Properties

Label 43.10.a.a.1.6
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 1.6
Root \(15.9205\) 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

\(f(q)\) \(=\) \(q-17.9205 q^{2} -172.986 q^{3} -190.855 q^{4} +238.484 q^{5} +3099.99 q^{6} -4524.82 q^{7} +12595.5 q^{8} +10241.0 q^{9} +O(q^{10})\) \(q-17.9205 q^{2} -172.986 q^{3} -190.855 q^{4} +238.484 q^{5} +3099.99 q^{6} -4524.82 q^{7} +12595.5 q^{8} +10241.0 q^{9} -4273.75 q^{10} +66772.1 q^{11} +33015.1 q^{12} +45566.0 q^{13} +81087.1 q^{14} -41254.2 q^{15} -128001. q^{16} +494574. q^{17} -183524. q^{18} -1.08372e6 q^{19} -45515.8 q^{20} +782728. q^{21} -1.19659e6 q^{22} +1.35193e6 q^{23} -2.17884e6 q^{24} -1.89625e6 q^{25} -816566. q^{26} +1.63333e6 q^{27} +863584. q^{28} -4.85463e6 q^{29} +739298. q^{30} +5.69784e6 q^{31} -4.15507e6 q^{32} -1.15506e7 q^{33} -8.86302e6 q^{34} -1.07910e6 q^{35} -1.95454e6 q^{36} -1.59658e7 q^{37} +1.94208e7 q^{38} -7.88225e6 q^{39} +3.00383e6 q^{40} +1.10007e7 q^{41} -1.40269e7 q^{42} -3.41880e6 q^{43} -1.27438e7 q^{44} +2.44231e6 q^{45} -2.42274e7 q^{46} +2.98409e7 q^{47} +2.21423e7 q^{48} -1.98796e7 q^{49} +3.39818e7 q^{50} -8.55541e7 q^{51} -8.69648e6 q^{52} +7.43943e6 q^{53} -2.92702e7 q^{54} +1.59241e7 q^{55} -5.69925e7 q^{56} +1.87467e8 q^{57} +8.69975e7 q^{58} +1.08119e8 q^{59} +7.87357e6 q^{60} +1.99515e8 q^{61} -1.02108e8 q^{62} -4.63386e7 q^{63} +1.39997e8 q^{64} +1.08667e7 q^{65} +2.06993e8 q^{66} -2.61727e8 q^{67} -9.43918e7 q^{68} -2.33865e8 q^{69} +1.93380e7 q^{70} +6.58887e7 q^{71} +1.28991e8 q^{72} -4.01146e8 q^{73} +2.86115e8 q^{74} +3.28024e8 q^{75} +2.06832e8 q^{76} -3.02132e8 q^{77} +1.41254e8 q^{78} +1.23252e8 q^{79} -3.05261e7 q^{80} -4.84116e8 q^{81} -1.97139e8 q^{82} -3.81346e8 q^{83} -1.49388e8 q^{84} +1.17948e8 q^{85} +6.12667e7 q^{86} +8.39780e8 q^{87} +8.41030e8 q^{88} -3.81633e8 q^{89} -4.37675e7 q^{90} -2.06178e8 q^{91} -2.58023e8 q^{92} -9.85643e8 q^{93} -5.34765e8 q^{94} -2.58449e8 q^{95} +7.18767e8 q^{96} -5.93343e8 q^{97} +3.56253e8 q^{98} +6.83813e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.9205 −0.791983 −0.395991 0.918254i \(-0.629599\pi\)
−0.395991 + 0.918254i \(0.629599\pi\)
\(3\) −172.986 −1.23300 −0.616501 0.787354i \(-0.711451\pi\)
−0.616501 + 0.787354i \(0.711451\pi\)
\(4\) −190.855 −0.372763
\(5\) 238.484 0.170645 0.0853226 0.996353i \(-0.472808\pi\)
0.0853226 + 0.996353i \(0.472808\pi\)
\(6\) 3099.99 0.976517
\(7\) −4524.82 −0.712295 −0.356148 0.934430i \(-0.615910\pi\)
−0.356148 + 0.934430i \(0.615910\pi\)
\(8\) 12595.5 1.08720
\(9\) 10241.0 0.520296
\(10\) −4273.75 −0.135148
\(11\) 66772.1 1.37508 0.687540 0.726146i \(-0.258690\pi\)
0.687540 + 0.726146i \(0.258690\pi\)
\(12\) 33015.1 0.459618
\(13\) 45566.0 0.442482 0.221241 0.975219i \(-0.428989\pi\)
0.221241 + 0.975219i \(0.428989\pi\)
\(14\) 81087.1 0.564126
\(15\) −41254.2 −0.210406
\(16\) −128001. −0.488284
\(17\) 494574. 1.43619 0.718094 0.695947i \(-0.245015\pi\)
0.718094 + 0.695947i \(0.245015\pi\)
\(18\) −183524. −0.412065
\(19\) −1.08372e6 −1.90776 −0.953881 0.300184i \(-0.902952\pi\)
−0.953881 + 0.300184i \(0.902952\pi\)
\(20\) −45515.8 −0.0636103
\(21\) 782728. 0.878262
\(22\) −1.19659e6 −1.08904
\(23\) 1.35193e6 1.00735 0.503675 0.863893i \(-0.331981\pi\)
0.503675 + 0.863893i \(0.331981\pi\)
\(24\) −2.17884e6 −1.34053
\(25\) −1.89625e6 −0.970880
\(26\) −816566. −0.350438
\(27\) 1.63333e6 0.591476
\(28\) 863584. 0.265518
\(29\) −4.85463e6 −1.27457 −0.637287 0.770627i \(-0.719943\pi\)
−0.637287 + 0.770627i \(0.719943\pi\)
\(30\) 739298. 0.166638
\(31\) 5.69784e6 1.10811 0.554054 0.832481i \(-0.313080\pi\)
0.554054 + 0.832481i \(0.313080\pi\)
\(32\) −4.15507e6 −0.700492
\(33\) −1.15506e7 −1.69548
\(34\) −8.86302e6 −1.13744
\(35\) −1.07910e6 −0.121550
\(36\) −1.95454e6 −0.193947
\(37\) −1.59658e7 −1.40050 −0.700249 0.713898i \(-0.746928\pi\)
−0.700249 + 0.713898i \(0.746928\pi\)
\(38\) 1.94208e7 1.51091
\(39\) −7.88225e6 −0.545581
\(40\) 3.00383e6 0.185526
\(41\) 1.10007e7 0.607988 0.303994 0.952674i \(-0.401680\pi\)
0.303994 + 0.952674i \(0.401680\pi\)
\(42\) −1.40269e7 −0.695568
\(43\) −3.41880e6 −0.152499
\(44\) −1.27438e7 −0.512580
\(45\) 2.44231e6 0.0887860
\(46\) −2.42274e7 −0.797803
\(47\) 2.98409e7 0.892014 0.446007 0.895029i \(-0.352846\pi\)
0.446007 + 0.895029i \(0.352846\pi\)
\(48\) 2.21423e7 0.602056
\(49\) −1.98796e7 −0.492635
\(50\) 3.39818e7 0.768920
\(51\) −8.55541e7 −1.77082
\(52\) −8.69648e6 −0.164941
\(53\) 7.43943e6 0.129509 0.0647543 0.997901i \(-0.479374\pi\)
0.0647543 + 0.997901i \(0.479374\pi\)
\(54\) −2.92702e7 −0.468439
\(55\) 1.59241e7 0.234651
\(56\) −5.69925e7 −0.774411
\(57\) 1.87467e8 2.35228
\(58\) 8.69975e7 1.00944
\(59\) 1.08119e8 1.16163 0.580815 0.814036i \(-0.302734\pi\)
0.580815 + 0.814036i \(0.302734\pi\)
\(60\) 7.87357e6 0.0784316
\(61\) 1.99515e8 1.84497 0.922487 0.386028i \(-0.126153\pi\)
0.922487 + 0.386028i \(0.126153\pi\)
\(62\) −1.02108e8 −0.877603
\(63\) −4.63386e7 −0.370604
\(64\) 1.39997e8 1.04306
\(65\) 1.08667e7 0.0755073
\(66\) 2.06993e8 1.34279
\(67\) −2.61727e8 −1.58677 −0.793383 0.608723i \(-0.791682\pi\)
−0.793383 + 0.608723i \(0.791682\pi\)
\(68\) −9.43918e7 −0.535358
\(69\) −2.33865e8 −1.24206
\(70\) 1.93380e7 0.0962653
\(71\) 6.58887e7 0.307715 0.153857 0.988093i \(-0.450830\pi\)
0.153857 + 0.988093i \(0.450830\pi\)
\(72\) 1.28991e8 0.565668
\(73\) −4.01146e8 −1.65329 −0.826646 0.562722i \(-0.809754\pi\)
−0.826646 + 0.562722i \(0.809754\pi\)
\(74\) 2.86115e8 1.10917
\(75\) 3.28024e8 1.19710
\(76\) 2.06832e8 0.711144
\(77\) −3.02132e8 −0.979464
\(78\) 1.41254e8 0.432091
\(79\) 1.23252e8 0.356019 0.178009 0.984029i \(-0.443034\pi\)
0.178009 + 0.984029i \(0.443034\pi\)
\(80\) −3.05261e7 −0.0833233
\(81\) −4.84116e8 −1.24959
\(82\) −1.97139e8 −0.481516
\(83\) −3.81346e8 −0.881998 −0.440999 0.897507i \(-0.645376\pi\)
−0.440999 + 0.897507i \(0.645376\pi\)
\(84\) −1.49388e8 −0.327384
\(85\) 1.17948e8 0.245078
\(86\) 6.12667e7 0.120776
\(87\) 8.39780e8 1.57155
\(88\) 8.41030e8 1.49499
\(89\) −3.81633e8 −0.644749 −0.322375 0.946612i \(-0.604481\pi\)
−0.322375 + 0.946612i \(0.604481\pi\)
\(90\) −4.37675e7 −0.0703169
\(91\) −2.06178e8 −0.315178
\(92\) −2.58023e8 −0.375503
\(93\) −9.85643e8 −1.36630
\(94\) −5.34765e8 −0.706460
\(95\) −2.58449e8 −0.325550
\(96\) 7.18767e8 0.863709
\(97\) −5.93343e8 −0.680507 −0.340254 0.940334i \(-0.610513\pi\)
−0.340254 + 0.940334i \(0.610513\pi\)
\(98\) 3.56253e8 0.390159
\(99\) 6.83813e8 0.715449
\(100\) 3.61909e8 0.361909
\(101\) 1.22816e8 0.117438 0.0587190 0.998275i \(-0.481298\pi\)
0.0587190 + 0.998275i \(0.481298\pi\)
\(102\) 1.53317e9 1.40246
\(103\) −1.45390e9 −1.27282 −0.636408 0.771353i \(-0.719580\pi\)
−0.636408 + 0.771353i \(0.719580\pi\)
\(104\) 5.73927e8 0.481068
\(105\) 1.86668e8 0.149871
\(106\) −1.33319e8 −0.102569
\(107\) −1.05553e9 −0.778475 −0.389237 0.921138i \(-0.627261\pi\)
−0.389237 + 0.921138i \(0.627261\pi\)
\(108\) −3.11729e8 −0.220481
\(109\) −1.48849e9 −1.01001 −0.505006 0.863116i \(-0.668510\pi\)
−0.505006 + 0.863116i \(0.668510\pi\)
\(110\) −2.85368e8 −0.185839
\(111\) 2.76185e9 1.72682
\(112\) 5.79180e8 0.347802
\(113\) −2.40872e9 −1.38974 −0.694869 0.719137i \(-0.744538\pi\)
−0.694869 + 0.719137i \(0.744538\pi\)
\(114\) −3.35951e9 −1.86296
\(115\) 3.22414e8 0.171899
\(116\) 9.26530e8 0.475115
\(117\) 4.66640e8 0.230221
\(118\) −1.93755e9 −0.919991
\(119\) −2.23786e9 −1.02299
\(120\) −5.19619e8 −0.228754
\(121\) 2.10057e9 0.890847
\(122\) −3.57541e9 −1.46119
\(123\) −1.90297e9 −0.749651
\(124\) −1.08746e9 −0.413062
\(125\) −9.18014e8 −0.336321
\(126\) 8.30412e8 0.293512
\(127\) 1.63138e9 0.556464 0.278232 0.960514i \(-0.410252\pi\)
0.278232 + 0.960514i \(0.410252\pi\)
\(128\) −3.81431e8 −0.125595
\(129\) 5.91403e8 0.188031
\(130\) −1.94738e8 −0.0598005
\(131\) −2.34159e9 −0.694690 −0.347345 0.937737i \(-0.612917\pi\)
−0.347345 + 0.937737i \(0.612917\pi\)
\(132\) 2.20449e9 0.632012
\(133\) 4.90362e9 1.35889
\(134\) 4.69029e9 1.25669
\(135\) 3.89523e8 0.100933
\(136\) 6.22942e9 1.56143
\(137\) 4.87195e9 1.18157 0.590786 0.806828i \(-0.298818\pi\)
0.590786 + 0.806828i \(0.298818\pi\)
\(138\) 4.19098e9 0.983693
\(139\) 3.89560e9 0.885131 0.442566 0.896736i \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(140\) 2.05951e8 0.0453093
\(141\) −5.16205e9 −1.09986
\(142\) −1.18076e9 −0.243705
\(143\) 3.04254e9 0.608448
\(144\) −1.31085e9 −0.254052
\(145\) −1.15775e9 −0.217500
\(146\) 7.18875e9 1.30938
\(147\) 3.43888e9 0.607421
\(148\) 3.04715e9 0.522055
\(149\) 3.97719e9 0.661056 0.330528 0.943796i \(-0.392773\pi\)
0.330528 + 0.943796i \(0.392773\pi\)
\(150\) −5.87836e9 −0.948081
\(151\) 1.04756e9 0.163977 0.0819886 0.996633i \(-0.473873\pi\)
0.0819886 + 0.996633i \(0.473873\pi\)
\(152\) −1.36500e10 −2.07413
\(153\) 5.06492e9 0.747242
\(154\) 5.41436e9 0.775718
\(155\) 1.35884e9 0.189093
\(156\) 1.50437e9 0.203373
\(157\) −6.06273e9 −0.796379 −0.398189 0.917303i \(-0.630361\pi\)
−0.398189 + 0.917303i \(0.630361\pi\)
\(158\) −2.20874e9 −0.281961
\(159\) −1.28691e9 −0.159684
\(160\) −9.90917e8 −0.119536
\(161\) −6.11726e9 −0.717530
\(162\) 8.67561e9 0.989652
\(163\) 4.36837e9 0.484702 0.242351 0.970189i \(-0.422081\pi\)
0.242351 + 0.970189i \(0.422081\pi\)
\(164\) −2.09955e9 −0.226636
\(165\) −2.75463e9 −0.289325
\(166\) 6.83392e9 0.698528
\(167\) −9.06833e9 −0.902200 −0.451100 0.892473i \(-0.648968\pi\)
−0.451100 + 0.892473i \(0.648968\pi\)
\(168\) 9.85888e9 0.954851
\(169\) −8.52824e9 −0.804210
\(170\) −2.11369e9 −0.194098
\(171\) −1.10983e10 −0.992601
\(172\) 6.52495e8 0.0568459
\(173\) −1.41700e10 −1.20272 −0.601359 0.798979i \(-0.705374\pi\)
−0.601359 + 0.798979i \(0.705374\pi\)
\(174\) −1.50493e10 −1.24464
\(175\) 8.58019e9 0.691554
\(176\) −8.54688e9 −0.671430
\(177\) −1.87030e10 −1.43229
\(178\) 6.83906e9 0.510630
\(179\) 2.05552e10 1.49652 0.748260 0.663405i \(-0.230889\pi\)
0.748260 + 0.663405i \(0.230889\pi\)
\(180\) −4.66127e8 −0.0330962
\(181\) −1.98258e10 −1.37302 −0.686511 0.727119i \(-0.740859\pi\)
−0.686511 + 0.727119i \(0.740859\pi\)
\(182\) 3.69481e9 0.249615
\(183\) −3.45131e10 −2.27486
\(184\) 1.70283e10 1.09519
\(185\) −3.80758e9 −0.238988
\(186\) 1.76632e10 1.08209
\(187\) 3.30238e10 1.97487
\(188\) −5.69528e9 −0.332510
\(189\) −7.39053e9 −0.421306
\(190\) 4.63154e9 0.257830
\(191\) −1.93769e10 −1.05350 −0.526750 0.850020i \(-0.676590\pi\)
−0.526750 + 0.850020i \(0.676590\pi\)
\(192\) −2.42175e10 −1.28610
\(193\) 2.44314e10 1.26748 0.633740 0.773546i \(-0.281519\pi\)
0.633740 + 0.773546i \(0.281519\pi\)
\(194\) 1.06330e10 0.538950
\(195\) −1.87979e9 −0.0931008
\(196\) 3.79412e9 0.183636
\(197\) −1.22473e10 −0.579354 −0.289677 0.957124i \(-0.593548\pi\)
−0.289677 + 0.957124i \(0.593548\pi\)
\(198\) −1.22543e10 −0.566623
\(199\) −2.12982e10 −0.962731 −0.481366 0.876520i \(-0.659859\pi\)
−0.481366 + 0.876520i \(0.659859\pi\)
\(200\) −2.38843e10 −1.05555
\(201\) 4.52751e10 1.95649
\(202\) −2.20092e9 −0.0930088
\(203\) 2.19663e10 0.907873
\(204\) 1.63284e10 0.660098
\(205\) 2.62350e9 0.103750
\(206\) 2.60546e10 1.00805
\(207\) 1.38451e10 0.524120
\(208\) −5.83247e9 −0.216057
\(209\) −7.23620e10 −2.62333
\(210\) −3.34519e9 −0.118695
\(211\) −3.57352e10 −1.24115 −0.620576 0.784146i \(-0.713101\pi\)
−0.620576 + 0.784146i \(0.713101\pi\)
\(212\) −1.41985e9 −0.0482761
\(213\) −1.13978e10 −0.379413
\(214\) 1.89157e10 0.616538
\(215\) −8.15329e8 −0.0260231
\(216\) 2.05727e10 0.643056
\(217\) −2.57817e10 −0.789301
\(218\) 2.66745e10 0.799912
\(219\) 6.93925e10 2.03851
\(220\) −3.03919e9 −0.0874692
\(221\) 2.25357e10 0.635487
\(222\) −4.94938e10 −1.36761
\(223\) 1.10106e10 0.298153 0.149076 0.988826i \(-0.452370\pi\)
0.149076 + 0.988826i \(0.452370\pi\)
\(224\) 1.88009e10 0.498958
\(225\) −1.94195e10 −0.505145
\(226\) 4.31655e10 1.10065
\(227\) 1.50858e10 0.377096 0.188548 0.982064i \(-0.439622\pi\)
0.188548 + 0.982064i \(0.439622\pi\)
\(228\) −3.57790e10 −0.876843
\(229\) 1.14580e10 0.275327 0.137664 0.990479i \(-0.456041\pi\)
0.137664 + 0.990479i \(0.456041\pi\)
\(230\) −5.77783e9 −0.136141
\(231\) 5.22644e10 1.20768
\(232\) −6.11466e10 −1.38572
\(233\) −1.02658e10 −0.228187 −0.114094 0.993470i \(-0.536396\pi\)
−0.114094 + 0.993470i \(0.536396\pi\)
\(234\) −8.36244e9 −0.182331
\(235\) 7.11658e9 0.152218
\(236\) −2.06350e10 −0.433013
\(237\) −2.13208e10 −0.438972
\(238\) 4.01036e10 0.810190
\(239\) 3.71284e10 0.736064 0.368032 0.929813i \(-0.380032\pi\)
0.368032 + 0.929813i \(0.380032\pi\)
\(240\) 5.28057e9 0.102738
\(241\) 4.83626e10 0.923492 0.461746 0.887012i \(-0.347223\pi\)
0.461746 + 0.887012i \(0.347223\pi\)
\(242\) −3.76433e10 −0.705536
\(243\) 5.15962e10 0.949269
\(244\) −3.80783e10 −0.687739
\(245\) −4.74096e9 −0.0840658
\(246\) 3.41022e10 0.593710
\(247\) −4.93806e10 −0.844150
\(248\) 7.17672e10 1.20474
\(249\) 6.59673e10 1.08751
\(250\) 1.64513e10 0.266360
\(251\) 3.34140e10 0.531369 0.265684 0.964060i \(-0.414402\pi\)
0.265684 + 0.964060i \(0.414402\pi\)
\(252\) 8.84395e9 0.138148
\(253\) 9.02715e10 1.38519
\(254\) −2.92351e10 −0.440710
\(255\) −2.04033e10 −0.302182
\(256\) −6.48432e10 −0.943593
\(257\) −8.20412e10 −1.17309 −0.586547 0.809915i \(-0.699513\pi\)
−0.586547 + 0.809915i \(0.699513\pi\)
\(258\) −1.05983e10 −0.148917
\(259\) 7.22423e10 0.997569
\(260\) −2.07397e9 −0.0281464
\(261\) −4.97162e10 −0.663156
\(262\) 4.19626e10 0.550182
\(263\) 4.19397e10 0.540536 0.270268 0.962785i \(-0.412888\pi\)
0.270268 + 0.962785i \(0.412888\pi\)
\(264\) −1.45486e11 −1.84333
\(265\) 1.77418e9 0.0221000
\(266\) −8.78754e10 −1.07622
\(267\) 6.60170e10 0.794977
\(268\) 4.99520e10 0.591488
\(269\) 1.06374e11 1.23866 0.619328 0.785132i \(-0.287405\pi\)
0.619328 + 0.785132i \(0.287405\pi\)
\(270\) −6.98046e9 −0.0799368
\(271\) −8.12651e10 −0.915255 −0.457628 0.889144i \(-0.651301\pi\)
−0.457628 + 0.889144i \(0.651301\pi\)
\(272\) −6.33058e10 −0.701267
\(273\) 3.56658e10 0.388615
\(274\) −8.73079e10 −0.935785
\(275\) −1.26617e11 −1.33504
\(276\) 4.46343e10 0.462996
\(277\) 2.01089e10 0.205225 0.102612 0.994721i \(-0.467280\pi\)
0.102612 + 0.994721i \(0.467280\pi\)
\(278\) −6.98112e10 −0.701009
\(279\) 5.83514e10 0.576544
\(280\) −1.35918e10 −0.132149
\(281\) −1.16967e11 −1.11914 −0.559572 0.828782i \(-0.689035\pi\)
−0.559572 + 0.828782i \(0.689035\pi\)
\(282\) 9.25066e10 0.871067
\(283\) −9.10743e10 −0.844028 −0.422014 0.906589i \(-0.638677\pi\)
−0.422014 + 0.906589i \(0.638677\pi\)
\(284\) −1.25752e10 −0.114705
\(285\) 4.47079e10 0.401404
\(286\) −5.45238e10 −0.481880
\(287\) −4.97764e10 −0.433067
\(288\) −4.25520e10 −0.364463
\(289\) 1.26015e11 1.06263
\(290\) 2.07475e10 0.172256
\(291\) 1.02640e11 0.839067
\(292\) 7.65607e10 0.616287
\(293\) −7.51554e10 −0.595739 −0.297869 0.954607i \(-0.596276\pi\)
−0.297869 + 0.954607i \(0.596276\pi\)
\(294\) −6.16266e10 −0.481067
\(295\) 2.57846e10 0.198226
\(296\) −2.01098e11 −1.52263
\(297\) 1.09061e11 0.813328
\(298\) −7.12733e10 −0.523545
\(299\) 6.16021e10 0.445734
\(300\) −6.26049e10 −0.446234
\(301\) 1.54695e10 0.108624
\(302\) −1.87729e10 −0.129867
\(303\) −2.12454e10 −0.144801
\(304\) 1.38716e11 0.931530
\(305\) 4.75810e10 0.314836
\(306\) −9.07661e10 −0.591803
\(307\) 2.57354e10 0.165352 0.0826759 0.996576i \(-0.473653\pi\)
0.0826759 + 0.996576i \(0.473653\pi\)
\(308\) 5.76634e10 0.365108
\(309\) 2.51503e11 1.56939
\(310\) −2.43511e10 −0.149759
\(311\) 5.97715e10 0.362304 0.181152 0.983455i \(-0.442017\pi\)
0.181152 + 0.983455i \(0.442017\pi\)
\(312\) −9.92811e10 −0.593159
\(313\) −6.89838e10 −0.406254 −0.203127 0.979152i \(-0.565110\pi\)
−0.203127 + 0.979152i \(0.565110\pi\)
\(314\) 1.08647e11 0.630718
\(315\) −1.10510e10 −0.0632418
\(316\) −2.35233e10 −0.132711
\(317\) −3.11421e11 −1.73213 −0.866067 0.499928i \(-0.833360\pi\)
−0.866067 + 0.499928i \(0.833360\pi\)
\(318\) 2.30622e10 0.126467
\(319\) −3.24154e11 −1.75264
\(320\) 3.33871e10 0.177993
\(321\) 1.82592e11 0.959861
\(322\) 1.09624e11 0.568271
\(323\) −5.35978e11 −2.73990
\(324\) 9.23959e10 0.465801
\(325\) −8.64045e10 −0.429597
\(326\) −7.82835e10 −0.383876
\(327\) 2.57487e11 1.24535
\(328\) 1.38560e11 0.661007
\(329\) −1.35025e11 −0.635378
\(330\) 4.93645e10 0.229140
\(331\) −1.00401e11 −0.459738 −0.229869 0.973222i \(-0.573830\pi\)
−0.229869 + 0.973222i \(0.573830\pi\)
\(332\) 7.27818e10 0.328777
\(333\) −1.63505e11 −0.728674
\(334\) 1.62509e11 0.714527
\(335\) −6.24178e10 −0.270774
\(336\) −1.00190e11 −0.428841
\(337\) −3.46652e11 −1.46406 −0.732030 0.681273i \(-0.761427\pi\)
−0.732030 + 0.681273i \(0.761427\pi\)
\(338\) 1.52831e11 0.636920
\(339\) 4.16673e11 1.71355
\(340\) −2.25109e10 −0.0913562
\(341\) 3.80457e11 1.52374
\(342\) 1.98888e11 0.786123
\(343\) 2.72544e11 1.06320
\(344\) −4.30616e10 −0.165797
\(345\) −5.57730e10 −0.211952
\(346\) 2.53935e11 0.952531
\(347\) 4.59014e10 0.169959 0.0849794 0.996383i \(-0.472918\pi\)
0.0849794 + 0.996383i \(0.472918\pi\)
\(348\) −1.60276e11 −0.585818
\(349\) −5.52760e11 −1.99445 −0.997224 0.0744645i \(-0.976275\pi\)
−0.997224 + 0.0744645i \(0.976275\pi\)
\(350\) −1.53762e11 −0.547698
\(351\) 7.44243e10 0.261718
\(352\) −2.77443e11 −0.963234
\(353\) −3.24055e11 −1.11079 −0.555395 0.831586i \(-0.687433\pi\)
−0.555395 + 0.831586i \(0.687433\pi\)
\(354\) 3.35168e11 1.13435
\(355\) 1.57134e10 0.0525100
\(356\) 7.28365e10 0.240339
\(357\) 3.87117e11 1.26135
\(358\) −3.68360e11 −1.18522
\(359\) 2.90617e11 0.923413 0.461707 0.887033i \(-0.347237\pi\)
0.461707 + 0.887033i \(0.347237\pi\)
\(360\) 3.07622e10 0.0965285
\(361\) 8.51753e11 2.63956
\(362\) 3.55289e11 1.08741
\(363\) −3.63368e11 −1.09842
\(364\) 3.93500e10 0.117487
\(365\) −9.56668e10 −0.282126
\(366\) 6.18493e11 1.80165
\(367\) −1.08398e11 −0.311906 −0.155953 0.987764i \(-0.549845\pi\)
−0.155953 + 0.987764i \(0.549845\pi\)
\(368\) −1.73048e11 −0.491872
\(369\) 1.12658e11 0.316334
\(370\) 6.82339e10 0.189275
\(371\) −3.36621e10 −0.0922483
\(372\) 1.88115e11 0.509307
\(373\) −4.17071e11 −1.11563 −0.557815 0.829965i \(-0.688360\pi\)
−0.557815 + 0.829965i \(0.688360\pi\)
\(374\) −5.91803e11 −1.56407
\(375\) 1.58803e11 0.414685
\(376\) 3.75862e11 0.969802
\(377\) −2.21206e11 −0.563976
\(378\) 1.32442e11 0.333667
\(379\) −2.63908e11 −0.657015 −0.328508 0.944501i \(-0.606546\pi\)
−0.328508 + 0.944501i \(0.606546\pi\)
\(380\) 4.93262e10 0.121353
\(381\) −2.82204e11 −0.686122
\(382\) 3.47245e11 0.834354
\(383\) 8.16196e11 1.93821 0.969103 0.246658i \(-0.0793325\pi\)
0.969103 + 0.246658i \(0.0793325\pi\)
\(384\) 6.59820e10 0.154858
\(385\) −7.20536e10 −0.167141
\(386\) −4.37824e11 −1.00382
\(387\) −3.50119e10 −0.0793444
\(388\) 1.13242e11 0.253668
\(389\) 6.95765e11 1.54060 0.770299 0.637683i \(-0.220107\pi\)
0.770299 + 0.637683i \(0.220107\pi\)
\(390\) 3.36868e10 0.0737342
\(391\) 6.68631e11 1.44674
\(392\) −2.50394e11 −0.535595
\(393\) 4.05062e11 0.856554
\(394\) 2.19479e11 0.458838
\(395\) 2.93937e10 0.0607528
\(396\) −1.30509e11 −0.266693
\(397\) 2.39129e10 0.0483143 0.0241571 0.999708i \(-0.492310\pi\)
0.0241571 + 0.999708i \(0.492310\pi\)
\(398\) 3.81676e11 0.762466
\(399\) −8.48255e11 −1.67552
\(400\) 2.42721e11 0.474065
\(401\) −1.31846e11 −0.254634 −0.127317 0.991862i \(-0.540637\pi\)
−0.127317 + 0.991862i \(0.540637\pi\)
\(402\) −8.11353e11 −1.54950
\(403\) 2.59627e11 0.490318
\(404\) −2.34400e10 −0.0437766
\(405\) −1.15454e11 −0.213236
\(406\) −3.93648e11 −0.719020
\(407\) −1.06607e12 −1.92580
\(408\) −1.07760e12 −1.92525
\(409\) 2.22821e11 0.393732 0.196866 0.980430i \(-0.436924\pi\)
0.196866 + 0.980430i \(0.436924\pi\)
\(410\) −4.70145e10 −0.0821683
\(411\) −8.42777e11 −1.45688
\(412\) 2.77483e11 0.474459
\(413\) −4.89219e11 −0.827423
\(414\) −2.48112e11 −0.415094
\(415\) −9.09449e10 −0.150509
\(416\) −1.89330e11 −0.309955
\(417\) −6.73882e11 −1.09137
\(418\) 1.29677e12 2.07763
\(419\) −8.93132e11 −1.41564 −0.707820 0.706393i \(-0.750321\pi\)
−0.707820 + 0.706393i \(0.750321\pi\)
\(420\) −3.56265e10 −0.0558665
\(421\) 4.93105e11 0.765014 0.382507 0.923953i \(-0.375061\pi\)
0.382507 + 0.923953i \(0.375061\pi\)
\(422\) 6.40393e11 0.982971
\(423\) 3.05600e11 0.464111
\(424\) 9.37036e10 0.140802
\(425\) −9.37836e11 −1.39437
\(426\) 2.04254e11 0.300489
\(427\) −9.02768e11 −1.31417
\(428\) 2.01453e11 0.290187
\(429\) −5.26315e11 −0.750218
\(430\) 1.46111e10 0.0206099
\(431\) 1.09242e12 1.52490 0.762449 0.647048i \(-0.223997\pi\)
0.762449 + 0.647048i \(0.223997\pi\)
\(432\) −2.09068e11 −0.288808
\(433\) −6.73980e11 −0.921406 −0.460703 0.887554i \(-0.652403\pi\)
−0.460703 + 0.887554i \(0.652403\pi\)
\(434\) 4.62021e11 0.625112
\(435\) 2.00274e11 0.268178
\(436\) 2.84085e11 0.376495
\(437\) −1.46511e12 −1.92178
\(438\) −1.24355e12 −1.61447
\(439\) 2.78225e11 0.357524 0.178762 0.983892i \(-0.442791\pi\)
0.178762 + 0.983892i \(0.442791\pi\)
\(440\) 2.00572e11 0.255114
\(441\) −2.03587e11 −0.256316
\(442\) −4.03852e11 −0.503294
\(443\) 3.25574e11 0.401636 0.200818 0.979629i \(-0.435640\pi\)
0.200818 + 0.979629i \(0.435640\pi\)
\(444\) −5.27113e11 −0.643695
\(445\) −9.10133e10 −0.110023
\(446\) −1.97316e11 −0.236132
\(447\) −6.87996e11 −0.815083
\(448\) −6.33463e11 −0.742968
\(449\) 9.27680e11 1.07718 0.538592 0.842567i \(-0.318956\pi\)
0.538592 + 0.842567i \(0.318956\pi\)
\(450\) 3.48007e11 0.400066
\(451\) 7.34544e11 0.836032
\(452\) 4.59715e11 0.518043
\(453\) −1.81213e11 −0.202184
\(454\) −2.70346e11 −0.298654
\(455\) −4.91700e10 −0.0537835
\(456\) 2.36125e12 2.55741
\(457\) 1.08416e12 1.16271 0.581355 0.813650i \(-0.302523\pi\)
0.581355 + 0.813650i \(0.302523\pi\)
\(458\) −2.05333e11 −0.218055
\(459\) 8.07803e11 0.849471
\(460\) −6.15343e10 −0.0640777
\(461\) 5.03578e11 0.519293 0.259647 0.965704i \(-0.416394\pi\)
0.259647 + 0.965704i \(0.416394\pi\)
\(462\) −9.36606e11 −0.956463
\(463\) −1.27085e12 −1.28523 −0.642613 0.766191i \(-0.722150\pi\)
−0.642613 + 0.766191i \(0.722150\pi\)
\(464\) 6.21396e11 0.622354
\(465\) −2.35060e11 −0.233153
\(466\) 1.83968e11 0.180720
\(467\) 1.68815e12 1.64243 0.821214 0.570621i \(-0.193297\pi\)
0.821214 + 0.570621i \(0.193297\pi\)
\(468\) −8.90606e10 −0.0858181
\(469\) 1.18427e12 1.13025
\(470\) −1.27533e11 −0.120554
\(471\) 1.04876e12 0.981937
\(472\) 1.36181e12 1.26293
\(473\) −2.28281e11 −0.209698
\(474\) 3.82081e11 0.347658
\(475\) 2.05500e12 1.85221
\(476\) 4.27106e11 0.381333
\(477\) 7.61871e10 0.0673828
\(478\) −6.65360e11 −0.582950
\(479\) −1.34770e12 −1.16972 −0.584862 0.811133i \(-0.698851\pi\)
−0.584862 + 0.811133i \(0.698851\pi\)
\(480\) 1.71414e11 0.147388
\(481\) −7.27496e11 −0.619695
\(482\) −8.66684e11 −0.731390
\(483\) 1.05820e12 0.884717
\(484\) −4.00904e11 −0.332075
\(485\) −1.41503e11 −0.116125
\(486\) −9.24631e11 −0.751805
\(487\) 4.15547e11 0.334765 0.167382 0.985892i \(-0.446469\pi\)
0.167382 + 0.985892i \(0.446469\pi\)
\(488\) 2.51299e12 2.00587
\(489\) −7.55665e11 −0.597640
\(490\) 8.49606e10 0.0665787
\(491\) 2.51440e11 0.195239 0.0976197 0.995224i \(-0.468877\pi\)
0.0976197 + 0.995224i \(0.468877\pi\)
\(492\) 3.63191e11 0.279442
\(493\) −2.40097e12 −1.83053
\(494\) 8.84925e11 0.668552
\(495\) 1.63078e11 0.122088
\(496\) −7.29327e11 −0.541072
\(497\) −2.98134e11 −0.219184
\(498\) −1.18217e12 −0.861286
\(499\) −1.18872e12 −0.858273 −0.429137 0.903240i \(-0.641182\pi\)
−0.429137 + 0.903240i \(0.641182\pi\)
\(500\) 1.75207e11 0.125368
\(501\) 1.56869e12 1.11242
\(502\) −5.98796e11 −0.420835
\(503\) −2.74830e10 −0.0191430 −0.00957148 0.999954i \(-0.503047\pi\)
−0.00957148 + 0.999954i \(0.503047\pi\)
\(504\) −5.83659e11 −0.402923
\(505\) 2.92896e10 0.0200402
\(506\) −1.61771e12 −1.09704
\(507\) 1.47526e12 0.991593
\(508\) −3.11356e11 −0.207430
\(509\) −2.56804e12 −1.69579 −0.847894 0.530165i \(-0.822130\pi\)
−0.847894 + 0.530165i \(0.822130\pi\)
\(510\) 3.65637e11 0.239323
\(511\) 1.81511e12 1.17763
\(512\) 1.35732e12 0.872904
\(513\) −1.77007e12 −1.12840
\(514\) 1.47022e12 0.929070
\(515\) −3.46730e11 −0.217200
\(516\) −1.12872e11 −0.0700911
\(517\) 1.99254e12 1.22659
\(518\) −1.29462e12 −0.790057
\(519\) 2.45121e12 1.48295
\(520\) 1.36872e11 0.0820920
\(521\) −5.58805e11 −0.332269 −0.166135 0.986103i \(-0.553129\pi\)
−0.166135 + 0.986103i \(0.553129\pi\)
\(522\) 8.90940e11 0.525208
\(523\) −1.74645e12 −1.02070 −0.510350 0.859967i \(-0.670484\pi\)
−0.510350 + 0.859967i \(0.670484\pi\)
\(524\) 4.46905e11 0.258955
\(525\) −1.48425e12 −0.852687
\(526\) −7.51581e11 −0.428095
\(527\) 2.81800e12 1.59145
\(528\) 1.47849e12 0.827875
\(529\) 2.65711e10 0.0147523
\(530\) −3.17943e10 −0.0175028
\(531\) 1.10724e12 0.604391
\(532\) −9.35880e11 −0.506545
\(533\) 5.01260e11 0.269024
\(534\) −1.18306e12 −0.629608
\(535\) −2.51727e11 −0.132843
\(536\) −3.29660e12 −1.72514
\(537\) −3.55575e12 −1.84521
\(538\) −1.90628e12 −0.980994
\(539\) −1.32740e12 −0.677413
\(540\) −7.43424e10 −0.0376240
\(541\) 1.23556e12 0.620122 0.310061 0.950717i \(-0.399650\pi\)
0.310061 + 0.950717i \(0.399650\pi\)
\(542\) 1.45631e12 0.724866
\(543\) 3.42958e12 1.69294
\(544\) −2.05499e12 −1.00604
\(545\) −3.54981e11 −0.172354
\(546\) −6.39149e11 −0.307776
\(547\) −1.66722e11 −0.0796251 −0.0398126 0.999207i \(-0.512676\pi\)
−0.0398126 + 0.999207i \(0.512676\pi\)
\(548\) −9.29835e11 −0.440447
\(549\) 2.04323e12 0.959933
\(550\) 2.26904e12 1.05733
\(551\) 5.26104e12 2.43158
\(552\) −2.94565e12 −1.35038
\(553\) −5.57694e11 −0.253590
\(554\) −3.60362e11 −0.162534
\(555\) 6.58656e11 0.294673
\(556\) −7.43494e11 −0.329945
\(557\) 5.10355e11 0.224659 0.112330 0.993671i \(-0.464169\pi\)
0.112330 + 0.993671i \(0.464169\pi\)
\(558\) −1.04569e12 −0.456613
\(559\) −1.55781e11 −0.0674778
\(560\) 1.38125e11 0.0593508
\(561\) −5.71263e12 −2.43502
\(562\) 2.09612e12 0.886343
\(563\) 6.39701e11 0.268342 0.134171 0.990958i \(-0.457163\pi\)
0.134171 + 0.990958i \(0.457163\pi\)
\(564\) 9.85202e11 0.409986
\(565\) −5.74440e11 −0.237152
\(566\) 1.63210e12 0.668456
\(567\) 2.19054e12 0.890076
\(568\) 8.29903e11 0.334549
\(569\) 2.50583e12 1.00218 0.501090 0.865395i \(-0.332933\pi\)
0.501090 + 0.865395i \(0.332933\pi\)
\(570\) −8.01189e11 −0.317905
\(571\) −2.06826e12 −0.814220 −0.407110 0.913379i \(-0.633463\pi\)
−0.407110 + 0.913379i \(0.633463\pi\)
\(572\) −5.80683e11 −0.226807
\(573\) 3.35193e12 1.29897
\(574\) 8.92019e11 0.342982
\(575\) −2.56360e12 −0.978015
\(576\) 1.43371e12 0.542701
\(577\) −1.61115e12 −0.605124 −0.302562 0.953130i \(-0.597842\pi\)
−0.302562 + 0.953130i \(0.597842\pi\)
\(578\) −2.25826e12 −0.841587
\(579\) −4.22629e12 −1.56281
\(580\) 2.20962e11 0.0810760
\(581\) 1.72552e12 0.628243
\(582\) −1.83936e12 −0.664527
\(583\) 4.96747e11 0.178085
\(584\) −5.05265e12 −1.79747
\(585\) 1.11286e11 0.0392862
\(586\) 1.34682e12 0.471815
\(587\) 2.54415e12 0.884446 0.442223 0.896905i \(-0.354190\pi\)
0.442223 + 0.896905i \(0.354190\pi\)
\(588\) −6.56328e11 −0.226424
\(589\) −6.17484e12 −2.11401
\(590\) −4.62074e11 −0.156992
\(591\) 2.11861e12 0.714345
\(592\) 2.04363e12 0.683841
\(593\) 1.32235e11 0.0439137 0.0219569 0.999759i \(-0.493010\pi\)
0.0219569 + 0.999759i \(0.493010\pi\)
\(594\) −1.95443e12 −0.644142
\(595\) −5.33693e11 −0.174568
\(596\) −7.59066e11 −0.246417
\(597\) 3.68429e12 1.18705
\(598\) −1.10394e12 −0.353013
\(599\) 2.32737e12 0.738660 0.369330 0.929298i \(-0.379587\pi\)
0.369330 + 0.929298i \(0.379587\pi\)
\(600\) 4.13163e12 1.30149
\(601\) −5.87945e12 −1.83824 −0.919119 0.393981i \(-0.871098\pi\)
−0.919119 + 0.393981i \(0.871098\pi\)
\(602\) −2.77221e11 −0.0860284
\(603\) −2.68035e12 −0.825588
\(604\) −1.99932e11 −0.0611247
\(605\) 5.00952e11 0.152019
\(606\) 3.80728e11 0.114680
\(607\) −2.28556e12 −0.683351 −0.341675 0.939818i \(-0.610994\pi\)
−0.341675 + 0.939818i \(0.610994\pi\)
\(608\) 4.50292e12 1.33637
\(609\) −3.79986e12 −1.11941
\(610\) −8.52676e11 −0.249345
\(611\) 1.35973e12 0.394700
\(612\) −9.66665e11 −0.278545
\(613\) −2.72675e12 −0.779960 −0.389980 0.920823i \(-0.627518\pi\)
−0.389980 + 0.920823i \(0.627518\pi\)
\(614\) −4.61192e11 −0.130956
\(615\) −4.53828e11 −0.127924
\(616\) −3.80551e12 −1.06488
\(617\) 1.68755e12 0.468784 0.234392 0.972142i \(-0.424690\pi\)
0.234392 + 0.972142i \(0.424690\pi\)
\(618\) −4.50706e12 −1.24293
\(619\) −5.29229e12 −1.44889 −0.724445 0.689332i \(-0.757904\pi\)
−0.724445 + 0.689332i \(0.757904\pi\)
\(620\) −2.59341e11 −0.0704871
\(621\) 2.20816e12 0.595823
\(622\) −1.07114e12 −0.286938
\(623\) 1.72682e12 0.459252
\(624\) 1.00893e12 0.266399
\(625\) 3.48468e12 0.913489
\(626\) 1.23623e12 0.321746
\(627\) 1.25176e13 3.23457
\(628\) 1.15710e12 0.296861
\(629\) −7.89626e12 −2.01138
\(630\) 1.98040e11 0.0500864
\(631\) −2.78836e12 −0.700191 −0.350095 0.936714i \(-0.613851\pi\)
−0.350095 + 0.936714i \(0.613851\pi\)
\(632\) 1.55243e12 0.387065
\(633\) 6.18167e12 1.53034
\(634\) 5.58083e12 1.37182
\(635\) 3.89057e11 0.0949579
\(636\) 2.45614e11 0.0595245
\(637\) −9.05833e11 −0.217982
\(638\) 5.80901e12 1.38806
\(639\) 6.74765e11 0.160103
\(640\) −9.09650e10 −0.0214321
\(641\) 5.16942e12 1.20943 0.604715 0.796442i \(-0.293287\pi\)
0.604715 + 0.796442i \(0.293287\pi\)
\(642\) −3.27214e12 −0.760194
\(643\) −3.58293e12 −0.826589 −0.413294 0.910598i \(-0.635622\pi\)
−0.413294 + 0.910598i \(0.635622\pi\)
\(644\) 1.16751e12 0.267469
\(645\) 1.41040e11 0.0320866
\(646\) 9.60500e12 2.16996
\(647\) 3.89598e12 0.874074 0.437037 0.899444i \(-0.356028\pi\)
0.437037 + 0.899444i \(0.356028\pi\)
\(648\) −6.09770e12 −1.35856
\(649\) 7.21933e12 1.59733
\(650\) 1.54841e12 0.340233
\(651\) 4.45986e12 0.973210
\(652\) −8.33725e11 −0.180679
\(653\) −6.62700e10 −0.0142629 −0.00713145 0.999975i \(-0.502270\pi\)
−0.00713145 + 0.999975i \(0.502270\pi\)
\(654\) −4.61430e12 −0.986294
\(655\) −5.58432e11 −0.118545
\(656\) −1.40810e12 −0.296871
\(657\) −4.10813e12 −0.860201
\(658\) 2.41971e12 0.503208
\(659\) 1.84962e12 0.382030 0.191015 0.981587i \(-0.438822\pi\)
0.191015 + 0.981587i \(0.438822\pi\)
\(660\) 5.25735e11 0.107850
\(661\) −3.74337e12 −0.762705 −0.381352 0.924430i \(-0.624542\pi\)
−0.381352 + 0.924430i \(0.624542\pi\)
\(662\) 1.79923e12 0.364104
\(663\) −3.89835e12 −0.783557
\(664\) −4.80325e12 −0.958913
\(665\) 1.16943e12 0.231888
\(666\) 2.93010e12 0.577097
\(667\) −6.56314e12 −1.28394
\(668\) 1.73073e12 0.336307
\(669\) −1.90467e12 −0.367623
\(670\) 1.11856e12 0.214448
\(671\) 1.33220e13 2.53699
\(672\) −3.25229e12 −0.615216
\(673\) 2.59774e12 0.488122 0.244061 0.969760i \(-0.421520\pi\)
0.244061 + 0.969760i \(0.421520\pi\)
\(674\) 6.21218e12 1.15951
\(675\) −3.09721e12 −0.574253
\(676\) 1.62766e12 0.299780
\(677\) −7.57984e12 −1.38679 −0.693396 0.720557i \(-0.743886\pi\)
−0.693396 + 0.720557i \(0.743886\pi\)
\(678\) −7.46700e12 −1.35710
\(679\) 2.68477e12 0.484722
\(680\) 1.48562e12 0.266450
\(681\) −2.60963e12 −0.464961
\(682\) −6.81798e12 −1.20677
\(683\) −9.09946e12 −1.60001 −0.800004 0.599994i \(-0.795170\pi\)
−0.800004 + 0.599994i \(0.795170\pi\)
\(684\) 2.11817e12 0.370005
\(685\) 1.16188e12 0.201630
\(686\) −4.88414e12 −0.842034
\(687\) −1.98207e12 −0.339479
\(688\) 4.37609e11 0.0744626
\(689\) 3.38985e11 0.0573052
\(690\) 9.99481e11 0.167862
\(691\) 1.40526e10 0.00234480 0.00117240 0.999999i \(-0.499627\pi\)
0.00117240 + 0.999999i \(0.499627\pi\)
\(692\) 2.70442e12 0.448329
\(693\) −3.09413e12 −0.509611
\(694\) −8.22578e11 −0.134604
\(695\) 9.29037e11 0.151043
\(696\) 1.05775e13 1.70860
\(697\) 5.44068e12 0.873184
\(698\) 9.90576e12 1.57957
\(699\) 1.77583e12 0.281355
\(700\) −1.63757e12 −0.257786
\(701\) −9.00853e12 −1.40904 −0.704519 0.709685i \(-0.748837\pi\)
−0.704519 + 0.709685i \(0.748837\pi\)
\(702\) −1.33372e12 −0.207276
\(703\) 1.73024e13 2.67182
\(704\) 9.34793e12 1.43429
\(705\) −1.23106e12 −0.187685
\(706\) 5.80723e12 0.879727
\(707\) −5.55720e11 −0.0836505
\(708\) 3.56956e12 0.533906
\(709\) −1.15494e13 −1.71653 −0.858264 0.513209i \(-0.828457\pi\)
−0.858264 + 0.513209i \(0.828457\pi\)
\(710\) −2.81592e11 −0.0415870
\(711\) 1.26222e12 0.185235
\(712\) −4.80687e12 −0.700974
\(713\) 7.70309e12 1.11625
\(714\) −6.93734e12 −0.998966
\(715\) 7.25596e11 0.103829
\(716\) −3.92306e12 −0.557848
\(717\) −6.42268e12 −0.907569
\(718\) −5.20801e12 −0.731327
\(719\) −2.80308e12 −0.391161 −0.195580 0.980688i \(-0.562659\pi\)
−0.195580 + 0.980688i \(0.562659\pi\)
\(720\) −3.12617e11 −0.0433528
\(721\) 6.57861e12 0.906621
\(722\) −1.52639e13 −2.09048
\(723\) −8.36603e12 −1.13867
\(724\) 3.78386e12 0.511813
\(725\) 9.20559e12 1.23746
\(726\) 6.51175e12 0.869927
\(727\) −5.45780e12 −0.724625 −0.362312 0.932057i \(-0.618013\pi\)
−0.362312 + 0.932057i \(0.618013\pi\)
\(728\) −2.59692e12 −0.342663
\(729\) 6.03463e11 0.0791366
\(730\) 1.71440e12 0.223439
\(731\) −1.69085e12 −0.219016
\(732\) 6.58700e12 0.847984
\(733\) −3.95238e12 −0.505697 −0.252849 0.967506i \(-0.581367\pi\)
−0.252849 + 0.967506i \(0.581367\pi\)
\(734\) 1.94255e12 0.247024
\(735\) 8.20118e11 0.103653
\(736\) −5.61738e12 −0.705640
\(737\) −1.74761e13 −2.18193
\(738\) −2.01890e12 −0.250531
\(739\) 2.12821e11 0.0262491 0.0131246 0.999914i \(-0.495822\pi\)
0.0131246 + 0.999914i \(0.495822\pi\)
\(740\) 7.26696e11 0.0890861
\(741\) 8.54212e12 1.04084
\(742\) 6.03242e11 0.0730591
\(743\) −4.10800e12 −0.494516 −0.247258 0.968950i \(-0.579530\pi\)
−0.247258 + 0.968950i \(0.579530\pi\)
\(744\) −1.24147e13 −1.48545
\(745\) 9.48495e11 0.112806
\(746\) 7.47413e12 0.883560
\(747\) −3.90536e12 −0.458900
\(748\) −6.30274e12 −0.736161
\(749\) 4.77609e12 0.554504
\(750\) −2.84583e12 −0.328423
\(751\) 5.19165e12 0.595560 0.297780 0.954634i \(-0.403754\pi\)
0.297780 + 0.954634i \(0.403754\pi\)
\(752\) −3.81966e12 −0.435556
\(753\) −5.78013e12 −0.655179
\(754\) 3.96412e12 0.446659
\(755\) 2.49826e11 0.0279819
\(756\) 1.41052e12 0.157047
\(757\) −1.20035e13 −1.32854 −0.664272 0.747491i \(-0.731258\pi\)
−0.664272 + 0.747491i \(0.731258\pi\)
\(758\) 4.72936e12 0.520345
\(759\) −1.56157e13 −1.70794
\(760\) −3.25530e12 −0.353940
\(761\) −1.27185e12 −0.137469 −0.0687345 0.997635i \(-0.521896\pi\)
−0.0687345 + 0.997635i \(0.521896\pi\)
\(762\) 5.05725e12 0.543397
\(763\) 6.73515e12 0.719427
\(764\) 3.69818e12 0.392706
\(765\) 1.20790e12 0.127513
\(766\) −1.46267e13 −1.53502
\(767\) 4.92654e12 0.514000
\(768\) 1.12169e13 1.16345
\(769\) 1.49703e13 1.54370 0.771849 0.635806i \(-0.219332\pi\)
0.771849 + 0.635806i \(0.219332\pi\)
\(770\) 1.29124e12 0.132373
\(771\) 1.41919e13 1.44643
\(772\) −4.66286e12 −0.472470
\(773\) 1.59529e13 1.60706 0.803529 0.595266i \(-0.202953\pi\)
0.803529 + 0.595266i \(0.202953\pi\)
\(774\) 6.27431e11 0.0628394
\(775\) −1.08045e13 −1.07584
\(776\) −7.47346e12 −0.739851
\(777\) −1.24969e13 −1.23000
\(778\) −1.24685e13 −1.22013
\(779\) −1.19217e13 −1.15990
\(780\) 3.58767e11 0.0347046
\(781\) 4.39953e12 0.423133
\(782\) −1.19822e13 −1.14579
\(783\) −7.92922e12 −0.753881
\(784\) 2.54460e12 0.240546
\(785\) −1.44586e12 −0.135898
\(786\) −7.25892e12 −0.678376
\(787\) −4.64442e12 −0.431564 −0.215782 0.976442i \(-0.569230\pi\)
−0.215782 + 0.976442i \(0.569230\pi\)
\(788\) 2.33747e12 0.215962
\(789\) −7.25496e12 −0.666482
\(790\) −5.26750e11 −0.0481152
\(791\) 1.08990e13 0.989904
\(792\) 8.61298e12 0.777840
\(793\) 9.09107e12 0.816368
\(794\) −4.28532e11 −0.0382641
\(795\) −3.06908e11 −0.0272494
\(796\) 4.06487e12 0.358871
\(797\) 1.24077e13 1.08925 0.544625 0.838680i \(-0.316672\pi\)
0.544625 + 0.838680i \(0.316672\pi\)
\(798\) 1.52012e13 1.32698
\(799\) 1.47585e13 1.28110
\(800\) 7.87905e12 0.680094
\(801\) −3.90830e12 −0.335460
\(802\) 2.36275e12 0.201666
\(803\) −2.67854e13 −2.27341
\(804\) −8.64097e12 −0.729306
\(805\) −1.45887e12 −0.122443
\(806\) −4.65266e12 −0.388323
\(807\) −1.84012e13 −1.52727
\(808\) 1.54693e12 0.127679
\(809\) −9.50544e12 −0.780196 −0.390098 0.920773i \(-0.627559\pi\)
−0.390098 + 0.920773i \(0.627559\pi\)
\(810\) 2.06899e12 0.168879
\(811\) −6.56385e12 −0.532801 −0.266400 0.963862i \(-0.585834\pi\)
−0.266400 + 0.963862i \(0.585834\pi\)
\(812\) −4.19238e12 −0.338422
\(813\) 1.40577e13 1.12851
\(814\) 1.91045e13 1.52520
\(815\) 1.04179e12 0.0827121
\(816\) 1.09510e13 0.864664
\(817\) 3.70501e12 0.290931
\(818\) −3.99306e12 −0.311829
\(819\) −2.11146e12 −0.163986
\(820\) −5.00708e11 −0.0386743
\(821\) 3.96939e12 0.304915 0.152458 0.988310i \(-0.451281\pi\)
0.152458 + 0.988310i \(0.451281\pi\)
\(822\) 1.51030e13 1.15383
\(823\) 2.57352e13 1.95537 0.977683 0.210086i \(-0.0673743\pi\)
0.977683 + 0.210086i \(0.0673743\pi\)
\(824\) −1.83126e13 −1.38381
\(825\) 2.19029e13 1.64611
\(826\) 8.76706e12 0.655305
\(827\) −1.59944e13 −1.18903 −0.594516 0.804084i \(-0.702656\pi\)
−0.594516 + 0.804084i \(0.702656\pi\)
\(828\) −2.64241e12 −0.195373
\(829\) 5.84546e12 0.429856 0.214928 0.976630i \(-0.431048\pi\)
0.214928 + 0.976630i \(0.431048\pi\)
\(830\) 1.62978e12 0.119200
\(831\) −3.47855e12 −0.253043
\(832\) 6.37911e12 0.461536
\(833\) −9.83194e12 −0.707516
\(834\) 1.20763e13 0.864346
\(835\) −2.16265e12 −0.153956
\(836\) 1.38106e13 0.977881
\(837\) 9.30646e12 0.655420
\(838\) 1.60054e13 1.12116
\(839\) −2.45685e12 −0.171178 −0.0855892 0.996331i \(-0.527277\pi\)
−0.0855892 + 0.996331i \(0.527277\pi\)
\(840\) 2.35118e12 0.162941
\(841\) 9.06028e12 0.624539
\(842\) −8.83669e12 −0.605878
\(843\) 2.02337e13 1.37991
\(844\) 6.82023e12 0.462656
\(845\) −2.03385e12 −0.137234
\(846\) −5.47652e12 −0.367568
\(847\) −9.50471e12 −0.634546
\(848\) −9.52253e11 −0.0632370
\(849\) 1.57545e13 1.04069
\(850\) 1.68065e13 1.10431
\(851\) −2.15847e13 −1.41079
\(852\) 2.17532e12 0.141431
\(853\) 1.38158e11 0.00893523 0.00446761 0.999990i \(-0.498578\pi\)
0.00446761 + 0.999990i \(0.498578\pi\)
\(854\) 1.61781e13 1.04080
\(855\) −2.64677e12 −0.169383
\(856\) −1.32950e13 −0.846361
\(857\) −4.58334e12 −0.290248 −0.145124 0.989414i \(-0.546358\pi\)
−0.145124 + 0.989414i \(0.546358\pi\)
\(858\) 9.43183e12 0.594160
\(859\) 1.34992e13 0.845938 0.422969 0.906144i \(-0.360988\pi\)
0.422969 + 0.906144i \(0.360988\pi\)
\(860\) 1.55609e11 0.00970047
\(861\) 8.61060e12 0.533973
\(862\) −1.95767e13 −1.20769
\(863\) 1.87111e13 1.14829 0.574144 0.818754i \(-0.305335\pi\)
0.574144 + 0.818754i \(0.305335\pi\)
\(864\) −6.78661e12 −0.414325
\(865\) −3.37932e12 −0.205238
\(866\) 1.20781e13 0.729738
\(867\) −2.17988e13 −1.31023
\(868\) 4.92056e12 0.294222
\(869\) 8.22981e12 0.489554
\(870\) −3.58902e12 −0.212392
\(871\) −1.19259e13 −0.702115
\(872\) −1.87483e13 −1.09809
\(873\) −6.07641e12 −0.354065
\(874\) 2.62556e13 1.52202
\(875\) 4.15385e12 0.239560
\(876\) −1.32439e13 −0.759884
\(877\) −2.55346e13 −1.45757 −0.728786 0.684741i \(-0.759915\pi\)
−0.728786 + 0.684741i \(0.759915\pi\)
\(878\) −4.98593e12 −0.283153
\(879\) 1.30008e13 0.734548
\(880\) −2.03829e12 −0.114576
\(881\) 3.99929e12 0.223662 0.111831 0.993727i \(-0.464329\pi\)
0.111831 + 0.993727i \(0.464329\pi\)
\(882\) 3.64838e12 0.202998
\(883\) −1.35158e12 −0.0748204 −0.0374102 0.999300i \(-0.511911\pi\)
−0.0374102 + 0.999300i \(0.511911\pi\)
\(884\) −4.30105e12 −0.236886
\(885\) −4.46036e12 −0.244414
\(886\) −5.83445e12 −0.318089
\(887\) −1.75151e13 −0.950074 −0.475037 0.879966i \(-0.657565\pi\)
−0.475037 + 0.879966i \(0.657565\pi\)
\(888\) 3.47870e13 1.87741
\(889\) −7.38168e12 −0.396367
\(890\) 1.63101e12 0.0871365
\(891\) −3.23255e13 −1.71828
\(892\) −2.10143e12 −0.111140
\(893\) −3.23391e13 −1.70175
\(894\) 1.23292e13 0.645532
\(895\) 4.90208e12 0.255374
\(896\) 1.72591e12 0.0894604
\(897\) −1.06563e13 −0.549591
\(898\) −1.66245e13 −0.853111
\(899\) −2.76609e13 −1.41237
\(900\) 3.70630e12 0.188300
\(901\) 3.67935e12 0.185999
\(902\) −1.31634e13 −0.662123
\(903\) −2.67599e12 −0.133934
\(904\) −3.03391e13 −1.51093
\(905\) −4.72814e12 −0.234300
\(906\) 3.24743e12 0.160126
\(907\) 3.20478e13 1.57241 0.786204 0.617967i \(-0.212044\pi\)
0.786204 + 0.617967i \(0.212044\pi\)
\(908\) −2.87920e12 −0.140568
\(909\) 1.25776e12 0.0611025
\(910\) 8.81153e11 0.0425956
\(911\) 3.17963e13 1.52948 0.764740 0.644340i \(-0.222868\pi\)
0.764740 + 0.644340i \(0.222868\pi\)
\(912\) −2.39959e13 −1.14858
\(913\) −2.54633e13 −1.21282
\(914\) −1.94287e13 −0.920846
\(915\) −8.23082e12 −0.388194
\(916\) −2.18682e12 −0.102632
\(917\) 1.05953e13 0.494824
\(918\) −1.44763e13 −0.672766
\(919\) 6.46550e12 0.299008 0.149504 0.988761i \(-0.452232\pi\)
0.149504 + 0.988761i \(0.452232\pi\)
\(920\) 4.06098e12 0.186890
\(921\) −4.45186e12 −0.203879
\(922\) −9.02438e12 −0.411271
\(923\) 3.00228e12 0.136158
\(924\) −9.97492e12 −0.450180
\(925\) 3.02751e13 1.35972
\(926\) 2.27743e13 1.01788
\(927\) −1.48893e13 −0.662241
\(928\) 2.01713e13 0.892829
\(929\) 3.58746e12 0.158022 0.0790109 0.996874i \(-0.474824\pi\)
0.0790109 + 0.996874i \(0.474824\pi\)
\(930\) 4.21240e12 0.184653
\(931\) 2.15439e13 0.939831
\(932\) 1.95928e12 0.0850598
\(933\) −1.03396e13 −0.446721
\(934\) −3.02526e13 −1.30077
\(935\) 7.87563e12 0.337002
\(936\) 5.87758e12 0.250298
\(937\) 4.03654e13 1.71073 0.855364 0.518027i \(-0.173333\pi\)
0.855364 + 0.518027i \(0.173333\pi\)
\(938\) −2.12227e13 −0.895135
\(939\) 1.19332e13 0.500912
\(940\) −1.35823e12 −0.0567413
\(941\) −1.34172e13 −0.557840 −0.278920 0.960314i \(-0.589977\pi\)
−0.278920 + 0.960314i \(0.589977\pi\)
\(942\) −1.87944e13 −0.777677
\(943\) 1.48723e13 0.612456
\(944\) −1.38393e13 −0.567205
\(945\) −1.76252e12 −0.0718938
\(946\) 4.09091e12 0.166077
\(947\) −8.00627e12 −0.323486 −0.161743 0.986833i \(-0.551712\pi\)
−0.161743 + 0.986833i \(0.551712\pi\)
\(948\) 4.06919e12 0.163633
\(949\) −1.82786e13 −0.731552
\(950\) −3.68266e13 −1.46692
\(951\) 5.38714e13 2.13573
\(952\) −2.81870e13 −1.11220
\(953\) −2.74035e13 −1.07619 −0.538094 0.842885i \(-0.680855\pi\)
−0.538094 + 0.842885i \(0.680855\pi\)
\(954\) −1.36531e12 −0.0533660
\(955\) −4.62108e12 −0.179775
\(956\) −7.08614e12 −0.274378
\(957\) 5.60739e13 2.16101
\(958\) 2.41515e13 0.926402
\(959\) −2.20447e13 −0.841628
\(960\) −5.77549e12 −0.219466
\(961\) 6.02571e12 0.227904
\(962\) 1.30371e13 0.490788
\(963\) −1.08097e13 −0.405037
\(964\) −9.23024e12 −0.344244
\(965\) 5.82650e12 0.216289
\(966\) −1.89634e13 −0.700680
\(967\) −1.14021e13 −0.419341 −0.209671 0.977772i \(-0.567239\pi\)
−0.209671 + 0.977772i \(0.567239\pi\)
\(968\) 2.64578e13 0.968534
\(969\) 9.27164e13 3.37831
\(970\) 2.53580e12 0.0919692
\(971\) 3.11540e13 1.12467 0.562337 0.826908i \(-0.309902\pi\)
0.562337 + 0.826908i \(0.309902\pi\)
\(972\) −9.84738e12 −0.353853
\(973\) −1.76269e13 −0.630475
\(974\) −7.44682e12 −0.265128
\(975\) 1.49467e13 0.529694
\(976\) −2.55380e13 −0.900872
\(977\) −4.42114e13 −1.55242 −0.776209 0.630475i \(-0.782860\pi\)
−0.776209 + 0.630475i \(0.782860\pi\)
\(978\) 1.35419e13 0.473320
\(979\) −2.54824e13 −0.886582
\(980\) 9.04836e11 0.0313367
\(981\) −1.52436e13 −0.525505
\(982\) −4.50594e12 −0.154626
\(983\) 1.16477e13 0.397877 0.198939 0.980012i \(-0.436251\pi\)
0.198939 + 0.980012i \(0.436251\pi\)
\(984\) −2.39689e13 −0.815024
\(985\) −2.92079e12 −0.0988639
\(986\) 4.30267e13 1.44975
\(987\) 2.33573e13 0.783422
\(988\) 9.42452e12 0.314668
\(989\) −4.62199e12 −0.153619
\(990\) −2.92245e12 −0.0966915
\(991\) 4.19672e12 0.138222 0.0691112 0.997609i \(-0.477984\pi\)
0.0691112 + 0.997609i \(0.477984\pi\)
\(992\) −2.36749e13 −0.776222
\(993\) 1.73678e13 0.566858
\(994\) 5.34273e12 0.173590
\(995\) −5.07929e12 −0.164285
\(996\) −1.25902e13 −0.405383
\(997\) 3.39392e13 1.08786 0.543930 0.839131i \(-0.316936\pi\)
0.543930 + 0.839131i \(0.316936\pi\)
\(998\) 2.13024e13 0.679737
\(999\) −2.60774e13 −0.828362
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.1.6 15
3.2 odd 2 387.10.a.c.1.10 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.6 15 1.1 even 1 trivial
387.10.a.c.1.10 15 3.2 odd 2