Properties

Label 15.10.a.c.1.1
Level 15
Weight 10
Character 15.1
Self dual yes
Analytic conductor 7.726
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.72553754246\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4729}) \)
Defining polynomial: \(x^{2} - x - 1182\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(34.8839\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.8839 q^{2} +81.0000 q^{3} +107.207 q^{4} -625.000 q^{5} -2015.59 q^{6} -4010.50 q^{7} +10072.8 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-24.8839 q^{2} +81.0000 q^{3} +107.207 q^{4} -625.000 q^{5} -2015.59 q^{6} -4010.50 q^{7} +10072.8 q^{8} +6561.00 q^{9} +15552.4 q^{10} +84861.3 q^{11} +8683.74 q^{12} +119425. q^{13} +99796.8 q^{14} -50625.0 q^{15} -305541. q^{16} +116934. q^{17} -163263. q^{18} -234932. q^{19} -67004.1 q^{20} -324851. q^{21} -2.11168e6 q^{22} +2.34570e6 q^{23} +815899. q^{24} +390625. q^{25} -2.97176e6 q^{26} +531441. q^{27} -429953. q^{28} -464196. q^{29} +1.25975e6 q^{30} -5.11766e6 q^{31} +2.44574e6 q^{32} +6.87377e6 q^{33} -2.90976e6 q^{34} +2.50656e6 q^{35} +703383. q^{36} +8.69354e6 q^{37} +5.84601e6 q^{38} +9.67345e6 q^{39} -6.29551e6 q^{40} -9.05805e6 q^{41} +8.08354e6 q^{42} +8.63491e6 q^{43} +9.09769e6 q^{44} -4.10063e6 q^{45} -5.83701e7 q^{46} +3.31511e7 q^{47} -2.47488e7 q^{48} -2.42695e7 q^{49} -9.72026e6 q^{50} +9.47161e6 q^{51} +1.28032e7 q^{52} -6.41254e7 q^{53} -1.32243e7 q^{54} -5.30383e7 q^{55} -4.03971e7 q^{56} -1.90295e7 q^{57} +1.15510e7 q^{58} +1.49407e8 q^{59} -5.42733e6 q^{60} +1.54634e8 q^{61} +1.27347e8 q^{62} -2.63129e7 q^{63} +9.55772e7 q^{64} -7.46408e7 q^{65} -1.71046e8 q^{66} +2.72755e8 q^{67} +1.25360e7 q^{68} +1.90002e8 q^{69} -6.23730e7 q^{70} -3.56924e8 q^{71} +6.60878e7 q^{72} +2.06253e8 q^{73} -2.16329e8 q^{74} +3.16406e7 q^{75} -2.51862e7 q^{76} -3.40337e8 q^{77} -2.40713e8 q^{78} -4.04380e8 q^{79} +1.90963e8 q^{80} +4.30467e7 q^{81} +2.25399e8 q^{82} -5.17034e6 q^{83} -3.48262e7 q^{84} -7.30834e7 q^{85} -2.14870e8 q^{86} -3.75999e7 q^{87} +8.54793e8 q^{88} +4.32242e8 q^{89} +1.02039e8 q^{90} -4.78955e8 q^{91} +2.51475e8 q^{92} -4.14530e8 q^{93} -8.24928e8 q^{94} +1.46832e8 q^{95} +1.98105e8 q^{96} -1.32066e9 q^{97} +6.03918e8 q^{98} +5.56775e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} - 11875q^{10} + 35488q^{11} + 123201q^{12} + 143676q^{13} - 245196q^{14} - 101250q^{15} + 707265q^{16} + 385156q^{17} + 124659q^{18} - 403296q^{19} - 950625q^{20} - 961632q^{21} - 4278368q^{22} + 223704q^{23} + 4021407q^{24} + 781250q^{25} - 1907546q^{26} + 1062882q^{27} - 11544484q^{28} - 74572q^{29} - 961875q^{30} - 5027128q^{31} + 26629583q^{32} + 2874528q^{33} + 8860882q^{34} + 7420000q^{35} + 9979281q^{36} + 5373628q^{37} - 1542476q^{38} + 11637756q^{39} - 31029375q^{40} + 14211332q^{41} - 19860876q^{42} + 27748920q^{43} - 60705952q^{44} - 8201250q^{45} - 151491648q^{46} + 95966440q^{47} + 57288465q^{48} - 2819950q^{49} + 7421875q^{50} + 31197636q^{51} + 47088706q^{52} - 64305596q^{53} + 10097379q^{54} - 22180000q^{55} - 351509340q^{56} - 32666976q^{57} + 28649198q^{58} + 187863136q^{59} - 77000625q^{60} + 154080060q^{61} + 131320056q^{62} - 77892192q^{63} + 638301089q^{64} - 89797500q^{65} - 346547808q^{66} + 33592376q^{67} + 391747238q^{68} + 18120024q^{69} + 153247500q^{70} - 228270976q^{71} + 325733967q^{72} - 33122316q^{73} - 362019226q^{74} + 63281250q^{75} - 263218724q^{76} + 47811456q^{77} - 154511226q^{78} - 932406760q^{79} - 442040625q^{80} + 86093442q^{81} + 1246549646q^{82} + 207040152q^{83} - 935103204q^{84} - 240722500q^{85} + 623926708q^{86} - 6040332q^{87} - 1099114848q^{88} + 224518164q^{89} - 77911875q^{90} - 669602528q^{91} - 2748592992q^{92} - 407197368q^{93} + 1931650816q^{94} + 252060000q^{95} + 2156996223q^{96} + 387134596q^{97} + 1545205739q^{98} + 232836768q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.8839 −1.09972 −0.549861 0.835256i \(-0.685319\pi\)
−0.549861 + 0.835256i \(0.685319\pi\)
\(3\) 81.0000 0.577350
\(4\) 107.207 0.209388
\(5\) −625.000 −0.447214
\(6\) −2015.59 −0.634925
\(7\) −4010.50 −0.631332 −0.315666 0.948870i \(-0.602228\pi\)
−0.315666 + 0.948870i \(0.602228\pi\)
\(8\) 10072.8 0.869453
\(9\) 6561.00 0.333333
\(10\) 15552.4 0.491811
\(11\) 84861.3 1.74760 0.873801 0.486283i \(-0.161648\pi\)
0.873801 + 0.486283i \(0.161648\pi\)
\(12\) 8683.74 0.120890
\(13\) 119425. 1.15971 0.579857 0.814718i \(-0.303108\pi\)
0.579857 + 0.814718i \(0.303108\pi\)
\(14\) 99796.8 0.694289
\(15\) −50625.0 −0.258199
\(16\) −305541. −1.16554
\(17\) 116934. 0.339562 0.169781 0.985482i \(-0.445694\pi\)
0.169781 + 0.985482i \(0.445694\pi\)
\(18\) −163263. −0.366574
\(19\) −234932. −0.413571 −0.206786 0.978386i \(-0.566300\pi\)
−0.206786 + 0.978386i \(0.566300\pi\)
\(20\) −67004.1 −0.0936411
\(21\) −324851. −0.364500
\(22\) −2.11168e6 −1.92188
\(23\) 2.34570e6 1.74782 0.873912 0.486084i \(-0.161575\pi\)
0.873912 + 0.486084i \(0.161575\pi\)
\(24\) 815899. 0.501979
\(25\) 390625. 0.200000
\(26\) −2.97176e6 −1.27536
\(27\) 531441. 0.192450
\(28\) −429953. −0.132193
\(29\) −464196. −0.121874 −0.0609369 0.998142i \(-0.519409\pi\)
−0.0609369 + 0.998142i \(0.519409\pi\)
\(30\) 1.25975e6 0.283947
\(31\) −5.11766e6 −0.995277 −0.497638 0.867385i \(-0.665799\pi\)
−0.497638 + 0.867385i \(0.665799\pi\)
\(32\) 2.44574e6 0.412321
\(33\) 6.87377e6 1.00898
\(34\) −2.90976e6 −0.373423
\(35\) 2.50656e6 0.282340
\(36\) 703383. 0.0697960
\(37\) 8.69354e6 0.762586 0.381293 0.924454i \(-0.375479\pi\)
0.381293 + 0.924454i \(0.375479\pi\)
\(38\) 5.84601e6 0.454813
\(39\) 9.67345e6 0.669562
\(40\) −6.29551e6 −0.388831
\(41\) −9.05805e6 −0.500619 −0.250310 0.968166i \(-0.580532\pi\)
−0.250310 + 0.968166i \(0.580532\pi\)
\(42\) 8.08354e6 0.400848
\(43\) 8.63491e6 0.385168 0.192584 0.981281i \(-0.438313\pi\)
0.192584 + 0.981281i \(0.438313\pi\)
\(44\) 9.09769e6 0.365927
\(45\) −4.10063e6 −0.149071
\(46\) −5.83701e7 −1.92212
\(47\) 3.31511e7 0.990964 0.495482 0.868618i \(-0.334991\pi\)
0.495482 + 0.868618i \(0.334991\pi\)
\(48\) −2.47488e7 −0.672927
\(49\) −2.42695e7 −0.601420
\(50\) −9.72026e6 −0.219944
\(51\) 9.47161e6 0.196046
\(52\) 1.28032e7 0.242830
\(53\) −6.41254e7 −1.11632 −0.558160 0.829733i \(-0.688492\pi\)
−0.558160 + 0.829733i \(0.688492\pi\)
\(54\) −1.32243e7 −0.211642
\(55\) −5.30383e7 −0.781551
\(56\) −4.03971e7 −0.548914
\(57\) −1.90295e7 −0.238775
\(58\) 1.15510e7 0.134027
\(59\) 1.49407e8 1.60523 0.802613 0.596500i \(-0.203442\pi\)
0.802613 + 0.596500i \(0.203442\pi\)
\(60\) −5.42733e6 −0.0540637
\(61\) 1.54634e8 1.42995 0.714973 0.699152i \(-0.246439\pi\)
0.714973 + 0.699152i \(0.246439\pi\)
\(62\) 1.27347e8 1.09453
\(63\) −2.63129e7 −0.210444
\(64\) 9.55772e7 0.712106
\(65\) −7.46408e7 −0.518640
\(66\) −1.71046e8 −1.10960
\(67\) 2.72755e8 1.65362 0.826811 0.562479i \(-0.190152\pi\)
0.826811 + 0.562479i \(0.190152\pi\)
\(68\) 1.25360e7 0.0711001
\(69\) 1.90002e8 1.00911
\(70\) −6.23730e7 −0.310496
\(71\) −3.56924e8 −1.66691 −0.833457 0.552584i \(-0.813642\pi\)
−0.833457 + 0.552584i \(0.813642\pi\)
\(72\) 6.60878e7 0.289818
\(73\) 2.06253e8 0.850057 0.425029 0.905180i \(-0.360264\pi\)
0.425029 + 0.905180i \(0.360264\pi\)
\(74\) −2.16329e8 −0.838632
\(75\) 3.16406e7 0.115470
\(76\) −2.51862e7 −0.0865968
\(77\) −3.40337e8 −1.10332
\(78\) −2.40713e8 −0.736331
\(79\) −4.04380e8 −1.16807 −0.584033 0.811730i \(-0.698526\pi\)
−0.584033 + 0.811730i \(0.698526\pi\)
\(80\) 1.90963e8 0.521247
\(81\) 4.30467e7 0.111111
\(82\) 2.25399e8 0.550542
\(83\) −5.17034e6 −0.0119582 −0.00597912 0.999982i \(-0.501903\pi\)
−0.00597912 + 0.999982i \(0.501903\pi\)
\(84\) −3.48262e7 −0.0763218
\(85\) −7.30834e7 −0.151857
\(86\) −2.14870e8 −0.423577
\(87\) −3.75999e7 −0.0703639
\(88\) 8.54793e8 1.51946
\(89\) 4.32242e8 0.730250 0.365125 0.930958i \(-0.381026\pi\)
0.365125 + 0.930958i \(0.381026\pi\)
\(90\) 1.02039e8 0.163937
\(91\) −4.78955e8 −0.732165
\(92\) 2.51475e8 0.365973
\(93\) −4.14530e8 −0.574623
\(94\) −8.24928e8 −1.08978
\(95\) 1.46832e8 0.184955
\(96\) 1.98105e8 0.238054
\(97\) −1.32066e9 −1.51468 −0.757338 0.653023i \(-0.773501\pi\)
−0.757338 + 0.653023i \(0.773501\pi\)
\(98\) 6.03918e8 0.661395
\(99\) 5.56775e8 0.582534
\(100\) 4.18776e7 0.0418776
\(101\) 5.30458e8 0.507230 0.253615 0.967305i \(-0.418380\pi\)
0.253615 + 0.967305i \(0.418380\pi\)
\(102\) −2.35690e8 −0.215596
\(103\) −6.07207e7 −0.0531580 −0.0265790 0.999647i \(-0.508461\pi\)
−0.0265790 + 0.999647i \(0.508461\pi\)
\(104\) 1.20295e9 1.00832
\(105\) 2.03032e8 0.163009
\(106\) 1.59569e9 1.22764
\(107\) −1.00828e9 −0.743625 −0.371812 0.928308i \(-0.621264\pi\)
−0.371812 + 0.928308i \(0.621264\pi\)
\(108\) 5.69740e7 0.0402967
\(109\) −1.77424e9 −1.20391 −0.601954 0.798531i \(-0.705611\pi\)
−0.601954 + 0.798531i \(0.705611\pi\)
\(110\) 1.31980e9 0.859489
\(111\) 7.04177e8 0.440279
\(112\) 1.22537e9 0.735846
\(113\) 9.45495e8 0.545514 0.272757 0.962083i \(-0.412064\pi\)
0.272757 + 0.962083i \(0.412064\pi\)
\(114\) 4.73526e8 0.262586
\(115\) −1.46606e9 −0.781651
\(116\) −4.97649e7 −0.0255189
\(117\) 7.83549e8 0.386572
\(118\) −3.71782e9 −1.76530
\(119\) −4.68962e8 −0.214376
\(120\) −5.09937e8 −0.224492
\(121\) 4.84349e9 2.05411
\(122\) −3.84788e9 −1.57254
\(123\) −7.33702e8 −0.289033
\(124\) −5.48647e8 −0.208399
\(125\) −2.44141e8 −0.0894427
\(126\) 6.54767e8 0.231430
\(127\) −5.19758e9 −1.77290 −0.886450 0.462825i \(-0.846836\pi\)
−0.886450 + 0.462825i \(0.846836\pi\)
\(128\) −3.63055e9 −1.19544
\(129\) 6.99428e8 0.222377
\(130\) 1.85735e9 0.570360
\(131\) 5.28408e8 0.156765 0.0783824 0.996923i \(-0.475024\pi\)
0.0783824 + 0.996923i \(0.475024\pi\)
\(132\) 7.36913e8 0.211268
\(133\) 9.42194e8 0.261101
\(134\) −6.78720e9 −1.81852
\(135\) −3.32151e8 −0.0860663
\(136\) 1.17785e9 0.295233
\(137\) 5.01761e9 1.21690 0.608449 0.793593i \(-0.291792\pi\)
0.608449 + 0.793593i \(0.291792\pi\)
\(138\) −4.72798e9 −1.10974
\(139\) 3.51872e9 0.799499 0.399750 0.916624i \(-0.369097\pi\)
0.399750 + 0.916624i \(0.369097\pi\)
\(140\) 2.68720e8 0.0591186
\(141\) 2.68524e9 0.572133
\(142\) 8.88165e9 1.83314
\(143\) 1.01346e10 2.02672
\(144\) −2.00465e9 −0.388515
\(145\) 2.90123e8 0.0545036
\(146\) −5.13238e9 −0.934827
\(147\) −1.96583e9 −0.347230
\(148\) 9.32005e8 0.159676
\(149\) 4.32815e9 0.719390 0.359695 0.933070i \(-0.382881\pi\)
0.359695 + 0.933070i \(0.382881\pi\)
\(150\) −7.87341e8 −0.126985
\(151\) −5.61832e9 −0.879448 −0.439724 0.898133i \(-0.644924\pi\)
−0.439724 + 0.898133i \(0.644924\pi\)
\(152\) −2.36642e9 −0.359581
\(153\) 7.67201e8 0.113187
\(154\) 8.46889e9 1.21334
\(155\) 3.19854e9 0.445101
\(156\) 1.03706e9 0.140198
\(157\) 1.55603e9 0.204394 0.102197 0.994764i \(-0.467413\pi\)
0.102197 + 0.994764i \(0.467413\pi\)
\(158\) 1.00625e10 1.28455
\(159\) −5.19416e9 −0.644507
\(160\) −1.52859e9 −0.184396
\(161\) −9.40745e9 −1.10346
\(162\) −1.07117e9 −0.122191
\(163\) −1.15580e10 −1.28245 −0.641224 0.767354i \(-0.721573\pi\)
−0.641224 + 0.767354i \(0.721573\pi\)
\(164\) −9.71083e8 −0.104824
\(165\) −4.29610e9 −0.451229
\(166\) 1.28658e8 0.0131507
\(167\) −1.50486e10 −1.49717 −0.748585 0.663039i \(-0.769266\pi\)
−0.748585 + 0.663039i \(0.769266\pi\)
\(168\) −3.27216e9 −0.316915
\(169\) 3.65789e9 0.344938
\(170\) 1.81860e9 0.167000
\(171\) −1.54139e9 −0.137857
\(172\) 9.25719e8 0.0806494
\(173\) 2.23157e10 1.89410 0.947052 0.321081i \(-0.104046\pi\)
0.947052 + 0.321081i \(0.104046\pi\)
\(174\) 9.35630e8 0.0773807
\(175\) −1.56660e9 −0.126266
\(176\) −2.59286e10 −2.03691
\(177\) 1.21019e10 0.926778
\(178\) −1.07558e10 −0.803072
\(179\) −1.73543e10 −1.26348 −0.631742 0.775179i \(-0.717660\pi\)
−0.631742 + 0.775179i \(0.717660\pi\)
\(180\) −4.39614e8 −0.0312137
\(181\) −1.34040e10 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(182\) 1.19183e10 0.805178
\(183\) 1.25253e10 0.825580
\(184\) 2.36278e10 1.51965
\(185\) −5.43346e9 −0.341039
\(186\) 1.03151e10 0.631926
\(187\) 9.92313e9 0.593419
\(188\) 3.55402e9 0.207496
\(189\) −2.13135e9 −0.121500
\(190\) −3.65375e9 −0.203399
\(191\) −6.01312e9 −0.326926 −0.163463 0.986549i \(-0.552266\pi\)
−0.163463 + 0.986549i \(0.552266\pi\)
\(192\) 7.74175e9 0.411134
\(193\) −6.91844e9 −0.358922 −0.179461 0.983765i \(-0.557435\pi\)
−0.179461 + 0.983765i \(0.557435\pi\)
\(194\) 3.28632e10 1.66572
\(195\) −6.04590e9 −0.299437
\(196\) −2.60185e9 −0.125930
\(197\) 1.66139e10 0.785909 0.392955 0.919558i \(-0.371453\pi\)
0.392955 + 0.919558i \(0.371453\pi\)
\(198\) −1.38547e10 −0.640625
\(199\) −3.06711e10 −1.38641 −0.693204 0.720742i \(-0.743801\pi\)
−0.693204 + 0.720742i \(0.743801\pi\)
\(200\) 3.93470e9 0.173891
\(201\) 2.20932e10 0.954719
\(202\) −1.31998e10 −0.557812
\(203\) 1.86166e9 0.0769428
\(204\) 1.01542e9 0.0410497
\(205\) 5.66128e9 0.223884
\(206\) 1.51096e9 0.0584591
\(207\) 1.53902e10 0.582608
\(208\) −3.64893e10 −1.35170
\(209\) −1.99366e10 −0.722758
\(210\) −5.05221e9 −0.179265
\(211\) 3.85598e10 1.33926 0.669628 0.742697i \(-0.266454\pi\)
0.669628 + 0.742697i \(0.266454\pi\)
\(212\) −6.87467e9 −0.233744
\(213\) −2.89108e10 −0.962393
\(214\) 2.50899e10 0.817780
\(215\) −5.39682e9 −0.172252
\(216\) 5.35311e9 0.167326
\(217\) 2.05244e10 0.628350
\(218\) 4.41500e10 1.32396
\(219\) 1.67065e10 0.490781
\(220\) −5.68606e9 −0.163647
\(221\) 1.39648e10 0.393795
\(222\) −1.75226e10 −0.484185
\(223\) −3.11357e10 −0.843115 −0.421558 0.906802i \(-0.638516\pi\)
−0.421558 + 0.906802i \(0.638516\pi\)
\(224\) −9.80866e9 −0.260312
\(225\) 2.56289e9 0.0666667
\(226\) −2.35276e10 −0.599914
\(227\) −1.14589e10 −0.286436 −0.143218 0.989691i \(-0.545745\pi\)
−0.143218 + 0.989691i \(0.545745\pi\)
\(228\) −2.04008e9 −0.0499967
\(229\) −3.04556e10 −0.731825 −0.365913 0.930649i \(-0.619243\pi\)
−0.365913 + 0.930649i \(0.619243\pi\)
\(230\) 3.64813e10 0.859598
\(231\) −2.75673e10 −0.637000
\(232\) −4.67576e9 −0.105964
\(233\) 2.83630e9 0.0630451 0.0315225 0.999503i \(-0.489964\pi\)
0.0315225 + 0.999503i \(0.489964\pi\)
\(234\) −1.94977e10 −0.425121
\(235\) −2.07195e10 −0.443173
\(236\) 1.60174e10 0.336115
\(237\) −3.27548e10 −0.674384
\(238\) 1.16696e10 0.235754
\(239\) −6.25862e10 −1.24076 −0.620381 0.784301i \(-0.713022\pi\)
−0.620381 + 0.784301i \(0.713022\pi\)
\(240\) 1.54680e10 0.300942
\(241\) 7.24015e10 1.38252 0.691259 0.722607i \(-0.257056\pi\)
0.691259 + 0.722607i \(0.257056\pi\)
\(242\) −1.20525e11 −2.25895
\(243\) 3.48678e9 0.0641500
\(244\) 1.65778e10 0.299414
\(245\) 1.51684e10 0.268963
\(246\) 1.82573e10 0.317855
\(247\) −2.80568e10 −0.479624
\(248\) −5.15493e10 −0.865347
\(249\) −4.18797e8 −0.00690410
\(250\) 6.07516e9 0.0983621
\(251\) −5.65927e10 −0.899971 −0.449986 0.893036i \(-0.648571\pi\)
−0.449986 + 0.893036i \(0.648571\pi\)
\(252\) −2.82092e9 −0.0440644
\(253\) 1.99059e11 3.05450
\(254\) 1.29336e11 1.94970
\(255\) −5.91976e9 −0.0876745
\(256\) 4.14066e10 0.602545
\(257\) 4.95688e10 0.708777 0.354388 0.935098i \(-0.384689\pi\)
0.354388 + 0.935098i \(0.384689\pi\)
\(258\) −1.74045e10 −0.244552
\(259\) −3.48655e10 −0.481445
\(260\) −8.00199e9 −0.108597
\(261\) −3.04559e9 −0.0406246
\(262\) −1.31488e10 −0.172398
\(263\) 3.59498e10 0.463336 0.231668 0.972795i \(-0.425582\pi\)
0.231668 + 0.972795i \(0.425582\pi\)
\(264\) 6.92382e10 0.877260
\(265\) 4.00784e10 0.499233
\(266\) −2.34454e10 −0.287138
\(267\) 3.50116e10 0.421610
\(268\) 2.92412e10 0.346249
\(269\) −7.09650e10 −0.826340 −0.413170 0.910654i \(-0.635579\pi\)
−0.413170 + 0.910654i \(0.635579\pi\)
\(270\) 8.26519e9 0.0946490
\(271\) 1.47201e11 1.65786 0.828930 0.559352i \(-0.188950\pi\)
0.828930 + 0.559352i \(0.188950\pi\)
\(272\) −3.57279e10 −0.395774
\(273\) −3.87954e10 −0.422716
\(274\) −1.24857e11 −1.33825
\(275\) 3.31489e10 0.349520
\(276\) 2.03695e10 0.211295
\(277\) −2.87336e10 −0.293245 −0.146622 0.989193i \(-0.546840\pi\)
−0.146622 + 0.989193i \(0.546840\pi\)
\(278\) −8.75593e10 −0.879227
\(279\) −3.35770e10 −0.331759
\(280\) 2.52482e10 0.245482
\(281\) 5.17071e10 0.494734 0.247367 0.968922i \(-0.420435\pi\)
0.247367 + 0.968922i \(0.420435\pi\)
\(282\) −6.68192e10 −0.629187
\(283\) −3.22434e9 −0.0298814 −0.0149407 0.999888i \(-0.504756\pi\)
−0.0149407 + 0.999888i \(0.504756\pi\)
\(284\) −3.82646e10 −0.349032
\(285\) 1.18934e10 0.106784
\(286\) −2.52188e11 −2.22883
\(287\) 3.63273e10 0.316057
\(288\) 1.60465e10 0.137440
\(289\) −1.04914e11 −0.884698
\(290\) −7.21937e9 −0.0599388
\(291\) −1.06974e11 −0.874499
\(292\) 2.21117e10 0.177992
\(293\) −1.13591e11 −0.900409 −0.450204 0.892926i \(-0.648649\pi\)
−0.450204 + 0.892926i \(0.648649\pi\)
\(294\) 4.89174e10 0.381856
\(295\) −9.33792e10 −0.717879
\(296\) 8.75685e10 0.663033
\(297\) 4.50988e10 0.336326
\(298\) −1.07701e11 −0.791129
\(299\) 2.80136e11 2.02698
\(300\) 3.39208e9 0.0241780
\(301\) −3.46303e10 −0.243169
\(302\) 1.39805e11 0.967148
\(303\) 4.29671e10 0.292849
\(304\) 7.17811e10 0.482036
\(305\) −9.66460e10 −0.639492
\(306\) −1.90909e10 −0.124474
\(307\) −2.30543e10 −0.148125 −0.0740627 0.997254i \(-0.523597\pi\)
−0.0740627 + 0.997254i \(0.523597\pi\)
\(308\) −3.64863e10 −0.231021
\(309\) −4.91837e9 −0.0306908
\(310\) −7.95920e10 −0.489488
\(311\) 7.71709e10 0.467769 0.233885 0.972264i \(-0.424856\pi\)
0.233885 + 0.972264i \(0.424856\pi\)
\(312\) 9.74389e10 0.582152
\(313\) −4.68832e10 −0.276101 −0.138050 0.990425i \(-0.544084\pi\)
−0.138050 + 0.990425i \(0.544084\pi\)
\(314\) −3.87200e10 −0.224777
\(315\) 1.64456e10 0.0941134
\(316\) −4.33522e10 −0.244579
\(317\) 2.19420e11 1.22042 0.610210 0.792240i \(-0.291085\pi\)
0.610210 + 0.792240i \(0.291085\pi\)
\(318\) 1.29251e11 0.708779
\(319\) −3.93923e10 −0.212987
\(320\) −5.97358e10 −0.318463
\(321\) −8.16706e10 −0.429332
\(322\) 2.34094e11 1.21350
\(323\) −2.74714e10 −0.140433
\(324\) 4.61489e9 0.0232653
\(325\) 4.66505e10 0.231943
\(326\) 2.87609e11 1.41034
\(327\) −1.43714e11 −0.695077
\(328\) −9.12401e10 −0.435265
\(329\) −1.32953e11 −0.625627
\(330\) 1.06904e11 0.496226
\(331\) 3.89075e9 0.0178159 0.00890794 0.999960i \(-0.497164\pi\)
0.00890794 + 0.999960i \(0.497164\pi\)
\(332\) −5.54295e8 −0.00250391
\(333\) 5.70383e10 0.254195
\(334\) 3.74466e11 1.64647
\(335\) −1.70472e11 −0.739522
\(336\) 9.92551e10 0.424841
\(337\) −1.48259e11 −0.626163 −0.313082 0.949726i \(-0.601361\pi\)
−0.313082 + 0.949726i \(0.601361\pi\)
\(338\) −9.10225e10 −0.379336
\(339\) 7.65851e10 0.314953
\(340\) −7.83503e9 −0.0317969
\(341\) −4.34291e11 −1.73935
\(342\) 3.83556e10 0.151604
\(343\) 2.59171e11 1.01103
\(344\) 8.69779e10 0.334885
\(345\) −1.18751e11 −0.451286
\(346\) −5.55302e11 −2.08299
\(347\) −1.63695e11 −0.606114 −0.303057 0.952972i \(-0.598007\pi\)
−0.303057 + 0.952972i \(0.598007\pi\)
\(348\) −4.03096e9 −0.0147333
\(349\) 7.02057e10 0.253313 0.126657 0.991947i \(-0.459575\pi\)
0.126657 + 0.991947i \(0.459575\pi\)
\(350\) 3.89831e10 0.138858
\(351\) 6.34675e10 0.223187
\(352\) 2.07549e11 0.720574
\(353\) −1.55727e10 −0.0533798 −0.0266899 0.999644i \(-0.508497\pi\)
−0.0266899 + 0.999644i \(0.508497\pi\)
\(354\) −3.01143e11 −1.01920
\(355\) 2.23078e11 0.745467
\(356\) 4.63392e10 0.152906
\(357\) −3.79859e10 −0.123770
\(358\) 4.31843e11 1.38948
\(359\) 3.28525e10 0.104386 0.0521931 0.998637i \(-0.483379\pi\)
0.0521931 + 0.998637i \(0.483379\pi\)
\(360\) −4.13049e10 −0.129610
\(361\) −2.67495e11 −0.828959
\(362\) 3.33543e11 1.02085
\(363\) 3.92323e11 1.18594
\(364\) −5.13472e10 −0.153306
\(365\) −1.28908e11 −0.380157
\(366\) −3.11679e11 −0.907908
\(367\) −4.76030e11 −1.36974 −0.684868 0.728667i \(-0.740140\pi\)
−0.684868 + 0.728667i \(0.740140\pi\)
\(368\) −7.16707e11 −2.03717
\(369\) −5.94299e10 −0.166873
\(370\) 1.35205e11 0.375048
\(371\) 2.57175e11 0.704768
\(372\) −4.44404e10 −0.120319
\(373\) −9.72745e10 −0.260201 −0.130101 0.991501i \(-0.541530\pi\)
−0.130101 + 0.991501i \(0.541530\pi\)
\(374\) −2.46926e11 −0.652596
\(375\) −1.97754e10 −0.0516398
\(376\) 3.33925e11 0.861597
\(377\) −5.54367e10 −0.141339
\(378\) 5.30361e10 0.133616
\(379\) 4.45171e11 1.10828 0.554141 0.832423i \(-0.313047\pi\)
0.554141 + 0.832423i \(0.313047\pi\)
\(380\) 1.57414e10 0.0387273
\(381\) −4.21004e11 −1.02358
\(382\) 1.49630e11 0.359528
\(383\) 2.19414e11 0.521039 0.260519 0.965469i \(-0.416106\pi\)
0.260519 + 0.965469i \(0.416106\pi\)
\(384\) −2.94075e11 −0.690187
\(385\) 2.12710e11 0.493418
\(386\) 1.72157e11 0.394714
\(387\) 5.66536e10 0.128389
\(388\) −1.41584e11 −0.317155
\(389\) −3.71956e11 −0.823603 −0.411801 0.911274i \(-0.635100\pi\)
−0.411801 + 0.911274i \(0.635100\pi\)
\(390\) 1.50445e11 0.329297
\(391\) 2.74291e11 0.593494
\(392\) −2.44462e11 −0.522907
\(393\) 4.28011e10 0.0905082
\(394\) −4.13417e11 −0.864281
\(395\) 2.52737e11 0.522375
\(396\) 5.96900e10 0.121976
\(397\) −1.38786e11 −0.280406 −0.140203 0.990123i \(-0.544776\pi\)
−0.140203 + 0.990123i \(0.544776\pi\)
\(398\) 7.63216e11 1.52466
\(399\) 7.63177e10 0.150747
\(400\) −1.19352e11 −0.233109
\(401\) 6.95127e11 1.34250 0.671251 0.741231i \(-0.265757\pi\)
0.671251 + 0.741231i \(0.265757\pi\)
\(402\) −5.49763e11 −1.04993
\(403\) −6.11178e11 −1.15424
\(404\) 5.68686e10 0.106208
\(405\) −2.69042e10 −0.0496904
\(406\) −4.63253e10 −0.0846157
\(407\) 7.37745e11 1.33270
\(408\) 9.54059e10 0.170453
\(409\) 9.10800e11 1.60942 0.804708 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(410\) −1.40875e11 −0.246210
\(411\) 4.06426e11 0.702577
\(412\) −6.50966e9 −0.0111307
\(413\) −5.99196e11 −1.01343
\(414\) −3.82967e11 −0.640707
\(415\) 3.23146e9 0.00534789
\(416\) 2.92084e11 0.478175
\(417\) 2.85016e11 0.461591
\(418\) 4.96100e11 0.794832
\(419\) 3.11152e11 0.493184 0.246592 0.969119i \(-0.420689\pi\)
0.246592 + 0.969119i \(0.420689\pi\)
\(420\) 2.17663e10 0.0341322
\(421\) 7.63941e11 1.18520 0.592598 0.805498i \(-0.298102\pi\)
0.592598 + 0.805498i \(0.298102\pi\)
\(422\) −9.59516e11 −1.47281
\(423\) 2.17505e11 0.330321
\(424\) −6.45924e11 −0.970588
\(425\) 4.56771e10 0.0679124
\(426\) 7.19414e11 1.05836
\(427\) −6.20159e11 −0.902771
\(428\) −1.08094e11 −0.155706
\(429\) 8.20901e11 1.17013
\(430\) 1.34294e11 0.189429
\(431\) 6.54629e10 0.0913792 0.0456896 0.998956i \(-0.485451\pi\)
0.0456896 + 0.998956i \(0.485451\pi\)
\(432\) −1.62377e11 −0.224309
\(433\) −7.87434e11 −1.07651 −0.538256 0.842781i \(-0.680917\pi\)
−0.538256 + 0.842781i \(0.680917\pi\)
\(434\) −5.10726e11 −0.691010
\(435\) 2.34999e10 0.0314677
\(436\) −1.90210e11 −0.252084
\(437\) −5.51080e11 −0.722850
\(438\) −4.15723e11 −0.539722
\(439\) −8.39211e11 −1.07840 −0.539201 0.842177i \(-0.681274\pi\)
−0.539201 + 0.842177i \(0.681274\pi\)
\(440\) −5.34246e11 −0.679523
\(441\) −1.59232e11 −0.200473
\(442\) −3.47499e11 −0.433065
\(443\) −1.28252e12 −1.58215 −0.791074 0.611720i \(-0.790478\pi\)
−0.791074 + 0.611720i \(0.790478\pi\)
\(444\) 7.54924e10 0.0921891
\(445\) −2.70151e11 −0.326578
\(446\) 7.74777e11 0.927192
\(447\) 3.50581e11 0.415340
\(448\) −3.83313e11 −0.449575
\(449\) 8.02030e10 0.0931284 0.0465642 0.998915i \(-0.485173\pi\)
0.0465642 + 0.998915i \(0.485173\pi\)
\(450\) −6.37746e10 −0.0733148
\(451\) −7.68678e11 −0.874883
\(452\) 1.01363e11 0.114224
\(453\) −4.55084e11 −0.507749
\(454\) 2.85143e11 0.315000
\(455\) 2.99347e11 0.327434
\(456\) −1.91680e11 −0.207604
\(457\) 1.27085e11 0.136292 0.0681461 0.997675i \(-0.478292\pi\)
0.0681461 + 0.997675i \(0.478292\pi\)
\(458\) 7.57853e11 0.804804
\(459\) 6.21433e10 0.0653487
\(460\) −1.57172e11 −0.163668
\(461\) −7.39880e11 −0.762970 −0.381485 0.924375i \(-0.624587\pi\)
−0.381485 + 0.924375i \(0.624587\pi\)
\(462\) 6.85980e11 0.700523
\(463\) 5.71025e11 0.577485 0.288742 0.957407i \(-0.406763\pi\)
0.288742 + 0.957407i \(0.406763\pi\)
\(464\) 1.41831e11 0.142049
\(465\) 2.59082e11 0.256979
\(466\) −7.05782e10 −0.0693320
\(467\) 1.19075e12 1.15850 0.579249 0.815150i \(-0.303346\pi\)
0.579249 + 0.815150i \(0.303346\pi\)
\(468\) 8.40017e10 0.0809434
\(469\) −1.09389e12 −1.04398
\(470\) 5.15580e11 0.487367
\(471\) 1.26038e11 0.118007
\(472\) 1.50495e12 1.39567
\(473\) 7.32770e11 0.673120
\(474\) 8.15065e11 0.741634
\(475\) −9.17701e10 −0.0827142
\(476\) −5.02759e10 −0.0448878
\(477\) −4.20727e11 −0.372107
\(478\) 1.55739e12 1.36449
\(479\) 1.46074e12 1.26784 0.633919 0.773400i \(-0.281445\pi\)
0.633919 + 0.773400i \(0.281445\pi\)
\(480\) −1.23816e11 −0.106461
\(481\) 1.03823e12 0.884382
\(482\) −1.80163e12 −1.52038
\(483\) −7.62003e11 −0.637081
\(484\) 5.19254e11 0.430107
\(485\) 8.25416e11 0.677384
\(486\) −8.67647e10 −0.0705472
\(487\) −1.46931e12 −1.18368 −0.591839 0.806056i \(-0.701598\pi\)
−0.591839 + 0.806056i \(0.701598\pi\)
\(488\) 1.55760e12 1.24327
\(489\) −9.36201e11 −0.740422
\(490\) −3.77449e11 −0.295785
\(491\) −1.76138e12 −1.36769 −0.683843 0.729629i \(-0.739693\pi\)
−0.683843 + 0.729629i \(0.739693\pi\)
\(492\) −7.86577e10 −0.0605199
\(493\) −5.42801e10 −0.0413837
\(494\) 6.98161e11 0.527453
\(495\) −3.47984e11 −0.260517
\(496\) 1.56365e12 1.16004
\(497\) 1.43145e12 1.05238
\(498\) 1.04213e10 0.00759259
\(499\) −1.63016e12 −1.17700 −0.588502 0.808496i \(-0.700282\pi\)
−0.588502 + 0.808496i \(0.700282\pi\)
\(500\) −2.61735e10 −0.0187282
\(501\) −1.21893e12 −0.864391
\(502\) 1.40825e12 0.989718
\(503\) 9.47625e11 0.660056 0.330028 0.943971i \(-0.392942\pi\)
0.330028 + 0.943971i \(0.392942\pi\)
\(504\) −2.65045e11 −0.182971
\(505\) −3.31536e11 −0.226840
\(506\) −4.95337e12 −3.35910
\(507\) 2.96289e11 0.199150
\(508\) −5.57214e11 −0.371224
\(509\) −2.90709e11 −0.191968 −0.0959838 0.995383i \(-0.530600\pi\)
−0.0959838 + 0.995383i \(0.530600\pi\)
\(510\) 1.47306e11 0.0964175
\(511\) −8.27180e11 −0.536668
\(512\) 8.28486e11 0.532808
\(513\) −1.24852e11 −0.0795918
\(514\) −1.23346e12 −0.779457
\(515\) 3.79504e10 0.0237730
\(516\) 7.49833e10 0.0465630
\(517\) 2.81325e12 1.73181
\(518\) 8.67587e11 0.529455
\(519\) 1.80757e12 1.09356
\(520\) −7.51843e11 −0.450933
\(521\) −2.04811e12 −1.21782 −0.608910 0.793240i \(-0.708393\pi\)
−0.608910 + 0.793240i \(0.708393\pi\)
\(522\) 7.57860e10 0.0446758
\(523\) 8.49299e10 0.0496367 0.0248184 0.999692i \(-0.492099\pi\)
0.0248184 + 0.999692i \(0.492099\pi\)
\(524\) 5.66488e10 0.0328247
\(525\) −1.26895e11 −0.0728999
\(526\) −8.94570e11 −0.509540
\(527\) −5.98426e11 −0.337958
\(528\) −2.10021e12 −1.17601
\(529\) 3.70117e12 2.05489
\(530\) −9.97305e11 −0.549018
\(531\) 9.80258e11 0.535075
\(532\) 1.01009e11 0.0546713
\(533\) −1.08176e12 −0.580575
\(534\) −8.71224e11 −0.463654
\(535\) 6.30175e11 0.332559
\(536\) 2.74741e12 1.43775
\(537\) −1.40570e12 −0.729472
\(538\) 1.76588e12 0.908745
\(539\) −2.05954e12 −1.05104
\(540\) −3.56087e10 −0.0180212
\(541\) −1.29101e12 −0.647949 −0.323974 0.946066i \(-0.605019\pi\)
−0.323974 + 0.946066i \(0.605019\pi\)
\(542\) −3.66292e12 −1.82318
\(543\) −1.08572e12 −0.535944
\(544\) 2.85989e11 0.140009
\(545\) 1.10890e12 0.538404
\(546\) 9.65379e11 0.464870
\(547\) −5.30222e11 −0.253230 −0.126615 0.991952i \(-0.540411\pi\)
−0.126615 + 0.991952i \(0.540411\pi\)
\(548\) 5.37921e11 0.254804
\(549\) 1.01455e12 0.476649
\(550\) −8.24874e11 −0.384375
\(551\) 1.09054e11 0.0504035
\(552\) 1.91386e12 0.877371
\(553\) 1.62177e12 0.737438
\(554\) 7.15002e11 0.322488
\(555\) −4.40110e11 −0.196899
\(556\) 3.77230e11 0.167405
\(557\) 8.17386e11 0.359815 0.179907 0.983684i \(-0.442420\pi\)
0.179907 + 0.983684i \(0.442420\pi\)
\(558\) 8.35525e11 0.364842
\(559\) 1.03123e12 0.446684
\(560\) −7.65857e11 −0.329080
\(561\) 8.03773e11 0.342611
\(562\) −1.28667e12 −0.544069
\(563\) −1.42197e12 −0.596488 −0.298244 0.954490i \(-0.596401\pi\)
−0.298244 + 0.954490i \(0.596401\pi\)
\(564\) 2.87876e11 0.119798
\(565\) −5.90934e11 −0.243961
\(566\) 8.02339e10 0.0328613
\(567\) −1.72639e11 −0.0701480
\(568\) −3.59523e12 −1.44930
\(569\) −1.04614e12 −0.418394 −0.209197 0.977874i \(-0.567085\pi\)
−0.209197 + 0.977874i \(0.567085\pi\)
\(570\) −2.95954e11 −0.117432
\(571\) 2.09508e12 0.824780 0.412390 0.911007i \(-0.364694\pi\)
0.412390 + 0.911007i \(0.364694\pi\)
\(572\) 1.08649e12 0.424371
\(573\) −4.87063e11 −0.188751
\(574\) −9.03965e11 −0.347575
\(575\) 9.16290e11 0.349565
\(576\) 6.27082e11 0.237369
\(577\) 2.68760e12 1.00942 0.504711 0.863288i \(-0.331599\pi\)
0.504711 + 0.863288i \(0.331599\pi\)
\(578\) 2.61068e12 0.972921
\(579\) −5.60394e11 −0.207224
\(580\) 3.11031e10 0.0114124
\(581\) 2.07357e10 0.00754962
\(582\) 2.66192e12 0.961705
\(583\) −5.44176e12 −1.95088
\(584\) 2.07755e12 0.739085
\(585\) −4.89718e11 −0.172880
\(586\) 2.82658e12 0.990199
\(587\) 1.61925e12 0.562913 0.281457 0.959574i \(-0.409182\pi\)
0.281457 + 0.959574i \(0.409182\pi\)
\(588\) −2.10750e11 −0.0727058
\(589\) 1.20230e12 0.411618
\(590\) 2.32364e12 0.789467
\(591\) 1.34572e12 0.453745
\(592\) −2.65623e12 −0.888828
\(593\) 5.82677e12 1.93500 0.967502 0.252863i \(-0.0813722\pi\)
0.967502 + 0.252863i \(0.0813722\pi\)
\(594\) −1.12223e12 −0.369865
\(595\) 2.93101e11 0.0958719
\(596\) 4.64007e11 0.150632
\(597\) −2.48436e12 −0.800442
\(598\) −6.97087e12 −2.22911
\(599\) −3.89086e12 −1.23488 −0.617441 0.786617i \(-0.711831\pi\)
−0.617441 + 0.786617i \(0.711831\pi\)
\(600\) 3.18710e11 0.100396
\(601\) 2.13051e12 0.666115 0.333058 0.942907i \(-0.391920\pi\)
0.333058 + 0.942907i \(0.391920\pi\)
\(602\) 8.61737e11 0.267418
\(603\) 1.78955e12 0.551207
\(604\) −6.02321e11 −0.184146
\(605\) −3.02718e12 −0.918628
\(606\) −1.06919e12 −0.322053
\(607\) −2.52733e12 −0.755637 −0.377818 0.925880i \(-0.623326\pi\)
−0.377818 + 0.925880i \(0.623326\pi\)
\(608\) −5.74582e11 −0.170524
\(609\) 1.50794e11 0.0444230
\(610\) 2.40493e12 0.703263
\(611\) 3.95908e12 1.14924
\(612\) 8.22490e10 0.0237000
\(613\) 1.89788e12 0.542870 0.271435 0.962457i \(-0.412502\pi\)
0.271435 + 0.962457i \(0.412502\pi\)
\(614\) 5.73681e11 0.162897
\(615\) 4.58564e11 0.129259
\(616\) −3.42815e12 −0.959283
\(617\) −1.40480e12 −0.390240 −0.195120 0.980779i \(-0.562510\pi\)
−0.195120 + 0.980779i \(0.562510\pi\)
\(618\) 1.22388e11 0.0337514
\(619\) −2.81250e12 −0.769989 −0.384995 0.922919i \(-0.625797\pi\)
−0.384995 + 0.922919i \(0.625797\pi\)
\(620\) 3.42904e11 0.0931988
\(621\) 1.24660e12 0.336369
\(622\) −1.92031e12 −0.514416
\(623\) −1.73351e12 −0.461030
\(624\) −2.95563e12 −0.780404
\(625\) 1.52588e11 0.0400000
\(626\) 1.16664e12 0.303634
\(627\) −1.61486e12 −0.417284
\(628\) 1.66817e11 0.0427977
\(629\) 1.01657e12 0.258945
\(630\) −4.09229e11 −0.103499
\(631\) −1.92653e12 −0.483775 −0.241888 0.970304i \(-0.577767\pi\)
−0.241888 + 0.970304i \(0.577767\pi\)
\(632\) −4.07325e12 −1.01558
\(633\) 3.12334e12 0.773219
\(634\) −5.46001e12 −1.34212
\(635\) 3.24848e12 0.792865
\(636\) −5.56848e11 −0.134952
\(637\) −2.89839e12 −0.697476
\(638\) 9.80232e11 0.234226
\(639\) −2.34178e12 −0.555638
\(640\) 2.26909e12 0.534617
\(641\) −7.10318e12 −1.66185 −0.830924 0.556385i \(-0.812188\pi\)
−0.830924 + 0.556385i \(0.812188\pi\)
\(642\) 2.03228e12 0.472146
\(643\) 3.40733e12 0.786077 0.393038 0.919522i \(-0.371424\pi\)
0.393038 + 0.919522i \(0.371424\pi\)
\(644\) −1.00854e12 −0.231051
\(645\) −4.37142e11 −0.0994498
\(646\) 6.83594e11 0.154437
\(647\) −4.31347e12 −0.967737 −0.483868 0.875141i \(-0.660769\pi\)
−0.483868 + 0.875141i \(0.660769\pi\)
\(648\) 4.33602e11 0.0966059
\(649\) 1.26789e13 2.80530
\(650\) −1.16084e12 −0.255073
\(651\) 1.66248e12 0.362778
\(652\) −1.23910e12 −0.268529
\(653\) −2.53169e12 −0.544880 −0.272440 0.962173i \(-0.587831\pi\)
−0.272440 + 0.962173i \(0.587831\pi\)
\(654\) 3.57615e12 0.764391
\(655\) −3.30255e11 −0.0701074
\(656\) 2.76760e12 0.583494
\(657\) 1.35323e12 0.283352
\(658\) 3.30838e12 0.688016
\(659\) −4.43263e12 −0.915540 −0.457770 0.889071i \(-0.651352\pi\)
−0.457770 + 0.889071i \(0.651352\pi\)
\(660\) −4.60571e11 −0.0944819
\(661\) 9.79827e11 0.199638 0.0998189 0.995006i \(-0.468174\pi\)
0.0998189 + 0.995006i \(0.468174\pi\)
\(662\) −9.68168e10 −0.0195925
\(663\) 1.13115e12 0.227358
\(664\) −5.20799e10 −0.0103971
\(665\) −5.88871e11 −0.116768
\(666\) −1.41933e12 −0.279544
\(667\) −1.08887e12 −0.213014
\(668\) −1.61331e12 −0.313489
\(669\) −2.52199e12 −0.486773
\(670\) 4.24200e12 0.813269
\(671\) 1.31224e13 2.49898
\(672\) −7.94502e11 −0.150291
\(673\) 7.69434e12 1.44578 0.722892 0.690961i \(-0.242812\pi\)
0.722892 + 0.690961i \(0.242812\pi\)
\(674\) 3.68927e12 0.688605
\(675\) 2.07594e11 0.0384900
\(676\) 3.92151e11 0.0722258
\(677\) 2.45111e12 0.448451 0.224225 0.974537i \(-0.428015\pi\)
0.224225 + 0.974537i \(0.428015\pi\)
\(678\) −1.90573e12 −0.346360
\(679\) 5.29653e12 0.956264
\(680\) −7.36157e11 −0.132032
\(681\) −9.28174e11 −0.165374
\(682\) 1.08068e13 1.91280
\(683\) 3.13070e12 0.550488 0.275244 0.961374i \(-0.411241\pi\)
0.275244 + 0.961374i \(0.411241\pi\)
\(684\) −1.65247e11 −0.0288656
\(685\) −3.13601e12 −0.544213
\(686\) −6.44918e12 −1.11185
\(687\) −2.46690e12 −0.422519
\(688\) −2.63831e12 −0.448930
\(689\) −7.65819e12 −1.29461
\(690\) 2.95499e12 0.496289
\(691\) −8.92815e12 −1.48974 −0.744870 0.667210i \(-0.767488\pi\)
−0.744870 + 0.667210i \(0.767488\pi\)
\(692\) 2.39239e12 0.396602
\(693\) −2.23295e12 −0.367772
\(694\) 4.07338e12 0.666556
\(695\) −2.19920e12 −0.357547
\(696\) −3.78737e11 −0.0611781
\(697\) −1.05919e12 −0.169991
\(698\) −1.74699e12 −0.278574
\(699\) 2.29741e11 0.0363991
\(700\) −1.67950e11 −0.0264387
\(701\) −1.44295e12 −0.225695 −0.112847 0.993612i \(-0.535997\pi\)
−0.112847 + 0.993612i \(0.535997\pi\)
\(702\) −1.57932e12 −0.245444
\(703\) −2.04239e12 −0.315383
\(704\) 8.11081e12 1.24448
\(705\) −1.67828e12 −0.255866
\(706\) 3.87509e11 0.0587030
\(707\) −2.12740e12 −0.320230
\(708\) 1.29741e12 0.194056
\(709\) −7.40550e12 −1.10064 −0.550321 0.834953i \(-0.685495\pi\)
−0.550321 + 0.834953i \(0.685495\pi\)
\(710\) −5.55103e12 −0.819806
\(711\) −2.65314e12 −0.389355
\(712\) 4.35390e12 0.634919
\(713\) −1.20045e13 −1.73957
\(714\) 9.45237e11 0.136113
\(715\) −6.33411e12 −0.906377
\(716\) −1.86050e12 −0.264558
\(717\) −5.06949e12 −0.716354
\(718\) −8.17496e11 −0.114796
\(719\) −3.90289e12 −0.544636 −0.272318 0.962207i \(-0.587790\pi\)
−0.272318 + 0.962207i \(0.587790\pi\)
\(720\) 1.25291e12 0.173749
\(721\) 2.43520e11 0.0335604
\(722\) 6.65631e12 0.911624
\(723\) 5.86452e12 0.798197
\(724\) −1.43699e12 −0.194371
\(725\) −1.81327e11 −0.0243748
\(726\) −9.76251e12 −1.30421
\(727\) 1.33674e12 0.177477 0.0887384 0.996055i \(-0.471716\pi\)
0.0887384 + 0.996055i \(0.471716\pi\)
\(728\) −4.82443e12 −0.636583
\(729\) 2.82430e11 0.0370370
\(730\) 3.20774e12 0.418067
\(731\) 1.00971e12 0.130788
\(732\) 1.34280e12 0.172866
\(733\) −5.98863e11 −0.0766231 −0.0383116 0.999266i \(-0.512198\pi\)
−0.0383116 + 0.999266i \(0.512198\pi\)
\(734\) 1.18455e13 1.50633
\(735\) 1.22864e12 0.155286
\(736\) 5.73699e12 0.720665
\(737\) 2.31464e13 2.88987
\(738\) 1.47884e12 0.183514
\(739\) −1.40697e13 −1.73534 −0.867671 0.497138i \(-0.834384\pi\)
−0.867671 + 0.497138i \(0.834384\pi\)
\(740\) −5.82503e11 −0.0714094
\(741\) −2.27260e12 −0.276911
\(742\) −6.39951e12 −0.775049
\(743\) −2.52604e12 −0.304082 −0.152041 0.988374i \(-0.548585\pi\)
−0.152041 + 0.988374i \(0.548585\pi\)
\(744\) −4.17549e12 −0.499608
\(745\) −2.70510e12 −0.321721
\(746\) 2.42056e12 0.286149
\(747\) −3.39226e10 −0.00398608
\(748\) 1.06383e12 0.124255
\(749\) 4.04371e12 0.469474
\(750\) 4.92088e11 0.0567894
\(751\) 3.78704e12 0.434430 0.217215 0.976124i \(-0.430303\pi\)
0.217215 + 0.976124i \(0.430303\pi\)
\(752\) −1.01290e13 −1.15501
\(753\) −4.58401e12 −0.519599
\(754\) 1.37948e12 0.155433
\(755\) 3.51145e12 0.393301
\(756\) −2.28494e11 −0.0254406
\(757\) 3.55646e12 0.393629 0.196814 0.980441i \(-0.436940\pi\)
0.196814 + 0.980441i \(0.436940\pi\)
\(758\) −1.10776e13 −1.21880
\(759\) 1.61238e13 1.76352
\(760\) 1.47902e12 0.160809
\(761\) 1.67100e13 1.80612 0.903058 0.429519i \(-0.141317\pi\)
0.903058 + 0.429519i \(0.141317\pi\)
\(762\) 1.04762e13 1.12566
\(763\) 7.11560e12 0.760066
\(764\) −6.44647e11 −0.0684544
\(765\) −4.79500e11 −0.0506189
\(766\) −5.45987e12 −0.572997
\(767\) 1.78429e13 1.86160
\(768\) 3.35393e12 0.347880
\(769\) −7.51512e12 −0.774939 −0.387469 0.921883i \(-0.626651\pi\)
−0.387469 + 0.921883i \(0.626651\pi\)
\(770\) −5.29306e12 −0.542623
\(771\) 4.01507e12 0.409213
\(772\) −7.41702e11 −0.0751540
\(773\) −8.52580e12 −0.858870 −0.429435 0.903098i \(-0.641287\pi\)
−0.429435 + 0.903098i \(0.641287\pi\)
\(774\) −1.40976e12 −0.141192
\(775\) −1.99909e12 −0.199055
\(776\) −1.33028e13 −1.31694
\(777\) −2.82410e12 −0.277962
\(778\) 9.25569e12 0.905734
\(779\) 2.12802e12 0.207042
\(780\) −6.48161e11 −0.0626985
\(781\) −3.02890e13 −2.91310
\(782\) −6.82543e12 −0.652679
\(783\) −2.46693e11 −0.0234546
\(784\) 7.41531e12 0.700982
\(785\) −9.72518e11 −0.0914080
\(786\) −1.06506e12 −0.0995338
\(787\) 3.20742e12 0.298037 0.149018 0.988834i \(-0.452389\pi\)
0.149018 + 0.988834i \(0.452389\pi\)
\(788\) 1.78112e12 0.164560
\(789\) 2.91194e12 0.267507
\(790\) −6.28908e12 −0.574467
\(791\) −3.79191e12 −0.344401
\(792\) 5.60830e12 0.506486
\(793\) 1.84672e13 1.65833
\(794\) 3.45353e12 0.308369
\(795\) 3.24635e12 0.288233
\(796\) −3.28815e12 −0.290297
\(797\) −1.83036e13 −1.60684 −0.803421 0.595412i \(-0.796989\pi\)
−0.803421 + 0.595412i \(0.796989\pi\)
\(798\) −1.89908e12 −0.165779
\(799\) 3.87648e12 0.336494
\(800\) 9.55368e11 0.0824643
\(801\) 2.83594e12 0.243417
\(802\) −1.72975e13 −1.47638
\(803\) 1.75029e13 1.48556
\(804\) 2.36853e12 0.199907
\(805\) 5.87966e12 0.493481
\(806\) 1.52085e13 1.26934
\(807\) −5.74817e12 −0.477088
\(808\) 5.34321e12 0.441013
\(809\) 9.38919e12 0.770655 0.385327 0.922780i \(-0.374089\pi\)
0.385327 + 0.922780i \(0.374089\pi\)
\(810\) 6.69480e11 0.0546456
\(811\) 2.03949e13 1.65549 0.827746 0.561103i \(-0.189623\pi\)
0.827746 + 0.561103i \(0.189623\pi\)
\(812\) 1.99582e11 0.0161109
\(813\) 1.19232e13 0.957166
\(814\) −1.83579e13 −1.46560
\(815\) 7.22377e12 0.573528
\(816\) −2.89396e12 −0.228500
\(817\) −2.02861e12 −0.159294
\(818\) −2.26642e13 −1.76991
\(819\) −3.14243e12 −0.244055
\(820\) 6.06927e11 0.0468785
\(821\) −1.52259e13 −1.16961 −0.584803 0.811176i \(-0.698828\pi\)
−0.584803 + 0.811176i \(0.698828\pi\)
\(822\) −1.01135e13 −0.772639
\(823\) −2.70705e12 −0.205682 −0.102841 0.994698i \(-0.532793\pi\)
−0.102841 + 0.994698i \(0.532793\pi\)
\(824\) −6.11629e11 −0.0462184
\(825\) 2.68506e12 0.201796
\(826\) 1.49103e13 1.11449
\(827\) 5.67203e10 0.00421661 0.00210831 0.999998i \(-0.499329\pi\)
0.00210831 + 0.999998i \(0.499329\pi\)
\(828\) 1.64993e12 0.121991
\(829\) −1.43206e13 −1.05309 −0.526547 0.850146i \(-0.676513\pi\)
−0.526547 + 0.850146i \(0.676513\pi\)
\(830\) −8.04113e10 −0.00588119
\(831\) −2.32742e12 −0.169305
\(832\) 1.14143e13 0.825839
\(833\) −2.83791e12 −0.204219
\(834\) −7.09231e12 −0.507622
\(835\) 9.40535e12 0.669555
\(836\) −2.13734e12 −0.151337
\(837\) −2.71973e12 −0.191541
\(838\) −7.74265e12 −0.542365
\(839\) −2.13693e13 −1.48889 −0.744443 0.667686i \(-0.767285\pi\)
−0.744443 + 0.667686i \(0.767285\pi\)
\(840\) 2.04510e12 0.141729
\(841\) −1.42917e13 −0.985147
\(842\) −1.90098e13 −1.30339
\(843\) 4.18827e12 0.285635
\(844\) 4.13386e12 0.280424
\(845\) −2.28618e12 −0.154261
\(846\) −5.41235e12 −0.363262
\(847\) −1.94248e13 −1.29683
\(848\) 1.95929e13 1.30112
\(849\) −2.61171e11 −0.0172521
\(850\) −1.13662e12 −0.0746847
\(851\) 2.03925e13 1.33287
\(852\) −3.09943e12 −0.201514
\(853\) −1.18123e13 −0.763948 −0.381974 0.924173i \(-0.624756\pi\)
−0.381974 + 0.924173i \(0.624756\pi\)
\(854\) 1.54319e13 0.992797
\(855\) 9.63366e11 0.0616515
\(856\) −1.01562e13 −0.646547
\(857\) 2.69459e13 1.70640 0.853198 0.521587i \(-0.174660\pi\)
0.853198 + 0.521587i \(0.174660\pi\)
\(858\) −2.04272e13 −1.28681
\(859\) 9.66669e12 0.605771 0.302885 0.953027i \(-0.402050\pi\)
0.302885 + 0.953027i \(0.402050\pi\)
\(860\) −5.78575e11 −0.0360675
\(861\) 2.94251e12 0.182475
\(862\) −1.62897e12 −0.100492
\(863\) 2.80982e12 0.172437 0.0862185 0.996276i \(-0.472522\pi\)
0.0862185 + 0.996276i \(0.472522\pi\)
\(864\) 1.29977e12 0.0793513
\(865\) −1.39473e13 −0.847069
\(866\) 1.95944e13 1.18386
\(867\) −8.49807e12 −0.510781
\(868\) 2.20035e12 0.131569
\(869\) −3.43162e13 −2.04132
\(870\) −5.84769e11 −0.0346057
\(871\) 3.25739e13 1.91773
\(872\) −1.78716e13 −1.04674
\(873\) −8.66488e12 −0.504892
\(874\) 1.37130e13 0.794933
\(875\) 9.79127e11 0.0564680
\(876\) 1.79105e12 0.102764
\(877\) 3.37045e13 1.92393 0.961966 0.273169i \(-0.0880718\pi\)
0.961966 + 0.273169i \(0.0880718\pi\)
\(878\) 2.08828e13 1.18594
\(879\) −9.20087e12 −0.519851
\(880\) 1.62054e13 0.910933
\(881\) −1.20571e13 −0.674297 −0.337148 0.941452i \(-0.609462\pi\)
−0.337148 + 0.941452i \(0.609462\pi\)
\(882\) 3.96231e12 0.220465
\(883\) −1.91595e13 −1.06062 −0.530311 0.847803i \(-0.677925\pi\)
−0.530311 + 0.847803i \(0.677925\pi\)
\(884\) 1.49712e12 0.0824559
\(885\) −7.56372e12 −0.414468
\(886\) 3.19140e13 1.73992
\(887\) −2.24991e13 −1.22042 −0.610209 0.792241i \(-0.708914\pi\)
−0.610209 + 0.792241i \(0.708914\pi\)
\(888\) 7.09305e12 0.382802
\(889\) 2.08449e13 1.11929
\(890\) 6.72241e12 0.359145
\(891\) 3.65300e12 0.194178
\(892\) −3.33795e12 −0.176538
\(893\) −7.78825e12 −0.409834
\(894\) −8.72380e12 −0.456759
\(895\) 1.08465e13 0.565047
\(896\) 1.45603e13 0.754719
\(897\) 2.26910e13 1.17028
\(898\) −1.99576e12 −0.102415
\(899\) 2.37560e12 0.121298
\(900\) 2.74759e11 0.0139592
\(901\) −7.49841e12 −0.379060
\(902\) 1.91277e13 0.962128
\(903\) −2.80506e12 −0.140393
\(904\) 9.52380e12 0.474299
\(905\) 8.37748e12 0.415140
\(906\) 1.13242e13 0.558383
\(907\) −2.39724e13 −1.17619 −0.588096 0.808791i \(-0.700122\pi\)
−0.588096 + 0.808791i \(0.700122\pi\)
\(908\) −1.22847e12 −0.0599763
\(909\) 3.48033e12 0.169077
\(910\) −7.44891e12 −0.360086
\(911\) −6.54470e12 −0.314816 −0.157408 0.987534i \(-0.550314\pi\)
−0.157408 + 0.987534i \(0.550314\pi\)
\(912\) 5.81427e12 0.278303
\(913\) −4.38762e11 −0.0208983
\(914\) −3.16236e12 −0.149884
\(915\) −7.82833e12 −0.369211
\(916\) −3.26504e12 −0.153235
\(917\) −2.11918e12 −0.0989706
\(918\) −1.54636e12 −0.0718654
\(919\) 2.63348e12 0.121789 0.0608947 0.998144i \(-0.480605\pi\)
0.0608947 + 0.998144i \(0.480605\pi\)
\(920\) −1.47674e13 −0.679609
\(921\) −1.86740e12 −0.0855203
\(922\) 1.84111e13 0.839055
\(923\) −4.26258e13 −1.93314
\(924\) −2.95539e12 −0.133380
\(925\) 3.39591e12 0.152517
\(926\) −1.42093e13 −0.635073
\(927\) −3.98388e11 −0.0177193
\(928\) −1.13530e12 −0.0502512
\(929\) 2.94271e13 1.29621 0.648106 0.761550i \(-0.275561\pi\)
0.648106 + 0.761550i \(0.275561\pi\)
\(930\) −6.44695e12 −0.282606
\(931\) 5.70166e12 0.248730
\(932\) 3.04070e11 0.0132009
\(933\) 6.25084e12 0.270067
\(934\) −2.96305e13 −1.27403
\(935\) −6.20196e12 −0.265385
\(936\) 7.89255e12 0.336106
\(937\) −1.71769e13 −0.727974 −0.363987 0.931404i \(-0.618585\pi\)
−0.363987 + 0.931404i \(0.618585\pi\)
\(938\) 2.72201e13 1.14809
\(939\) −3.79754e12 −0.159407
\(940\) −2.22126e12 −0.0927950
\(941\) −2.46944e13 −1.02670 −0.513352 0.858178i \(-0.671596\pi\)
−0.513352 + 0.858178i \(0.671596\pi\)
\(942\) −3.13632e12 −0.129775
\(943\) −2.12475e13 −0.874994
\(944\) −4.56498e13 −1.87096
\(945\) 1.33209e12 0.0543364
\(946\) −1.82341e13 −0.740244
\(947\) −1.19428e12 −0.0482537 −0.0241268 0.999709i \(-0.507681\pi\)
−0.0241268 + 0.999709i \(0.507681\pi\)
\(948\) −3.51153e12 −0.141208
\(949\) 2.46319e13 0.985824
\(950\) 2.28360e12 0.0909626
\(951\) 1.77730e13 0.704610
\(952\) −4.72377e12 −0.186390
\(953\) 1.47694e13 0.580023 0.290011 0.957023i \(-0.406341\pi\)
0.290011 + 0.957023i \(0.406341\pi\)
\(954\) 1.04693e13 0.409214
\(955\) 3.75820e12 0.146206
\(956\) −6.70966e12 −0.259800
\(957\) −3.19077e12 −0.122968
\(958\) −3.63489e13 −1.39427
\(959\) −2.01231e13 −0.768267
\(960\) −4.83860e12 −0.183865
\(961\) −2.49176e11 −0.00942435
\(962\) −2.58351e13 −0.972574
\(963\) −6.61532e12 −0.247875
\(964\) 7.76192e12 0.289483
\(965\) 4.32402e12 0.160515
\(966\) 1.89616e13 0.700612
\(967\) −8.95584e12 −0.329372 −0.164686 0.986346i \(-0.552661\pi\)
−0.164686 + 0.986346i \(0.552661\pi\)
\(968\) 4.87876e13 1.78596
\(969\) −2.22518e12 −0.0810790
\(970\) −2.05395e13 −0.744934
\(971\) −2.20587e13 −0.796330 −0.398165 0.917314i \(-0.630353\pi\)
−0.398165 + 0.917314i \(0.630353\pi\)
\(972\) 3.73806e11 0.0134322
\(973\) −1.41118e13 −0.504749
\(974\) 3.65621e13 1.30172
\(975\) 3.77869e12 0.133912
\(976\) −4.72469e13 −1.66667
\(977\) 3.56972e13 1.25345 0.626727 0.779239i \(-0.284394\pi\)
0.626727 + 0.779239i \(0.284394\pi\)
\(978\) 2.32963e13 0.814258
\(979\) 3.66806e13 1.27619
\(980\) 1.62615e12 0.0563176
\(981\) −1.16408e13 −0.401303
\(982\) 4.38300e13 1.50408
\(983\) 3.19525e13 1.09147 0.545737 0.837956i \(-0.316250\pi\)
0.545737 + 0.837956i \(0.316250\pi\)
\(984\) −7.39045e12 −0.251300
\(985\) −1.03837e13 −0.351469
\(986\) 1.35070e12 0.0455105
\(987\) −1.07692e13 −0.361206
\(988\) −3.00787e12 −0.100428
\(989\) 2.02549e13 0.673205
\(990\) 8.65920e12 0.286496
\(991\) 1.56883e13 0.516706 0.258353 0.966051i \(-0.416820\pi\)
0.258353 + 0.966051i \(0.416820\pi\)
\(992\) −1.25165e13 −0.410374
\(993\) 3.15151e11 0.0102860
\(994\) −3.56199e13 −1.15732
\(995\) 1.91694e13 0.620020
\(996\) −4.48979e10 −0.00144563
\(997\) 6.52145e12 0.209033 0.104517 0.994523i \(-0.466670\pi\)
0.104517 + 0.994523i \(0.466670\pi\)
\(998\) 4.05647e13 1.29438
\(999\) 4.62010e12 0.146760
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 15.10.a.c.1.1 2
3.2 odd 2 45.10.a.e.1.2 2
4.3 odd 2 240.10.a.m.1.1 2
5.2 odd 4 75.10.b.e.49.2 4
5.3 odd 4 75.10.b.e.49.3 4
5.4 even 2 75.10.a.g.1.2 2
15.2 even 4 225.10.b.g.199.3 4
15.8 even 4 225.10.b.g.199.2 4
15.14 odd 2 225.10.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.1 2 1.1 even 1 trivial
45.10.a.e.1.2 2 3.2 odd 2
75.10.a.g.1.2 2 5.4 even 2
75.10.b.e.49.2 4 5.2 odd 4
75.10.b.e.49.3 4 5.3 odd 4
225.10.a.j.1.1 2 15.14 odd 2
225.10.b.g.199.2 4 15.8 even 4
225.10.b.g.199.3 4 15.2 even 4
240.10.a.m.1.1 2 4.3 odd 2