Properties

Label 43.10.a.a.1.8
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 1.8
Root \(2.38595\) 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

\(f(q)\) \(=\) \(q-4.38595 q^{2} +117.920 q^{3} -492.763 q^{4} -1237.15 q^{5} -517.190 q^{6} +12549.3 q^{7} +4406.84 q^{8} -5777.96 q^{9} +O(q^{10})\) \(q-4.38595 q^{2} +117.920 q^{3} -492.763 q^{4} -1237.15 q^{5} -517.190 q^{6} +12549.3 q^{7} +4406.84 q^{8} -5777.96 q^{9} +5426.06 q^{10} +27918.0 q^{11} -58106.5 q^{12} -139652. q^{13} -55040.4 q^{14} -145884. q^{15} +232967. q^{16} -408645. q^{17} +25341.8 q^{18} -157490. q^{19} +609620. q^{20} +1.47980e6 q^{21} -122447. q^{22} -730739. q^{23} +519653. q^{24} -422597. q^{25} +612508. q^{26} -3.00235e6 q^{27} -6.18382e6 q^{28} -6.14391e6 q^{29} +639839. q^{30} +6.18625e6 q^{31} -3.27808e6 q^{32} +3.29208e6 q^{33} +1.79230e6 q^{34} -1.55253e7 q^{35} +2.84717e6 q^{36} -2.16507e7 q^{37} +690744. q^{38} -1.64677e7 q^{39} -5.45190e6 q^{40} +2.42738e7 q^{41} -6.49035e6 q^{42} -3.41880e6 q^{43} -1.37570e7 q^{44} +7.14817e6 q^{45} +3.20499e6 q^{46} -3.21569e7 q^{47} +2.74713e7 q^{48} +1.17130e8 q^{49} +1.85349e6 q^{50} -4.81873e7 q^{51} +6.88155e7 q^{52} -6.96541e7 q^{53} +1.31681e7 q^{54} -3.45386e7 q^{55} +5.53026e7 q^{56} -1.85712e7 q^{57} +2.69469e7 q^{58} -1.12041e8 q^{59} +7.18862e7 q^{60} +8.03385e7 q^{61} -2.71326e7 q^{62} -7.25091e7 q^{63} -1.04901e8 q^{64} +1.72770e8 q^{65} -1.44389e7 q^{66} -5.78429e7 q^{67} +2.01366e8 q^{68} -8.61685e7 q^{69} +6.80930e7 q^{70} +3.11039e8 q^{71} -2.54625e7 q^{72} -3.23803e7 q^{73} +9.49590e7 q^{74} -4.98325e7 q^{75} +7.76054e7 q^{76} +3.50350e8 q^{77} +7.22267e7 q^{78} +2.36871e8 q^{79} -2.88214e8 q^{80} -2.40308e8 q^{81} -1.06464e8 q^{82} +4.61987e8 q^{83} -7.29193e8 q^{84} +5.05554e8 q^{85} +1.49947e7 q^{86} -7.24488e8 q^{87} +1.23030e8 q^{88} +6.16826e8 q^{89} -3.13515e7 q^{90} -1.75253e9 q^{91} +3.60082e8 q^{92} +7.29481e8 q^{93} +1.41038e8 q^{94} +1.94838e8 q^{95} -3.86551e8 q^{96} -1.00137e9 q^{97} -5.13728e8 q^{98} -1.61309e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.38595 −0.193833 −0.0969167 0.995292i \(-0.530898\pi\)
−0.0969167 + 0.995292i \(0.530898\pi\)
\(3\) 117.920 0.840505 0.420253 0.907407i \(-0.361941\pi\)
0.420253 + 0.907407i \(0.361941\pi\)
\(4\) −492.763 −0.962429
\(5\) −1237.15 −0.885229 −0.442614 0.896712i \(-0.645949\pi\)
−0.442614 + 0.896712i \(0.645949\pi\)
\(6\) −517.190 −0.162918
\(7\) 12549.3 1.97550 0.987750 0.156046i \(-0.0498748\pi\)
0.987750 + 0.156046i \(0.0498748\pi\)
\(8\) 4406.84 0.380384
\(9\) −5777.96 −0.293551
\(10\) 5426.06 0.171587
\(11\) 27918.0 0.574933 0.287467 0.957791i \(-0.407187\pi\)
0.287467 + 0.957791i \(0.407187\pi\)
\(12\) −58106.5 −0.808926
\(13\) −139652. −1.35613 −0.678067 0.735000i \(-0.737182\pi\)
−0.678067 + 0.735000i \(0.737182\pi\)
\(14\) −55040.4 −0.382918
\(15\) −145884. −0.744040
\(16\) 232967. 0.888697
\(17\) −408645. −1.18666 −0.593330 0.804959i \(-0.702187\pi\)
−0.593330 + 0.804959i \(0.702187\pi\)
\(18\) 25341.8 0.0568999
\(19\) −157490. −0.277244 −0.138622 0.990345i \(-0.544267\pi\)
−0.138622 + 0.990345i \(0.544267\pi\)
\(20\) 609620. 0.851970
\(21\) 1.47980e6 1.66042
\(22\) −122447. −0.111441
\(23\) −730739. −0.544487 −0.272243 0.962228i \(-0.587766\pi\)
−0.272243 + 0.962228i \(0.587766\pi\)
\(24\) 519653. 0.319715
\(25\) −422597. −0.216370
\(26\) 612508. 0.262864
\(27\) −3.00235e6 −1.08724
\(28\) −6.18382e6 −1.90128
\(29\) −6.14391e6 −1.61307 −0.806537 0.591184i \(-0.798661\pi\)
−0.806537 + 0.591184i \(0.798661\pi\)
\(30\) 639839. 0.144220
\(31\) 6.18625e6 1.20309 0.601547 0.798837i \(-0.294551\pi\)
0.601547 + 0.798837i \(0.294551\pi\)
\(32\) −3.27808e6 −0.552644
\(33\) 3.29208e6 0.483234
\(34\) 1.79230e6 0.230015
\(35\) −1.55253e7 −1.74877
\(36\) 2.84717e6 0.282521
\(37\) −2.16507e7 −1.89917 −0.949587 0.313503i \(-0.898497\pi\)
−0.949587 + 0.313503i \(0.898497\pi\)
\(38\) 690744. 0.0537392
\(39\) −1.64677e7 −1.13984
\(40\) −5.45190e6 −0.336727
\(41\) 2.42738e7 1.34156 0.670780 0.741657i \(-0.265960\pi\)
0.670780 + 0.741657i \(0.265960\pi\)
\(42\) −6.49035e6 −0.321845
\(43\) −3.41880e6 −0.152499
\(44\) −1.37570e7 −0.553332
\(45\) 7.14817e6 0.259859
\(46\) 3.20499e6 0.105540
\(47\) −3.21569e7 −0.961243 −0.480622 0.876928i \(-0.659589\pi\)
−0.480622 + 0.876928i \(0.659589\pi\)
\(48\) 2.74713e7 0.746955
\(49\) 1.17130e8 2.90260
\(50\) 1.85349e6 0.0419397
\(51\) −4.81873e7 −0.997395
\(52\) 6.88155e7 1.30518
\(53\) −6.96541e7 −1.21257 −0.606283 0.795249i \(-0.707340\pi\)
−0.606283 + 0.795249i \(0.707340\pi\)
\(54\) 1.31681e7 0.210743
\(55\) −3.45386e7 −0.508947
\(56\) 5.53026e7 0.751449
\(57\) −1.85712e7 −0.233025
\(58\) 2.69469e7 0.312668
\(59\) −1.12041e8 −1.20377 −0.601886 0.798582i \(-0.705584\pi\)
−0.601886 + 0.798582i \(0.705584\pi\)
\(60\) 7.18862e7 0.716085
\(61\) 8.03385e7 0.742915 0.371458 0.928450i \(-0.378858\pi\)
0.371458 + 0.928450i \(0.378858\pi\)
\(62\) −2.71326e7 −0.233200
\(63\) −7.25091e7 −0.579909
\(64\) −1.04901e8 −0.781577
\(65\) 1.72770e8 1.20049
\(66\) −1.44389e7 −0.0936670
\(67\) −5.78429e7 −0.350682 −0.175341 0.984508i \(-0.556103\pi\)
−0.175341 + 0.984508i \(0.556103\pi\)
\(68\) 2.01366e8 1.14208
\(69\) −8.61685e7 −0.457644
\(70\) 6.80930e7 0.338970
\(71\) 3.11039e8 1.45262 0.726311 0.687366i \(-0.241233\pi\)
0.726311 + 0.687366i \(0.241233\pi\)
\(72\) −2.54625e7 −0.111662
\(73\) −3.23803e7 −0.133453 −0.0667265 0.997771i \(-0.521255\pi\)
−0.0667265 + 0.997771i \(0.521255\pi\)
\(74\) 9.49590e7 0.368124
\(75\) −4.98325e7 −0.181860
\(76\) 7.76054e7 0.266828
\(77\) 3.50350e8 1.13578
\(78\) 7.22267e7 0.220939
\(79\) 2.36871e8 0.684210 0.342105 0.939662i \(-0.388860\pi\)
0.342105 + 0.939662i \(0.388860\pi\)
\(80\) −2.88214e8 −0.786701
\(81\) −2.40308e8 −0.620277
\(82\) −1.06464e8 −0.260039
\(83\) 4.61987e8 1.06851 0.534255 0.845324i \(-0.320592\pi\)
0.534255 + 0.845324i \(0.320592\pi\)
\(84\) −7.29193e8 −1.59803
\(85\) 5.05554e8 1.05047
\(86\) 1.49947e7 0.0295593
\(87\) −7.24488e8 −1.35580
\(88\) 1.23030e8 0.218696
\(89\) 6.16826e8 1.04210 0.521048 0.853527i \(-0.325541\pi\)
0.521048 + 0.853527i \(0.325541\pi\)
\(90\) −3.13515e7 −0.0503695
\(91\) −1.75253e9 −2.67904
\(92\) 3.60082e8 0.524029
\(93\) 7.29481e8 1.01121
\(94\) 1.41038e8 0.186321
\(95\) 1.94838e8 0.245425
\(96\) −3.86551e8 −0.464500
\(97\) −1.00137e9 −1.14848 −0.574240 0.818687i \(-0.694702\pi\)
−0.574240 + 0.818687i \(0.694702\pi\)
\(98\) −5.13728e8 −0.562621
\(99\) −1.61309e8 −0.168772
\(100\) 2.08240e8 0.208240
\(101\) −3.05585e8 −0.292203 −0.146102 0.989270i \(-0.546673\pi\)
−0.146102 + 0.989270i \(0.546673\pi\)
\(102\) 2.11347e8 0.193328
\(103\) −3.81600e8 −0.334073 −0.167036 0.985951i \(-0.553420\pi\)
−0.167036 + 0.985951i \(0.553420\pi\)
\(104\) −6.15425e8 −0.515852
\(105\) −1.83073e9 −1.46985
\(106\) 3.05499e8 0.235036
\(107\) −1.99829e9 −1.47378 −0.736888 0.676015i \(-0.763705\pi\)
−0.736888 + 0.676015i \(0.763705\pi\)
\(108\) 1.47945e9 1.04639
\(109\) 5.64232e8 0.382859 0.191429 0.981506i \(-0.438688\pi\)
0.191429 + 0.981506i \(0.438688\pi\)
\(110\) 1.51485e8 0.0986510
\(111\) −2.55305e9 −1.59627
\(112\) 2.92356e9 1.75562
\(113\) 9.01181e8 0.519947 0.259973 0.965616i \(-0.416286\pi\)
0.259973 + 0.965616i \(0.416286\pi\)
\(114\) 8.14523e7 0.0451681
\(115\) 9.04030e8 0.481995
\(116\) 3.02750e9 1.55247
\(117\) 8.06904e8 0.398094
\(118\) 4.91407e8 0.233331
\(119\) −5.12820e9 −2.34425
\(120\) −6.42887e8 −0.283021
\(121\) −1.57853e9 −0.669452
\(122\) −3.52361e8 −0.144002
\(123\) 2.86235e9 1.12759
\(124\) −3.04836e9 −1.15789
\(125\) 2.93911e9 1.07677
\(126\) 3.18021e8 0.112406
\(127\) 3.22590e9 1.10036 0.550179 0.835047i \(-0.314559\pi\)
0.550179 + 0.835047i \(0.314559\pi\)
\(128\) 2.13847e9 0.704139
\(129\) −4.03144e8 −0.128176
\(130\) −7.57761e8 −0.232695
\(131\) −3.22175e9 −0.955808 −0.477904 0.878412i \(-0.658603\pi\)
−0.477904 + 0.878412i \(0.658603\pi\)
\(132\) −1.62222e9 −0.465079
\(133\) −1.97639e9 −0.547696
\(134\) 2.53696e8 0.0679739
\(135\) 3.71434e9 0.962453
\(136\) −1.80084e9 −0.451387
\(137\) −4.02495e9 −0.976153 −0.488077 0.872801i \(-0.662301\pi\)
−0.488077 + 0.872801i \(0.662301\pi\)
\(138\) 3.77931e8 0.0887067
\(139\) 2.36428e9 0.537196 0.268598 0.963252i \(-0.413440\pi\)
0.268598 + 0.963252i \(0.413440\pi\)
\(140\) 7.65028e9 1.68307
\(141\) −3.79193e9 −0.807930
\(142\) −1.36420e9 −0.281567
\(143\) −3.89881e9 −0.779686
\(144\) −1.34607e9 −0.260878
\(145\) 7.60091e9 1.42794
\(146\) 1.42018e8 0.0258676
\(147\) 1.38120e10 2.43965
\(148\) 1.06687e10 1.82782
\(149\) 3.40025e9 0.565161 0.282581 0.959244i \(-0.408810\pi\)
0.282581 + 0.959244i \(0.408810\pi\)
\(150\) 2.18563e8 0.0352505
\(151\) −2.71186e9 −0.424494 −0.212247 0.977216i \(-0.568078\pi\)
−0.212247 + 0.977216i \(0.568078\pi\)
\(152\) −6.94035e8 −0.105459
\(153\) 2.36114e9 0.348345
\(154\) −1.53662e9 −0.220152
\(155\) −7.65329e9 −1.06501
\(156\) 8.11470e9 1.09701
\(157\) 2.25484e9 0.296188 0.148094 0.988973i \(-0.452686\pi\)
0.148094 + 0.988973i \(0.452686\pi\)
\(158\) −1.03890e9 −0.132623
\(159\) −8.21359e9 −1.01917
\(160\) 4.05547e9 0.489216
\(161\) −9.17024e9 −1.07563
\(162\) 1.05398e9 0.120231
\(163\) 2.12327e9 0.235592 0.117796 0.993038i \(-0.462417\pi\)
0.117796 + 0.993038i \(0.462417\pi\)
\(164\) −1.19612e10 −1.29116
\(165\) −4.07278e9 −0.427773
\(166\) −2.02625e9 −0.207113
\(167\) −7.94637e9 −0.790578 −0.395289 0.918557i \(-0.629355\pi\)
−0.395289 + 0.918557i \(0.629355\pi\)
\(168\) 6.52127e9 0.631597
\(169\) 8.89824e9 0.839100
\(170\) −2.21733e9 −0.203616
\(171\) 9.09972e8 0.0813852
\(172\) 1.68466e9 0.146769
\(173\) −1.13404e10 −0.962548 −0.481274 0.876570i \(-0.659826\pi\)
−0.481274 + 0.876570i \(0.659826\pi\)
\(174\) 3.17757e9 0.262799
\(175\) −5.30328e9 −0.427438
\(176\) 6.50396e9 0.510941
\(177\) −1.32119e10 −1.01178
\(178\) −2.70537e9 −0.201993
\(179\) −1.54500e10 −1.12484 −0.562420 0.826852i \(-0.690130\pi\)
−0.562420 + 0.826852i \(0.690130\pi\)
\(180\) −3.52236e9 −0.250096
\(181\) 1.12851e10 0.781540 0.390770 0.920488i \(-0.372209\pi\)
0.390770 + 0.920488i \(0.372209\pi\)
\(182\) 7.68652e9 0.519288
\(183\) 9.47348e9 0.624424
\(184\) −3.22025e9 −0.207114
\(185\) 2.67851e10 1.68120
\(186\) −3.19947e9 −0.196006
\(187\) −1.14086e10 −0.682250
\(188\) 1.58457e10 0.925128
\(189\) −3.76772e10 −2.14783
\(190\) −8.54551e8 −0.0475715
\(191\) 1.87758e10 1.02082 0.510409 0.859932i \(-0.329494\pi\)
0.510409 + 0.859932i \(0.329494\pi\)
\(192\) −1.23699e10 −0.656919
\(193\) 2.54039e9 0.131793 0.0658966 0.997826i \(-0.479009\pi\)
0.0658966 + 0.997826i \(0.479009\pi\)
\(194\) 4.39197e9 0.222614
\(195\) 2.03730e10 1.00902
\(196\) −5.77175e10 −2.79354
\(197\) −2.66438e10 −1.26037 −0.630186 0.776444i \(-0.717021\pi\)
−0.630186 + 0.776444i \(0.717021\pi\)
\(198\) 7.07493e8 0.0327137
\(199\) 2.48904e10 1.12511 0.562553 0.826761i \(-0.309819\pi\)
0.562553 + 0.826761i \(0.309819\pi\)
\(200\) −1.86232e9 −0.0823036
\(201\) −6.82081e9 −0.294750
\(202\) 1.34028e9 0.0566388
\(203\) −7.71016e10 −3.18663
\(204\) 2.37450e10 0.959921
\(205\) −3.00302e10 −1.18759
\(206\) 1.67368e9 0.0647545
\(207\) 4.22218e9 0.159834
\(208\) −3.25343e10 −1.20519
\(209\) −4.39681e9 −0.159397
\(210\) 8.02950e9 0.284906
\(211\) 2.83390e10 0.984267 0.492134 0.870520i \(-0.336217\pi\)
0.492134 + 0.870520i \(0.336217\pi\)
\(212\) 3.43230e10 1.16701
\(213\) 3.66777e10 1.22094
\(214\) 8.76440e9 0.285667
\(215\) 4.22955e9 0.134996
\(216\) −1.32309e10 −0.413568
\(217\) 7.76329e10 2.37671
\(218\) −2.47470e9 −0.0742109
\(219\) −3.81827e9 −0.112168
\(220\) 1.70194e10 0.489826
\(221\) 5.70682e10 1.60927
\(222\) 1.11975e10 0.309410
\(223\) −8.66457e9 −0.234626 −0.117313 0.993095i \(-0.537428\pi\)
−0.117313 + 0.993095i \(0.537428\pi\)
\(224\) −4.11375e10 −1.09175
\(225\) 2.44175e9 0.0635154
\(226\) −3.95254e9 −0.100783
\(227\) −1.92752e10 −0.481817 −0.240908 0.970548i \(-0.577445\pi\)
−0.240908 + 0.970548i \(0.577445\pi\)
\(228\) 9.15120e9 0.224270
\(229\) −2.30675e9 −0.0554294 −0.0277147 0.999616i \(-0.508823\pi\)
−0.0277147 + 0.999616i \(0.508823\pi\)
\(230\) −3.96503e9 −0.0934268
\(231\) 4.13132e10 0.954629
\(232\) −2.70753e10 −0.613588
\(233\) −2.20050e10 −0.489125 −0.244562 0.969634i \(-0.578644\pi\)
−0.244562 + 0.969634i \(0.578644\pi\)
\(234\) −3.53904e9 −0.0771639
\(235\) 3.97827e10 0.850920
\(236\) 5.52098e10 1.15854
\(237\) 2.79317e10 0.575083
\(238\) 2.24920e10 0.454394
\(239\) −3.65120e10 −0.723845 −0.361922 0.932208i \(-0.617879\pi\)
−0.361922 + 0.932208i \(0.617879\pi\)
\(240\) −3.39860e10 −0.661226
\(241\) 8.05560e10 1.53823 0.769115 0.639110i \(-0.220697\pi\)
0.769115 + 0.639110i \(0.220697\pi\)
\(242\) 6.92337e9 0.129762
\(243\) 3.07581e10 0.565890
\(244\) −3.95879e10 −0.715003
\(245\) −1.44907e11 −2.56946
\(246\) −1.25541e10 −0.218564
\(247\) 2.19939e10 0.375980
\(248\) 2.72618e10 0.457638
\(249\) 5.44773e10 0.898088
\(250\) −1.28908e10 −0.208713
\(251\) 2.34339e10 0.372661 0.186330 0.982487i \(-0.440341\pi\)
0.186330 + 0.982487i \(0.440341\pi\)
\(252\) 3.57298e10 0.558121
\(253\) −2.04008e10 −0.313043
\(254\) −1.41486e10 −0.213286
\(255\) 5.96147e10 0.882923
\(256\) 4.43303e10 0.645091
\(257\) −1.48746e10 −0.212689 −0.106345 0.994329i \(-0.533915\pi\)
−0.106345 + 0.994329i \(0.533915\pi\)
\(258\) 1.76817e9 0.0248448
\(259\) −2.71701e11 −3.75182
\(260\) −8.51348e10 −1.15539
\(261\) 3.54993e10 0.473519
\(262\) 1.41304e10 0.185268
\(263\) −1.29482e10 −0.166882 −0.0834410 0.996513i \(-0.526591\pi\)
−0.0834410 + 0.996513i \(0.526591\pi\)
\(264\) 1.45077e10 0.183815
\(265\) 8.61722e10 1.07340
\(266\) 8.66833e9 0.106162
\(267\) 7.27359e10 0.875887
\(268\) 2.85028e10 0.337506
\(269\) 1.38611e11 1.61403 0.807016 0.590529i \(-0.201081\pi\)
0.807016 + 0.590529i \(0.201081\pi\)
\(270\) −1.62909e10 −0.186556
\(271\) −3.55761e10 −0.400679 −0.200339 0.979727i \(-0.564204\pi\)
−0.200339 + 0.979727i \(0.564204\pi\)
\(272\) −9.52008e10 −1.05458
\(273\) −2.06658e11 −2.25175
\(274\) 1.76532e10 0.189211
\(275\) −1.17981e10 −0.124398
\(276\) 4.24607e10 0.440450
\(277\) 1.13489e11 1.15823 0.579114 0.815247i \(-0.303399\pi\)
0.579114 + 0.815247i \(0.303399\pi\)
\(278\) −1.03696e10 −0.104126
\(279\) −3.57439e10 −0.353169
\(280\) −6.84174e10 −0.665205
\(281\) 2.84629e10 0.272334 0.136167 0.990686i \(-0.456522\pi\)
0.136167 + 0.990686i \(0.456522\pi\)
\(282\) 1.66312e10 0.156604
\(283\) 1.03614e11 0.960243 0.480121 0.877202i \(-0.340593\pi\)
0.480121 + 0.877202i \(0.340593\pi\)
\(284\) −1.53269e11 −1.39805
\(285\) 2.29753e10 0.206281
\(286\) 1.71000e10 0.151129
\(287\) 3.04618e11 2.65025
\(288\) 1.89406e10 0.162229
\(289\) 4.84032e10 0.408163
\(290\) −3.33372e10 −0.276782
\(291\) −1.18082e11 −0.965303
\(292\) 1.59558e10 0.128439
\(293\) −1.91800e11 −1.52035 −0.760176 0.649717i \(-0.774888\pi\)
−0.760176 + 0.649717i \(0.774888\pi\)
\(294\) −6.05786e10 −0.472886
\(295\) 1.38611e11 1.06561
\(296\) −9.54114e10 −0.722416
\(297\) −8.38195e10 −0.625088
\(298\) −1.49133e10 −0.109547
\(299\) 1.02049e11 0.738397
\(300\) 2.45556e10 0.175027
\(301\) −4.29034e10 −0.301261
\(302\) 1.18941e10 0.0822811
\(303\) −3.60344e10 −0.245599
\(304\) −3.66900e10 −0.246386
\(305\) −9.93904e10 −0.657650
\(306\) −1.03558e10 −0.0675209
\(307\) −5.93480e10 −0.381315 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(308\) −1.72640e11 −1.09311
\(309\) −4.49982e10 −0.280790
\(310\) 3.35670e10 0.206435
\(311\) −1.46203e11 −0.886205 −0.443102 0.896471i \(-0.646122\pi\)
−0.443102 + 0.896471i \(0.646122\pi\)
\(312\) −7.25707e10 −0.433577
\(313\) −1.38227e11 −0.814036 −0.407018 0.913420i \(-0.633431\pi\)
−0.407018 + 0.913420i \(0.633431\pi\)
\(314\) −9.88963e9 −0.0574112
\(315\) 8.97042e10 0.513352
\(316\) −1.16721e11 −0.658504
\(317\) 5.56681e10 0.309628 0.154814 0.987944i \(-0.450522\pi\)
0.154814 + 0.987944i \(0.450522\pi\)
\(318\) 3.60244e10 0.197549
\(319\) −1.71526e11 −0.927409
\(320\) 1.29778e11 0.691874
\(321\) −2.35638e11 −1.23872
\(322\) 4.02202e10 0.208494
\(323\) 6.43577e10 0.328995
\(324\) 1.18415e11 0.596973
\(325\) 5.90166e10 0.293426
\(326\) −9.31255e9 −0.0456656
\(327\) 6.65341e10 0.321795
\(328\) 1.06971e11 0.510308
\(329\) −4.03545e11 −1.89894
\(330\) 1.78630e10 0.0829167
\(331\) 2.09845e11 0.960888 0.480444 0.877025i \(-0.340476\pi\)
0.480444 + 0.877025i \(0.340476\pi\)
\(332\) −2.27650e11 −1.02836
\(333\) 1.25097e11 0.557504
\(334\) 3.48524e10 0.153240
\(335\) 7.15600e10 0.310434
\(336\) 3.44745e11 1.47561
\(337\) 2.40308e11 1.01493 0.507463 0.861673i \(-0.330583\pi\)
0.507463 + 0.861673i \(0.330583\pi\)
\(338\) −3.90272e10 −0.162646
\(339\) 1.06267e11 0.437018
\(340\) −2.49118e11 −1.01100
\(341\) 1.72708e11 0.691699
\(342\) −3.99109e9 −0.0157752
\(343\) 9.63491e11 3.75858
\(344\) −1.50661e10 −0.0580081
\(345\) 1.06603e11 0.405120
\(346\) 4.97386e10 0.186574
\(347\) 1.62927e10 0.0603266 0.0301633 0.999545i \(-0.490397\pi\)
0.0301633 + 0.999545i \(0.490397\pi\)
\(348\) 3.57001e11 1.30486
\(349\) −1.41443e11 −0.510347 −0.255174 0.966895i \(-0.582133\pi\)
−0.255174 + 0.966895i \(0.582133\pi\)
\(350\) 2.32599e10 0.0828518
\(351\) 4.19284e11 1.47444
\(352\) −9.15176e10 −0.317733
\(353\) 3.15790e10 0.108246 0.0541230 0.998534i \(-0.482764\pi\)
0.0541230 + 0.998534i \(0.482764\pi\)
\(354\) 5.79466e10 0.196116
\(355\) −3.84801e11 −1.28590
\(356\) −3.03949e11 −1.00294
\(357\) −6.04715e11 −1.97035
\(358\) 6.77631e10 0.218032
\(359\) 9.18482e10 0.291840 0.145920 0.989296i \(-0.453386\pi\)
0.145920 + 0.989296i \(0.453386\pi\)
\(360\) 3.15009e10 0.0988465
\(361\) −2.97885e11 −0.923136
\(362\) −4.94958e10 −0.151489
\(363\) −1.86140e11 −0.562678
\(364\) 8.63584e11 2.57839
\(365\) 4.00591e10 0.118136
\(366\) −4.15502e10 −0.121034
\(367\) 2.61752e11 0.753169 0.376584 0.926382i \(-0.377098\pi\)
0.376584 + 0.926382i \(0.377098\pi\)
\(368\) −1.70238e11 −0.483884
\(369\) −1.40253e11 −0.393816
\(370\) −1.17478e11 −0.325874
\(371\) −8.74107e11 −2.39542
\(372\) −3.59461e11 −0.973215
\(373\) 2.92272e11 0.781803 0.390902 0.920433i \(-0.372163\pi\)
0.390902 + 0.920433i \(0.372163\pi\)
\(374\) 5.00374e10 0.132243
\(375\) 3.46579e11 0.905027
\(376\) −1.41710e11 −0.365642
\(377\) 8.58011e11 2.18754
\(378\) 1.65251e11 0.416322
\(379\) 4.04776e11 1.00772 0.503859 0.863786i \(-0.331913\pi\)
0.503859 + 0.863786i \(0.331913\pi\)
\(380\) −9.60092e10 −0.236204
\(381\) 3.80397e11 0.924857
\(382\) −8.23497e10 −0.197869
\(383\) −6.03567e11 −1.43328 −0.716640 0.697443i \(-0.754321\pi\)
−0.716640 + 0.697443i \(0.754321\pi\)
\(384\) 2.52168e11 0.591833
\(385\) −4.33434e11 −1.00543
\(386\) −1.11420e10 −0.0255459
\(387\) 1.97537e10 0.0447660
\(388\) 4.93440e11 1.10533
\(389\) −4.00613e11 −0.887057 −0.443529 0.896260i \(-0.646274\pi\)
−0.443529 + 0.896260i \(0.646274\pi\)
\(390\) −8.93549e10 −0.195581
\(391\) 2.98613e11 0.646121
\(392\) 5.16175e11 1.10410
\(393\) −3.79907e11 −0.803362
\(394\) 1.16859e11 0.244302
\(395\) −2.93044e11 −0.605683
\(396\) 7.94872e10 0.162431
\(397\) −7.65107e11 −1.54584 −0.772920 0.634503i \(-0.781205\pi\)
−0.772920 + 0.634503i \(0.781205\pi\)
\(398\) −1.09168e11 −0.218083
\(399\) −2.33055e11 −0.460341
\(400\) −9.84510e10 −0.192287
\(401\) 5.13976e11 0.992643 0.496322 0.868139i \(-0.334684\pi\)
0.496322 + 0.868139i \(0.334684\pi\)
\(402\) 2.99157e10 0.0571324
\(403\) −8.63924e11 −1.63156
\(404\) 1.50581e11 0.281225
\(405\) 2.97296e11 0.549088
\(406\) 3.38164e11 0.617675
\(407\) −6.04445e11 −1.09190
\(408\) −2.12354e11 −0.379393
\(409\) 3.29917e11 0.582975 0.291488 0.956575i \(-0.405850\pi\)
0.291488 + 0.956575i \(0.405850\pi\)
\(410\) 1.31711e11 0.230194
\(411\) −4.74621e11 −0.820462
\(412\) 1.88039e11 0.321521
\(413\) −1.40603e12 −2.37805
\(414\) −1.85183e10 −0.0309812
\(415\) −5.71545e11 −0.945875
\(416\) 4.57792e11 0.749459
\(417\) 2.78795e11 0.451516
\(418\) 1.92842e10 0.0308964
\(419\) 1.13273e11 0.179541 0.0897707 0.995962i \(-0.471387\pi\)
0.0897707 + 0.995962i \(0.471387\pi\)
\(420\) 9.02118e11 1.41463
\(421\) −5.70807e11 −0.885564 −0.442782 0.896629i \(-0.646008\pi\)
−0.442782 + 0.896629i \(0.646008\pi\)
\(422\) −1.24293e11 −0.190784
\(423\) 1.85801e11 0.282173
\(424\) −3.06955e11 −0.461241
\(425\) 1.72692e11 0.256757
\(426\) −1.60866e11 −0.236658
\(427\) 1.00819e12 1.46763
\(428\) 9.84684e11 1.41840
\(429\) −4.59746e11 −0.655331
\(430\) −1.85506e10 −0.0261668
\(431\) −5.38814e11 −0.752127 −0.376064 0.926594i \(-0.622723\pi\)
−0.376064 + 0.926594i \(0.622723\pi\)
\(432\) −6.99447e11 −0.966224
\(433\) −8.64588e11 −1.18199 −0.590995 0.806675i \(-0.701265\pi\)
−0.590995 + 0.806675i \(0.701265\pi\)
\(434\) −3.40494e11 −0.460687
\(435\) 8.96297e11 1.20019
\(436\) −2.78033e11 −0.368474
\(437\) 1.15084e11 0.150956
\(438\) 1.67468e10 0.0217419
\(439\) −3.81449e10 −0.0490170 −0.0245085 0.999700i \(-0.507802\pi\)
−0.0245085 + 0.999700i \(0.507802\pi\)
\(440\) −1.52206e11 −0.193596
\(441\) −6.76774e11 −0.852059
\(442\) −2.50298e11 −0.311931
\(443\) 4.04411e11 0.498891 0.249446 0.968389i \(-0.419752\pi\)
0.249446 + 0.968389i \(0.419752\pi\)
\(444\) 1.25805e12 1.53629
\(445\) −7.63103e11 −0.922494
\(446\) 3.80024e10 0.0454783
\(447\) 4.00956e11 0.475021
\(448\) −1.31644e12 −1.54400
\(449\) 5.23002e11 0.607288 0.303644 0.952786i \(-0.401797\pi\)
0.303644 + 0.952786i \(0.401797\pi\)
\(450\) −1.07094e10 −0.0123114
\(451\) 6.77675e11 0.771307
\(452\) −4.44069e11 −0.500412
\(453\) −3.19782e11 −0.356789
\(454\) 8.45400e10 0.0933923
\(455\) 2.16814e12 2.37157
\(456\) −8.18403e10 −0.0886392
\(457\) −1.62214e12 −1.73966 −0.869832 0.493347i \(-0.835773\pi\)
−0.869832 + 0.493347i \(0.835773\pi\)
\(458\) 1.01173e10 0.0107441
\(459\) 1.22690e12 1.29018
\(460\) −4.45473e11 −0.463886
\(461\) −5.64486e11 −0.582102 −0.291051 0.956708i \(-0.594005\pi\)
−0.291051 + 0.956708i \(0.594005\pi\)
\(462\) −1.81198e11 −0.185039
\(463\) −1.76040e11 −0.178031 −0.0890156 0.996030i \(-0.528372\pi\)
−0.0890156 + 0.996030i \(0.528372\pi\)
\(464\) −1.43133e12 −1.43353
\(465\) −9.02473e11 −0.895150
\(466\) 9.65129e10 0.0948088
\(467\) 1.36802e11 0.133097 0.0665483 0.997783i \(-0.478801\pi\)
0.0665483 + 0.997783i \(0.478801\pi\)
\(468\) −3.97613e11 −0.383137
\(469\) −7.25885e11 −0.692772
\(470\) −1.74485e11 −0.164937
\(471\) 2.65890e11 0.248948
\(472\) −4.93748e11 −0.457896
\(473\) −9.54461e10 −0.0876765
\(474\) −1.22507e11 −0.111470
\(475\) 6.65549e10 0.0599872
\(476\) 2.52699e12 2.25617
\(477\) 4.02458e11 0.355949
\(478\) 1.60140e11 0.140305
\(479\) −4.86473e11 −0.422230 −0.211115 0.977461i \(-0.567709\pi\)
−0.211115 + 0.977461i \(0.567709\pi\)
\(480\) 4.78219e11 0.411189
\(481\) 3.02357e12 2.57554
\(482\) −3.53315e11 −0.298160
\(483\) −1.08135e12 −0.904075
\(484\) 7.77843e11 0.644300
\(485\) 1.23884e12 1.01667
\(486\) −1.34904e11 −0.109688
\(487\) −8.10600e11 −0.653020 −0.326510 0.945194i \(-0.605873\pi\)
−0.326510 + 0.945194i \(0.605873\pi\)
\(488\) 3.54039e11 0.282593
\(489\) 2.50375e11 0.198016
\(490\) 6.35556e11 0.498048
\(491\) 1.12030e12 0.869893 0.434947 0.900456i \(-0.356767\pi\)
0.434947 + 0.900456i \(0.356767\pi\)
\(492\) −1.41046e12 −1.08522
\(493\) 2.51068e12 1.91417
\(494\) −9.64640e10 −0.0728776
\(495\) 1.99563e11 0.149402
\(496\) 1.44119e12 1.06919
\(497\) 3.90331e12 2.86966
\(498\) −2.38935e11 −0.174079
\(499\) 1.93337e12 1.39593 0.697963 0.716134i \(-0.254090\pi\)
0.697963 + 0.716134i \(0.254090\pi\)
\(500\) −1.44829e12 −1.03631
\(501\) −9.37033e11 −0.664485
\(502\) −1.02780e11 −0.0722341
\(503\) −9.53953e11 −0.664463 −0.332232 0.943198i \(-0.607802\pi\)
−0.332232 + 0.943198i \(0.607802\pi\)
\(504\) −3.19536e11 −0.220588
\(505\) 3.78052e11 0.258667
\(506\) 8.94768e10 0.0606783
\(507\) 1.04928e12 0.705268
\(508\) −1.58961e12 −1.05902
\(509\) −2.27539e12 −1.50254 −0.751270 0.659995i \(-0.770558\pi\)
−0.751270 + 0.659995i \(0.770558\pi\)
\(510\) −2.61467e11 −0.171140
\(511\) −4.06349e11 −0.263636
\(512\) −1.28933e12 −0.829180
\(513\) 4.72840e11 0.301430
\(514\) 6.52392e10 0.0412263
\(515\) 4.72095e11 0.295731
\(516\) 1.98655e11 0.123360
\(517\) −8.97755e11 −0.552650
\(518\) 1.19167e12 0.727228
\(519\) −1.33726e12 −0.809027
\(520\) 7.61370e11 0.456647
\(521\) 8.63866e11 0.513661 0.256830 0.966456i \(-0.417322\pi\)
0.256830 + 0.966456i \(0.417322\pi\)
\(522\) −1.55698e11 −0.0917837
\(523\) −3.96246e11 −0.231583 −0.115792 0.993274i \(-0.536940\pi\)
−0.115792 + 0.993274i \(0.536940\pi\)
\(524\) 1.58756e12 0.919897
\(525\) −6.25361e11 −0.359264
\(526\) 5.67903e10 0.0323473
\(527\) −2.52798e12 −1.42767
\(528\) 7.66945e11 0.429449
\(529\) −1.26717e12 −0.703534
\(530\) −3.77947e11 −0.208061
\(531\) 6.47369e11 0.353368
\(532\) 9.73891e11 0.527118
\(533\) −3.38989e12 −1.81933
\(534\) −3.19016e11 −0.169776
\(535\) 2.47217e12 1.30463
\(536\) −2.54904e11 −0.133394
\(537\) −1.82186e12 −0.945434
\(538\) −6.07941e11 −0.312854
\(539\) 3.27004e12 1.66880
\(540\) −1.83029e12 −0.926292
\(541\) −3.00813e12 −1.50976 −0.754881 0.655862i \(-0.772305\pi\)
−0.754881 + 0.655862i \(0.772305\pi\)
\(542\) 1.56035e11 0.0776650
\(543\) 1.33073e12 0.656888
\(544\) 1.33957e12 0.655800
\(545\) −6.98037e11 −0.338918
\(546\) 9.06392e11 0.436465
\(547\) −1.67014e12 −0.797645 −0.398823 0.917028i \(-0.630581\pi\)
−0.398823 + 0.917028i \(0.630581\pi\)
\(548\) 1.98335e12 0.939478
\(549\) −4.64192e11 −0.218083
\(550\) 5.17457e10 0.0241125
\(551\) 9.67606e11 0.447215
\(552\) −3.79731e11 −0.174081
\(553\) 2.97255e12 1.35166
\(554\) −4.97756e11 −0.224503
\(555\) 3.15849e12 1.41306
\(556\) −1.16503e12 −0.517012
\(557\) −3.27102e12 −1.43991 −0.719954 0.694022i \(-0.755837\pi\)
−0.719954 + 0.694022i \(0.755837\pi\)
\(558\) 1.56771e11 0.0684560
\(559\) 4.77443e11 0.206809
\(560\) −3.61687e12 −1.55413
\(561\) −1.34529e12 −0.573435
\(562\) −1.24837e11 −0.0527874
\(563\) −3.16570e12 −1.32795 −0.663976 0.747754i \(-0.731132\pi\)
−0.663976 + 0.747754i \(0.731132\pi\)
\(564\) 1.86852e12 0.777575
\(565\) −1.11489e12 −0.460272
\(566\) −4.54448e11 −0.186127
\(567\) −3.01569e12 −1.22536
\(568\) 1.37070e12 0.552555
\(569\) −3.41258e12 −1.36483 −0.682414 0.730966i \(-0.739070\pi\)
−0.682414 + 0.730966i \(0.739070\pi\)
\(570\) −1.00768e11 −0.0399841
\(571\) 5.10789e11 0.201085 0.100542 0.994933i \(-0.467942\pi\)
0.100542 + 0.994933i \(0.467942\pi\)
\(572\) 1.92119e12 0.750393
\(573\) 2.21403e12 0.858002
\(574\) −1.33604e12 −0.513707
\(575\) 3.08808e11 0.117810
\(576\) 6.06116e11 0.229432
\(577\) 3.29567e12 1.23781 0.618903 0.785467i \(-0.287577\pi\)
0.618903 + 0.785467i \(0.287577\pi\)
\(578\) −2.12294e11 −0.0791157
\(579\) 2.99562e11 0.110773
\(580\) −3.74545e12 −1.37429
\(581\) 5.79759e12 2.11084
\(582\) 5.17900e11 0.187108
\(583\) −1.94460e12 −0.697144
\(584\) −1.42695e11 −0.0507634
\(585\) −9.98258e11 −0.352404
\(586\) 8.41226e11 0.294695
\(587\) 3.75472e12 1.30529 0.652644 0.757665i \(-0.273660\pi\)
0.652644 + 0.757665i \(0.273660\pi\)
\(588\) −6.80603e12 −2.34799
\(589\) −9.74274e11 −0.333551
\(590\) −6.07942e11 −0.206551
\(591\) −3.14183e12 −1.05935
\(592\) −5.04390e12 −1.68779
\(593\) −7.28844e11 −0.242041 −0.121020 0.992650i \(-0.538617\pi\)
−0.121020 + 0.992650i \(0.538617\pi\)
\(594\) 3.67628e11 0.121163
\(595\) 6.34432e12 2.07520
\(596\) −1.67552e12 −0.543927
\(597\) 2.93507e12 0.945658
\(598\) −4.47583e11 −0.143126
\(599\) 1.79149e12 0.568582 0.284291 0.958738i \(-0.408242\pi\)
0.284291 + 0.958738i \(0.408242\pi\)
\(600\) −2.19604e11 −0.0691767
\(601\) 3.67032e11 0.114754 0.0573772 0.998353i \(-0.481726\pi\)
0.0573772 + 0.998353i \(0.481726\pi\)
\(602\) 1.88172e11 0.0583944
\(603\) 3.34213e11 0.102943
\(604\) 1.33631e12 0.408545
\(605\) 1.95287e12 0.592618
\(606\) 1.58045e11 0.0476052
\(607\) −3.38377e12 −1.01170 −0.505850 0.862622i \(-0.668821\pi\)
−0.505850 + 0.862622i \(0.668821\pi\)
\(608\) 5.16266e11 0.153217
\(609\) −9.09179e12 −2.67838
\(610\) 4.35921e11 0.127475
\(611\) 4.49078e12 1.30357
\(612\) −1.16348e12 −0.335257
\(613\) 3.01980e11 0.0863786 0.0431893 0.999067i \(-0.486248\pi\)
0.0431893 + 0.999067i \(0.486248\pi\)
\(614\) 2.60298e11 0.0739116
\(615\) −3.54115e12 −0.998174
\(616\) 1.54394e12 0.432033
\(617\) 3.65251e12 1.01463 0.507315 0.861761i \(-0.330638\pi\)
0.507315 + 0.861761i \(0.330638\pi\)
\(618\) 1.97360e11 0.0544265
\(619\) 1.83928e10 0.00503546 0.00251773 0.999997i \(-0.499199\pi\)
0.00251773 + 0.999997i \(0.499199\pi\)
\(620\) 3.77126e12 1.02500
\(621\) 2.19393e12 0.591986
\(622\) 6.41239e11 0.171776
\(623\) 7.74071e12 2.05866
\(624\) −3.83643e12 −1.01297
\(625\) −2.81072e12 −0.736815
\(626\) 6.06257e11 0.157787
\(627\) −5.18471e11 −0.133974
\(628\) −1.11110e12 −0.285060
\(629\) 8.84747e12 2.25367
\(630\) −3.93438e11 −0.0995049
\(631\) −3.45921e12 −0.868650 −0.434325 0.900756i \(-0.643013\pi\)
−0.434325 + 0.900756i \(0.643013\pi\)
\(632\) 1.04385e12 0.260263
\(633\) 3.34172e12 0.827282
\(634\) −2.44157e11 −0.0600162
\(635\) −3.99091e12 −0.974069
\(636\) 4.04736e12 0.980877
\(637\) −1.63575e13 −3.93631
\(638\) 7.52304e11 0.179763
\(639\) −1.79717e12 −0.426418
\(640\) −2.64560e12 −0.623325
\(641\) −1.71702e12 −0.401710 −0.200855 0.979621i \(-0.564372\pi\)
−0.200855 + 0.979621i \(0.564372\pi\)
\(642\) 1.03349e12 0.240105
\(643\) −5.95110e11 −0.137293 −0.0686464 0.997641i \(-0.521868\pi\)
−0.0686464 + 0.997641i \(0.521868\pi\)
\(644\) 4.51876e12 1.03522
\(645\) 4.98747e11 0.113465
\(646\) −2.82270e11 −0.0637702
\(647\) −2.77664e12 −0.622946 −0.311473 0.950255i \(-0.600822\pi\)
−0.311473 + 0.950255i \(0.600822\pi\)
\(648\) −1.05900e12 −0.235944
\(649\) −3.12797e12 −0.692088
\(650\) −2.58844e11 −0.0568758
\(651\) 9.15444e12 1.99764
\(652\) −1.04627e12 −0.226740
\(653\) 2.92497e12 0.629523 0.314761 0.949171i \(-0.398075\pi\)
0.314761 + 0.949171i \(0.398075\pi\)
\(654\) −2.91815e11 −0.0623747
\(655\) 3.98577e12 0.846109
\(656\) 5.65498e12 1.19224
\(657\) 1.87092e11 0.0391752
\(658\) 1.76993e12 0.368077
\(659\) −3.06130e12 −0.632299 −0.316149 0.948709i \(-0.602390\pi\)
−0.316149 + 0.948709i \(0.602390\pi\)
\(660\) 2.00692e12 0.411701
\(661\) 7.09295e12 1.44517 0.722587 0.691280i \(-0.242953\pi\)
0.722587 + 0.691280i \(0.242953\pi\)
\(662\) −9.20370e11 −0.186252
\(663\) 6.72947e12 1.35260
\(664\) 2.03590e12 0.406444
\(665\) 2.44508e12 0.484836
\(666\) −5.48669e11 −0.108063
\(667\) 4.48960e12 0.878297
\(668\) 3.91568e12 0.760875
\(669\) −1.02172e12 −0.197204
\(670\) −3.13859e11 −0.0601724
\(671\) 2.24289e12 0.427127
\(672\) −4.85092e12 −0.917620
\(673\) 3.65617e12 0.687003 0.343502 0.939152i \(-0.388387\pi\)
0.343502 + 0.939152i \(0.388387\pi\)
\(674\) −1.05398e12 −0.196727
\(675\) 1.26878e12 0.235245
\(676\) −4.38473e12 −0.807574
\(677\) −1.69929e12 −0.310898 −0.155449 0.987844i \(-0.549682\pi\)
−0.155449 + 0.987844i \(0.549682\pi\)
\(678\) −4.66082e11 −0.0847088
\(679\) −1.25665e13 −2.26882
\(680\) 2.22790e12 0.399581
\(681\) −2.27292e12 −0.404970
\(682\) −7.57488e11 −0.134074
\(683\) −3.57423e12 −0.628477 −0.314239 0.949344i \(-0.601749\pi\)
−0.314239 + 0.949344i \(0.601749\pi\)
\(684\) −4.48401e11 −0.0783274
\(685\) 4.97945e12 0.864119
\(686\) −4.22582e12 −0.728539
\(687\) −2.72011e11 −0.0465887
\(688\) −7.96467e11 −0.135525
\(689\) 9.72735e12 1.64440
\(690\) −4.67555e11 −0.0785257
\(691\) 9.20422e11 0.153580 0.0767902 0.997047i \(-0.475533\pi\)
0.0767902 + 0.997047i \(0.475533\pi\)
\(692\) 5.58815e12 0.926384
\(693\) −2.02431e12 −0.333409
\(694\) −7.14588e10 −0.0116933
\(695\) −2.92496e12 −0.475541
\(696\) −3.19271e12 −0.515724
\(697\) −9.91937e12 −1.59198
\(698\) 6.20360e11 0.0989224
\(699\) −2.59482e12 −0.411112
\(700\) 2.61326e12 0.411379
\(701\) 3.78617e11 0.0592200 0.0296100 0.999562i \(-0.490573\pi\)
0.0296100 + 0.999562i \(0.490573\pi\)
\(702\) −1.83896e12 −0.285796
\(703\) 3.40978e12 0.526535
\(704\) −2.92864e12 −0.449354
\(705\) 4.69116e12 0.715203
\(706\) −1.38504e11 −0.0209817
\(707\) −3.83486e12 −0.577248
\(708\) 6.51032e12 0.973762
\(709\) −4.49168e12 −0.667576 −0.333788 0.942648i \(-0.608327\pi\)
−0.333788 + 0.942648i \(0.608327\pi\)
\(710\) 1.68772e12 0.249251
\(711\) −1.36863e12 −0.200850
\(712\) 2.71826e12 0.396397
\(713\) −4.52054e12 −0.655069
\(714\) 2.65225e12 0.381920
\(715\) 4.82339e12 0.690201
\(716\) 7.61321e12 1.08258
\(717\) −4.30549e12 −0.608395
\(718\) −4.02842e11 −0.0565684
\(719\) −8.71449e12 −1.21608 −0.608039 0.793907i \(-0.708044\pi\)
−0.608039 + 0.793907i \(0.708044\pi\)
\(720\) 1.66529e12 0.230936
\(721\) −4.78880e12 −0.659961
\(722\) 1.30651e12 0.178935
\(723\) 9.49914e12 1.29289
\(724\) −5.56087e12 −0.752176
\(725\) 2.59640e12 0.349020
\(726\) 8.16401e11 0.109066
\(727\) 1.20858e13 1.60461 0.802306 0.596913i \(-0.203606\pi\)
0.802306 + 0.596913i \(0.203606\pi\)
\(728\) −7.72313e12 −1.01907
\(729\) 8.35698e12 1.09591
\(730\) −1.75697e11 −0.0228988
\(731\) 1.39708e12 0.180964
\(732\) −4.66819e12 −0.600964
\(733\) −2.17693e12 −0.278534 −0.139267 0.990255i \(-0.544475\pi\)
−0.139267 + 0.990255i \(0.544475\pi\)
\(734\) −1.14803e12 −0.145989
\(735\) −1.70874e13 −2.15965
\(736\) 2.39542e12 0.300907
\(737\) −1.61486e12 −0.201619
\(738\) 6.15142e11 0.0763346
\(739\) 1.41514e13 1.74542 0.872709 0.488240i \(-0.162361\pi\)
0.872709 + 0.488240i \(0.162361\pi\)
\(740\) −1.31987e13 −1.61804
\(741\) 2.59351e12 0.316014
\(742\) 3.83379e12 0.464313
\(743\) −1.36261e13 −1.64030 −0.820148 0.572152i \(-0.806109\pi\)
−0.820148 + 0.572152i \(0.806109\pi\)
\(744\) 3.21471e12 0.384648
\(745\) −4.20660e12 −0.500297
\(746\) −1.28189e12 −0.151540
\(747\) −2.66934e12 −0.313661
\(748\) 5.62172e12 0.656617
\(749\) −2.50770e13 −2.91144
\(750\) −1.52008e12 −0.175425
\(751\) −1.39601e13 −1.60143 −0.800715 0.599046i \(-0.795547\pi\)
−0.800715 + 0.599046i \(0.795547\pi\)
\(752\) −7.49148e12 −0.854254
\(753\) 2.76332e12 0.313223
\(754\) −3.76319e12 −0.424019
\(755\) 3.35497e12 0.375774
\(756\) 1.85660e13 2.06714
\(757\) 6.22034e12 0.688467 0.344233 0.938884i \(-0.388139\pi\)
0.344233 + 0.938884i \(0.388139\pi\)
\(758\) −1.77533e12 −0.195329
\(759\) −2.40565e12 −0.263115
\(760\) 8.58622e11 0.0933557
\(761\) −7.99871e12 −0.864548 −0.432274 0.901742i \(-0.642289\pi\)
−0.432274 + 0.901742i \(0.642289\pi\)
\(762\) −1.66840e12 −0.179268
\(763\) 7.08070e12 0.756338
\(764\) −9.25202e12 −0.982464
\(765\) −2.92107e12 −0.308365
\(766\) 2.64721e12 0.277818
\(767\) 1.56468e13 1.63248
\(768\) 5.22741e12 0.542202
\(769\) −1.87310e13 −1.93149 −0.965745 0.259493i \(-0.916444\pi\)
−0.965745 + 0.259493i \(0.916444\pi\)
\(770\) 1.90102e12 0.194885
\(771\) −1.75401e12 −0.178767
\(772\) −1.25181e12 −0.126842
\(773\) −7.45839e12 −0.751342 −0.375671 0.926753i \(-0.622588\pi\)
−0.375671 + 0.926753i \(0.622588\pi\)
\(774\) −8.66387e10 −0.00867716
\(775\) −2.61429e12 −0.260313
\(776\) −4.41289e12 −0.436863
\(777\) −3.20388e13 −3.15342
\(778\) 1.75707e12 0.171941
\(779\) −3.82288e12 −0.371939
\(780\) −1.00391e13 −0.971108
\(781\) 8.68360e12 0.835161
\(782\) −1.30970e12 −0.125240
\(783\) 1.84462e13 1.75379
\(784\) 2.72875e13 2.57953
\(785\) −2.78957e12 −0.262194
\(786\) 1.66626e12 0.155718
\(787\) −1.24079e13 −1.15295 −0.576476 0.817114i \(-0.695573\pi\)
−0.576476 + 0.817114i \(0.695573\pi\)
\(788\) 1.31291e13 1.21302
\(789\) −1.52685e12 −0.140265
\(790\) 1.28527e12 0.117402
\(791\) 1.13092e13 1.02715
\(792\) −7.10863e11 −0.0641982
\(793\) −1.12194e13 −1.00749
\(794\) 3.35572e12 0.299636
\(795\) 1.01614e13 0.902197
\(796\) −1.22651e13 −1.08283
\(797\) 4.11416e12 0.361176 0.180588 0.983559i \(-0.442200\pi\)
0.180588 + 0.983559i \(0.442200\pi\)
\(798\) 1.02217e12 0.0892295
\(799\) 1.31408e13 1.14067
\(800\) 1.38531e12 0.119575
\(801\) −3.56399e12 −0.305908
\(802\) −2.25427e12 −0.192408
\(803\) −9.03993e11 −0.0767265
\(804\) 3.36105e12 0.283676
\(805\) 1.13449e13 0.952181
\(806\) 3.78913e12 0.316251
\(807\) 1.63450e13 1.35660
\(808\) −1.34666e12 −0.111150
\(809\) 1.87181e13 1.53636 0.768182 0.640231i \(-0.221162\pi\)
0.768182 + 0.640231i \(0.221162\pi\)
\(810\) −1.30393e12 −0.106432
\(811\) −1.17522e13 −0.953949 −0.476975 0.878917i \(-0.658267\pi\)
−0.476975 + 0.878917i \(0.658267\pi\)
\(812\) 3.79928e13 3.06690
\(813\) −4.19512e12 −0.336773
\(814\) 2.65107e12 0.211646
\(815\) −2.62679e12 −0.208553
\(816\) −1.12260e13 −0.886382
\(817\) 5.38428e11 0.0422793
\(818\) −1.44700e12 −0.113000
\(819\) 1.01261e13 0.786435
\(820\) 1.47978e13 1.14297
\(821\) 2.29795e13 1.76521 0.882605 0.470115i \(-0.155787\pi\)
0.882605 + 0.470115i \(0.155787\pi\)
\(822\) 2.08166e12 0.159033
\(823\) 7.89246e12 0.599671 0.299836 0.953991i \(-0.403068\pi\)
0.299836 + 0.953991i \(0.403068\pi\)
\(824\) −1.68165e12 −0.127076
\(825\) −1.39122e12 −0.104557
\(826\) 6.16680e12 0.460946
\(827\) −1.43874e13 −1.06957 −0.534784 0.844989i \(-0.679607\pi\)
−0.534784 + 0.844989i \(0.679607\pi\)
\(828\) −2.08054e12 −0.153829
\(829\) −1.40139e12 −0.103053 −0.0515267 0.998672i \(-0.516409\pi\)
−0.0515267 + 0.998672i \(0.516409\pi\)
\(830\) 2.50677e12 0.183342
\(831\) 1.33826e13 0.973496
\(832\) 1.46497e13 1.05992
\(833\) −4.78648e13 −3.44440
\(834\) −1.22278e12 −0.0875189
\(835\) 9.83081e12 0.699842
\(836\) 2.16659e12 0.153408
\(837\) −1.85733e13 −1.30805
\(838\) −4.96811e11 −0.0348011
\(839\) −2.01539e13 −1.40420 −0.702101 0.712078i \(-0.747754\pi\)
−0.702101 + 0.712078i \(0.747754\pi\)
\(840\) −8.06775e12 −0.559108
\(841\) 2.32405e13 1.60200
\(842\) 2.50353e12 0.171652
\(843\) 3.35634e12 0.228898
\(844\) −1.39644e13 −0.947287
\(845\) −1.10084e13 −0.742796
\(846\) −8.14914e11 −0.0546947
\(847\) −1.98094e13 −1.32250
\(848\) −1.62271e13 −1.07760
\(849\) 1.22182e13 0.807089
\(850\) −7.57420e11 −0.0497682
\(851\) 1.58210e13 1.03407
\(852\) −1.80734e13 −1.17506
\(853\) −1.31259e13 −0.848907 −0.424453 0.905450i \(-0.639534\pi\)
−0.424453 + 0.905450i \(0.639534\pi\)
\(854\) −4.42187e12 −0.284476
\(855\) −1.12577e12 −0.0720445
\(856\) −8.80615e12 −0.560601
\(857\) −8.66664e12 −0.548829 −0.274414 0.961612i \(-0.588484\pi\)
−0.274414 + 0.961612i \(0.588484\pi\)
\(858\) 2.01643e12 0.127025
\(859\) 1.00716e13 0.631143 0.315571 0.948902i \(-0.397804\pi\)
0.315571 + 0.948902i \(0.397804\pi\)
\(860\) −2.08417e12 −0.129924
\(861\) 3.59204e13 2.22755
\(862\) 2.36321e12 0.145787
\(863\) −8.56376e12 −0.525552 −0.262776 0.964857i \(-0.584638\pi\)
−0.262776 + 0.964857i \(0.584638\pi\)
\(864\) 9.84195e12 0.600854
\(865\) 1.40298e13 0.852075
\(866\) 3.79204e12 0.229109
\(867\) 5.70769e12 0.343063
\(868\) −3.82546e13 −2.28742
\(869\) 6.61296e12 0.393375
\(870\) −3.93111e12 −0.232637
\(871\) 8.07788e12 0.475571
\(872\) 2.48648e12 0.145634
\(873\) 5.78589e12 0.337137
\(874\) −5.04754e11 −0.0292603
\(875\) 3.68837e13 2.12715
\(876\) 1.88151e12 0.107954
\(877\) −1.86361e12 −0.106379 −0.0531896 0.998584i \(-0.516939\pi\)
−0.0531896 + 0.998584i \(0.516939\pi\)
\(878\) 1.67302e11 0.00950113
\(879\) −2.26170e13 −1.27786
\(880\) −8.04635e12 −0.452300
\(881\) 1.12984e13 0.631868 0.315934 0.948781i \(-0.397682\pi\)
0.315934 + 0.948781i \(0.397682\pi\)
\(882\) 2.96830e12 0.165158
\(883\) −1.69207e13 −0.936690 −0.468345 0.883546i \(-0.655150\pi\)
−0.468345 + 0.883546i \(0.655150\pi\)
\(884\) −2.81211e13 −1.54881
\(885\) 1.63450e13 0.895654
\(886\) −1.77373e12 −0.0967018
\(887\) 1.15592e13 0.627004 0.313502 0.949588i \(-0.398498\pi\)
0.313502 + 0.949588i \(0.398498\pi\)
\(888\) −1.12509e13 −0.607195
\(889\) 4.04827e13 2.17376
\(890\) 3.34693e12 0.178810
\(891\) −6.70893e12 −0.356618
\(892\) 4.26959e12 0.225810
\(893\) 5.06439e12 0.266499
\(894\) −1.75857e12 −0.0920750
\(895\) 1.91139e13 0.995741
\(896\) 2.68362e13 1.39103
\(897\) 1.20336e13 0.620627
\(898\) −2.29386e12 −0.117713
\(899\) −3.80078e13 −1.94068
\(900\) −1.20320e12 −0.0611291
\(901\) 2.84638e13 1.43890
\(902\) −2.97225e12 −0.149505
\(903\) −5.05916e12 −0.253211
\(904\) 3.97136e12 0.197780
\(905\) −1.39613e13 −0.691842
\(906\) 1.40255e12 0.0691577
\(907\) −8.82407e12 −0.432948 −0.216474 0.976288i \(-0.569456\pi\)
−0.216474 + 0.976288i \(0.569456\pi\)
\(908\) 9.49810e12 0.463714
\(909\) 1.76565e12 0.0857765
\(910\) −9.50934e12 −0.459689
\(911\) −9.12800e12 −0.439079 −0.219540 0.975604i \(-0.570456\pi\)
−0.219540 + 0.975604i \(0.570456\pi\)
\(912\) −4.32647e12 −0.207089
\(913\) 1.28978e13 0.614321
\(914\) 7.11463e12 0.337205
\(915\) −1.17201e13 −0.552759
\(916\) 1.13668e12 0.0533468
\(917\) −4.04306e13 −1.88820
\(918\) −5.38110e12 −0.250080
\(919\) −1.89826e13 −0.877881 −0.438940 0.898516i \(-0.644646\pi\)
−0.438940 + 0.898516i \(0.644646\pi\)
\(920\) 3.98392e12 0.183343
\(921\) −6.99830e12 −0.320497
\(922\) 2.47581e12 0.112831
\(923\) −4.34373e13 −1.96995
\(924\) −2.03576e13 −0.918763
\(925\) 9.14953e12 0.410924
\(926\) 7.72101e11 0.0345084
\(927\) 2.20487e12 0.0980673
\(928\) 2.01403e13 0.891455
\(929\) 1.66439e13 0.733136 0.366568 0.930391i \(-0.380533\pi\)
0.366568 + 0.930391i \(0.380533\pi\)
\(930\) 3.95820e12 0.173510
\(931\) −1.84469e13 −0.804729
\(932\) 1.08433e13 0.470748
\(933\) −1.72402e13 −0.744860
\(934\) −6.00008e11 −0.0257986
\(935\) 1.41141e13 0.603948
\(936\) 3.55590e12 0.151429
\(937\) 1.20784e13 0.511896 0.255948 0.966691i \(-0.417612\pi\)
0.255948 + 0.966691i \(0.417612\pi\)
\(938\) 3.18370e12 0.134282
\(939\) −1.62997e13 −0.684201
\(940\) −1.96035e13 −0.818950
\(941\) −1.09033e13 −0.453320 −0.226660 0.973974i \(-0.572781\pi\)
−0.226660 + 0.973974i \(0.572781\pi\)
\(942\) −1.16618e12 −0.0482544
\(943\) −1.77378e13 −0.730461
\(944\) −2.61019e13 −1.06979
\(945\) 4.66122e13 1.90133
\(946\) 4.18622e11 0.0169946
\(947\) 2.49583e13 1.00842 0.504209 0.863582i \(-0.331784\pi\)
0.504209 + 0.863582i \(0.331784\pi\)
\(948\) −1.37637e13 −0.553476
\(949\) 4.52198e12 0.180980
\(950\) −2.91907e11 −0.0116275
\(951\) 6.56436e12 0.260244
\(952\) −2.25992e13 −0.891715
\(953\) −1.44768e13 −0.568530 −0.284265 0.958746i \(-0.591750\pi\)
−0.284265 + 0.958746i \(0.591750\pi\)
\(954\) −1.76516e12 −0.0689949
\(955\) −2.32284e13 −0.903657
\(956\) 1.79918e13 0.696649
\(957\) −2.02263e13 −0.779492
\(958\) 2.13365e12 0.0818423
\(959\) −5.05102e13 −1.92839
\(960\) 1.53034e13 0.581524
\(961\) 1.18301e13 0.447437
\(962\) −1.32612e13 −0.499225
\(963\) 1.15460e13 0.432628
\(964\) −3.96951e13 −1.48044
\(965\) −3.14284e12 −0.116667
\(966\) 4.74275e12 0.175240
\(967\) −1.31877e12 −0.0485007 −0.0242504 0.999706i \(-0.507720\pi\)
−0.0242504 + 0.999706i \(0.507720\pi\)
\(968\) −6.95635e12 −0.254649
\(969\) 7.58903e12 0.276522
\(970\) −5.43351e12 −0.197064
\(971\) −5.24865e13 −1.89479 −0.947394 0.320069i \(-0.896294\pi\)
−0.947394 + 0.320069i \(0.896294\pi\)
\(972\) −1.51565e13 −0.544628
\(973\) 2.96700e13 1.06123
\(974\) 3.55525e12 0.126577
\(975\) 6.95922e12 0.246626
\(976\) 1.87162e13 0.660227
\(977\) 5.15657e13 1.81065 0.905327 0.424714i \(-0.139625\pi\)
0.905327 + 0.424714i \(0.139625\pi\)
\(978\) −1.09813e12 −0.0383822
\(979\) 1.72206e13 0.599135
\(980\) 7.14050e13 2.47293
\(981\) −3.26011e12 −0.112388
\(982\) −4.91356e12 −0.168614
\(983\) 2.11520e13 0.722537 0.361269 0.932462i \(-0.382344\pi\)
0.361269 + 0.932462i \(0.382344\pi\)
\(984\) 1.26139e13 0.428917
\(985\) 3.29623e13 1.11572
\(986\) −1.10117e13 −0.371030
\(987\) −4.75859e13 −1.59607
\(988\) −1.08378e13 −0.361854
\(989\) 2.49825e12 0.0830334
\(990\) −8.75272e11 −0.0289591
\(991\) −4.73443e13 −1.55932 −0.779661 0.626201i \(-0.784609\pi\)
−0.779661 + 0.626201i \(0.784609\pi\)
\(992\) −2.02791e13 −0.664883
\(993\) 2.47449e13 0.807632
\(994\) −1.71197e13 −0.556235
\(995\) −3.07931e13 −0.995977
\(996\) −2.68444e13 −0.864345
\(997\) −1.49776e13 −0.480081 −0.240041 0.970763i \(-0.577161\pi\)
−0.240041 + 0.970763i \(0.577161\pi\)
\(998\) −8.47966e12 −0.270577
\(999\) 6.50030e13 2.06485
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.1.8 15
3.2 odd 2 387.10.a.c.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.8 15 1.1 even 1 trivial
387.10.a.c.1.8 15 3.2 odd 2