 −0.222462
\(305\) −6.97768e10 −0.461702
\(306\) −3.58883e10 −0.233995
\(307\) 1.52485e11 0.979725 0.489863 0.871800i \(-0.337047\pi\)
0.489863 + 0.871800i \(0.337047\pi\)
\(308\) −9.91209e10 −0.627606
\(309\) −6.51843e10 −0.406752
\(310\) 3.95991e10 0.243533
\(311\) −2.83289e11 −1.71715 −0.858574 0.512690i \(-0.828649\pi\)
−0.858574 + 0.512690i \(0.828649\pi\)
\(312\) −9.62290e10 −0.574924
\(313\) −2.44015e11 −1.43704 −0.718518 0.695508i \(-0.755179\pi\)
−0.718518 + 0.695508i \(0.755179\pi\)
\(314\) −6.08284e10 −0.353120
\(315\) −3.42931e10 −0.196250
\(316\) −1.87524e11 −1.05795
\(317\) −1.70636e11 −0.949084 −0.474542 0.880233i \(-0.657386\pi\)
−0.474542 + 0.880233i \(0.657386\pi\)
\(318\) −5.52556e10 −0.303008
\(319\) −4.08694e10 −0.220974
\(320\) −6.09440e9 −0.0324905
\(321\) 3.02843e11 1.59201
\(322\) −6.50623e9 −0.0337270
\(323\) −5.87209e10 −0.300180
\(324\) 1.85079e11 0.933052
\(325\) 8.74684e10 0.434887
\(326\) −1.35823e11 −0.666030
\(327\) −2.58875e10 −0.125206
\(328\) −1.21826e11 −0.581177
\(329\) −7.05149e10 −0.331818
\(330\) 8.36736e10 0.388397
\(331\) 3.68117e11 1.68562 0.842811 0.538210i \(-0.180899\pi\)
0.842811 + 0.538210i \(0.180899\pi\)
\(332\) 2.22758e11 1.00626
\(333\) −2.31469e11 −1.03156
\(334\) 9.37673e10 0.412280
\(335\) 7.90873e10 0.343088
\(336\) −8.01322e10 −0.342989
\(337\) −4.28944e10 −0.181162 −0.0905808 0.995889i \(-0.528872\pi\)
−0.0905808 + 0.995889i \(0.528872\pi\)
\(338\) 7.13801e10 0.297476
\(339\) 1.23382e11 0.507402
\(340\) −6.06808e10 −0.246261
\(341\) 4.64540e11 1.86050
\(342\) 4.07435e10 0.161043
\(343\) 2.29843e11 0.896620
\(344\) −3.06597e10 −0.118047
\(345\) −2.48313e10 −0.0943656
\(346\) 2.67233e10 0.100242
\(347\) 1.60170e11 0.593059 0.296529 0.955024i \(-0.404171\pi\)
0.296529 + 0.955024i \(0.404171\pi\)
\(348\) −4.52668e10 −0.165453
\(349\) 3.04550e11 1.09887 0.549433 0.835538i \(-0.314844\pi\)
0.549433 + 0.835538i \(0.314844\pi\)
\(350\) −4.90264e10 −0.174632
\(351\) 3.53715e10 0.124386
\(352\) 4.18923e11 1.45443
\(353\) −1.58317e11 −0.542676 −0.271338 0.962484i \(-0.587466\pi\)
−0.271338 + 0.962484i \(0.587466\pi\)
\(354\) −2.12835e11 −0.720324
\(355\) 1.30075e11 0.434676
\(356\) −9.04566e9 −0.0298480
\(357\) −1.42041e11 −0.462813
\(358\) 1.45988e11 0.469726
\(359\) −3.29410e11 −1.04668 −0.523338 0.852125i \(-0.675314\pi\)
−0.523338 + 0.852125i \(0.675314\pi\)
\(360\) 9.35194e10 0.293454
\(361\) −2.56023e11 −0.793406
\(362\) 1.73466e11 0.530916
\(363\) 5.33760e11 1.61349
\(364\) −7.78984e10 −0.232580
\(365\) 7.38207e10 0.217701
\(366\) 2.00532e11 0.584142
\(367\) 5.66994e11 1.63148 0.815738 0.578421i \(-0.196331\pi\)
0.815738 + 0.578421i \(0.196331\pi\)
\(368\) −2.63602e10 −0.0749261
\(369\) −2.22605e11 −0.625052
\(370\) 8.65653e10 0.240124
\(371\) −9.93536e10 −0.272271
\(372\) 5.14524e11 1.39304
\(373\) −4.36684e11 −1.16809 −0.584046 0.811720i \(-0.698531\pi\)
−0.584046 + 0.811720i \(0.698531\pi\)
\(374\) 1.57450e11 0.416121
\(375\) −4.23170e11 −1.10503
\(376\) 1.92299e11 0.496171
\(377\) −3.21190e10 −0.0818890
\(378\) −1.98259e10 −0.0499481
\(379\) −5.72159e11 −1.42443 −0.712213 0.701963i \(-0.752307\pi\)
−0.712213 + 0.701963i \(0.752307\pi\)
\(380\) 6.88903e10 0.169485
\(381\) 1.66238e11 0.404174
\(382\) −1.87080e10 −0.0449513
\(383\) −6.60393e11 −1.56822 −0.784111 0.620620i \(-0.786881\pi\)
−0.784111 + 0.620620i \(0.786881\pi\)
\(384\) 5.84141e11 1.37097
\(385\) 1.50451e11 0.348997
\(386\) 4.06454e9 0.00931899
\(387\) −5.60225e10 −0.126959
\(388\) −3.93861e11 −0.882268
\(389\) −5.55324e11 −1.22963 −0.614814 0.788672i \(-0.710769\pi\)
−0.614814 + 0.788672i \(0.710769\pi\)
\(390\) 6.57584e10 0.143933
\(391\) −4.67254e10 −0.101102
\(392\) −2.64908e11 −0.566641
\(393\) −9.82561e11 −2.07775
\(394\) −2.30728e11 −0.482355
\(395\) 2.84634e11 0.588301
\(396\) 4.93918e11 1.00932
\(397\) −7.47227e11 −1.50972 −0.754858 0.655889i \(-0.772294\pi\)
−0.754858 + 0.655889i \(0.772294\pi\)
\(398\) 3.15855e11 0.630978
\(399\) 1.61257e11 0.318523
\(400\) −1.98632e11 −0.387953
\(401\) −4.59550e11 −0.887530 −0.443765 0.896143i \(-0.646357\pi\)
−0.443765 + 0.896143i \(0.646357\pi\)
\(402\) −2.27289e11 −0.434072
\(403\) 3.65079e11 0.689468
\(404\) −8.31936e10 −0.155372
\(405\) −2.80924e11 −0.518849
\(406\) 1.80028e10 0.0328831
\(407\) 1.01550e12 1.83445
\(408\) 3.87354e11 0.692050
\(409\) 4.85966e11 0.858718 0.429359 0.903134i \(-0.358739\pi\)
0.429359 + 0.903134i \(0.358739\pi\)
\(410\) 8.32504e10 0.145499
\(411\) 5.27166e11 0.911296
\(412\) 1.43900e11 0.246050
\(413\) −3.82692e11 −0.647253
\(414\) 3.24205e10 0.0542397
\(415\) −3.38114e11 −0.559560
\(416\) 3.29228e11 0.538985
\(417\) −1.10487e12 −1.78936
\(418\) −1.78751e11 −0.286388
\(419\) 6.23327e11 0.987991 0.493995 0.869465i \(-0.335536\pi\)
0.493995 + 0.869465i \(0.335536\pi\)
\(420\) 1.66639e11 0.261310
\(421\) −1.20079e11 −0.186294 −0.0931468 0.995652i \(-0.529693\pi\)
−0.0931468 + 0.995652i \(0.529693\pi\)
\(422\) 7.59311e10 0.116550
\(423\) 3.51375e11 0.533628
\(424\) 2.70944e11 0.407130
\(425\) −3.52090e11 −0.523484
\(426\) −3.73822e11 −0.549948
\(427\) 3.60571e11 0.524886
\(428\) −6.68553e11 −0.963028
\(429\) 7.71417e11 1.09959
\(430\) 2.09514e10 0.0295533
\(431\) −1.24998e12 −1.74485 −0.872423 0.488752i \(-0.837452\pi\)
−0.872423 + 0.488752i \(0.837452\pi\)
\(432\) −8.03251e10 −0.110962
\(433\) 3.88881e10 0.0531644 0.0265822 0.999647i \(-0.491538\pi\)
0.0265822 + 0.999647i \(0.491538\pi\)
\(434\) −2.04628e11 −0.276861
\(435\) 6.87085e10 0.0920044
\(436\) 5.71490e10 0.0757390
\(437\) 5.30468e10 0.0695814
\(438\) −2.12153e11 −0.275433
\(439\) −5.52309e11 −0.709728 −0.354864 0.934918i \(-0.615473\pi\)
−0.354864 + 0.934918i \(0.615473\pi\)
\(440\) −4.10290e11 −0.521860
\(441\) −4.84049e11 −0.609418
\(442\) 1.23739e11 0.154207
\(443\) −1.47315e12 −1.81731 −0.908656 0.417546i \(-0.862890\pi\)
−0.908656 + 0.417546i \(0.862890\pi\)
\(444\) 1.12477e12 1.37354
\(445\) 1.37300e10 0.0165978
\(446\) −1.45078e11 −0.173618
\(447\) −1.74216e12 −2.06397
\(448\) 3.14927e10 0.0369368
\(449\) 7.85918e11 0.912575 0.456288 0.889832i \(-0.349179\pi\)
0.456288 + 0.889832i \(0.349179\pi\)
\(450\) 2.44298e11 0.280843
\(451\) 9.76616e11 1.11155
\(452\) −2.72377e11 −0.306935
\(453\) 2.06986e12 2.30940
\(454\) −2.71617e11 −0.300059
\(455\) 1.18238e11 0.129332
\(456\) −4.39759e11 −0.476291
\(457\) −1.62167e12 −1.73916 −0.869579 0.493794i \(-0.835610\pi\)
−0.869579 + 0.493794i \(0.835610\pi\)
\(458\) −2.75326e11 −0.292383
\(459\) −1.42382e11 −0.149727
\(460\) 5.48174e10 0.0570831
\(461\) −8.89063e11 −0.916808 −0.458404 0.888744i \(-0.651579\pi\)
−0.458404 + 0.888744i \(0.651579\pi\)
\(462\) −4.32382e11 −0.441549
\(463\) 4.38923e11 0.443889 0.221944 0.975059i \(-0.428760\pi\)
0.221944 + 0.975059i \(0.428760\pi\)
\(464\) 7.29388e10 0.0730513
\(465\) −7.80972e11 −0.774635
\(466\) −3.47224e11 −0.341093
\(467\) 6.08342e11 0.591864 0.295932 0.955209i \(-0.404370\pi\)
0.295932 + 0.955209i \(0.404370\pi\)
\(468\) 3.88167e11 0.374035
\(469\) −4.08683e11 −0.390039
\(470\) −1.31408e11 −0.124217
\(471\) 1.19965e12 1.12321
\(472\) 1.04363e12 0.967846
\(473\) 2.45783e11 0.225775
\(474\) −8.18009e11 −0.744313
\(475\) 3.99724e11 0.360279
\(476\) 3.13567e11 0.279962
\(477\) 4.95078e11 0.437865
\(478\) 6.57296e11 0.575884
\(479\) 2.16565e12 1.87965 0.939827 0.341650i \(-0.110986\pi\)
0.939827 + 0.341650i \(0.110986\pi\)
\(480\) −7.04281e11 −0.605564
\(481\) 7.98077e11 0.679817
\(482\) 3.99020e11 0.336731
\(483\) 1.28315e11 0.107280
\(484\) −1.17832e12 −0.976023
\(485\) 5.97824e11 0.490609
\(486\) 6.88682e11 0.559958
\(487\) 1.64181e12 1.32264 0.661320 0.750104i \(-0.269996\pi\)
0.661320 + 0.750104i \(0.269996\pi\)
\(488\) −9.83301e11 −0.784869
\(489\) 2.67869e12 2.11852
\(490\) 1.81026e11 0.141860
\(491\) 1.52108e12 1.18109 0.590547 0.807003i \(-0.298912\pi\)
0.590547 + 0.807003i \(0.298912\pi\)
\(492\) 1.08170e12 0.832267
\(493\) 1.29290e11 0.0985718
\(494\) −1.40479e11 −0.106130
\(495\) −7.49696e11 −0.561257
\(496\) −8.29056e11 −0.615058
\(497\) −6.72160e11 −0.494161
\(498\) 9.71707e11 0.707951
\(499\) −4.02982e11 −0.290960 −0.145480 0.989361i \(-0.546473\pi\)
−0.145480 + 0.989361i \(0.546473\pi\)
\(500\) 9.34186e11 0.668449
\(501\) −1.84927e12 −1.31139
\(502\) 4.66232e11 0.327669
\(503\) −4.10239e11 −0.285747 −0.142873 0.989741i \(-0.545634\pi\)
−0.142873 + 0.989741i \(0.545634\pi\)
\(504\) −4.83260e11 −0.333614
\(505\) 1.26276e11 0.0863990
\(506\) −1.42236e11 −0.0964564
\(507\) −1.40775e12 −0.946218
\(508\) −3.66986e11 −0.244491
\(509\) 1.26933e12 0.838192 0.419096 0.907942i \(-0.362347\pi\)
0.419096 + 0.907942i \(0.362347\pi\)
\(510\) −2.64700e11 −0.173256
\(511\) −3.81467e11 −0.247493
\(512\) −1.33676e12 −0.859684
\(513\) 1.61645e11 0.103047
\(514\) −1.64050e11 −0.103667
\(515\) −2.18420e11 −0.136823
\(516\) 2.72228e11 0.169048
\(517\) −1.54156e12 −0.948970
\(518\) −4.47325e11 −0.272985
\(519\) −5.27036e11 −0.318850
\(520\) −3.22444e11 −0.193392
\(521\) 2.92628e12 1.73999 0.869994 0.493062i \(-0.164122\pi\)
0.869994 + 0.493062i \(0.164122\pi\)
\(522\) −8.97077e10 −0.0528826
\(523\) 2.35078e12 1.37390 0.686950 0.726705i \(-0.258949\pi\)
0.686950 + 0.726705i \(0.258949\pi\)
\(524\) 2.16909e12 1.25686
\(525\) 9.66895e11 0.555473
\(526\) −5.71964e11 −0.325786
\(527\) −1.46957e12 −0.829930
\(528\) −1.75181e12 −0.980921
\(529\) −1.75894e12 −0.976565
\(530\) −1.85151e11 −0.101926
\(531\) 1.90695e12 1.04091
\(532\) −3.55989e11 −0.192679
\(533\) 7.67516e11 0.411922
\(534\) −3.94587e10 −0.0209994
\(535\) 1.01477e12 0.535518
\(536\) 1.11450e12 0.583231
\(537\) −2.87917e12 −1.49411
\(538\) 1.03725e12 0.533782
\(539\) 2.12363e12 1.08375
\(540\) 1.67040e11 0.0845375
\(541\) 3.12960e12 1.57073 0.785364 0.619034i \(-0.212476\pi\)
0.785364 + 0.619034i \(0.212476\pi\)
\(542\) −1.07322e12 −0.534187
\(543\) −3.42109e12 −1.68875
\(544\) −1.32526e12 −0.648790
\(545\) −8.67439e10 −0.0421167
\(546\) −3.39806e11 −0.163630
\(547\) −2.95721e12 −1.41234 −0.706169 0.708043i \(-0.749578\pi\)
−0.706169 + 0.708043i \(0.749578\pi\)
\(548\) −1.16377e12 −0.551257
\(549\) −1.79672e12 −0.844121
\(550\) −1.07179e12 −0.499433
\(551\) −1.46781e11 −0.0678403
\(552\) −3.49925e11 −0.160416
\(553\) −1.47084e12 −0.668810
\(554\) −1.41720e11 −0.0639201
\(555\) −1.70724e12 −0.763792
\(556\) 2.43909e12 1.08241
\(557\) −1.07910e12 −0.475021 −0.237511 0.971385i \(-0.576331\pi\)
−0.237511 + 0.971385i \(0.576331\pi\)
\(558\) 1.01966e12 0.445247
\(559\) 1.93159e11 0.0836684
\(560\) −2.68507e11 −0.115375
\(561\) −3.10522e12 −1.32361
\(562\) −1.23697e12 −0.523055
\(563\) 1.53793e12 0.645132 0.322566 0.946547i \(-0.395455\pi\)
0.322566 + 0.946547i \(0.395455\pi\)
\(564\) −1.70743e12 −0.710536
\(565\) 4.13428e11 0.170680
\(566\) 1.50093e11 0.0614735
\(567\) 1.45167e12 0.589853
\(568\) 1.83302e12 0.738925
\(569\) 2.07875e12 0.831377 0.415688 0.909507i \(-0.363541\pi\)
0.415688 + 0.909507i \(0.363541\pi\)
\(570\) 3.00511e11 0.119240
\(571\) −1.16515e12 −0.458689 −0.229345 0.973345i \(-0.573658\pi\)
−0.229345 + 0.973345i \(0.573658\pi\)
\(572\) −1.70297e12 −0.665159
\(573\) 3.68958e11 0.142982
\(574\) −4.30195e11 −0.165410
\(575\) 3.18068e11 0.121343
\(576\) −1.56928e11 −0.0594017
\(577\) 7.45248e11 0.279904 0.139952 0.990158i \(-0.455305\pi\)
0.139952 + 0.990158i \(0.455305\pi\)
\(578\) 6.43897e11 0.239961
\(579\) −8.01607e10 −0.0296420
\(580\) −1.51680e11 −0.0556548
\(581\) 1.74720e12 0.636136
\(582\) −1.71809e12 −0.620715
\(583\) −2.17202e12 −0.778672
\(584\) 1.04029e12 0.370080
\(585\) −5.89181e11 −0.207992
\(586\) 1.71529e12 0.600895
\(587\) −2.49286e11 −0.0866615 −0.0433308 0.999061i \(-0.513797\pi\)
−0.0433308 + 0.999061i \(0.513797\pi\)
\(588\) 2.35212e12 0.811451
\(589\) 1.66838e12 0.571184
\(590\) −7.13167e11 −0.242302
\(591\) 4.55040e12 1.53428
\(592\) −1.81235e12 −0.606449
\(593\) −3.10748e12 −1.03196 −0.515980 0.856601i \(-0.672572\pi\)
−0.515980 + 0.856601i \(0.672572\pi\)
\(594\) −4.33423e11 −0.142847
\(595\) −4.75950e11 −0.155681
\(596\) 3.84597e12 1.24853
\(597\) −6.22928e12 −2.00703
\(598\) −1.11782e11 −0.0357451
\(599\) 2.25538e12 0.715811 0.357906 0.933758i \(-0.383491\pi\)
0.357906 + 0.933758i \(0.383491\pi\)
\(600\) −2.63679e12 −0.830605
\(601\) 6.65807e11 0.208168 0.104084 0.994569i \(-0.466809\pi\)
0.104084 + 0.994569i \(0.466809\pi\)
\(602\) −1.08266e11 −0.0335977
\(603\) 2.03646e12 0.627260
\(604\) −4.56940e12 −1.39699
\(605\) 1.78852e12 0.542744
\(606\) −3.62904e11 −0.109311
\(607\) 7.48544e11 0.223804 0.111902 0.993719i \(-0.464306\pi\)
0.111902 + 0.993719i \(0.464306\pi\)
\(608\) 1.50455e12 0.446519
\(609\) −3.55050e11 −0.104595
\(610\) 6.71943e11 0.196493
\(611\) −1.21150e12 −0.351672
\(612\) −1.56250e12 −0.450235
\(613\) −2.50001e12 −0.715103 −0.357552 0.933893i \(-0.616388\pi\)
−0.357552 + 0.933893i \(0.616388\pi\)
\(614\) −1.46841e12 −0.416956
\(615\) −1.64186e12 −0.462805
\(616\) 2.12017e12 0.593277
\(617\) 1.56036e12 0.433451 0.216726 0.976233i \(-0.430462\pi\)
0.216726 + 0.976233i \(0.430462\pi\)
\(618\) 6.27717e11 0.173107
\(619\) −1.65116e12 −0.452045 −0.226023 0.974122i \(-0.572572\pi\)
−0.226023 + 0.974122i \(0.572572\pi\)
\(620\) 1.72407e12 0.468588
\(621\) 1.28624e11 0.0347065
\(622\) 2.72804e12 0.730792
\(623\) −7.09496e10 −0.0188692
\(624\) −1.37673e12 −0.363512
\(625\) 1.60575e12 0.420939
\(626\) 2.34984e12 0.611580
\(627\) 3.52531e12 0.910949
\(628\) −2.64834e12 −0.679448
\(629\) −3.21253e12 −0.818313
\(630\) 3.30238e11 0.0835208
\(631\) −9.48276e9 −0.00238124 −0.00119062 0.999999i \(-0.500379\pi\)
−0.00119062 + 0.999999i \(0.500379\pi\)
\(632\) 4.01108e12 1.00008
\(633\) −1.49751e12 −0.370726
\(634\) 1.64321e12 0.403916
\(635\) 5.57031e11 0.135956
\(636\) −2.40572e12 −0.583026
\(637\) 1.66894e12 0.401619
\(638\) 3.93568e11 0.0940429
\(639\) 3.34936e12 0.794709
\(640\) 1.95734e12 0.461165
\(641\) 2.83779e12 0.663925 0.331963 0.943293i \(-0.392289\pi\)
0.331963 + 0.943293i \(0.392289\pi\)
\(642\) −2.91634e12 −0.677533
\(643\) 6.64160e12 1.53223 0.766114 0.642704i \(-0.222188\pi\)
0.766114 + 0.642704i \(0.222188\pi\)
\(644\) −2.83268e11 −0.0648950
\(645\) −4.13203e11 −0.0940037
\(646\) 5.65475e11 0.127752
\(647\) −1.93866e12 −0.434943 −0.217471 0.976067i \(-0.569781\pi\)
−0.217471 + 0.976067i \(0.569781\pi\)
\(648\) −3.95880e12 −0.882015
\(649\) −8.36621e12 −1.85109
\(650\) −8.42310e11 −0.185081
\(651\) 4.03566e12 0.880644
\(652\) −5.91346e12 −1.28153
\(653\) 3.90359e11 0.0840147 0.0420073 0.999117i \(-0.486625\pi\)
0.0420073 + 0.999117i \(0.486625\pi\)
\(654\) 2.49293e11 0.0532857
\(655\) −3.29237e12 −0.698912
\(656\) −1.74295e12 −0.367466
\(657\) 1.90085e12 0.398018
\(658\) 6.79050e11 0.141216
\(659\) 4.87974e12 1.00789 0.503944 0.863737i \(-0.331882\pi\)
0.503944 + 0.863737i \(0.331882\pi\)
\(660\) 3.64298e12 0.747324
\(661\) −8.13003e12 −1.65648 −0.828239 0.560376i \(-0.810657\pi\)
−0.828239 + 0.560376i \(0.810657\pi\)
\(662\) −3.54492e12 −0.717375
\(663\) −2.44037e12 −0.490506
\(664\) −4.76473e12 −0.951222
\(665\) 5.40340e11 0.107145
\(666\) 2.22902e12 0.439015
\(667\) −1.16797e11 −0.0228488
\(668\) 4.08244e12 0.793279
\(669\) 2.86122e12 0.552248
\(670\) −7.61601e11 −0.146013
\(671\) 7.88261e12 1.50113
\(672\) 3.63936e12 0.688436
\(673\) 7.83754e12 1.47269 0.736346 0.676606i \(-0.236550\pi\)
0.736346 + 0.676606i \(0.236550\pi\)
\(674\) 4.13068e11 0.0770996
\(675\) 9.69222e11 0.179703
\(676\) 3.10775e12 0.572381
\(677\) 3.51888e12 0.643807 0.321903 0.946773i \(-0.395677\pi\)
0.321903 + 0.946773i \(0.395677\pi\)
\(678\) −1.18815e12 −0.215943
\(679\) −3.08925e12 −0.557749
\(680\) 1.29795e12 0.232791
\(681\) 5.35682e12 0.954433
\(682\) −4.47347e12 −0.791798
\(683\) 9.44474e12 1.66072 0.830361 0.557226i \(-0.188134\pi\)
0.830361 + 0.557226i \(0.188134\pi\)
\(684\) 1.77389e12 0.309866
\(685\) 1.76643e12 0.306541
\(686\) −2.21336e12 −0.381588
\(687\) 5.42997e12 0.930019
\(688\) −4.38644e11 −0.0746387
\(689\) −1.70697e12 −0.288562
\(690\) 2.39123e11 0.0401605
\(691\) −3.85564e12 −0.643346 −0.321673 0.946851i \(-0.604245\pi\)
−0.321673 + 0.946851i \(0.604245\pi\)
\(692\) 1.16348e12 0.192877
\(693\) 3.87405e12 0.638065
\(694\) −1.54241e12 −0.252397
\(695\) −3.70219e12 −0.601903
\(696\) 9.68245e11 0.156403
\(697\) −3.08951e12 −0.495841
\(698\) −2.93278e12 −0.467660
\(699\) 6.84793e12 1.08496
\(700\) −2.13451e12 −0.336014
\(701\) −2.65363e12 −0.415059 −0.207529 0.978229i \(-0.566542\pi\)
−0.207529 + 0.978229i \(0.566542\pi\)
\(702\) −3.40624e11 −0.0529368
\(703\) 3.64715e12 0.563189
\(704\) 6.88477e11 0.105636
\(705\) 2.59162e12 0.395113
\(706\) 1.52457e12 0.230954
\(707\) −6.52528e11 −0.0982228
\(708\) −9.26639e12 −1.38599
\(709\) −8.34085e12 −1.23966 −0.619830 0.784736i \(-0.712798\pi\)
−0.619830 + 0.784736i \(0.712798\pi\)
\(710\) −1.25260e12 −0.184991
\(711\) 7.32918e12 1.07558
\(712\) 1.93484e11 0.0282154
\(713\) 1.32756e12 0.192377
\(714\) 1.36783e12 0.196966
\(715\) 2.58487e12 0.369880
\(716\) 6.35603e12 0.903811
\(717\) −1.29631e13 −1.83178
\(718\) 3.17218e12 0.445449
\(719\) −1.03986e13 −1.45110 −0.725549 0.688170i \(-0.758414\pi\)
−0.725549 + 0.688170i \(0.758414\pi\)
\(720\) 1.33797e12 0.185545
\(721\) 1.12868e12 0.155547
\(722\) 2.46547e12 0.337662
\(723\) −7.86944e12 −1.07108
\(724\) 7.55236e12 1.02155
\(725\) −8.80098e11 −0.118307
\(726\) −5.14004e12 −0.686676
\(727\) −5.23334e12 −0.694822 −0.347411 0.937713i \(-0.612939\pi\)
−0.347411 + 0.937713i \(0.612939\pi\)
\(728\) 1.66623e12 0.219858
\(729\) −4.89333e12 −0.641698
\(730\) −7.10884e11 −0.0926501
\(731\) −7.77530e11 −0.100714
\(732\) 8.73075e12 1.12396
\(733\) −1.15223e12 −0.147425 −0.0737123 0.997280i \(-0.523485\pi\)
−0.0737123 + 0.997280i \(0.523485\pi\)
\(734\) −5.46008e12 −0.694331
\(735\) −3.57018e12 −0.451229
\(736\) 1.19720e12 0.150389
\(737\) −8.93440e12 −1.11548
\(738\) 2.14366e12 0.266012
\(739\) −5.82737e12 −0.718741 −0.359371 0.933195i \(-0.617009\pi\)
−0.359371 + 0.933195i \(0.617009\pi\)
\(740\) 3.76888e12 0.462029
\(741\) 2.77052e12 0.337582
\(742\) 9.56763e11 0.115874
\(743\) −6.73156e12 −0.810338 −0.405169 0.914242i \(-0.632787\pi\)
−0.405169 + 0.914242i \(0.632787\pi\)
\(744\) −1.10055e13 −1.31684
\(745\) −5.83763e12 −0.694278
\(746\) 4.20521e12 0.497122
\(747\) −8.70627e12 −1.02303
\(748\) 6.85504e12 0.800669
\(749\) −5.24379e12 −0.608804
\(750\) 4.07508e12 0.470284
\(751\) −1.31942e13 −1.51357 −0.756784 0.653665i \(-0.773231\pi\)
−0.756784 + 0.653665i \(0.773231\pi\)
\(752\) 2.75119e12 0.313719
\(753\) −9.19500e12 −1.04226
\(754\) 3.09302e11 0.0348507
\(755\) 6.93569e12 0.776834
\(756\) −8.63179e11 −0.0961065
\(757\) −7.77025e12 −0.860010 −0.430005 0.902826i \(-0.641488\pi\)
−0.430005 + 0.902826i \(0.641488\pi\)
\(758\) 5.50982e12 0.606214
\(759\) 2.80516e12 0.306811
\(760\) −1.47354e12 −0.160215
\(761\) −1.84489e13 −1.99407 −0.997035 0.0769500i \(-0.975482\pi\)
−0.997035 + 0.0769500i \(0.975482\pi\)
\(762\) −1.60085e12 −0.172010
\(763\) 4.48248e11 0.0478804
\(764\) −8.14509e11 −0.0864920
\(765\) 2.37165e12 0.250365
\(766\) 6.35950e12 0.667411
\(767\) −6.57494e12 −0.685982
\(768\) −4.69399e12 −0.486874
\(769\) 1.24516e13 1.28397 0.641985 0.766717i \(-0.278111\pi\)
0.641985 + 0.766717i \(0.278111\pi\)
\(770\) −1.44883e12 −0.148528
\(771\) 3.23538e12 0.329747
\(772\) 1.76962e11 0.0179309
\(773\) −1.48775e13 −1.49873 −0.749365 0.662157i \(-0.769641\pi\)
−0.749365 + 0.662157i \(0.769641\pi\)
\(774\) 5.39489e11 0.0540317
\(775\) 1.00036e13 0.996090
\(776\) 8.42459e12 0.834009
\(777\) 8.82212e12 0.868317
\(778\) 5.34771e12 0.523310
\(779\) 3.50748e12 0.341254
\(780\) 2.86299e12 0.276945
\(781\) −1.46944e13 −1.41326
\(782\) 4.49960e11 0.0430273
\(783\) −3.55905e11 −0.0338381
\(784\) −3.79000e12 −0.358275
\(785\) 4.01980e12 0.377825
\(786\) 9.46195e12 0.884258
\(787\) 1.10622e13 1.02791 0.513956 0.857816i \(-0.328179\pi\)
0.513956 + 0.857816i \(0.328179\pi\)
\(788\) −1.00454e13 −0.928112
\(789\) 1.12802e13 1.03627
\(790\) −2.74099e12 −0.250372
\(791\) −2.13638e12 −0.194037
\(792\) −1.05648e13 −0.954107
\(793\) 6.19488e12 0.556293
\(794\) 7.19570e12 0.642512
\(795\) 3.65153e12 0.324207
\(796\) 1.37517e13 1.21408
\(797\) −5.04498e12 −0.442891 −0.221446 0.975173i \(-0.571077\pi\)
−0.221446 + 0.975173i \(0.571077\pi\)
\(798\) −1.55289e12 −0.135558
\(799\) 4.87670e12 0.423316
\(800\) 9.02124e12 0.778684
\(801\) 3.53541e11 0.0303454
\(802\) 4.42541e12 0.377719
\(803\) −8.33943e12 −0.707810
\(804\) −9.89571e12 −0.835208
\(805\) 4.29960e11 0.0360866
\(806\) −3.51566e12 −0.293427
\(807\) −2.04566e13 −1.69787
\(808\) 1.77949e12 0.146874
\(809\) 1.42809e11 0.0117216 0.00586079 0.999983i \(-0.498134\pi\)
0.00586079 + 0.999983i \(0.498134\pi\)
\(810\) 2.70526e12 0.220814
\(811\) 2.03598e13 1.65264 0.826322 0.563197i \(-0.190429\pi\)
0.826322 + 0.563197i \(0.190429\pi\)
\(812\) 7.83805e11 0.0632712
\(813\) 2.11660e13 1.69915
\(814\) −9.77918e12 −0.780715
\(815\) 8.97577e12 0.712627
\(816\) 5.54182e12 0.437569
\(817\) 8.82721e11 0.0693145
\(818\) −4.67979e12 −0.365457
\(819\) 3.04458e12 0.236456
\(820\) 3.62455e12 0.279958
\(821\) −1.84922e11 −0.0142051 −0.00710255 0.999975i \(-0.502261\pi\)
−0.00710255 + 0.999975i \(0.502261\pi\)
\(822\) −5.07655e12 −0.387834
\(823\) 2.87069e12 0.218116 0.109058 0.994035i \(-0.465217\pi\)
0.109058 + 0.994035i \(0.465217\pi\)
\(824\) −3.07799e12 −0.232592
\(825\) 2.11377e13 1.58861
\(826\) 3.68528e12 0.275461
\(827\) 2.25945e13 1.67969 0.839844 0.542828i \(-0.182647\pi\)
0.839844 + 0.542828i \(0.182647\pi\)
\(828\) 1.41152e12 0.104364
\(829\) −1.25049e12 −0.0919570 −0.0459785 0.998942i \(-0.514641\pi\)
−0.0459785 + 0.998942i \(0.514641\pi\)
\(830\) 3.25600e12 0.238140
\(831\) 2.79500e12 0.203318
\(832\) 5.41069e11 0.0391469
\(833\) −6.71806e12 −0.483439
\(834\) 1.06397e13 0.761523
\(835\) −6.19655e12 −0.441124
\(836\) −7.78245e12 −0.551046
\(837\) 4.04537e12 0.284901
\(838\) −6.00256e12 −0.420474
\(839\) −9.40360e12 −0.655187 −0.327594 0.944819i \(-0.606238\pi\)
−0.327594 + 0.944819i \(0.606238\pi\)
\(840\) −3.56437e12 −0.247016
\(841\) −1.41840e13 −0.977723
\(842\) 1.15635e12 0.0792837
\(843\) 2.43955e13 1.66374
\(844\) 3.30588e12 0.224257
\(845\) −4.71711e12 −0.318288
\(846\) −3.38370e12 −0.227104
\(847\) −9.24217e12 −0.617019
\(848\) 3.87635e12 0.257420
\(849\) −2.96013e12 −0.195536
\(850\) 3.39058e12 0.222787
\(851\) 2.90211e12 0.189684
\(852\) −1.62755e13 −1.05817
\(853\) 4.97590e12 0.321811 0.160905 0.986970i \(-0.448559\pi\)
0.160905 + 0.986970i \(0.448559\pi\)
\(854\) −3.47225e12 −0.223384
\(855\) −2.69251e12 −0.172310
\(856\) 1.43002e13 0.910352
\(857\) −2.45954e13 −1.55755 −0.778773 0.627306i \(-0.784158\pi\)
−0.778773 + 0.627306i \(0.784158\pi\)
\(858\) −7.42865e12 −0.467969
\(859\) 4.12964e12 0.258787 0.129394 0.991593i \(-0.458697\pi\)
0.129394 + 0.991593i \(0.458697\pi\)
\(860\) 9.12183e11 0.0568642
\(861\) 8.48429e12 0.526140
\(862\) 1.20372e13 0.742579
\(863\) −5.44656e11 −0.0334252 −0.0167126 0.999860i \(-0.505320\pi\)
−0.0167126 + 0.999860i \(0.505320\pi\)
\(864\) 3.64812e12 0.222719
\(865\) −1.76599e12 −0.107255
\(866\) −3.74488e11 −0.0226260
\(867\) −1.26989e13 −0.763274
\(868\) −8.90909e12 −0.532715
\(869\) −3.21547e13 −1.91274
\(870\) −6.61654e11 −0.0391557
\(871\) −7.02148e12 −0.413377
\(872\) −1.22240e12 −0.0715961
\(873\) 1.53937e13 0.896971
\(874\) −5.10835e11 −0.0296128
\(875\) 7.32728e12 0.422578
\(876\) −9.23673e12 −0.529968
\(877\) 2.79747e13 1.59686 0.798431 0.602087i \(-0.205664\pi\)
0.798431 + 0.602087i \(0.205664\pi\)
\(878\) 5.31867e12 0.302049
\(879\) −3.38289e13 −1.91134
\(880\) −5.86996e12 −0.329961
\(881\) 2.31406e13 1.29415 0.647073 0.762428i \(-0.275993\pi\)
0.647073 + 0.762428i \(0.275993\pi\)
\(882\) 4.66133e12 0.259359
\(883\) −2.00961e13 −1.11247 −0.556235 0.831025i \(-0.687754\pi\)
−0.556235 + 0.831025i \(0.687754\pi\)
\(884\) 5.38733e12 0.296714
\(885\) 1.40650e13 0.770719
\(886\) 1.41862e13 0.773420
\(887\) 1.62218e12 0.0879916 0.0439958 0.999032i \(-0.485991\pi\)
0.0439958 + 0.999032i \(0.485991\pi\)
\(888\) −2.40585e13 −1.29841
\(889\) −2.87845e12 −0.154561
\(890\) −1.32218e11 −0.00706376
\(891\) 3.17356e13 1.68693
\(892\) −6.31641e12 −0.334063
\(893\) −5.53646e12 −0.291340
\(894\) 1.67768e13 0.878395
\(895\) −9.64754e12 −0.502589
\(896\) −1.01145e13 −0.524276
\(897\) 2.20456e12 0.113699
\(898\) −7.56829e12 −0.388378
\(899\) −3.67338e12 −0.187563
\(900\) 1.06362e13 0.540377
\(901\) 6.87113e12 0.347350
\(902\) −9.40470e12 −0.473059
\(903\) 2.13522e12 0.106868
\(904\) 5.82606e12 0.290146
\(905\) −1.14634e13 −0.568060
\(906\) −1.99325e13 −0.982844
\(907\) 3.75549e13 1.84261 0.921306 0.388839i \(-0.127124\pi\)
0.921306 + 0.388839i \(0.127124\pi\)
\(908\) −1.18257e13 −0.577350
\(909\) 3.25154e12 0.157962
\(910\) −1.13862e12 −0.0550419
\(911\) −1.18903e13 −0.571951 −0.285975 0.958237i \(-0.592318\pi\)
−0.285975 + 0.958237i \(0.592318\pi\)
\(912\) −6.29156e12 −0.301149
\(913\) 3.81963e13 1.81929
\(914\) 1.56165e13 0.740159
\(915\) −1.32520e13 −0.625010
\(916\) −1.19871e13 −0.562582
\(917\) 1.70133e13 0.794558
\(918\) 1.37113e12 0.0637214
\(919\) −2.35907e11 −0.0109099 −0.00545495 0.999985i \(-0.501736\pi\)
−0.00545495 + 0.999985i \(0.501736\pi\)
\(920\) −1.17253e12 −0.0539608
\(921\) 2.89599e13 1.32626
\(922\) 8.56157e12 0.390179
\(923\) −1.15482e13 −0.523730
\(924\) −1.88250e13 −0.849595
\(925\) 2.18683e13 0.982147
\(926\) −4.22678e12 −0.188912
\(927\) −5.62420e12 −0.250151
\(928\) −3.31266e12 −0.146626
\(929\) −1.46785e11 −0.00646562 −0.00323281 0.999995i \(-0.501029\pi\)
−0.00323281 + 0.999995i \(0.501029\pi\)
\(930\) 7.52066e12 0.329673
\(931\) 7.62694e12 0.332718
\(932\) −1.51174e13 −0.656305
\(933\) −5.38022e13 −2.32452
\(934\) −5.85826e12 −0.251888
\(935\) −1.04050e13 −0.445234
\(936\) −8.30278e12 −0.353576
\(937\) 2.39861e13 1.01656 0.508279 0.861193i \(-0.330282\pi\)
0.508279 + 0.861193i \(0.330282\pi\)
\(938\) 3.93556e12 0.165995
\(939\) −4.63434e13 −1.94533
\(940\) −5.72124e12 −0.239010
\(941\) −1.94273e12 −0.0807716 −0.0403858 0.999184i \(-0.512859\pi\)
−0.0403858 + 0.999184i \(0.512859\pi\)
\(942\) −1.15525e13 −0.478022
\(943\) 2.79098e12 0.114935
\(944\) 1.49310e13 0.611949
\(945\) 1.31018e12 0.0534426
\(946\) −2.36686e12 −0.0960865
\(947\) 3.34211e12 0.135035 0.0675173 0.997718i \(-0.478492\pi\)
0.0675173 + 0.997718i \(0.478492\pi\)
\(948\) −3.56145e13 −1.43215
\(949\) −6.55390e12 −0.262302
\(950\) −3.84929e12 −0.153329
\(951\) −3.24072e13 −1.28478
\(952\) −6.70712e12 −0.264649
\(953\) −2.86874e13 −1.12661 −0.563304 0.826250i \(-0.690470\pi\)
−0.563304 + 0.826250i \(0.690470\pi\)
\(954\) −4.76754e12 −0.186349
\(955\) 1.23631e12 0.0480962
\(956\) 2.86173e13 1.10807
\(957\) −7.76192e12 −0.299134
\(958\) −2.08549e13 −0.799952
\(959\) −9.12800e12 −0.348491
\(960\) −1.15745e12 −0.0439826
\(961\) 1.53137e13 0.579196
\(962\) −7.68539e12 −0.289320
\(963\) 2.61297e13 0.979077
\(964\) 1.73725e13 0.647912
\(965\) −2.68603e11 −0.00997097
\(966\) −1.23566e12 −0.0456565
\(967\) −3.93409e13 −1.44685 −0.723427 0.690401i \(-0.757434\pi\)
−0.723427 + 0.690401i \(0.757434\pi\)
\(968\) 2.52040e13 0.922636
\(969\) −1.11523e13 −0.406356
\(970\) −5.75697e12 −0.208796
\(971\) −1.16162e13 −0.419353 −0.209676 0.977771i \(-0.567241\pi\)
−0.209676 + 0.977771i \(0.567241\pi\)
\(972\) 2.99838e13 1.07743
\(973\) 1.91310e13 0.684274
\(974\) −1.58104e13 −0.562895
\(975\) 1.66120e13 0.588710
\(976\) −1.40679e13 −0.496256
\(977\) 3.35215e13 1.17706 0.588530 0.808476i \(-0.299707\pi\)
0.588530 + 0.808476i \(0.299707\pi\)
\(978\) −2.57955e13 −0.901610
\(979\) −1.55106e12 −0.0539643
\(980\) 7.88150e12 0.272955
\(981\) −2.23361e12 −0.0770012
\(982\) −1.46478e13 −0.502655
\(983\) −5.05172e13 −1.72563 −0.862817 0.505516i \(-0.831302\pi\)
−0.862817 + 0.505516i \(0.831302\pi\)
\(984\) −2.31372e13 −0.786744
\(985\) 1.52475e13 0.516102
\(986\) −1.24504e12 −0.0419507
\(987\) −1.33922e13 −0.449184
\(988\) −6.11617e12 −0.204208
\(989\) 7.02399e11 0.0233454
\(990\) 7.21948e12 0.238862
\(991\) 5.31629e13 1.75096 0.875482 0.483250i \(-0.160544\pi\)
0.875482 + 0.483250i \(0.160544\pi\)
\(992\) 3.76532e13 1.23452
\(993\) 6.99128e13 2.28184
\(994\) 6.47282e12 0.210307
\(995\) −2.08731e13 −0.675122
\(996\) 4.23061e13 1.36219
\(997\) −4.26532e13 −1.36717 −0.683586 0.729870i \(-0.739581\pi\)
−0.683586 + 0.729870i \(0.739581\pi\)
\(998\) 3.88067e12 0.123828
\(999\) 8.84335e12 0.280913
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 43.10.a.a.1.7 15
3.2 odd 2 387.10.a.c.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.7 15 1.1 even 1 trivial
387.10.a.c.1.9 15 3.2 odd 2