Properties

Label 43.10.a.a.1.2
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 1.2
Root \(39.2075\) 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

\(f(q)\) \(=\) \(q-41.2075 q^{2} -186.815 q^{3} +1186.05 q^{4} -1998.59 q^{5} +7698.15 q^{6} -10690.1 q^{7} -27776.1 q^{8} +15216.7 q^{9} +O(q^{10})\) \(q-41.2075 q^{2} -186.815 q^{3} +1186.05 q^{4} -1998.59 q^{5} +7698.15 q^{6} -10690.1 q^{7} -27776.1 q^{8} +15216.7 q^{9} +82356.9 q^{10} +36286.0 q^{11} -221572. q^{12} +164599. q^{13} +440512. q^{14} +373366. q^{15} +537322. q^{16} -397627. q^{17} -627041. q^{18} +501060. q^{19} -2.37044e6 q^{20} +1.99706e6 q^{21} -1.49525e6 q^{22} +342901. q^{23} +5.18898e6 q^{24} +2.04125e6 q^{25} -6.78271e6 q^{26} +834374. q^{27} -1.26790e7 q^{28} -205168. q^{29} -1.53855e7 q^{30} -4.44691e6 q^{31} -7.92032e6 q^{32} -6.77875e6 q^{33} +1.63852e7 q^{34} +2.13651e7 q^{35} +1.80478e7 q^{36} +9.29370e6 q^{37} -2.06474e7 q^{38} -3.07495e7 q^{39} +5.55131e7 q^{40} -2.39332e6 q^{41} -8.22940e7 q^{42} -3.41880e6 q^{43} +4.30372e7 q^{44} -3.04119e7 q^{45} -1.41301e7 q^{46} -4.72891e7 q^{47} -1.00380e8 q^{48} +7.39245e7 q^{49} -8.41146e7 q^{50} +7.42824e7 q^{51} +1.95224e8 q^{52} -9.18336e7 q^{53} -3.43824e7 q^{54} -7.25209e7 q^{55} +2.96929e8 q^{56} -9.36053e7 q^{57} +8.45447e6 q^{58} +1.40816e8 q^{59} +4.42833e8 q^{60} +8.99461e7 q^{61} +1.83246e8 q^{62} -1.62668e8 q^{63} +5.12674e7 q^{64} -3.28967e8 q^{65} +2.79335e8 q^{66} +2.51089e8 q^{67} -4.71607e8 q^{68} -6.40589e7 q^{69} -8.80403e8 q^{70} +3.30330e8 q^{71} -4.22660e8 q^{72} -1.51643e8 q^{73} -3.82970e8 q^{74} -3.81335e8 q^{75} +5.94285e8 q^{76} -3.87900e8 q^{77} +1.26711e9 q^{78} +2.48990e8 q^{79} -1.07389e9 q^{80} -4.55383e8 q^{81} +9.86226e7 q^{82} -2.98476e8 q^{83} +2.36863e9 q^{84} +7.94694e8 q^{85} +1.40880e8 q^{86} +3.83284e7 q^{87} -1.00788e9 q^{88} -8.14630e8 q^{89} +1.25320e9 q^{90} -1.75958e9 q^{91} +4.06699e8 q^{92} +8.30747e8 q^{93} +1.94866e9 q^{94} -1.00142e9 q^{95} +1.47963e9 q^{96} +9.42482e8 q^{97} -3.04624e9 q^{98} +5.52152e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −41.2075 −1.82113 −0.910565 0.413366i \(-0.864353\pi\)
−0.910565 + 0.413366i \(0.864353\pi\)
\(3\) −186.815 −1.33157 −0.665787 0.746142i \(-0.731904\pi\)
−0.665787 + 0.746142i \(0.731904\pi\)
\(4\) 1186.05 2.31651
\(5\) −1998.59 −1.43008 −0.715038 0.699085i \(-0.753591\pi\)
−0.715038 + 0.699085i \(0.753591\pi\)
\(6\) 7698.15 2.42497
\(7\) −10690.1 −1.68283 −0.841415 0.540389i \(-0.818277\pi\)
−0.841415 + 0.540389i \(0.818277\pi\)
\(8\) −27776.1 −2.39754
\(9\) 15216.7 0.773087
\(10\) 82356.9 2.60435
\(11\) 36286.0 0.747260 0.373630 0.927578i \(-0.378113\pi\)
0.373630 + 0.927578i \(0.378113\pi\)
\(12\) −221572. −3.08461
\(13\) 164599. 1.59839 0.799194 0.601073i \(-0.205260\pi\)
0.799194 + 0.601073i \(0.205260\pi\)
\(14\) 440512. 3.06465
\(15\) 373366. 1.90425
\(16\) 537322. 2.04972
\(17\) −397627. −1.15466 −0.577332 0.816510i \(-0.695906\pi\)
−0.577332 + 0.816510i \(0.695906\pi\)
\(18\) −627041. −1.40789
\(19\) 501060. 0.882061 0.441031 0.897492i \(-0.354613\pi\)
0.441031 + 0.897492i \(0.354613\pi\)
\(20\) −2.37044e6 −3.31279
\(21\) 1.99706e6 2.24081
\(22\) −1.49525e6 −1.36086
\(23\) 342901. 0.255501 0.127751 0.991806i \(-0.459224\pi\)
0.127751 + 0.991806i \(0.459224\pi\)
\(24\) 5.18898e6 3.19250
\(25\) 2.04125e6 1.04512
\(26\) −6.78271e6 −2.91087
\(27\) 834374. 0.302151
\(28\) −1.26790e7 −3.89830
\(29\) −205168. −0.0538666 −0.0269333 0.999637i \(-0.508574\pi\)
−0.0269333 + 0.999637i \(0.508574\pi\)
\(30\) −1.53855e7 −3.46789
\(31\) −4.44691e6 −0.864829 −0.432415 0.901675i \(-0.642338\pi\)
−0.432415 + 0.901675i \(0.642338\pi\)
\(32\) −7.92032e6 −1.33527
\(33\) −6.77875e6 −0.995032
\(34\) 1.63852e7 2.10279
\(35\) 2.13651e7 2.40658
\(36\) 1.80478e7 1.79087
\(37\) 9.29370e6 0.815231 0.407616 0.913154i \(-0.366360\pi\)
0.407616 + 0.913154i \(0.366360\pi\)
\(38\) −2.06474e7 −1.60635
\(39\) −3.07495e7 −2.12837
\(40\) 5.55131e7 3.42867
\(41\) −2.39332e6 −0.132274 −0.0661368 0.997811i \(-0.521067\pi\)
−0.0661368 + 0.997811i \(0.521067\pi\)
\(42\) −8.22940e7 −4.08081
\(43\) −3.41880e6 −0.152499
\(44\) 4.30372e7 1.73104
\(45\) −3.04119e7 −1.10557
\(46\) −1.41301e7 −0.465301
\(47\) −4.72891e7 −1.41358 −0.706790 0.707423i \(-0.749857\pi\)
−0.706790 + 0.707423i \(0.749857\pi\)
\(48\) −1.00380e8 −2.72935
\(49\) 7.39245e7 1.83192
\(50\) −8.41146e7 −1.90330
\(51\) 7.42824e7 1.53752
\(52\) 1.95224e8 3.70269
\(53\) −9.18336e7 −1.59868 −0.799338 0.600882i \(-0.794816\pi\)
−0.799338 + 0.600882i \(0.794816\pi\)
\(54\) −3.43824e7 −0.550256
\(55\) −7.25209e7 −1.06864
\(56\) 2.96929e8 4.03466
\(57\) −9.36053e7 −1.17453
\(58\) 8.45447e6 0.0980980
\(59\) 1.40816e8 1.51293 0.756464 0.654035i \(-0.226925\pi\)
0.756464 + 0.654035i \(0.226925\pi\)
\(60\) 4.42833e8 4.41122
\(61\) 8.99461e7 0.831760 0.415880 0.909420i \(-0.363474\pi\)
0.415880 + 0.909420i \(0.363474\pi\)
\(62\) 1.83246e8 1.57497
\(63\) −1.62668e8 −1.30097
\(64\) 5.12674e7 0.381972
\(65\) −3.28967e8 −2.28582
\(66\) 2.79335e8 1.81208
\(67\) 2.51089e8 1.52227 0.761135 0.648593i \(-0.224642\pi\)
0.761135 + 0.648593i \(0.224642\pi\)
\(68\) −4.71607e8 −2.67479
\(69\) −6.40589e7 −0.340219
\(70\) −8.80403e8 −4.38269
\(71\) 3.30330e8 1.54271 0.771357 0.636402i \(-0.219578\pi\)
0.771357 + 0.636402i \(0.219578\pi\)
\(72\) −4.22660e8 −1.85351
\(73\) −1.51643e8 −0.624986 −0.312493 0.949920i \(-0.601164\pi\)
−0.312493 + 0.949920i \(0.601164\pi\)
\(74\) −3.82970e8 −1.48464
\(75\) −3.81335e8 −1.39165
\(76\) 5.94285e8 2.04331
\(77\) −3.87900e8 −1.25751
\(78\) 1.26711e9 3.87604
\(79\) 2.48990e8 0.719217 0.359608 0.933103i \(-0.382910\pi\)
0.359608 + 0.933103i \(0.382910\pi\)
\(80\) −1.07389e9 −2.93126
\(81\) −4.55383e8 −1.17542
\(82\) 9.86226e7 0.240887
\(83\) −2.98476e8 −0.690331 −0.345166 0.938542i \(-0.612177\pi\)
−0.345166 + 0.938542i \(0.612177\pi\)
\(84\) 2.36863e9 5.19087
\(85\) 7.94694e8 1.65126
\(86\) 1.40880e8 0.277720
\(87\) 3.83284e7 0.0717273
\(88\) −1.00788e9 −1.79159
\(89\) −8.14630e8 −1.37628 −0.688138 0.725580i \(-0.741572\pi\)
−0.688138 + 0.725580i \(0.741572\pi\)
\(90\) 1.25320e9 2.01339
\(91\) −1.75958e9 −2.68982
\(92\) 4.06699e8 0.591872
\(93\) 8.30747e8 1.15158
\(94\) 1.94866e9 2.57431
\(95\) −1.00142e9 −1.26142
\(96\) 1.47963e9 1.77800
\(97\) 9.42482e8 1.08094 0.540468 0.841364i \(-0.318247\pi\)
0.540468 + 0.841364i \(0.318247\pi\)
\(98\) −3.04624e9 −3.33616
\(99\) 5.52152e8 0.577697
\(100\) 2.42103e9 2.42103
\(101\) −6.02282e8 −0.575909 −0.287954 0.957644i \(-0.592975\pi\)
−0.287954 + 0.957644i \(0.592975\pi\)
\(102\) −3.06099e9 −2.80002
\(103\) −5.70962e8 −0.499850 −0.249925 0.968265i \(-0.580406\pi\)
−0.249925 + 0.968265i \(0.580406\pi\)
\(104\) −4.57192e9 −3.83220
\(105\) −3.99132e9 −3.20453
\(106\) 3.78423e9 2.91140
\(107\) 5.23029e8 0.385743 0.192872 0.981224i \(-0.438220\pi\)
0.192872 + 0.981224i \(0.438220\pi\)
\(108\) 9.89613e8 0.699937
\(109\) 1.69448e9 1.14979 0.574895 0.818227i \(-0.305043\pi\)
0.574895 + 0.818227i \(0.305043\pi\)
\(110\) 2.98840e9 1.94613
\(111\) −1.73620e9 −1.08554
\(112\) −5.74402e9 −3.44933
\(113\) 3.15269e9 1.81898 0.909489 0.415727i \(-0.136473\pi\)
0.909489 + 0.415727i \(0.136473\pi\)
\(114\) 3.85724e9 2.13897
\(115\) −6.85319e8 −0.365387
\(116\) −2.43341e8 −0.124783
\(117\) 2.50465e9 1.23569
\(118\) −5.80267e9 −2.75524
\(119\) 4.25067e9 1.94310
\(120\) −1.03707e10 −4.56552
\(121\) −1.04128e9 −0.441602
\(122\) −3.70645e9 −1.51474
\(123\) 4.47107e8 0.176132
\(124\) −5.27428e9 −2.00339
\(125\) −1.76121e8 −0.0645233
\(126\) 6.70312e9 2.36924
\(127\) 9.03372e8 0.308141 0.154071 0.988060i \(-0.450762\pi\)
0.154071 + 0.988060i \(0.450762\pi\)
\(128\) 1.94260e9 0.639646
\(129\) 6.38682e8 0.203063
\(130\) 1.35559e10 4.16277
\(131\) 1.08637e9 0.322299 0.161149 0.986930i \(-0.448480\pi\)
0.161149 + 0.986930i \(0.448480\pi\)
\(132\) −8.03997e9 −2.30500
\(133\) −5.35638e9 −1.48436
\(134\) −1.03468e10 −2.77225
\(135\) −1.66757e9 −0.432099
\(136\) 1.10445e10 2.76835
\(137\) −5.85232e8 −0.141934 −0.0709668 0.997479i \(-0.522608\pi\)
−0.0709668 + 0.997479i \(0.522608\pi\)
\(138\) 2.63970e9 0.619583
\(139\) 6.47466e8 0.147113 0.0735564 0.997291i \(-0.476565\pi\)
0.0735564 + 0.997291i \(0.476565\pi\)
\(140\) 2.53402e10 5.57486
\(141\) 8.83429e9 1.88229
\(142\) −1.36121e10 −2.80948
\(143\) 5.97264e9 1.19441
\(144\) 8.17626e9 1.58461
\(145\) 4.10048e8 0.0770333
\(146\) 6.24883e9 1.13818
\(147\) −1.38102e10 −2.43933
\(148\) 1.10228e10 1.88849
\(149\) −5.65496e9 −0.939921 −0.469960 0.882687i \(-0.655732\pi\)
−0.469960 + 0.882687i \(0.655732\pi\)
\(150\) 1.57138e10 2.53438
\(151\) −5.68427e9 −0.889772 −0.444886 0.895587i \(-0.646756\pi\)
−0.444886 + 0.895587i \(0.646756\pi\)
\(152\) −1.39175e10 −2.11478
\(153\) −6.05056e9 −0.892655
\(154\) 1.59844e10 2.29009
\(155\) 8.88756e9 1.23677
\(156\) −3.64706e10 −4.93040
\(157\) −4.76925e9 −0.626471 −0.313236 0.949675i \(-0.601413\pi\)
−0.313236 + 0.949675i \(0.601413\pi\)
\(158\) −1.02602e10 −1.30979
\(159\) 1.71559e10 2.12875
\(160\) 1.58295e10 1.90953
\(161\) −3.66564e9 −0.429965
\(162\) 1.87652e10 2.14060
\(163\) −3.07695e9 −0.341411 −0.170705 0.985322i \(-0.554605\pi\)
−0.170705 + 0.985322i \(0.554605\pi\)
\(164\) −2.83861e9 −0.306414
\(165\) 1.35480e10 1.42297
\(166\) 1.22994e10 1.25718
\(167\) 1.83022e9 0.182087 0.0910437 0.995847i \(-0.470980\pi\)
0.0910437 + 0.995847i \(0.470980\pi\)
\(168\) −5.54706e10 −5.37244
\(169\) 1.64884e10 1.55485
\(170\) −3.27473e10 −3.00715
\(171\) 7.62447e9 0.681910
\(172\) −4.05489e9 −0.353265
\(173\) 7.61751e9 0.646555 0.323278 0.946304i \(-0.395215\pi\)
0.323278 + 0.946304i \(0.395215\pi\)
\(174\) −1.57942e9 −0.130625
\(175\) −2.18211e10 −1.75876
\(176\) 1.94973e10 1.53167
\(177\) −2.63065e10 −2.01457
\(178\) 3.35689e10 2.50638
\(179\) −2.69122e9 −0.195934 −0.0979672 0.995190i \(-0.531234\pi\)
−0.0979672 + 0.995190i \(0.531234\pi\)
\(180\) −3.60702e10 −2.56108
\(181\) −1.34253e10 −0.929761 −0.464881 0.885373i \(-0.653903\pi\)
−0.464881 + 0.885373i \(0.653903\pi\)
\(182\) 7.25078e10 4.89850
\(183\) −1.68032e10 −1.10755
\(184\) −9.52444e9 −0.612575
\(185\) −1.85743e10 −1.16584
\(186\) −3.42330e10 −2.09718
\(187\) −1.44283e10 −0.862834
\(188\) −5.60874e10 −3.27458
\(189\) −8.91954e9 −0.508469
\(190\) 4.12658e10 2.29720
\(191\) −2.22480e10 −1.20960 −0.604799 0.796378i \(-0.706746\pi\)
−0.604799 + 0.796378i \(0.706746\pi\)
\(192\) −9.57750e9 −0.508624
\(193\) −2.60547e10 −1.35169 −0.675847 0.737042i \(-0.736222\pi\)
−0.675847 + 0.737042i \(0.736222\pi\)
\(194\) −3.88373e10 −1.96853
\(195\) 6.14557e10 3.04373
\(196\) 8.76785e10 4.24366
\(197\) 8.60900e9 0.407244 0.203622 0.979050i \(-0.434729\pi\)
0.203622 + 0.979050i \(0.434729\pi\)
\(198\) −2.27528e10 −1.05206
\(199\) −4.77571e9 −0.215873 −0.107937 0.994158i \(-0.534424\pi\)
−0.107937 + 0.994158i \(0.534424\pi\)
\(200\) −5.66979e10 −2.50572
\(201\) −4.69072e10 −2.02701
\(202\) 2.48185e10 1.04880
\(203\) 2.19327e9 0.0906483
\(204\) 8.81031e10 3.56168
\(205\) 4.78327e9 0.189161
\(206\) 2.35279e10 0.910291
\(207\) 5.21781e9 0.197525
\(208\) 8.84427e10 3.27625
\(209\) 1.81815e10 0.659129
\(210\) 1.64472e11 5.83587
\(211\) 1.74070e10 0.604578 0.302289 0.953216i \(-0.402249\pi\)
0.302289 + 0.953216i \(0.402249\pi\)
\(212\) −1.08920e11 −3.70335
\(213\) −6.17105e10 −2.05424
\(214\) −2.15527e10 −0.702489
\(215\) 6.83279e9 0.218085
\(216\) −2.31757e10 −0.724419
\(217\) 4.75378e10 1.45536
\(218\) −6.98254e10 −2.09392
\(219\) 2.83292e10 0.832215
\(220\) −8.60138e10 −2.47552
\(221\) −6.54490e10 −1.84560
\(222\) 7.15443e10 1.97691
\(223\) 4.18882e10 1.13428 0.567140 0.823622i \(-0.308050\pi\)
0.567140 + 0.823622i \(0.308050\pi\)
\(224\) 8.46690e10 2.24703
\(225\) 3.10610e10 0.807968
\(226\) −1.29914e11 −3.31260
\(227\) 6.69191e9 0.167276 0.0836380 0.996496i \(-0.473346\pi\)
0.0836380 + 0.996496i \(0.473346\pi\)
\(228\) −1.11021e11 −2.72081
\(229\) −4.12032e10 −0.990081 −0.495041 0.868870i \(-0.664847\pi\)
−0.495041 + 0.868870i \(0.664847\pi\)
\(230\) 2.82403e10 0.665416
\(231\) 7.24655e10 1.67447
\(232\) 5.69877e9 0.129147
\(233\) −7.86029e10 −1.74718 −0.873588 0.486666i \(-0.838213\pi\)
−0.873588 + 0.486666i \(0.838213\pi\)
\(234\) −1.03210e11 −2.25036
\(235\) 9.45116e10 2.02153
\(236\) 1.67016e11 3.50472
\(237\) −4.65149e10 −0.957690
\(238\) −1.75159e11 −3.53864
\(239\) 3.00733e10 0.596198 0.298099 0.954535i \(-0.403647\pi\)
0.298099 + 0.954535i \(0.403647\pi\)
\(240\) 2.00618e11 3.90318
\(241\) 1.44474e10 0.275875 0.137937 0.990441i \(-0.455953\pi\)
0.137937 + 0.990441i \(0.455953\pi\)
\(242\) 4.29083e10 0.804215
\(243\) 6.86492e10 1.26301
\(244\) 1.06681e11 1.92678
\(245\) −1.47745e11 −2.61978
\(246\) −1.84241e10 −0.320759
\(247\) 8.24741e10 1.40988
\(248\) 1.23518e11 2.07346
\(249\) 5.57596e10 0.919227
\(250\) 7.25751e9 0.117505
\(251\) −4.66837e10 −0.742392 −0.371196 0.928555i \(-0.621052\pi\)
−0.371196 + 0.928555i \(0.621052\pi\)
\(252\) −1.92933e11 −3.01372
\(253\) 1.24425e10 0.190926
\(254\) −3.72256e10 −0.561165
\(255\) −1.48460e11 −2.19877
\(256\) −1.06299e11 −1.54685
\(257\) 6.18273e10 0.884059 0.442030 0.897000i \(-0.354259\pi\)
0.442030 + 0.897000i \(0.354259\pi\)
\(258\) −2.63185e10 −0.369804
\(259\) −9.93505e10 −1.37190
\(260\) −3.90172e11 −5.29513
\(261\) −3.12198e9 −0.0416436
\(262\) −4.47667e10 −0.586948
\(263\) 6.15683e10 0.793517 0.396759 0.917923i \(-0.370135\pi\)
0.396759 + 0.917923i \(0.370135\pi\)
\(264\) 1.88287e11 2.38563
\(265\) 1.83538e11 2.28623
\(266\) 2.20723e11 2.70321
\(267\) 1.52185e11 1.83261
\(268\) 2.97806e11 3.52636
\(269\) 5.70312e10 0.664091 0.332045 0.943263i \(-0.392261\pi\)
0.332045 + 0.943263i \(0.392261\pi\)
\(270\) 6.87165e10 0.786908
\(271\) −9.67574e10 −1.08974 −0.544870 0.838521i \(-0.683421\pi\)
−0.544870 + 0.838521i \(0.683421\pi\)
\(272\) −2.13654e11 −2.36674
\(273\) 3.28715e11 3.58169
\(274\) 2.41159e10 0.258480
\(275\) 7.40687e10 0.780976
\(276\) −7.59773e10 −0.788121
\(277\) −1.10031e11 −1.12294 −0.561471 0.827497i \(-0.689764\pi\)
−0.561471 + 0.827497i \(0.689764\pi\)
\(278\) −2.66804e10 −0.267911
\(279\) −6.76671e10 −0.668589
\(280\) −5.93440e11 −5.76987
\(281\) −4.71720e10 −0.451342 −0.225671 0.974204i \(-0.572457\pi\)
−0.225671 + 0.974204i \(0.572457\pi\)
\(282\) −3.64038e11 −3.42789
\(283\) 6.65135e10 0.616411 0.308206 0.951320i \(-0.400271\pi\)
0.308206 + 0.951320i \(0.400271\pi\)
\(284\) 3.91790e11 3.57372
\(285\) 1.87079e11 1.67967
\(286\) −2.46117e11 −2.17518
\(287\) 2.55848e10 0.222594
\(288\) −1.20521e11 −1.03228
\(289\) 3.95191e10 0.333247
\(290\) −1.68970e10 −0.140288
\(291\) −1.76069e11 −1.43935
\(292\) −1.79857e11 −1.44779
\(293\) 5.04654e10 0.400027 0.200013 0.979793i \(-0.435901\pi\)
0.200013 + 0.979793i \(0.435901\pi\)
\(294\) 5.69082e11 4.44234
\(295\) −2.81434e11 −2.16360
\(296\) −2.58143e11 −1.95455
\(297\) 3.02761e10 0.225785
\(298\) 2.33027e11 1.71172
\(299\) 5.64412e10 0.408391
\(300\) −4.52284e11 −3.22378
\(301\) 3.65473e10 0.256629
\(302\) 2.34234e11 1.62039
\(303\) 1.12515e11 0.766864
\(304\) 2.69231e11 1.80798
\(305\) −1.79766e11 −1.18948
\(306\) 2.49328e11 1.62564
\(307\) 9.46104e10 0.607878 0.303939 0.952692i \(-0.401698\pi\)
0.303939 + 0.952692i \(0.401698\pi\)
\(308\) −4.60071e11 −2.91304
\(309\) 1.06664e11 0.665586
\(310\) −3.66234e11 −2.25232
\(311\) 1.39290e10 0.0844302 0.0422151 0.999109i \(-0.486559\pi\)
0.0422151 + 0.999109i \(0.486559\pi\)
\(312\) 8.54101e11 5.10286
\(313\) −1.97862e11 −1.16523 −0.582617 0.812747i \(-0.697971\pi\)
−0.582617 + 0.812747i \(0.697971\pi\)
\(314\) 1.96529e11 1.14089
\(315\) 3.25106e11 1.86049
\(316\) 2.95316e11 1.66608
\(317\) −8.01606e10 −0.445856 −0.222928 0.974835i \(-0.571561\pi\)
−0.222928 + 0.974835i \(0.571561\pi\)
\(318\) −7.06949e11 −3.87674
\(319\) −7.44473e9 −0.0402523
\(320\) −1.02463e11 −0.546249
\(321\) −9.77094e10 −0.513646
\(322\) 1.51052e11 0.783023
\(323\) −1.99235e11 −1.01848
\(324\) −5.40109e11 −2.72288
\(325\) 3.35988e11 1.67051
\(326\) 1.26793e11 0.621753
\(327\) −3.16554e11 −1.53103
\(328\) 6.64771e10 0.317132
\(329\) 5.05524e11 2.37881
\(330\) −5.58277e11 −2.59142
\(331\) 3.39745e11 1.55570 0.777851 0.628448i \(-0.216310\pi\)
0.777851 + 0.628448i \(0.216310\pi\)
\(332\) −3.54009e11 −1.59916
\(333\) 1.41419e11 0.630245
\(334\) −7.54188e10 −0.331605
\(335\) −5.01825e11 −2.17696
\(336\) 1.07307e12 4.59304
\(337\) −1.99608e11 −0.843030 −0.421515 0.906821i \(-0.638502\pi\)
−0.421515 + 0.906821i \(0.638502\pi\)
\(338\) −6.79444e11 −2.83158
\(339\) −5.88968e11 −2.42210
\(340\) 9.42550e11 3.82516
\(341\) −1.61360e11 −0.646253
\(342\) −3.14185e11 −1.24185
\(343\) −3.58875e11 −1.39997
\(344\) 9.49609e10 0.365622
\(345\) 1.28028e11 0.486539
\(346\) −3.13898e11 −1.17746
\(347\) 3.63746e11 1.34684 0.673420 0.739260i \(-0.264825\pi\)
0.673420 + 0.739260i \(0.264825\pi\)
\(348\) 4.54596e10 0.166157
\(349\) −1.07178e11 −0.386715 −0.193357 0.981128i \(-0.561938\pi\)
−0.193357 + 0.981128i \(0.561938\pi\)
\(350\) 8.99193e11 3.20293
\(351\) 1.37337e11 0.482955
\(352\) −2.87397e11 −0.997791
\(353\) 1.04485e11 0.358152 0.179076 0.983835i \(-0.442689\pi\)
0.179076 + 0.983835i \(0.442689\pi\)
\(354\) 1.08402e12 3.66880
\(355\) −6.60195e11 −2.20620
\(356\) −9.66196e11 −3.18816
\(357\) −7.94086e11 −2.58738
\(358\) 1.10898e11 0.356822
\(359\) 1.05990e9 0.00336774 0.00168387 0.999999i \(-0.499464\pi\)
0.00168387 + 0.999999i \(0.499464\pi\)
\(360\) 8.44725e11 2.65066
\(361\) −7.16263e10 −0.221968
\(362\) 5.53224e11 1.69322
\(363\) 1.94525e11 0.588026
\(364\) −2.08696e12 −6.23100
\(365\) 3.03073e11 0.893778
\(366\) 6.92419e11 2.01699
\(367\) −9.86256e10 −0.283787 −0.141893 0.989882i \(-0.545319\pi\)
−0.141893 + 0.989882i \(0.545319\pi\)
\(368\) 1.84248e11 0.523707
\(369\) −3.64184e10 −0.102259
\(370\) 7.65401e11 2.12315
\(371\) 9.81710e11 2.69030
\(372\) 9.85311e11 2.66766
\(373\) −3.53431e11 −0.945398 −0.472699 0.881224i \(-0.656720\pi\)
−0.472699 + 0.881224i \(0.656720\pi\)
\(374\) 5.94553e11 1.57133
\(375\) 3.29020e10 0.0859175
\(376\) 1.31351e12 3.38912
\(377\) −3.37705e10 −0.0860997
\(378\) 3.67551e11 0.925987
\(379\) −8.14234e9 −0.0202709 −0.0101354 0.999949i \(-0.503226\pi\)
−0.0101354 + 0.999949i \(0.503226\pi\)
\(380\) −1.18773e12 −2.92209
\(381\) −1.68763e11 −0.410312
\(382\) 9.16784e11 2.20283
\(383\) 4.12171e11 0.978774 0.489387 0.872067i \(-0.337221\pi\)
0.489387 + 0.872067i \(0.337221\pi\)
\(384\) −3.62907e11 −0.851735
\(385\) 7.75255e11 1.79834
\(386\) 1.07365e12 2.46161
\(387\) −5.20228e10 −0.117895
\(388\) 1.11784e12 2.50400
\(389\) −5.55473e11 −1.22996 −0.614979 0.788544i \(-0.710835\pi\)
−0.614979 + 0.788544i \(0.710835\pi\)
\(390\) −2.53244e12 −5.54303
\(391\) −1.36346e11 −0.295018
\(392\) −2.05333e12 −4.39210
\(393\) −2.02951e11 −0.429165
\(394\) −3.54755e11 −0.741644
\(395\) −4.97629e11 −1.02854
\(396\) 6.54883e11 1.33824
\(397\) −6.23323e11 −1.25938 −0.629689 0.776847i \(-0.716818\pi\)
−0.629689 + 0.776847i \(0.716818\pi\)
\(398\) 1.96795e11 0.393134
\(399\) 1.00065e12 1.97653
\(400\) 1.09681e12 2.14220
\(401\) 5.35221e11 1.03367 0.516836 0.856084i \(-0.327110\pi\)
0.516836 + 0.856084i \(0.327110\pi\)
\(402\) 1.93292e12 3.69146
\(403\) −7.31957e11 −1.38233
\(404\) −7.14339e11 −1.33410
\(405\) 9.10125e11 1.68095
\(406\) −9.03790e10 −0.165082
\(407\) 3.37231e11 0.609190
\(408\) −2.06328e12 −3.68627
\(409\) −5.78560e11 −1.02234 −0.511168 0.859481i \(-0.670787\pi\)
−0.511168 + 0.859481i \(0.670787\pi\)
\(410\) −1.97106e11 −0.344488
\(411\) 1.09330e11 0.188995
\(412\) −6.77192e11 −1.15791
\(413\) −1.50534e12 −2.54600
\(414\) −2.15013e11 −0.359718
\(415\) 5.96531e11 0.987227
\(416\) −1.30368e12 −2.13427
\(417\) −1.20956e11 −0.195891
\(418\) −7.49212e11 −1.20036
\(419\) −8.98515e11 −1.42417 −0.712086 0.702093i \(-0.752249\pi\)
−0.712086 + 0.702093i \(0.752249\pi\)
\(420\) −4.73392e12 −7.42334
\(421\) 5.58393e11 0.866304 0.433152 0.901321i \(-0.357401\pi\)
0.433152 + 0.901321i \(0.357401\pi\)
\(422\) −7.17297e11 −1.10102
\(423\) −7.19582e11 −1.09282
\(424\) 2.55078e12 3.83289
\(425\) −8.11654e11 −1.20676
\(426\) 2.54293e12 3.74103
\(427\) −9.61532e11 −1.39971
\(428\) 6.20341e11 0.893580
\(429\) −1.11578e12 −1.59045
\(430\) −2.81562e11 −0.397160
\(431\) −3.49762e11 −0.488230 −0.244115 0.969746i \(-0.578497\pi\)
−0.244115 + 0.969746i \(0.578497\pi\)
\(432\) 4.48328e11 0.619325
\(433\) −9.15456e11 −1.25153 −0.625766 0.780011i \(-0.715214\pi\)
−0.625766 + 0.780011i \(0.715214\pi\)
\(434\) −1.95891e12 −2.65040
\(435\) −7.66029e10 −0.102576
\(436\) 2.00975e12 2.66350
\(437\) 1.71814e11 0.225368
\(438\) −1.16737e12 −1.51557
\(439\) −1.38571e12 −1.78066 −0.890332 0.455312i \(-0.849528\pi\)
−0.890332 + 0.455312i \(0.849528\pi\)
\(440\) 2.01435e12 2.56211
\(441\) 1.12488e12 1.41623
\(442\) 2.69699e12 3.36108
\(443\) 1.31854e12 1.62658 0.813292 0.581856i \(-0.197673\pi\)
0.813292 + 0.581856i \(0.197673\pi\)
\(444\) −2.05923e12 −2.51467
\(445\) 1.62811e12 1.96818
\(446\) −1.72611e12 −2.06567
\(447\) 1.05643e12 1.25157
\(448\) −5.48053e11 −0.642794
\(449\) −3.56492e11 −0.413944 −0.206972 0.978347i \(-0.566361\pi\)
−0.206972 + 0.978347i \(0.566361\pi\)
\(450\) −1.27995e12 −1.47141
\(451\) −8.68440e10 −0.0988428
\(452\) 3.73926e12 4.21369
\(453\) 1.06191e12 1.18480
\(454\) −2.75756e11 −0.304631
\(455\) 3.51668e12 3.84664
\(456\) 2.59999e12 2.81598
\(457\) 1.30599e12 1.40061 0.700307 0.713842i \(-0.253047\pi\)
0.700307 + 0.713842i \(0.253047\pi\)
\(458\) 1.69788e12 1.80307
\(459\) −3.31769e11 −0.348883
\(460\) −8.12826e11 −0.846423
\(461\) −8.59137e11 −0.885948 −0.442974 0.896534i \(-0.646077\pi\)
−0.442974 + 0.896534i \(0.646077\pi\)
\(462\) −2.98612e12 −3.04943
\(463\) 4.67927e11 0.473221 0.236610 0.971605i \(-0.423964\pi\)
0.236610 + 0.971605i \(0.423964\pi\)
\(464\) −1.10241e11 −0.110411
\(465\) −1.66032e12 −1.64685
\(466\) 3.23902e12 3.18183
\(467\) 1.06844e12 1.03950 0.519748 0.854320i \(-0.326026\pi\)
0.519748 + 0.854320i \(0.326026\pi\)
\(468\) 2.97065e12 2.86250
\(469\) −2.68417e12 −2.56172
\(470\) −3.89458e12 −3.68146
\(471\) 8.90965e11 0.834193
\(472\) −3.91132e12 −3.62731
\(473\) −1.24055e11 −0.113956
\(474\) 1.91676e12 1.74408
\(475\) 1.02279e12 0.921859
\(476\) 5.04152e12 4.50122
\(477\) −1.39740e12 −1.23592
\(478\) −1.23925e12 −1.08575
\(479\) 1.71944e12 1.49237 0.746185 0.665739i \(-0.231884\pi\)
0.746185 + 0.665739i \(0.231884\pi\)
\(480\) −2.95718e12 −2.54268
\(481\) 1.52973e12 1.30306
\(482\) −5.95340e11 −0.502404
\(483\) 6.84795e11 0.572530
\(484\) −1.23501e12 −1.02298
\(485\) −1.88364e12 −1.54582
\(486\) −2.82886e12 −2.30011
\(487\) 1.43644e12 1.15720 0.578598 0.815613i \(-0.303600\pi\)
0.578598 + 0.815613i \(0.303600\pi\)
\(488\) −2.49835e12 −1.99418
\(489\) 5.74820e11 0.454613
\(490\) 6.08819e12 4.77096
\(491\) −2.21628e12 −1.72091 −0.860453 0.509530i \(-0.829819\pi\)
−0.860453 + 0.509530i \(0.829819\pi\)
\(492\) 5.30293e11 0.408012
\(493\) 8.15804e10 0.0621978
\(494\) −3.39855e12 −2.56757
\(495\) −1.10353e12 −0.826151
\(496\) −2.38942e12 −1.77266
\(497\) −3.53126e12 −2.59613
\(498\) −2.29771e12 −1.67403
\(499\) 1.04835e12 0.756927 0.378464 0.925616i \(-0.376453\pi\)
0.378464 + 0.925616i \(0.376453\pi\)
\(500\) −2.08890e11 −0.149469
\(501\) −3.41912e11 −0.242463
\(502\) 1.92372e12 1.35199
\(503\) 2.02850e12 1.41292 0.706462 0.707751i \(-0.250290\pi\)
0.706462 + 0.707751i \(0.250290\pi\)
\(504\) 4.51827e12 3.11914
\(505\) 1.20372e12 0.823593
\(506\) −5.12724e11 −0.347701
\(507\) −3.08027e12 −2.07039
\(508\) 1.07145e12 0.713813
\(509\) −1.00903e12 −0.666303 −0.333152 0.942873i \(-0.608112\pi\)
−0.333152 + 0.942873i \(0.608112\pi\)
\(510\) 6.11767e12 4.00424
\(511\) 1.62108e12 1.05175
\(512\) 3.38569e12 2.17737
\(513\) 4.18072e11 0.266516
\(514\) −2.54775e12 −1.60999
\(515\) 1.14112e12 0.714823
\(516\) 7.57512e11 0.470398
\(517\) −1.71593e12 −1.05631
\(518\) 4.09398e12 2.49840
\(519\) −1.42306e12 −0.860935
\(520\) 9.13741e12 5.48034
\(521\) 9.68502e11 0.575878 0.287939 0.957649i \(-0.407030\pi\)
0.287939 + 0.957649i \(0.407030\pi\)
\(522\) 1.28649e11 0.0758383
\(523\) −2.51347e12 −1.46898 −0.734490 0.678619i \(-0.762579\pi\)
−0.734490 + 0.678619i \(0.762579\pi\)
\(524\) 1.28850e12 0.746610
\(525\) 4.07650e12 2.34191
\(526\) −2.53707e12 −1.44510
\(527\) 1.76821e12 0.998587
\(528\) −3.64237e12 −2.03954
\(529\) −1.68357e12 −0.934719
\(530\) −7.56314e12 −4.16352
\(531\) 2.14275e12 1.16963
\(532\) −6.35296e12 −3.43854
\(533\) −3.93938e11 −0.211425
\(534\) −6.27115e12 −3.33743
\(535\) −1.04532e12 −0.551643
\(536\) −6.97428e12 −3.64971
\(537\) 5.02759e11 0.260901
\(538\) −2.35011e12 −1.20940
\(539\) 2.68242e12 1.36892
\(540\) −1.97783e12 −1.00096
\(541\) 8.92784e11 0.448083 0.224042 0.974580i \(-0.428075\pi\)
0.224042 + 0.974580i \(0.428075\pi\)
\(542\) 3.98713e12 1.98456
\(543\) 2.50805e12 1.23805
\(544\) 3.14933e12 1.54178
\(545\) −3.38658e12 −1.64429
\(546\) −1.35455e13 −6.52272
\(547\) 9.58310e11 0.457681 0.228841 0.973464i \(-0.426507\pi\)
0.228841 + 0.973464i \(0.426507\pi\)
\(548\) −6.94117e11 −0.328791
\(549\) 1.36868e12 0.643023
\(550\) −3.05218e12 −1.42226
\(551\) −1.02802e11 −0.0475136
\(552\) 1.77930e12 0.815689
\(553\) −2.66172e12 −1.21032
\(554\) 4.53411e12 2.04502
\(555\) 3.46995e12 1.55241
\(556\) 7.67930e11 0.340789
\(557\) −1.35685e12 −0.597289 −0.298645 0.954364i \(-0.596535\pi\)
−0.298645 + 0.954364i \(0.596535\pi\)
\(558\) 2.78839e12 1.21759
\(559\) −5.62732e11 −0.243752
\(560\) 1.14800e13 4.93281
\(561\) 2.69541e12 1.14893
\(562\) 1.94384e12 0.821953
\(563\) −1.64809e12 −0.691341 −0.345670 0.938356i \(-0.612348\pi\)
−0.345670 + 0.938356i \(0.612348\pi\)
\(564\) 1.04779e13 4.36034
\(565\) −6.30093e12 −2.60128
\(566\) −2.74085e12 −1.12257
\(567\) 4.86809e12 1.97804
\(568\) −9.17528e12 −3.69872
\(569\) 1.46538e12 0.586066 0.293033 0.956102i \(-0.405335\pi\)
0.293033 + 0.956102i \(0.405335\pi\)
\(570\) −7.70905e12 −3.05889
\(571\) 7.35186e11 0.289424 0.144712 0.989474i \(-0.453774\pi\)
0.144712 + 0.989474i \(0.453774\pi\)
\(572\) 7.08388e12 2.76687
\(573\) 4.15625e12 1.61067
\(574\) −1.05428e12 −0.405373
\(575\) 6.99945e11 0.267029
\(576\) 7.80120e11 0.295298
\(577\) −4.52671e12 −1.70017 −0.850083 0.526649i \(-0.823448\pi\)
−0.850083 + 0.526649i \(0.823448\pi\)
\(578\) −1.62848e12 −0.606886
\(579\) 4.86740e12 1.79988
\(580\) 4.86339e11 0.178449
\(581\) 3.19073e12 1.16171
\(582\) 7.25537e12 2.62124
\(583\) −3.33227e12 −1.19463
\(584\) 4.21206e12 1.49843
\(585\) −5.00578e12 −1.76714
\(586\) −2.07955e12 −0.728501
\(587\) −1.38198e12 −0.480429 −0.240214 0.970720i \(-0.577218\pi\)
−0.240214 + 0.970720i \(0.577218\pi\)
\(588\) −1.63796e13 −5.65074
\(589\) −2.22817e12 −0.762833
\(590\) 1.15972e13 3.94020
\(591\) −1.60829e12 −0.542275
\(592\) 4.99371e12 1.67100
\(593\) 4.26421e12 1.41609 0.708047 0.706165i \(-0.249576\pi\)
0.708047 + 0.706165i \(0.249576\pi\)
\(594\) −1.24760e12 −0.411184
\(595\) −8.49535e12 −2.77878
\(596\) −6.70709e12 −2.17734
\(597\) 8.92172e11 0.287451
\(598\) −2.32580e12 −0.743732
\(599\) 1.35010e12 0.428493 0.214246 0.976780i \(-0.431270\pi\)
0.214246 + 0.976780i \(0.431270\pi\)
\(600\) 1.05920e13 3.33654
\(601\) 3.69329e12 1.15472 0.577362 0.816488i \(-0.304082\pi\)
0.577362 + 0.816488i \(0.304082\pi\)
\(602\) −1.50602e12 −0.467355
\(603\) 3.82075e12 1.17685
\(604\) −6.74186e12 −2.06117
\(605\) 2.08108e12 0.631525
\(606\) −4.63646e12 −1.39656
\(607\) −6.36583e12 −1.90329 −0.951647 0.307193i \(-0.900610\pi\)
−0.951647 + 0.307193i \(0.900610\pi\)
\(608\) −3.96856e12 −1.17779
\(609\) −4.09734e11 −0.120705
\(610\) 7.40768e12 2.16620
\(611\) −7.78374e12 −2.25945
\(612\) −7.17629e12 −2.06785
\(613\) −1.03194e12 −0.295178 −0.147589 0.989049i \(-0.547151\pi\)
−0.147589 + 0.989049i \(0.547151\pi\)
\(614\) −3.89866e12 −1.10702
\(615\) −8.93585e11 −0.251882
\(616\) 1.07744e13 3.01494
\(617\) 2.40238e12 0.667357 0.333679 0.942687i \(-0.391710\pi\)
0.333679 + 0.942687i \(0.391710\pi\)
\(618\) −4.39535e12 −1.21212
\(619\) 1.91546e12 0.524403 0.262201 0.965013i \(-0.415552\pi\)
0.262201 + 0.965013i \(0.415552\pi\)
\(620\) 1.05411e13 2.86500
\(621\) 2.86108e11 0.0772000
\(622\) −5.73978e11 −0.153758
\(623\) 8.70847e12 2.31604
\(624\) −1.65224e13 −4.36257
\(625\) −3.63482e12 −0.952845
\(626\) 8.15340e12 2.12204
\(627\) −3.39656e12 −0.877679
\(628\) −5.65659e12 −1.45123
\(629\) −3.69542e12 −0.941318
\(630\) −1.33968e13 −3.38820
\(631\) 4.65531e12 1.16901 0.584503 0.811391i \(-0.301289\pi\)
0.584503 + 0.811391i \(0.301289\pi\)
\(632\) −6.91597e12 −1.72435
\(633\) −3.25188e12 −0.805040
\(634\) 3.30322e12 0.811961
\(635\) −1.80547e12 −0.440665
\(636\) 2.03478e13 4.93129
\(637\) 1.21679e13 2.92812
\(638\) 3.06779e11 0.0733047
\(639\) 5.02653e12 1.19265
\(640\) −3.88248e12 −0.914742
\(641\) 5.52343e12 1.29225 0.646127 0.763230i \(-0.276388\pi\)
0.646127 + 0.763230i \(0.276388\pi\)
\(642\) 4.02635e12 0.935415
\(643\) 1.72571e12 0.398124 0.199062 0.979987i \(-0.436211\pi\)
0.199062 + 0.979987i \(0.436211\pi\)
\(644\) −4.34765e12 −0.996021
\(645\) −1.27646e12 −0.290396
\(646\) 8.20997e12 1.85479
\(647\) −5.22795e12 −1.17290 −0.586452 0.809984i \(-0.699476\pi\)
−0.586452 + 0.809984i \(0.699476\pi\)
\(648\) 1.26488e13 2.81813
\(649\) 5.10965e12 1.13055
\(650\) −1.38452e13 −3.04221
\(651\) −8.88076e12 −1.93792
\(652\) −3.64944e12 −0.790882
\(653\) −7.44348e12 −1.60202 −0.801008 0.598654i \(-0.795703\pi\)
−0.801008 + 0.598654i \(0.795703\pi\)
\(654\) 1.30444e13 2.78820
\(655\) −2.17122e12 −0.460912
\(656\) −1.28598e12 −0.271124
\(657\) −2.30751e12 −0.483169
\(658\) −2.08314e13 −4.33213
\(659\) −1.96795e11 −0.0406471 −0.0203236 0.999793i \(-0.506470\pi\)
−0.0203236 + 0.999793i \(0.506470\pi\)
\(660\) 1.60686e13 3.29633
\(661\) −5.13401e12 −1.04605 −0.523023 0.852319i \(-0.675196\pi\)
−0.523023 + 0.852319i \(0.675196\pi\)
\(662\) −1.40000e13 −2.83314
\(663\) 1.22268e13 2.45755
\(664\) 8.29049e12 1.65510
\(665\) 1.07052e13 2.12275
\(666\) −5.82753e12 −1.14776
\(667\) −7.03524e10 −0.0137630
\(668\) 2.17074e12 0.421808
\(669\) −7.82533e12 −1.51038
\(670\) 2.06790e13 3.96453
\(671\) 3.26378e12 0.621541
\(672\) −1.58174e13 −2.99208
\(673\) −8.52529e11 −0.160192 −0.0800961 0.996787i \(-0.525523\pi\)
−0.0800961 + 0.996787i \(0.525523\pi\)
\(674\) 8.22534e12 1.53527
\(675\) 1.70316e12 0.315784
\(676\) 1.95561e13 3.60182
\(677\) −1.41760e12 −0.259361 −0.129681 0.991556i \(-0.541395\pi\)
−0.129681 + 0.991556i \(0.541395\pi\)
\(678\) 2.42699e13 4.41096
\(679\) −1.00752e13 −1.81903
\(680\) −2.20735e13 −3.95896
\(681\) −1.25015e12 −0.222740
\(682\) 6.64925e12 1.17691
\(683\) −1.05881e12 −0.186176 −0.0930880 0.995658i \(-0.529674\pi\)
−0.0930880 + 0.995658i \(0.529674\pi\)
\(684\) 9.04304e12 1.57965
\(685\) 1.16964e12 0.202976
\(686\) 1.47883e13 2.54954
\(687\) 7.69735e12 1.31837
\(688\) −1.83700e12 −0.312580
\(689\) −1.51157e13 −2.55531
\(690\) −5.27569e12 −0.886050
\(691\) −2.62758e12 −0.438434 −0.219217 0.975676i \(-0.570350\pi\)
−0.219217 + 0.975676i \(0.570350\pi\)
\(692\) 9.03478e12 1.49775
\(693\) −5.90256e12 −0.972166
\(694\) −1.49891e13 −2.45277
\(695\) −1.29402e12 −0.210382
\(696\) −1.06461e12 −0.171969
\(697\) 9.51648e11 0.152732
\(698\) 4.41653e12 0.704258
\(699\) 1.46842e13 2.32649
\(700\) −2.58811e13 −4.07418
\(701\) 4.78305e11 0.0748124 0.0374062 0.999300i \(-0.488090\pi\)
0.0374062 + 0.999300i \(0.488090\pi\)
\(702\) −5.65932e12 −0.879523
\(703\) 4.65670e12 0.719084
\(704\) 1.86029e12 0.285432
\(705\) −1.76561e13 −2.69181
\(706\) −4.30555e12 −0.652241
\(707\) 6.43845e12 0.969156
\(708\) −3.12010e13 −4.66679
\(709\) −1.43959e12 −0.213959 −0.106979 0.994261i \(-0.534118\pi\)
−0.106979 + 0.994261i \(0.534118\pi\)
\(710\) 2.72050e13 4.01778
\(711\) 3.78880e12 0.556017
\(712\) 2.26272e13 3.29968
\(713\) −1.52485e12 −0.220965
\(714\) 3.27223e13 4.71196
\(715\) −1.19369e13 −1.70810
\(716\) −3.19193e12 −0.453885
\(717\) −5.61813e12 −0.793882
\(718\) −4.36757e10 −0.00613310
\(719\) −5.06222e12 −0.706417 −0.353209 0.935545i \(-0.614909\pi\)
−0.353209 + 0.935545i \(0.614909\pi\)
\(720\) −1.63410e13 −2.26612
\(721\) 6.10363e12 0.841162
\(722\) 2.95154e12 0.404232
\(723\) −2.69898e12 −0.367348
\(724\) −1.59232e13 −2.15380
\(725\) −4.18799e11 −0.0562970
\(726\) −8.01590e12 −1.07087
\(727\) 7.74749e12 1.02862 0.514311 0.857604i \(-0.328048\pi\)
0.514311 + 0.857604i \(0.328048\pi\)
\(728\) 4.88742e13 6.44895
\(729\) −3.86136e12 −0.506369
\(730\) −1.24889e13 −1.62769
\(731\) 1.35941e12 0.176084
\(732\) −1.99296e13 −2.56565
\(733\) 6.14525e10 0.00786270 0.00393135 0.999992i \(-0.498749\pi\)
0.00393135 + 0.999992i \(0.498749\pi\)
\(734\) 4.06411e12 0.516813
\(735\) 2.76009e13 3.48843
\(736\) −2.71588e12 −0.341162
\(737\) 9.11103e12 1.13753
\(738\) 1.50071e12 0.186227
\(739\) 1.45629e13 1.79617 0.898086 0.439819i \(-0.144957\pi\)
0.898086 + 0.439819i \(0.144957\pi\)
\(740\) −2.20302e13 −2.70069
\(741\) −1.54074e13 −1.87735
\(742\) −4.04538e13 −4.89938
\(743\) −3.15672e12 −0.380003 −0.190001 0.981784i \(-0.560849\pi\)
−0.190001 + 0.981784i \(0.560849\pi\)
\(744\) −2.30749e13 −2.76097
\(745\) 1.13020e13 1.34416
\(746\) 1.45640e13 1.72169
\(747\) −4.54181e12 −0.533686
\(748\) −1.71127e13 −1.99877
\(749\) −5.59122e12 −0.649141
\(750\) −1.35581e12 −0.156467
\(751\) −6.68968e12 −0.767407 −0.383704 0.923456i \(-0.625352\pi\)
−0.383704 + 0.923456i \(0.625352\pi\)
\(752\) −2.54095e13 −2.89744
\(753\) 8.72119e12 0.988549
\(754\) 1.39160e12 0.156799
\(755\) 1.13605e13 1.27244
\(756\) −1.05791e13 −1.17787
\(757\) 1.05129e12 0.116357 0.0581783 0.998306i \(-0.481471\pi\)
0.0581783 + 0.998306i \(0.481471\pi\)
\(758\) 3.35525e11 0.0369159
\(759\) −2.32444e12 −0.254232
\(760\) 2.78154e13 3.02430
\(761\) −9.91519e12 −1.07169 −0.535846 0.844315i \(-0.680007\pi\)
−0.535846 + 0.844315i \(0.680007\pi\)
\(762\) 6.95429e12 0.747232
\(763\) −1.81142e13 −1.93490
\(764\) −2.63873e13 −2.80205
\(765\) 1.20926e13 1.27657
\(766\) −1.69845e13 −1.78247
\(767\) 2.31782e13 2.41825
\(768\) 1.98581e13 2.05974
\(769\) 8.72311e12 0.899503 0.449752 0.893154i \(-0.351512\pi\)
0.449752 + 0.893154i \(0.351512\pi\)
\(770\) −3.19463e13 −3.27501
\(771\) −1.15502e13 −1.17719
\(772\) −3.09023e13 −3.13122
\(773\) 4.56418e12 0.459785 0.229893 0.973216i \(-0.426163\pi\)
0.229893 + 0.973216i \(0.426163\pi\)
\(774\) 2.14373e12 0.214702
\(775\) −9.07724e12 −0.903849
\(776\) −2.61785e13 −2.59159
\(777\) 1.85601e13 1.82678
\(778\) 2.28897e13 2.23991
\(779\) −1.19920e12 −0.116673
\(780\) 7.28899e13 7.05085
\(781\) 1.19864e13 1.15281
\(782\) 5.61849e12 0.537266
\(783\) −1.71187e11 −0.0162758
\(784\) 3.97212e13 3.75492
\(785\) 9.53178e12 0.895902
\(786\) 8.36308e12 0.781564
\(787\) 3.65924e12 0.340020 0.170010 0.985442i \(-0.445620\pi\)
0.170010 + 0.985442i \(0.445620\pi\)
\(788\) 1.02107e13 0.943386
\(789\) −1.15019e13 −1.05663
\(790\) 2.05060e13 1.87310
\(791\) −3.37025e13 −3.06103
\(792\) −1.53366e13 −1.38505
\(793\) 1.48050e13 1.32948
\(794\) 2.56856e13 2.29349
\(795\) −3.42876e13 −3.04428
\(796\) −5.66426e12 −0.500074
\(797\) 5.96161e12 0.523360 0.261680 0.965155i \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(798\) −4.12342e13 −3.59952
\(799\) 1.88034e13 1.63221
\(800\) −1.61673e13 −1.39551
\(801\) −1.23960e13 −1.06398
\(802\) −2.20551e13 −1.88245
\(803\) −5.50252e12 −0.467027
\(804\) −5.56345e13 −4.69561
\(805\) 7.32612e12 0.614883
\(806\) 3.01621e13 2.51741
\(807\) −1.06543e13 −0.884285
\(808\) 1.67290e13 1.38076
\(809\) 3.36558e12 0.276243 0.138121 0.990415i \(-0.455894\pi\)
0.138121 + 0.990415i \(0.455894\pi\)
\(810\) −3.75040e13 −3.06122
\(811\) −1.91615e13 −1.55537 −0.777687 0.628651i \(-0.783607\pi\)
−0.777687 + 0.628651i \(0.783607\pi\)
\(812\) 2.60134e12 0.209988
\(813\) 1.80757e13 1.45107
\(814\) −1.38964e13 −1.10941
\(815\) 6.14958e12 0.488243
\(816\) 3.99136e13 3.15148
\(817\) −1.71303e12 −0.134513
\(818\) 2.38410e13 1.86181
\(819\) −2.67750e13 −2.07946
\(820\) 5.67322e12 0.438195
\(821\) 2.01302e13 1.54634 0.773169 0.634200i \(-0.218670\pi\)
0.773169 + 0.634200i \(0.218670\pi\)
\(822\) −4.50520e12 −0.344185
\(823\) 2.08369e12 0.158319 0.0791596 0.996862i \(-0.474776\pi\)
0.0791596 + 0.996862i \(0.474776\pi\)
\(824\) 1.58591e13 1.19841
\(825\) −1.38371e13 −1.03993
\(826\) 6.20311e13 4.63660
\(827\) −1.11909e13 −0.831934 −0.415967 0.909380i \(-0.636557\pi\)
−0.415967 + 0.909380i \(0.636557\pi\)
\(828\) 6.18861e12 0.457569
\(829\) 1.03196e13 0.758868 0.379434 0.925219i \(-0.376119\pi\)
0.379434 + 0.925219i \(0.376119\pi\)
\(830\) −2.45815e13 −1.79787
\(831\) 2.05554e13 1.49528
\(832\) 8.43857e12 0.610540
\(833\) −2.93943e13 −2.11525
\(834\) 4.98429e12 0.356744
\(835\) −3.65787e12 −0.260399
\(836\) 2.15642e13 1.52688
\(837\) −3.71038e12 −0.261309
\(838\) 3.70255e13 2.59360
\(839\) −8.02040e12 −0.558814 −0.279407 0.960173i \(-0.590138\pi\)
−0.279407 + 0.960173i \(0.590138\pi\)
\(840\) 1.10863e14 7.68300
\(841\) −1.44651e13 −0.997098
\(842\) −2.30100e13 −1.57765
\(843\) 8.81242e12 0.600995
\(844\) 2.06456e13 1.40051
\(845\) −3.29535e13 −2.22355
\(846\) 2.96522e13 1.99017
\(847\) 1.11313e13 0.743142
\(848\) −4.93442e13 −3.27684
\(849\) −1.24257e13 −0.820797
\(850\) 3.34462e13 2.19767
\(851\) 3.18682e12 0.208293
\(852\) −7.31920e13 −4.75867
\(853\) −1.47888e13 −0.956448 −0.478224 0.878238i \(-0.658719\pi\)
−0.478224 + 0.878238i \(0.658719\pi\)
\(854\) 3.96223e13 2.54905
\(855\) −1.52382e13 −0.975184
\(856\) −1.45277e13 −0.924836
\(857\) −9.11573e12 −0.577268 −0.288634 0.957439i \(-0.593201\pi\)
−0.288634 + 0.957439i \(0.593201\pi\)
\(858\) 4.59783e13 2.89641
\(859\) −2.57161e13 −1.61152 −0.805760 0.592242i \(-0.798243\pi\)
−0.805760 + 0.592242i \(0.798243\pi\)
\(860\) 8.10406e12 0.505196
\(861\) −4.77961e12 −0.296400
\(862\) 1.44128e13 0.889130
\(863\) −1.98415e13 −1.21766 −0.608829 0.793302i \(-0.708360\pi\)
−0.608829 + 0.793302i \(0.708360\pi\)
\(864\) −6.60851e12 −0.403452
\(865\) −1.52243e13 −0.924623
\(866\) 3.77236e13 2.27920
\(867\) −7.38274e12 −0.443743
\(868\) 5.63825e13 3.37136
\(869\) 9.03484e12 0.537442
\(870\) 3.15661e12 0.186803
\(871\) 4.13291e13 2.43318
\(872\) −4.70662e13 −2.75667
\(873\) 1.43414e13 0.835659
\(874\) −7.08002e12 −0.410424
\(875\) 1.88275e12 0.108582
\(876\) 3.35999e13 1.92784
\(877\) 3.27261e13 1.86808 0.934041 0.357166i \(-0.116257\pi\)
0.934041 + 0.357166i \(0.116257\pi\)
\(878\) 5.71016e13 3.24282
\(879\) −9.42766e12 −0.532665
\(880\) −3.89671e13 −2.19041
\(881\) −1.23728e13 −0.691950 −0.345975 0.938244i \(-0.612452\pi\)
−0.345975 + 0.938244i \(0.612452\pi\)
\(882\) −4.63536e13 −2.57914
\(883\) −2.40830e13 −1.33318 −0.666589 0.745426i \(-0.732246\pi\)
−0.666589 + 0.745426i \(0.732246\pi\)
\(884\) −7.76261e13 −4.27536
\(885\) 5.25760e13 2.88100
\(886\) −5.43337e13 −2.96222
\(887\) −1.12077e13 −0.607941 −0.303971 0.952681i \(-0.598312\pi\)
−0.303971 + 0.952681i \(0.598312\pi\)
\(888\) 4.82248e13 2.60263
\(889\) −9.65712e12 −0.518549
\(890\) −6.70905e13 −3.58431
\(891\) −1.65240e13 −0.878347
\(892\) 4.96817e13 2.62757
\(893\) −2.36947e13 −1.24686
\(894\) −4.35327e13 −2.27928
\(895\) 5.37865e12 0.280201
\(896\) −2.07666e13 −1.07642
\(897\) −1.05440e13 −0.543802
\(898\) 1.46901e13 0.753845
\(899\) 9.12365e11 0.0465854
\(900\) 3.68401e13 1.87167
\(901\) 3.65155e13 1.84593
\(902\) 3.57862e12 0.180006
\(903\) −6.82757e12 −0.341721
\(904\) −8.75693e13 −4.36108
\(905\) 2.68318e13 1.32963
\(906\) −4.37584e13 −2.15767
\(907\) 9.35683e11 0.0459088 0.0229544 0.999737i \(-0.492693\pi\)
0.0229544 + 0.999737i \(0.492693\pi\)
\(908\) 7.93697e12 0.387497
\(909\) −9.16473e12 −0.445228
\(910\) −1.44914e14 −7.00524
\(911\) 1.46656e13 0.705452 0.352726 0.935727i \(-0.385255\pi\)
0.352726 + 0.935727i \(0.385255\pi\)
\(912\) −5.02962e13 −2.40746
\(913\) −1.08305e13 −0.515857
\(914\) −5.38167e13 −2.55070
\(915\) 3.35828e13 1.58388
\(916\) −4.88692e13 −2.29354
\(917\) −1.16134e13 −0.542374
\(918\) 1.36714e13 0.635360
\(919\) −2.92432e13 −1.35240 −0.676199 0.736719i \(-0.736374\pi\)
−0.676199 + 0.736719i \(0.736374\pi\)
\(920\) 1.90355e13 0.876029
\(921\) −1.76746e13 −0.809434
\(922\) 3.54029e13 1.61343
\(923\) 5.43720e13 2.46586
\(924\) 8.59480e13 3.87893
\(925\) 1.89707e13 0.852014
\(926\) −1.92821e13 −0.861796
\(927\) −8.68814e12 −0.386427
\(928\) 1.62500e12 0.0719262
\(929\) −7.23318e11 −0.0318609 −0.0159305 0.999873i \(-0.505071\pi\)
−0.0159305 + 0.999873i \(0.505071\pi\)
\(930\) 6.84178e13 2.99913
\(931\) 3.70406e13 1.61586
\(932\) −9.32273e13 −4.04736
\(933\) −2.60214e12 −0.112425
\(934\) −4.40276e13 −1.89306
\(935\) 2.88362e13 1.23392
\(936\) −6.95694e13 −2.96263
\(937\) 3.23620e13 1.37154 0.685768 0.727821i \(-0.259467\pi\)
0.685768 + 0.727821i \(0.259467\pi\)
\(938\) 1.10608e14 4.66523
\(939\) 3.69635e13 1.55159
\(940\) 1.12096e14 4.68290
\(941\) 3.92794e12 0.163309 0.0816547 0.996661i \(-0.473980\pi\)
0.0816547 + 0.996661i \(0.473980\pi\)
\(942\) −3.67144e13 −1.51917
\(943\) −8.20671e11 −0.0337961
\(944\) 7.56636e13 3.10108
\(945\) 1.78265e13 0.727149
\(946\) 5.11197e12 0.207529
\(947\) 4.49509e13 1.81620 0.908099 0.418755i \(-0.137533\pi\)
0.908099 + 0.418755i \(0.137533\pi\)
\(948\) −5.51693e13 −2.21850
\(949\) −2.49603e13 −0.998970
\(950\) −4.21465e13 −1.67882
\(951\) 1.49752e13 0.593690
\(952\) −1.18067e14 −4.65867
\(953\) −1.50521e13 −0.591126 −0.295563 0.955323i \(-0.595507\pi\)
−0.295563 + 0.955323i \(0.595507\pi\)
\(954\) 5.75834e13 2.25076
\(955\) 4.44647e13 1.72982
\(956\) 3.56686e13 1.38110
\(957\) 1.39078e12 0.0535989
\(958\) −7.08536e13 −2.71780
\(959\) 6.25618e12 0.238850
\(960\) 1.91415e13 0.727371
\(961\) −6.66464e12 −0.252070
\(962\) −6.30365e13 −2.37303
\(963\) 7.95876e12 0.298213
\(964\) 1.71354e13 0.639068
\(965\) 5.20728e13 1.93303
\(966\) −2.82187e13 −1.04265
\(967\) −7.83639e12 −0.288202 −0.144101 0.989563i \(-0.546029\pi\)
−0.144101 + 0.989563i \(0.546029\pi\)
\(968\) 2.89226e13 1.05876
\(969\) 3.72200e13 1.35619
\(970\) 7.76199e13 2.81514
\(971\) −1.42988e13 −0.516195 −0.258098 0.966119i \(-0.583096\pi\)
−0.258098 + 0.966119i \(0.583096\pi\)
\(972\) 8.14217e13 2.92578
\(973\) −6.92147e12 −0.247566
\(974\) −5.91920e13 −2.10741
\(975\) −6.27674e13 −2.22440
\(976\) 4.83300e13 1.70488
\(977\) 5.21673e13 1.83178 0.915888 0.401433i \(-0.131488\pi\)
0.915888 + 0.401433i \(0.131488\pi\)
\(978\) −2.36869e13 −0.827909
\(979\) −2.95597e13 −1.02844
\(980\) −1.75234e14 −6.06876
\(981\) 2.57844e13 0.888888
\(982\) 9.13271e13 3.13399
\(983\) −2.02089e13 −0.690322 −0.345161 0.938544i \(-0.612176\pi\)
−0.345161 + 0.938544i \(0.612176\pi\)
\(984\) −1.24189e13 −0.422284
\(985\) −1.72059e13 −0.582390
\(986\) −3.36172e12 −0.113270
\(987\) −9.44393e13 −3.16757
\(988\) 9.78188e13 3.26600
\(989\) −1.17231e12 −0.0389636
\(990\) 4.54736e13 1.50453
\(991\) 1.19883e13 0.394846 0.197423 0.980318i \(-0.436743\pi\)
0.197423 + 0.980318i \(0.436743\pi\)
\(992\) 3.52209e13 1.15478
\(993\) −6.34692e13 −2.07153
\(994\) 1.45514e14 4.72788
\(995\) 9.54470e12 0.308716
\(996\) 6.61340e13 2.12940
\(997\) −1.83297e13 −0.587525 −0.293763 0.955878i \(-0.594908\pi\)
−0.293763 + 0.955878i \(0.594908\pi\)
\(998\) −4.31999e13 −1.37846
\(999\) 7.75442e12 0.246323
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.1.2 15
3.2 odd 2 387.10.a.c.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.2 15 1.1 even 1 trivial
387.10.a.c.1.14 15 3.2 odd 2