Properties

Label 38.10.a.d.1.2
Level $38$
Weight $10$
Character 38.1
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(219.264\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} -66.6053 q^{3} +256.000 q^{4} -2418.56 q^{5} +1065.68 q^{6} -1533.95 q^{7} -4096.00 q^{8} -15246.7 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} -66.6053 q^{3} +256.000 q^{4} -2418.56 q^{5} +1065.68 q^{6} -1533.95 q^{7} -4096.00 q^{8} -15246.7 q^{9} +38696.9 q^{10} -5815.50 q^{11} -17051.0 q^{12} -135546. q^{13} +24543.2 q^{14} +161089. q^{15} +65536.0 q^{16} -448133. q^{17} +243948. q^{18} +130321. q^{19} -619151. q^{20} +102169. q^{21} +93048.1 q^{22} +2.07551e6 q^{23} +272815. q^{24} +3.89629e6 q^{25} +2.16874e6 q^{26} +2.32650e6 q^{27} -392691. q^{28} -643630. q^{29} -2.57742e6 q^{30} -194155. q^{31} -1.04858e6 q^{32} +387343. q^{33} +7.17012e6 q^{34} +3.70994e6 q^{35} -3.90316e6 q^{36} +9.46515e6 q^{37} -2.08514e6 q^{38} +9.02808e6 q^{39} +9.90641e6 q^{40} -2.34337e7 q^{41} -1.63471e6 q^{42} +3.80295e7 q^{43} -1.48877e6 q^{44} +3.68751e7 q^{45} -3.32081e7 q^{46} -2.28118e7 q^{47} -4.36504e6 q^{48} -3.80006e7 q^{49} -6.23407e7 q^{50} +2.98480e7 q^{51} -3.46998e7 q^{52} -6.65359e7 q^{53} -3.72241e7 q^{54} +1.40651e7 q^{55} +6.28306e6 q^{56} -8.68007e6 q^{57} +1.02981e7 q^{58} -9.14212e7 q^{59} +4.12387e7 q^{60} -4.02765e7 q^{61} +3.10649e6 q^{62} +2.33877e7 q^{63} +1.67772e7 q^{64} +3.27826e8 q^{65} -6.19749e6 q^{66} +2.42739e8 q^{67} -1.14722e8 q^{68} -1.38240e8 q^{69} -5.93591e7 q^{70} -1.83180e8 q^{71} +6.24506e7 q^{72} -1.39336e8 q^{73} -1.51442e8 q^{74} -2.59514e8 q^{75} +3.33622e7 q^{76} +8.92069e6 q^{77} -1.44449e8 q^{78} +2.51527e8 q^{79} -1.58503e8 q^{80} +1.45144e8 q^{81} +3.74939e8 q^{82} -1.30417e8 q^{83} +2.61553e7 q^{84} +1.08383e9 q^{85} -6.08473e8 q^{86} +4.28692e7 q^{87} +2.38203e7 q^{88} +7.10883e8 q^{89} -5.90002e8 q^{90} +2.07921e8 q^{91} +5.31330e8 q^{92} +1.29318e7 q^{93} +3.64988e8 q^{94} -3.15189e8 q^{95} +6.98407e7 q^{96} -1.31644e9 q^{97} +6.08010e8 q^{98} +8.86675e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + O(q^{10}) \) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + 9760585q^{25} - 1927776q^{26} + 12387978q^{27} + 3150592q^{28} + 6153240q^{29} + 9457440q^{30} + 12774024q^{31} - 4194304q^{32} - 3258022q^{33} + 6594224q^{34} + 9823425q^{35} + 4233728q^{36} + 20506048q^{37} - 8340544q^{38} + 69881444q^{39} + 5713920q^{40} + 11620300q^{41} - 38992096q^{42} + 7698327q^{43} - 26687744q^{44} - 124015815q^{45} - 48164800q^{46} - 31581083q^{47} + 5505024q^{48} + 18970383q^{49} - 156169360q^{50} - 8594812q^{51} + 30844416q^{52} + 72549422q^{53} - 198207648q^{54} + 21332505q^{55} - 50409472q^{56} + 10946964q^{57} - 98451840q^{58} - 149234120q^{59} - 151319040q^{60} + 129004373q^{61} - 204384384q^{62} + 102967551q^{63} + 67108864q^{64} + 124691700q^{65} + 52128352q^{66} + 132595266q^{67} - 105507584q^{68} - 45529972q^{69} - 157174800q^{70} - 47138482q^{71} - 67739648q^{72} - 39332795q^{73} - 328096768q^{74} + 824627010q^{75} + 133448704q^{76} - 165933719q^{77} - 1118103104q^{78} - 307010840q^{79} - 91422720q^{80} + 1305551744q^{81} - 185924800q^{82} - 746568232q^{83} + 623873536q^{84} - 105005985q^{85} - 123173232q^{86} - 82148208q^{87} + 427003904q^{88} + 286943482q^{89} + 1984253040q^{90} + 3155781114q^{91} + 770636800q^{92} + 1151901596q^{93} + 505297328q^{94} - 181797795q^{95} - 88080384q^{96} + 793519958q^{97} - 303526128q^{98} - 1681833809q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) −66.6053 −0.474748 −0.237374 0.971418i \(-0.576287\pi\)
−0.237374 + 0.971418i \(0.576287\pi\)
\(4\) 256.000 0.500000
\(5\) −2418.56 −1.73058 −0.865289 0.501273i \(-0.832865\pi\)
−0.865289 + 0.501273i \(0.832865\pi\)
\(6\) 1065.68 0.335697
\(7\) −1533.95 −0.241474 −0.120737 0.992685i \(-0.538526\pi\)
−0.120737 + 0.992685i \(0.538526\pi\)
\(8\) −4096.00 −0.353553
\(9\) −15246.7 −0.774615
\(10\) 38696.9 1.22370
\(11\) −5815.50 −0.119762 −0.0598812 0.998206i \(-0.519072\pi\)
−0.0598812 + 0.998206i \(0.519072\pi\)
\(12\) −17051.0 −0.237374
\(13\) −135546. −1.31626 −0.658130 0.752904i \(-0.728652\pi\)
−0.658130 + 0.752904i \(0.728652\pi\)
\(14\) 24543.2 0.170748
\(15\) 161089. 0.821588
\(16\) 65536.0 0.250000
\(17\) −448133. −1.30133 −0.650663 0.759366i \(-0.725509\pi\)
−0.650663 + 0.759366i \(0.725509\pi\)
\(18\) 243948. 0.547735
\(19\) 130321. 0.229416
\(20\) −619151. −0.865289
\(21\) 102169. 0.114639
\(22\) 93048.1 0.0846848
\(23\) 2.07551e6 1.54650 0.773249 0.634102i \(-0.218630\pi\)
0.773249 + 0.634102i \(0.218630\pi\)
\(24\) 272815. 0.167849
\(25\) 3.89629e6 1.99490
\(26\) 2.16874e6 0.930737
\(27\) 2.32650e6 0.842494
\(28\) −392691. −0.120737
\(29\) −643630. −0.168984 −0.0844920 0.996424i \(-0.526927\pi\)
−0.0844920 + 0.996424i \(0.526927\pi\)
\(30\) −2.57742e6 −0.580951
\(31\) −194155. −0.0377591 −0.0188796 0.999822i \(-0.506010\pi\)
−0.0188796 + 0.999822i \(0.506010\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) 387343. 0.0568569
\(34\) 7.17012e6 0.920177
\(35\) 3.70994e6 0.417889
\(36\) −3.90316e6 −0.387307
\(37\) 9.46515e6 0.830271 0.415135 0.909760i \(-0.363734\pi\)
0.415135 + 0.909760i \(0.363734\pi\)
\(38\) −2.08514e6 −0.162221
\(39\) 9.02808e6 0.624892
\(40\) 9.90641e6 0.611852
\(41\) −2.34337e7 −1.29513 −0.647565 0.762010i \(-0.724213\pi\)
−0.647565 + 0.762010i \(0.724213\pi\)
\(42\) −1.63471e6 −0.0810621
\(43\) 3.80295e7 1.69634 0.848170 0.529724i \(-0.177704\pi\)
0.848170 + 0.529724i \(0.177704\pi\)
\(44\) −1.48877e6 −0.0598812
\(45\) 3.68751e7 1.34053
\(46\) −3.32081e7 −1.09354
\(47\) −2.28118e7 −0.681897 −0.340948 0.940082i \(-0.610748\pi\)
−0.340948 + 0.940082i \(0.610748\pi\)
\(48\) −4.36504e6 −0.118687
\(49\) −3.80006e7 −0.941690
\(50\) −6.23407e7 −1.41061
\(51\) 2.98480e7 0.617802
\(52\) −3.46998e7 −0.658130
\(53\) −6.65359e7 −1.15828 −0.579141 0.815227i \(-0.696612\pi\)
−0.579141 + 0.815227i \(0.696612\pi\)
\(54\) −3.72241e7 −0.595733
\(55\) 1.40651e7 0.207258
\(56\) 6.28306e6 0.0853739
\(57\) −8.68007e6 −0.108915
\(58\) 1.02981e7 0.119490
\(59\) −9.14212e7 −0.982230 −0.491115 0.871095i \(-0.663410\pi\)
−0.491115 + 0.871095i \(0.663410\pi\)
\(60\) 4.12387e7 0.410794
\(61\) −4.02765e7 −0.372450 −0.186225 0.982507i \(-0.559625\pi\)
−0.186225 + 0.982507i \(0.559625\pi\)
\(62\) 3.10649e6 0.0266997
\(63\) 2.33877e7 0.187049
\(64\) 1.67772e7 0.125000
\(65\) 3.27826e8 2.27789
\(66\) −6.19749e6 −0.0402039
\(67\) 2.42739e8 1.47164 0.735822 0.677175i \(-0.236796\pi\)
0.735822 + 0.677175i \(0.236796\pi\)
\(68\) −1.14722e8 −0.650663
\(69\) −1.38240e8 −0.734197
\(70\) −5.93591e7 −0.295492
\(71\) −1.83180e8 −0.855491 −0.427745 0.903899i \(-0.640692\pi\)
−0.427745 + 0.903899i \(0.640692\pi\)
\(72\) 6.24506e7 0.273868
\(73\) −1.39336e8 −0.574264 −0.287132 0.957891i \(-0.592702\pi\)
−0.287132 + 0.957891i \(0.592702\pi\)
\(74\) −1.51442e8 −0.587090
\(75\) −2.59514e8 −0.947075
\(76\) 3.33622e7 0.114708
\(77\) 8.92069e6 0.0289195
\(78\) −1.44449e8 −0.441865
\(79\) 2.51527e8 0.726546 0.363273 0.931683i \(-0.381659\pi\)
0.363273 + 0.931683i \(0.381659\pi\)
\(80\) −1.58503e8 −0.432645
\(81\) 1.45144e8 0.374642
\(82\) 3.74939e8 0.915795
\(83\) −1.30417e8 −0.301636 −0.150818 0.988562i \(-0.548191\pi\)
−0.150818 + 0.988562i \(0.548191\pi\)
\(84\) 2.61553e7 0.0573196
\(85\) 1.08383e9 2.25205
\(86\) −6.08473e8 −1.19949
\(87\) 4.28692e7 0.0802247
\(88\) 2.38203e7 0.0423424
\(89\) 7.10883e8 1.20100 0.600500 0.799625i \(-0.294968\pi\)
0.600500 + 0.799625i \(0.294968\pi\)
\(90\) −5.90002e8 −0.947899
\(91\) 2.07921e8 0.317842
\(92\) 5.31330e8 0.773249
\(93\) 1.29318e7 0.0179261
\(94\) 3.64988e8 0.482174
\(95\) −3.15189e8 −0.397022
\(96\) 6.98407e7 0.0839244
\(97\) −1.31644e9 −1.50983 −0.754916 0.655822i \(-0.772322\pi\)
−0.754916 + 0.655822i \(0.772322\pi\)
\(98\) 6.08010e8 0.665876
\(99\) 8.86675e7 0.0927696
\(100\) 9.97451e8 0.997451
\(101\) 1.38054e9 1.32009 0.660044 0.751227i \(-0.270538\pi\)
0.660044 + 0.751227i \(0.270538\pi\)
\(102\) −4.77568e8 −0.436852
\(103\) −9.35066e8 −0.818606 −0.409303 0.912399i \(-0.634228\pi\)
−0.409303 + 0.912399i \(0.634228\pi\)
\(104\) 5.55197e8 0.465368
\(105\) −2.47102e8 −0.198392
\(106\) 1.06457e9 0.819030
\(107\) −4.27772e8 −0.315490 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(108\) 5.95585e8 0.421247
\(109\) −1.13801e8 −0.0772197 −0.0386098 0.999254i \(-0.512293\pi\)
−0.0386098 + 0.999254i \(0.512293\pi\)
\(110\) −2.25042e8 −0.146554
\(111\) −6.30429e8 −0.394169
\(112\) −1.00529e8 −0.0603684
\(113\) 1.37096e9 0.790990 0.395495 0.918468i \(-0.370573\pi\)
0.395495 + 0.918468i \(0.370573\pi\)
\(114\) 1.38881e8 0.0770143
\(115\) −5.01974e9 −2.67634
\(116\) −1.64769e8 −0.0844920
\(117\) 2.06663e9 1.01959
\(118\) 1.46274e9 0.694541
\(119\) 6.87413e8 0.314236
\(120\) −6.59819e8 −0.290475
\(121\) −2.32413e9 −0.985657
\(122\) 6.44424e8 0.263362
\(123\) 1.56081e9 0.614860
\(124\) −4.97038e7 −0.0188796
\(125\) −4.69966e9 −1.72176
\(126\) −3.74204e8 −0.132264
\(127\) 1.64706e9 0.561813 0.280906 0.959735i \(-0.409365\pi\)
0.280906 + 0.959735i \(0.409365\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −2.53297e9 −0.805334
\(130\) −5.24521e9 −1.61071
\(131\) −3.65669e9 −1.08485 −0.542423 0.840106i \(-0.682493\pi\)
−0.542423 + 0.840106i \(0.682493\pi\)
\(132\) 9.91599e7 0.0284285
\(133\) −1.99906e8 −0.0553979
\(134\) −3.88382e9 −1.04061
\(135\) −5.62678e9 −1.45800
\(136\) 1.83555e9 0.460088
\(137\) −3.24163e9 −0.786179 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(138\) 2.21184e9 0.519155
\(139\) 8.69038e9 1.97457 0.987285 0.158963i \(-0.0508151\pi\)
0.987285 + 0.158963i \(0.0508151\pi\)
\(140\) 9.49746e8 0.208945
\(141\) 1.51938e9 0.323729
\(142\) 2.93088e9 0.604923
\(143\) 7.88269e8 0.157638
\(144\) −9.99210e8 −0.193654
\(145\) 1.55666e9 0.292440
\(146\) 2.22938e9 0.406066
\(147\) 2.53104e9 0.447065
\(148\) 2.42308e9 0.415135
\(149\) −7.11665e9 −1.18287 −0.591435 0.806353i \(-0.701438\pi\)
−0.591435 + 0.806353i \(0.701438\pi\)
\(150\) 4.15222e9 0.669683
\(151\) 1.87879e9 0.294091 0.147046 0.989130i \(-0.453024\pi\)
0.147046 + 0.989130i \(0.453024\pi\)
\(152\) −5.33795e8 −0.0811107
\(153\) 6.83256e9 1.00803
\(154\) −1.42731e8 −0.0204491
\(155\) 4.69576e8 0.0653451
\(156\) 2.31119e9 0.312446
\(157\) 4.95963e9 0.651480 0.325740 0.945459i \(-0.394387\pi\)
0.325740 + 0.945459i \(0.394387\pi\)
\(158\) −4.02444e9 −0.513746
\(159\) 4.43164e9 0.549892
\(160\) 2.53604e9 0.305926
\(161\) −3.18373e9 −0.373439
\(162\) −2.32230e9 −0.264912
\(163\) 2.87539e9 0.319046 0.159523 0.987194i \(-0.449004\pi\)
0.159523 + 0.987194i \(0.449004\pi\)
\(164\) −5.99902e9 −0.647565
\(165\) −9.36812e8 −0.0983953
\(166\) 2.08668e9 0.213289
\(167\) 1.47841e10 1.47085 0.735427 0.677604i \(-0.236982\pi\)
0.735427 + 0.677604i \(0.236982\pi\)
\(168\) −4.18485e8 −0.0405310
\(169\) 7.76823e9 0.732541
\(170\) −1.73413e10 −1.59244
\(171\) −1.98697e9 −0.177709
\(172\) 9.73556e9 0.848170
\(173\) 1.45582e10 1.23566 0.617832 0.786310i \(-0.288011\pi\)
0.617832 + 0.786310i \(0.288011\pi\)
\(174\) −6.85907e8 −0.0567275
\(175\) −5.97672e9 −0.481716
\(176\) −3.81125e8 −0.0299406
\(177\) 6.08914e9 0.466311
\(178\) −1.13741e10 −0.849235
\(179\) −1.82403e10 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(180\) 9.44003e9 0.670266
\(181\) 2.81167e10 1.94720 0.973602 0.228252i \(-0.0733011\pi\)
0.973602 + 0.228252i \(0.0733011\pi\)
\(182\) −3.32673e9 −0.224748
\(183\) 2.68263e9 0.176820
\(184\) −8.50129e9 −0.546770
\(185\) −2.28920e10 −1.43685
\(186\) −2.06908e8 −0.0126756
\(187\) 2.60612e9 0.155850
\(188\) −5.83981e9 −0.340948
\(189\) −3.56874e9 −0.203440
\(190\) 5.04302e9 0.280737
\(191\) 2.82317e8 0.0153492 0.00767462 0.999971i \(-0.497557\pi\)
0.00767462 + 0.999971i \(0.497557\pi\)
\(192\) −1.11745e9 −0.0593435
\(193\) 1.43117e10 0.742477 0.371239 0.928538i \(-0.378933\pi\)
0.371239 + 0.928538i \(0.378933\pi\)
\(194\) 2.10630e10 1.06761
\(195\) −2.18349e10 −1.08142
\(196\) −9.72816e9 −0.470845
\(197\) 2.02116e10 0.956100 0.478050 0.878333i \(-0.341344\pi\)
0.478050 + 0.878333i \(0.341344\pi\)
\(198\) −1.41868e9 −0.0655980
\(199\) −3.95924e10 −1.78967 −0.894835 0.446398i \(-0.852707\pi\)
−0.894835 + 0.446398i \(0.852707\pi\)
\(200\) −1.59592e10 −0.705304
\(201\) −1.61677e10 −0.698660
\(202\) −2.20886e10 −0.933443
\(203\) 9.87296e8 0.0408052
\(204\) 7.64109e9 0.308901
\(205\) 5.66757e10 2.24132
\(206\) 1.49611e10 0.578842
\(207\) −3.16447e10 −1.19794
\(208\) −8.88315e9 −0.329065
\(209\) −7.57882e8 −0.0274754
\(210\) 3.95363e9 0.140284
\(211\) −1.31017e10 −0.455047 −0.227523 0.973773i \(-0.573063\pi\)
−0.227523 + 0.973773i \(0.573063\pi\)
\(212\) −1.70332e10 −0.579141
\(213\) 1.22008e10 0.406142
\(214\) 6.84436e9 0.223085
\(215\) −9.19766e10 −2.93565
\(216\) −9.52936e9 −0.297867
\(217\) 2.97825e8 0.00911784
\(218\) 1.82082e9 0.0546026
\(219\) 9.28054e9 0.272631
\(220\) 3.60067e9 0.103629
\(221\) 6.07426e10 1.71288
\(222\) 1.00869e10 0.278720
\(223\) 6.50346e10 1.76105 0.880526 0.473997i \(-0.157189\pi\)
0.880526 + 0.473997i \(0.157189\pi\)
\(224\) 1.60846e9 0.0426869
\(225\) −5.94058e10 −1.54528
\(226\) −2.19353e10 −0.559315
\(227\) 1.02990e10 0.257442 0.128721 0.991681i \(-0.458913\pi\)
0.128721 + 0.991681i \(0.458913\pi\)
\(228\) −2.22210e9 −0.0544573
\(229\) 1.71789e10 0.412797 0.206398 0.978468i \(-0.433826\pi\)
0.206398 + 0.978468i \(0.433826\pi\)
\(230\) 8.03158e10 1.89246
\(231\) −5.94165e8 −0.0137294
\(232\) 2.63631e9 0.0597448
\(233\) −6.52465e10 −1.45029 −0.725146 0.688595i \(-0.758228\pi\)
−0.725146 + 0.688595i \(0.758228\pi\)
\(234\) −3.30662e10 −0.720962
\(235\) 5.51716e10 1.18008
\(236\) −2.34038e10 −0.491115
\(237\) −1.67530e10 −0.344926
\(238\) −1.09986e10 −0.222199
\(239\) −1.43024e10 −0.283543 −0.141771 0.989899i \(-0.545280\pi\)
−0.141771 + 0.989899i \(0.545280\pi\)
\(240\) 1.05571e10 0.205397
\(241\) −4.79560e10 −0.915727 −0.457864 0.889022i \(-0.651385\pi\)
−0.457864 + 0.889022i \(0.651385\pi\)
\(242\) 3.71860e10 0.696965
\(243\) −5.54600e10 −1.02035
\(244\) −1.03108e10 −0.186225
\(245\) 9.19066e10 1.62967
\(246\) −2.49729e10 −0.434772
\(247\) −1.76645e10 −0.301971
\(248\) 7.95261e8 0.0133499
\(249\) 8.68647e9 0.143201
\(250\) 7.51946e10 1.21747
\(251\) −1.25023e10 −0.198819 −0.0994096 0.995047i \(-0.531695\pi\)
−0.0994096 + 0.995047i \(0.531695\pi\)
\(252\) 5.98726e9 0.0935245
\(253\) −1.20701e10 −0.185212
\(254\) −2.63529e10 −0.397261
\(255\) −7.21891e10 −1.06915
\(256\) 4.29497e9 0.0625000
\(257\) 1.83653e10 0.262602 0.131301 0.991343i \(-0.458085\pi\)
0.131301 + 0.991343i \(0.458085\pi\)
\(258\) 4.05275e10 0.569457
\(259\) −1.45191e10 −0.200489
\(260\) 8.39234e10 1.13895
\(261\) 9.81326e9 0.130897
\(262\) 5.85071e10 0.767101
\(263\) 8.39157e10 1.08154 0.540770 0.841171i \(-0.318133\pi\)
0.540770 + 0.841171i \(0.318133\pi\)
\(264\) −1.58656e9 −0.0201020
\(265\) 1.60921e11 2.00450
\(266\) 3.19849e9 0.0391722
\(267\) −4.73485e10 −0.570172
\(268\) 6.21412e10 0.735822
\(269\) 9.33233e9 0.108669 0.0543344 0.998523i \(-0.482696\pi\)
0.0543344 + 0.998523i \(0.482696\pi\)
\(270\) 9.00286e10 1.03096
\(271\) −1.57854e11 −1.77785 −0.888924 0.458055i \(-0.848546\pi\)
−0.888924 + 0.458055i \(0.848546\pi\)
\(272\) −2.93688e10 −0.325332
\(273\) −1.38486e10 −0.150895
\(274\) 5.18661e10 0.555912
\(275\) −2.26589e10 −0.238914
\(276\) −3.53894e10 −0.367098
\(277\) −1.12981e11 −1.15305 −0.576525 0.817080i \(-0.695592\pi\)
−0.576525 + 0.817080i \(0.695592\pi\)
\(278\) −1.39046e11 −1.39623
\(279\) 2.96024e9 0.0292488
\(280\) −1.51959e10 −0.147746
\(281\) 7.83548e10 0.749699 0.374849 0.927086i \(-0.377694\pi\)
0.374849 + 0.927086i \(0.377694\pi\)
\(282\) −2.43102e10 −0.228911
\(283\) 8.46004e10 0.784032 0.392016 0.919958i \(-0.371778\pi\)
0.392016 + 0.919958i \(0.371778\pi\)
\(284\) −4.68941e10 −0.427745
\(285\) 2.09932e10 0.188485
\(286\) −1.26123e10 −0.111467
\(287\) 3.59461e10 0.312740
\(288\) 1.59874e10 0.136934
\(289\) 8.22349e10 0.693451
\(290\) −2.49065e10 −0.206786
\(291\) 8.76819e10 0.716789
\(292\) −3.56701e10 −0.287132
\(293\) −1.06621e11 −0.845157 −0.422579 0.906326i \(-0.638875\pi\)
−0.422579 + 0.906326i \(0.638875\pi\)
\(294\) −4.04967e10 −0.316123
\(295\) 2.21107e11 1.69983
\(296\) −3.87693e10 −0.293545
\(297\) −1.35298e10 −0.100899
\(298\) 1.13866e11 0.836416
\(299\) −2.81327e11 −2.03559
\(300\) −6.64355e10 −0.473538
\(301\) −5.83354e10 −0.409622
\(302\) −3.00606e10 −0.207954
\(303\) −9.19513e10 −0.626709
\(304\) 8.54072e9 0.0573539
\(305\) 9.74110e10 0.644553
\(306\) −1.09321e11 −0.712782
\(307\) 1.65292e11 1.06201 0.531006 0.847368i \(-0.321814\pi\)
0.531006 + 0.847368i \(0.321814\pi\)
\(308\) 2.28370e9 0.0144597
\(309\) 6.22803e10 0.388631
\(310\) −7.51322e9 −0.0462060
\(311\) −1.08969e11 −0.660515 −0.330257 0.943891i \(-0.607136\pi\)
−0.330257 + 0.943891i \(0.607136\pi\)
\(312\) −3.69790e10 −0.220933
\(313\) 9.48622e10 0.558655 0.279328 0.960196i \(-0.409888\pi\)
0.279328 + 0.960196i \(0.409888\pi\)
\(314\) −7.93541e10 −0.460666
\(315\) −5.65645e10 −0.323703
\(316\) 6.43910e10 0.363273
\(317\) 2.75000e11 1.52956 0.764780 0.644291i \(-0.222848\pi\)
0.764780 + 0.644291i \(0.222848\pi\)
\(318\) −7.09063e10 −0.388833
\(319\) 3.74303e9 0.0202379
\(320\) −4.05767e10 −0.216322
\(321\) 2.84919e10 0.149778
\(322\) 5.09396e10 0.264061
\(323\) −5.84011e10 −0.298545
\(324\) 3.71569e10 0.187321
\(325\) −5.28127e11 −2.62581
\(326\) −4.60063e10 −0.225599
\(327\) 7.57977e9 0.0366599
\(328\) 9.59844e10 0.457898
\(329\) 3.49921e10 0.164660
\(330\) 1.49890e10 0.0695760
\(331\) −1.28057e11 −0.586378 −0.293189 0.956055i \(-0.594716\pi\)
−0.293189 + 0.956055i \(0.594716\pi\)
\(332\) −3.33868e10 −0.150818
\(333\) −1.44313e11 −0.643140
\(334\) −2.36545e11 −1.04005
\(335\) −5.87078e11 −2.54680
\(336\) 6.69576e9 0.0286598
\(337\) −3.03698e10 −0.128265 −0.0641324 0.997941i \(-0.520428\pi\)
−0.0641324 + 0.997941i \(0.520428\pi\)
\(338\) −1.24292e11 −0.517985
\(339\) −9.13130e10 −0.375521
\(340\) 2.77462e11 1.12602
\(341\) 1.12911e9 0.00452212
\(342\) 3.17915e10 0.125659
\(343\) 1.20191e11 0.468867
\(344\) −1.55769e11 −0.599747
\(345\) 3.34341e11 1.27059
\(346\) −2.32931e11 −0.873747
\(347\) 4.03811e11 1.49519 0.747594 0.664156i \(-0.231209\pi\)
0.747594 + 0.664156i \(0.231209\pi\)
\(348\) 1.09745e10 0.0401124
\(349\) 6.29420e10 0.227105 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 9.56275e10 0.340625
\(351\) −3.15349e11 −1.10894
\(352\) 6.09800e9 0.0211712
\(353\) 9.36033e10 0.320852 0.160426 0.987048i \(-0.448713\pi\)
0.160426 + 0.987048i \(0.448713\pi\)
\(354\) −9.74262e10 −0.329732
\(355\) 4.43031e11 1.48049
\(356\) 1.81986e11 0.600500
\(357\) −4.57853e10 −0.149183
\(358\) 2.91846e11 0.939030
\(359\) 3.18605e11 1.01234 0.506171 0.862433i \(-0.331060\pi\)
0.506171 + 0.862433i \(0.331060\pi\)
\(360\) −1.51040e11 −0.473949
\(361\) 1.69836e10 0.0526316
\(362\) −4.49868e11 −1.37688
\(363\) 1.54799e11 0.467939
\(364\) 5.32277e10 0.158921
\(365\) 3.36993e11 0.993810
\(366\) −4.29220e10 −0.125030
\(367\) 9.81213e10 0.282336 0.141168 0.989986i \(-0.454914\pi\)
0.141168 + 0.989986i \(0.454914\pi\)
\(368\) 1.36021e11 0.386625
\(369\) 3.57287e11 1.00323
\(370\) 3.66272e11 1.01601
\(371\) 1.02063e11 0.279695
\(372\) 3.31054e9 0.00896303
\(373\) 2.24989e11 0.601827 0.300913 0.953652i \(-0.402709\pi\)
0.300913 + 0.953652i \(0.402709\pi\)
\(374\) −4.16979e10 −0.110203
\(375\) 3.13022e11 0.817400
\(376\) 9.34370e10 0.241087
\(377\) 8.72415e10 0.222427
\(378\) 5.70999e10 0.143854
\(379\) 6.24000e11 1.55349 0.776744 0.629817i \(-0.216870\pi\)
0.776744 + 0.629817i \(0.216870\pi\)
\(380\) −8.06883e10 −0.198511
\(381\) −1.09703e11 −0.266719
\(382\) −4.51707e9 −0.0108536
\(383\) 5.36899e11 1.27497 0.637483 0.770465i \(-0.279976\pi\)
0.637483 + 0.770465i \(0.279976\pi\)
\(384\) 1.78792e10 0.0419622
\(385\) −2.15752e10 −0.0500474
\(386\) −2.28987e11 −0.525011
\(387\) −5.79826e11 −1.31401
\(388\) −3.37009e11 −0.754916
\(389\) −1.30155e11 −0.288195 −0.144097 0.989564i \(-0.546028\pi\)
−0.144097 + 0.989564i \(0.546028\pi\)
\(390\) 3.49359e11 0.764682
\(391\) −9.30103e11 −2.01250
\(392\) 1.55650e11 0.332938
\(393\) 2.43555e11 0.515028
\(394\) −3.23386e11 −0.676064
\(395\) −6.08333e11 −1.25735
\(396\) 2.26989e10 0.0463848
\(397\) −7.36706e11 −1.48846 −0.744230 0.667924i \(-0.767183\pi\)
−0.744230 + 0.667924i \(0.767183\pi\)
\(398\) 6.33478e11 1.26549
\(399\) 1.33148e10 0.0263000
\(400\) 2.55347e11 0.498726
\(401\) −5.90469e11 −1.14037 −0.570187 0.821515i \(-0.693129\pi\)
−0.570187 + 0.821515i \(0.693129\pi\)
\(402\) 2.58683e11 0.494027
\(403\) 2.63170e10 0.0497008
\(404\) 3.53418e11 0.660044
\(405\) −3.51039e11 −0.648348
\(406\) −1.57967e10 −0.0288536
\(407\) −5.50446e10 −0.0994352
\(408\) −1.22257e11 −0.218426
\(409\) −4.27579e11 −0.755547 −0.377773 0.925898i \(-0.623310\pi\)
−0.377773 + 0.925898i \(0.623310\pi\)
\(410\) −9.06811e11 −1.58486
\(411\) 2.15910e11 0.373237
\(412\) −2.39377e11 −0.409303
\(413\) 1.40236e11 0.237183
\(414\) 5.06316e11 0.847071
\(415\) 3.15421e11 0.522005
\(416\) 1.42130e11 0.232684
\(417\) −5.78825e11 −0.937422
\(418\) 1.21261e10 0.0194280
\(419\) 9.75902e11 1.54683 0.773416 0.633899i \(-0.218546\pi\)
0.773416 + 0.633899i \(0.218546\pi\)
\(420\) −6.32581e10 −0.0991960
\(421\) −6.82920e10 −0.105950 −0.0529750 0.998596i \(-0.516870\pi\)
−0.0529750 + 0.998596i \(0.516870\pi\)
\(422\) 2.09627e11 0.321766
\(423\) 3.47805e11 0.528207
\(424\) 2.72531e11 0.409515
\(425\) −1.74606e12 −2.59602
\(426\) −1.95212e11 −0.287186
\(427\) 6.17821e10 0.0899368
\(428\) −1.09510e11 −0.157745
\(429\) −5.25028e10 −0.0748385
\(430\) 1.47163e12 2.07582
\(431\) −2.97113e11 −0.414738 −0.207369 0.978263i \(-0.566490\pi\)
−0.207369 + 0.978263i \(0.566490\pi\)
\(432\) 1.52470e11 0.210624
\(433\) −4.51168e11 −0.616798 −0.308399 0.951257i \(-0.599793\pi\)
−0.308399 + 0.951257i \(0.599793\pi\)
\(434\) −4.76520e9 −0.00644729
\(435\) −1.03682e11 −0.138835
\(436\) −2.91331e10 −0.0386098
\(437\) 2.70482e11 0.354791
\(438\) −1.48489e11 −0.192779
\(439\) 5.55482e11 0.713805 0.356903 0.934142i \(-0.383833\pi\)
0.356903 + 0.934142i \(0.383833\pi\)
\(440\) −5.76108e10 −0.0732768
\(441\) 5.79385e11 0.729447
\(442\) −9.71881e11 −1.21119
\(443\) −1.61691e12 −1.99466 −0.997329 0.0730386i \(-0.976730\pi\)
−0.997329 + 0.0730386i \(0.976730\pi\)
\(444\) −1.61390e11 −0.197085
\(445\) −1.71931e12 −2.07842
\(446\) −1.04055e12 −1.24525
\(447\) 4.74006e11 0.561565
\(448\) −2.57354e10 −0.0301842
\(449\) 4.52893e11 0.525880 0.262940 0.964812i \(-0.415308\pi\)
0.262940 + 0.964812i \(0.415308\pi\)
\(450\) 9.50492e11 1.09268
\(451\) 1.36279e11 0.155108
\(452\) 3.50965e11 0.395495
\(453\) −1.25137e11 −0.139619
\(454\) −1.64784e11 −0.182039
\(455\) −5.02868e11 −0.550051
\(456\) 3.55536e10 0.0385071
\(457\) 1.10310e12 1.18302 0.591511 0.806297i \(-0.298532\pi\)
0.591511 + 0.806297i \(0.298532\pi\)
\(458\) −2.74863e11 −0.291891
\(459\) −1.04258e12 −1.09636
\(460\) −1.28505e12 −1.33817
\(461\) 2.22092e11 0.229023 0.114511 0.993422i \(-0.463470\pi\)
0.114511 + 0.993422i \(0.463470\pi\)
\(462\) 9.50664e9 0.00970819
\(463\) 9.97229e11 1.00851 0.504255 0.863555i \(-0.331767\pi\)
0.504255 + 0.863555i \(0.331767\pi\)
\(464\) −4.21809e10 −0.0422460
\(465\) −3.12762e10 −0.0310225
\(466\) 1.04394e12 1.02551
\(467\) 8.73582e11 0.849919 0.424959 0.905212i \(-0.360288\pi\)
0.424959 + 0.905212i \(0.360288\pi\)
\(468\) 5.29059e11 0.509797
\(469\) −3.72349e11 −0.355363
\(470\) −8.82745e11 −0.834440
\(471\) −3.30338e11 −0.309289
\(472\) 3.74461e11 0.347271
\(473\) −2.21161e11 −0.203158
\(474\) 2.68049e11 0.243900
\(475\) 5.07769e11 0.457662
\(476\) 1.75978e11 0.157118
\(477\) 1.01446e12 0.897223
\(478\) 2.28838e11 0.200495
\(479\) 1.60658e12 1.39442 0.697209 0.716868i \(-0.254425\pi\)
0.697209 + 0.716868i \(0.254425\pi\)
\(480\) −1.68914e11 −0.145238
\(481\) −1.28296e12 −1.09285
\(482\) 7.67296e11 0.647517
\(483\) 2.12053e11 0.177289
\(484\) −5.94977e11 −0.492828
\(485\) 3.18389e12 2.61288
\(486\) 8.87359e11 0.721500
\(487\) −9.25381e11 −0.745487 −0.372744 0.927934i \(-0.621583\pi\)
−0.372744 + 0.927934i \(0.621583\pi\)
\(488\) 1.64973e11 0.131681
\(489\) −1.91516e11 −0.151466
\(490\) −1.47051e12 −1.15235
\(491\) −9.28628e11 −0.721066 −0.360533 0.932746i \(-0.617405\pi\)
−0.360533 + 0.932746i \(0.617405\pi\)
\(492\) 3.99567e11 0.307430
\(493\) 2.88432e11 0.219903
\(494\) 2.82632e11 0.213526
\(495\) −2.14447e11 −0.160545
\(496\) −1.27242e10 −0.00943978
\(497\) 2.80989e11 0.206579
\(498\) −1.38984e11 −0.101259
\(499\) 1.19609e12 0.863601 0.431800 0.901969i \(-0.357878\pi\)
0.431800 + 0.901969i \(0.357878\pi\)
\(500\) −1.20311e12 −0.860878
\(501\) −9.84696e11 −0.698285
\(502\) 2.00037e11 0.140586
\(503\) 1.02039e12 0.710741 0.355370 0.934726i \(-0.384355\pi\)
0.355370 + 0.934726i \(0.384355\pi\)
\(504\) −9.57961e10 −0.0661318
\(505\) −3.33892e12 −2.28452
\(506\) 1.93122e11 0.130965
\(507\) −5.17405e11 −0.347772
\(508\) 4.21646e11 0.280906
\(509\) −5.49549e11 −0.362891 −0.181446 0.983401i \(-0.558078\pi\)
−0.181446 + 0.983401i \(0.558078\pi\)
\(510\) 1.15503e12 0.756007
\(511\) 2.13735e11 0.138670
\(512\) −6.87195e10 −0.0441942
\(513\) 3.03192e11 0.193281
\(514\) −2.93844e11 −0.185688
\(515\) 2.26151e12 1.41666
\(516\) −6.48440e11 −0.402667
\(517\) 1.32662e11 0.0816656
\(518\) 2.32305e11 0.141767
\(519\) −9.69654e11 −0.586629
\(520\) −1.34277e12 −0.805356
\(521\) 1.28517e12 0.764172 0.382086 0.924127i \(-0.375206\pi\)
0.382086 + 0.924127i \(0.375206\pi\)
\(522\) −1.57012e11 −0.0925584
\(523\) 1.65568e12 0.967654 0.483827 0.875164i \(-0.339246\pi\)
0.483827 + 0.875164i \(0.339246\pi\)
\(524\) −9.36114e11 −0.542423
\(525\) 3.98081e11 0.228694
\(526\) −1.34265e12 −0.764764
\(527\) 8.70074e10 0.0491370
\(528\) 2.53849e10 0.0142142
\(529\) 2.50659e12 1.39166
\(530\) −2.57473e12 −1.41740
\(531\) 1.39388e12 0.760849
\(532\) −5.11759e10 −0.0276989
\(533\) 3.17634e12 1.70473
\(534\) 7.57577e11 0.403172
\(535\) 1.03459e12 0.545980
\(536\) −9.94259e11 −0.520305
\(537\) 1.21490e12 0.630460
\(538\) −1.49317e11 −0.0768404
\(539\) 2.20993e11 0.112779
\(540\) −1.44046e12 −0.729001
\(541\) −1.91314e12 −0.960192 −0.480096 0.877216i \(-0.659398\pi\)
−0.480096 + 0.877216i \(0.659398\pi\)
\(542\) 2.52567e12 1.25713
\(543\) −1.87272e12 −0.924431
\(544\) 4.69901e11 0.230044
\(545\) 2.75235e11 0.133635
\(546\) 2.21578e11 0.106699
\(547\) −8.87680e11 −0.423949 −0.211974 0.977275i \(-0.567989\pi\)
−0.211974 + 0.977275i \(0.567989\pi\)
\(548\) −8.29858e11 −0.393089
\(549\) 6.14085e11 0.288505
\(550\) 3.62543e11 0.168938
\(551\) −8.38785e10 −0.0387676
\(552\) 5.66231e11 0.259578
\(553\) −3.85830e11 −0.175442
\(554\) 1.80770e12 0.815329
\(555\) 1.52473e12 0.682141
\(556\) 2.22474e12 0.987285
\(557\) −1.28351e12 −0.565001 −0.282501 0.959267i \(-0.591164\pi\)
−0.282501 + 0.959267i \(0.591164\pi\)
\(558\) −4.73638e10 −0.0206820
\(559\) −5.15475e12 −2.23283
\(560\) 2.43135e11 0.104472
\(561\) −1.73581e11 −0.0739894
\(562\) −1.25368e12 −0.530117
\(563\) 4.00016e12 1.67799 0.838995 0.544139i \(-0.183144\pi\)
0.838995 + 0.544139i \(0.183144\pi\)
\(564\) 3.88962e11 0.161865
\(565\) −3.31574e12 −1.36887
\(566\) −1.35361e12 −0.554394
\(567\) −2.22644e11 −0.0904662
\(568\) 7.50305e11 0.302462
\(569\) −1.23909e12 −0.495563 −0.247782 0.968816i \(-0.579702\pi\)
−0.247782 + 0.968816i \(0.579702\pi\)
\(570\) −3.35892e11 −0.133279
\(571\) −5.04024e12 −1.98421 −0.992107 0.125395i \(-0.959980\pi\)
−0.992107 + 0.125395i \(0.959980\pi\)
\(572\) 2.01797e11 0.0788192
\(573\) −1.88038e10 −0.00728702
\(574\) −5.75137e11 −0.221140
\(575\) 8.08679e12 3.08511
\(576\) −2.55798e11 −0.0968268
\(577\) −4.83898e12 −1.81745 −0.908725 0.417395i \(-0.862943\pi\)
−0.908725 + 0.417395i \(0.862943\pi\)
\(578\) −1.31576e12 −0.490344
\(579\) −9.53234e11 −0.352489
\(580\) 3.98504e11 0.146220
\(581\) 2.00053e11 0.0728372
\(582\) −1.40291e12 −0.506846
\(583\) 3.86940e11 0.138719
\(584\) 5.70722e11 0.203033
\(585\) −4.99827e12 −1.76449
\(586\) 1.70593e12 0.597617
\(587\) −1.95960e12 −0.681233 −0.340617 0.940202i \(-0.610636\pi\)
−0.340617 + 0.940202i \(0.610636\pi\)
\(588\) 6.47946e11 0.223533
\(589\) −2.53025e10 −0.00866254
\(590\) −3.53772e12 −1.20196
\(591\) −1.34620e12 −0.453906
\(592\) 6.20308e11 0.207568
\(593\) −4.89414e12 −1.62529 −0.812643 0.582762i \(-0.801972\pi\)
−0.812643 + 0.582762i \(0.801972\pi\)
\(594\) 2.16477e11 0.0713464
\(595\) −1.66255e12 −0.543810
\(596\) −1.82186e12 −0.591435
\(597\) 2.63706e12 0.849642
\(598\) 4.50123e12 1.43938
\(599\) 4.50082e12 1.42847 0.714235 0.699906i \(-0.246775\pi\)
0.714235 + 0.699906i \(0.246775\pi\)
\(600\) 1.06297e12 0.334842
\(601\) −5.43075e12 −1.69795 −0.848974 0.528435i \(-0.822779\pi\)
−0.848974 + 0.528435i \(0.822779\pi\)
\(602\) 9.33366e11 0.289646
\(603\) −3.70098e12 −1.13996
\(604\) 4.80970e11 0.147046
\(605\) 5.62104e12 1.70576
\(606\) 1.47122e12 0.443150
\(607\) 5.89921e12 1.76378 0.881891 0.471454i \(-0.156271\pi\)
0.881891 + 0.471454i \(0.156271\pi\)
\(608\) −1.36651e11 −0.0405554
\(609\) −6.57591e10 −0.0193722
\(610\) −1.55858e12 −0.455768
\(611\) 3.09205e12 0.897554
\(612\) 1.74914e12 0.504013
\(613\) −2.44465e11 −0.0699268 −0.0349634 0.999389i \(-0.511131\pi\)
−0.0349634 + 0.999389i \(0.511131\pi\)
\(614\) −2.64467e12 −0.750955
\(615\) −3.77490e12 −1.06406
\(616\) −3.65391e10 −0.0102246
\(617\) −6.79288e11 −0.188699 −0.0943497 0.995539i \(-0.530077\pi\)
−0.0943497 + 0.995539i \(0.530077\pi\)
\(618\) −9.96485e11 −0.274804
\(619\) 1.31874e12 0.361037 0.180519 0.983572i \(-0.442222\pi\)
0.180519 + 0.983572i \(0.442222\pi\)
\(620\) 1.20211e11 0.0326726
\(621\) 4.82868e12 1.30292
\(622\) 1.74351e12 0.467054
\(623\) −1.09046e12 −0.290010
\(624\) 5.91664e11 0.156223
\(625\) 3.75646e12 0.984732
\(626\) −1.51780e12 −0.395029
\(627\) 5.04790e10 0.0130439
\(628\) 1.26967e12 0.325740
\(629\) −4.24164e12 −1.08045
\(630\) 9.05033e11 0.228893
\(631\) −6.44908e12 −1.61944 −0.809721 0.586815i \(-0.800382\pi\)
−0.809721 + 0.586815i \(0.800382\pi\)
\(632\) −1.03026e12 −0.256873
\(633\) 8.72641e11 0.216032
\(634\) −4.40001e12 −1.08156
\(635\) −3.98350e12 −0.972261
\(636\) 1.13450e12 0.274946
\(637\) 5.15083e12 1.23951
\(638\) −5.98885e10 −0.0143104
\(639\) 2.79290e12 0.662676
\(640\) 6.49226e11 0.152963
\(641\) −5.38181e12 −1.25912 −0.629561 0.776951i \(-0.716765\pi\)
−0.629561 + 0.776951i \(0.716765\pi\)
\(642\) −4.55870e11 −0.105909
\(643\) −7.16838e12 −1.65376 −0.826878 0.562381i \(-0.809885\pi\)
−0.826878 + 0.562381i \(0.809885\pi\)
\(644\) −8.15034e11 −0.186719
\(645\) 6.12613e12 1.39369
\(646\) 9.34417e11 0.211103
\(647\) −1.24135e12 −0.278501 −0.139250 0.990257i \(-0.544469\pi\)
−0.139250 + 0.990257i \(0.544469\pi\)
\(648\) −5.94510e11 −0.132456
\(649\) 5.31661e11 0.117634
\(650\) 8.45003e12 1.85673
\(651\) −1.98367e10 −0.00432867
\(652\) 7.36100e11 0.159523
\(653\) 6.46167e12 1.39071 0.695353 0.718669i \(-0.255248\pi\)
0.695353 + 0.718669i \(0.255248\pi\)
\(654\) −1.21276e11 −0.0259224
\(655\) 8.84392e12 1.87741
\(656\) −1.53575e12 −0.323782
\(657\) 2.12443e12 0.444834
\(658\) −5.59874e11 −0.116432
\(659\) 3.48331e12 0.719462 0.359731 0.933056i \(-0.382869\pi\)
0.359731 + 0.933056i \(0.382869\pi\)
\(660\) −2.39824e11 −0.0491977
\(661\) −2.69783e12 −0.549678 −0.274839 0.961490i \(-0.588625\pi\)
−0.274839 + 0.961490i \(0.588625\pi\)
\(662\) 2.04891e12 0.414632
\(663\) −4.04578e12 −0.813188
\(664\) 5.34189e11 0.106645
\(665\) 4.83484e11 0.0958704
\(666\) 2.30900e12 0.454769
\(667\) −1.33586e12 −0.261333
\(668\) 3.78472e12 0.735427
\(669\) −4.33165e12 −0.836056
\(670\) 9.39325e12 1.80086
\(671\) 2.34228e11 0.0446054
\(672\) −1.07132e11 −0.0202655
\(673\) −7.15485e12 −1.34441 −0.672207 0.740364i \(-0.734653\pi\)
−0.672207 + 0.740364i \(0.734653\pi\)
\(674\) 4.85917e11 0.0906970
\(675\) 9.06474e12 1.68069
\(676\) 1.98867e12 0.366271
\(677\) 3.99069e12 0.730127 0.365064 0.930983i \(-0.381047\pi\)
0.365064 + 0.930983i \(0.381047\pi\)
\(678\) 1.46101e12 0.265533
\(679\) 2.01935e12 0.364585
\(680\) −4.43938e12 −0.796219
\(681\) −6.85969e11 −0.122220
\(682\) −1.80658e10 −0.00319762
\(683\) −7.70306e12 −1.35447 −0.677236 0.735766i \(-0.736823\pi\)
−0.677236 + 0.735766i \(0.736823\pi\)
\(684\) −5.08664e11 −0.0888544
\(685\) 7.84008e12 1.36054
\(686\) −1.92306e12 −0.331539
\(687\) −1.14421e12 −0.195974
\(688\) 2.49230e12 0.424085
\(689\) 9.01868e12 1.52460
\(690\) −5.34946e12 −0.898439
\(691\) −6.94842e12 −1.15940 −0.579702 0.814828i \(-0.696831\pi\)
−0.579702 + 0.814828i \(0.696831\pi\)
\(692\) 3.72690e12 0.617832
\(693\) −1.36011e11 −0.0224014
\(694\) −6.46098e12 −1.05726
\(695\) −2.10182e13 −3.41715
\(696\) −1.75592e11 −0.0283637
\(697\) 1.05014e13 1.68539
\(698\) −1.00707e12 −0.160587
\(699\) 4.34576e12 0.688523
\(700\) −1.53004e12 −0.240858
\(701\) 8.89057e12 1.39059 0.695294 0.718725i \(-0.255274\pi\)
0.695294 + 0.718725i \(0.255274\pi\)
\(702\) 5.04558e12 0.784140
\(703\) 1.23351e12 0.190477
\(704\) −9.75680e10 −0.0149703
\(705\) −3.67472e12 −0.560239
\(706\) −1.49765e12 −0.226877
\(707\) −2.11768e12 −0.318767
\(708\) 1.55882e12 0.233156
\(709\) 5.07939e12 0.754924 0.377462 0.926025i \(-0.376797\pi\)
0.377462 + 0.926025i \(0.376797\pi\)
\(710\) −7.08850e12 −1.04687
\(711\) −3.83497e12 −0.562793
\(712\) −2.91178e12 −0.424617
\(713\) −4.02972e11 −0.0583944
\(714\) 7.32565e11 0.105488
\(715\) −1.90647e12 −0.272806
\(716\) −4.66953e12 −0.663994
\(717\) 9.52615e11 0.134611
\(718\) −5.09768e12 −0.715834
\(719\) −2.62697e11 −0.0366585 −0.0183293 0.999832i \(-0.505835\pi\)
−0.0183293 + 0.999832i \(0.505835\pi\)
\(720\) 2.41665e12 0.335133
\(721\) 1.43434e12 0.197672
\(722\) −2.71737e11 −0.0372161
\(723\) 3.19412e12 0.434739
\(724\) 7.19789e12 0.973602
\(725\) −2.50777e12 −0.337106
\(726\) −2.47679e12 −0.330882
\(727\) −5.69375e12 −0.755951 −0.377976 0.925816i \(-0.623380\pi\)
−0.377976 + 0.925816i \(0.623380\pi\)
\(728\) −8.51644e11 −0.112374
\(729\) 8.37055e11 0.109769
\(730\) −5.39189e12 −0.702730
\(731\) −1.70423e13 −2.20749
\(732\) 6.86753e11 0.0884098
\(733\) −1.28415e13 −1.64304 −0.821521 0.570178i \(-0.806874\pi\)
−0.821521 + 0.570178i \(0.806874\pi\)
\(734\) −1.56994e12 −0.199642
\(735\) −6.12147e12 −0.773682
\(736\) −2.17633e12 −0.273385
\(737\) −1.41165e12 −0.176248
\(738\) −5.71660e12 −0.709388
\(739\) 2.66001e12 0.328083 0.164042 0.986453i \(-0.447547\pi\)
0.164042 + 0.986453i \(0.447547\pi\)
\(740\) −5.86035e12 −0.718425
\(741\) 1.17655e12 0.143360
\(742\) −1.63300e12 −0.197774
\(743\) 1.02254e13 1.23092 0.615462 0.788166i \(-0.288969\pi\)
0.615462 + 0.788166i \(0.288969\pi\)
\(744\) −5.29686e10 −0.00633782
\(745\) 1.72120e13 2.04705
\(746\) −3.59982e12 −0.425556
\(747\) 1.98844e12 0.233652
\(748\) 6.67166e11 0.0779250
\(749\) 6.56181e11 0.0761826
\(750\) −5.00836e12 −0.577989
\(751\) 7.08317e12 0.812546 0.406273 0.913752i \(-0.366828\pi\)
0.406273 + 0.913752i \(0.366828\pi\)
\(752\) −1.49499e12 −0.170474
\(753\) 8.32720e11 0.0943890
\(754\) −1.39586e12 −0.157280
\(755\) −4.54396e12 −0.508948
\(756\) −9.13598e11 −0.101720
\(757\) 1.55185e13 1.71758 0.858792 0.512325i \(-0.171216\pi\)
0.858792 + 0.512325i \(0.171216\pi\)
\(758\) −9.98399e12 −1.09848
\(759\) 8.03934e11 0.0879291
\(760\) 1.29101e12 0.140368
\(761\) 3.36663e12 0.363885 0.181943 0.983309i \(-0.441761\pi\)
0.181943 + 0.983309i \(0.441761\pi\)
\(762\) 1.75524e12 0.188599
\(763\) 1.74565e11 0.0186465
\(764\) 7.22732e10 0.00767462
\(765\) −1.65249e13 −1.74447
\(766\) −8.59039e12 −0.901537
\(767\) 1.23918e13 1.29287
\(768\) −2.86067e11 −0.0296717
\(769\) −9.31790e12 −0.960836 −0.480418 0.877040i \(-0.659515\pi\)
−0.480418 + 0.877040i \(0.659515\pi\)
\(770\) 3.45203e11 0.0353888
\(771\) −1.22322e12 −0.124670
\(772\) 3.66379e12 0.371239
\(773\) −7.33840e12 −0.739254 −0.369627 0.929180i \(-0.620514\pi\)
−0.369627 + 0.929180i \(0.620514\pi\)
\(774\) 9.27722e12 0.929145
\(775\) −7.56487e11 −0.0753258
\(776\) 5.39214e12 0.533806
\(777\) 9.67046e11 0.0951815
\(778\) 2.08247e12 0.203784
\(779\) −3.05390e12 −0.297123
\(780\) −5.58974e12 −0.540712
\(781\) 1.06528e12 0.102456
\(782\) 1.48817e13 1.42305
\(783\) −1.49741e12 −0.142368
\(784\) −2.49041e12 −0.235423
\(785\) −1.19951e13 −1.12744
\(786\) −3.89688e12 −0.364180
\(787\) 1.81540e13 1.68689 0.843443 0.537219i \(-0.180525\pi\)
0.843443 + 0.537219i \(0.180525\pi\)
\(788\) 5.17417e12 0.478050
\(789\) −5.58923e12 −0.513459
\(790\) 9.73333e12 0.889077
\(791\) −2.10298e12 −0.191003
\(792\) −3.63182e11 −0.0327990
\(793\) 5.45932e12 0.490241
\(794\) 1.17873e13 1.05250
\(795\) −1.07182e13 −0.951632
\(796\) −1.01356e13 −0.894835
\(797\) 5.44638e12 0.478129 0.239065 0.971004i \(-0.423159\pi\)
0.239065 + 0.971004i \(0.423159\pi\)
\(798\) −2.13037e11 −0.0185969
\(799\) 1.02227e13 0.887371
\(800\) −4.08556e12 −0.352652
\(801\) −1.08386e13 −0.930312
\(802\) 9.44751e12 0.806367
\(803\) 8.10312e11 0.0687753
\(804\) −4.13893e12 −0.349330
\(805\) 7.70002e12 0.646265
\(806\) −4.21072e11 −0.0351438
\(807\) −6.21582e11 −0.0515903
\(808\) −5.65469e12 −0.466722
\(809\) 2.44611e11 0.0200774 0.0100387 0.999950i \(-0.496805\pi\)
0.0100387 + 0.999950i \(0.496805\pi\)
\(810\) 5.61663e12 0.458451
\(811\) 1.86901e13 1.51711 0.758555 0.651609i \(-0.225906\pi\)
0.758555 + 0.651609i \(0.225906\pi\)
\(812\) 2.52748e11 0.0204026
\(813\) 1.05139e13 0.844030
\(814\) 8.80714e11 0.0703113
\(815\) −6.95430e12 −0.552134
\(816\) 1.95612e12 0.154450
\(817\) 4.95605e12 0.389167
\(818\) 6.84126e12 0.534252
\(819\) −3.17011e12 −0.246205
\(820\) 1.45090e13 1.12066
\(821\) 2.98096e12 0.228987 0.114494 0.993424i \(-0.463475\pi\)
0.114494 + 0.993424i \(0.463475\pi\)
\(822\) −3.45456e12 −0.263918
\(823\) −1.01707e13 −0.772774 −0.386387 0.922337i \(-0.626277\pi\)
−0.386387 + 0.922337i \(0.626277\pi\)
\(824\) 3.83003e12 0.289421
\(825\) 1.50920e12 0.113424
\(826\) −2.24377e12 −0.167713
\(827\) 1.99682e13 1.48445 0.742223 0.670153i \(-0.233771\pi\)
0.742223 + 0.670153i \(0.233771\pi\)
\(828\) −8.10105e12 −0.598970
\(829\) 2.35288e13 1.73023 0.865115 0.501574i \(-0.167245\pi\)
0.865115 + 0.501574i \(0.167245\pi\)
\(830\) −5.04674e12 −0.369113
\(831\) 7.52516e12 0.547408
\(832\) −2.27409e12 −0.164533
\(833\) 1.70293e13 1.22545
\(834\) 9.26121e12 0.662858
\(835\) −3.57561e13 −2.54543
\(836\) −1.94018e11 −0.0137377
\(837\) −4.51704e11 −0.0318119
\(838\) −1.56144e13 −1.09378
\(839\) 9.16607e12 0.638637 0.319319 0.947647i \(-0.396546\pi\)
0.319319 + 0.947647i \(0.396546\pi\)
\(840\) 1.01213e12 0.0701422
\(841\) −1.40929e13 −0.971444
\(842\) 1.09267e12 0.0749179
\(843\) −5.21884e12 −0.355918
\(844\) −3.35403e12 −0.227523
\(845\) −1.87879e13 −1.26772
\(846\) −5.56488e12 −0.373499
\(847\) 3.56509e12 0.238010
\(848\) −4.36050e12 −0.289571
\(849\) −5.63484e12 −0.372217
\(850\) 2.79369e13 1.83566
\(851\) 1.96450e13 1.28401
\(852\) 3.12339e12 0.203071
\(853\) 9.94411e12 0.643125 0.321562 0.946888i \(-0.395792\pi\)
0.321562 + 0.946888i \(0.395792\pi\)
\(854\) −9.88514e11 −0.0635949
\(855\) 4.80560e12 0.307539
\(856\) 1.75216e12 0.111543
\(857\) −1.90800e13 −1.20827 −0.604137 0.796880i \(-0.706482\pi\)
−0.604137 + 0.796880i \(0.706482\pi\)
\(858\) 8.40045e11 0.0529188
\(859\) 3.10124e12 0.194342 0.0971710 0.995268i \(-0.469021\pi\)
0.0971710 + 0.995268i \(0.469021\pi\)
\(860\) −2.35460e13 −1.46782
\(861\) −2.39420e12 −0.148473
\(862\) 4.75381e12 0.293264
\(863\) −2.40338e13 −1.47494 −0.737470 0.675380i \(-0.763980\pi\)
−0.737470 + 0.675380i \(0.763980\pi\)
\(864\) −2.43952e12 −0.148933
\(865\) −3.52099e13 −2.13842
\(866\) 7.21869e12 0.436142
\(867\) −5.47728e12 −0.329214
\(868\) 7.62431e10 0.00455892
\(869\) −1.46276e12 −0.0870129
\(870\) 1.65890e12 0.0981713
\(871\) −3.29023e13 −1.93707
\(872\) 4.66130e11 0.0273013
\(873\) 2.00714e13 1.16954
\(874\) −4.32772e12 −0.250875
\(875\) 7.20905e12 0.415759
\(876\) 2.37582e12 0.136315
\(877\) −5.20893e12 −0.297338 −0.148669 0.988887i \(-0.547499\pi\)
−0.148669 + 0.988887i \(0.547499\pi\)
\(878\) −8.88771e12 −0.504736
\(879\) 7.10151e12 0.401237
\(880\) 9.21772e11 0.0518145
\(881\) −7.36387e12 −0.411826 −0.205913 0.978570i \(-0.566016\pi\)
−0.205913 + 0.978570i \(0.566016\pi\)
\(882\) −9.27016e12 −0.515797
\(883\) 2.98715e13 1.65362 0.826808 0.562485i \(-0.190155\pi\)
0.826808 + 0.562485i \(0.190155\pi\)
\(884\) 1.55501e13 0.856442
\(885\) −1.47269e13 −0.806988
\(886\) 2.58705e13 1.41044
\(887\) −1.43208e13 −0.776802 −0.388401 0.921490i \(-0.626973\pi\)
−0.388401 + 0.921490i \(0.626973\pi\)
\(888\) 2.58224e12 0.139360
\(889\) −2.52650e12 −0.135663
\(890\) 2.75090e13 1.46967
\(891\) −8.44086e11 −0.0448680
\(892\) 1.66489e13 0.880526
\(893\) −2.97285e12 −0.156438
\(894\) −7.58410e12 −0.397086
\(895\) 4.41153e13 2.29819
\(896\) 4.11766e11 0.0213435
\(897\) 1.87379e13 0.966394
\(898\) −7.24628e12 −0.371853
\(899\) 1.24964e11 0.00638069
\(900\) −1.52079e13 −0.772640
\(901\) 2.98169e13 1.50730
\(902\) −2.18046e12 −0.109678
\(903\) 3.88544e12 0.194467
\(904\) −5.61544e12 −0.279657
\(905\) −6.80020e13 −3.36979
\(906\) 2.00220e12 0.0987257
\(907\) −3.98348e13 −1.95448 −0.977238 0.212146i \(-0.931955\pi\)
−0.977238 + 0.212146i \(0.931955\pi\)
\(908\) 2.63655e12 0.128721
\(909\) −2.10487e13 −1.02256
\(910\) 8.04589e12 0.388945
\(911\) −8.77892e12 −0.422288 −0.211144 0.977455i \(-0.567719\pi\)
−0.211144 + 0.977455i \(0.567719\pi\)
\(912\) −5.68857e11 −0.0272287
\(913\) 7.58442e11 0.0361247
\(914\) −1.76496e13 −0.836522
\(915\) −6.48809e12 −0.306000
\(916\) 4.39780e12 0.206398
\(917\) 5.60918e12 0.261962
\(918\) 1.66813e13 0.775244
\(919\) 2.89837e12 0.134040 0.0670200 0.997752i \(-0.478651\pi\)
0.0670200 + 0.997752i \(0.478651\pi\)
\(920\) 2.05608e13 0.946228
\(921\) −1.10093e13 −0.504188
\(922\) −3.55347e12 −0.161943
\(923\) 2.48293e13 1.12605
\(924\) −1.52106e11 −0.00686472
\(925\) 3.68790e13 1.65631
\(926\) −1.59557e13 −0.713125
\(927\) 1.42567e13 0.634104
\(928\) 6.74895e11 0.0298724
\(929\) 1.09141e13 0.480749 0.240375 0.970680i \(-0.422730\pi\)
0.240375 + 0.970680i \(0.422730\pi\)
\(930\) 5.00420e11 0.0219362
\(931\) −4.95228e12 −0.216039
\(932\) −1.67031e13 −0.725146
\(933\) 7.25793e12 0.313578
\(934\) −1.39773e13 −0.600983
\(935\) −6.30304e12 −0.269711
\(936\) −8.46494e12 −0.360481
\(937\) 3.31892e13 1.40659 0.703296 0.710897i \(-0.251711\pi\)
0.703296 + 0.710897i \(0.251711\pi\)
\(938\) 5.95759e12 0.251280
\(939\) −6.31832e12 −0.265220
\(940\) 1.41239e13 0.590038
\(941\) −4.06875e13 −1.69164 −0.845819 0.533471i \(-0.820888\pi\)
−0.845819 + 0.533471i \(0.820888\pi\)
\(942\) 5.28540e12 0.218700
\(943\) −4.86368e13 −2.00292
\(944\) −5.99138e12 −0.245557
\(945\) 8.63120e12 0.352069
\(946\) 3.53857e12 0.143654
\(947\) −1.90875e13 −0.771211 −0.385606 0.922664i \(-0.626007\pi\)
−0.385606 + 0.922664i \(0.626007\pi\)
\(948\) −4.28878e12 −0.172463
\(949\) 1.88865e13 0.755882
\(950\) −8.12430e12 −0.323616
\(951\) −1.83165e13 −0.726155
\(952\) −2.81564e12 −0.111099
\(953\) −2.79671e12 −0.109832 −0.0549161 0.998491i \(-0.517489\pi\)
−0.0549161 + 0.998491i \(0.517489\pi\)
\(954\) −1.62313e13 −0.634432
\(955\) −6.82800e11 −0.0265631
\(956\) −3.66141e12 −0.141771
\(957\) −2.49306e11 −0.00960790
\(958\) −2.57053e13 −0.986002
\(959\) 4.97250e12 0.189842
\(960\) 2.70262e12 0.102699
\(961\) −2.64019e13 −0.998574
\(962\) 2.05274e13 0.772764
\(963\) 6.52213e12 0.244383
\(964\) −1.22767e13 −0.457864
\(965\) −3.46136e13 −1.28492
\(966\) −3.39285e12 −0.125362
\(967\) −1.90809e10 −0.000701745 0 −0.000350873 1.00000i \(-0.500112\pi\)
−0.000350873 1.00000i \(0.500112\pi\)
\(968\) 9.51963e12 0.348482
\(969\) 3.88982e12 0.141733
\(970\) −5.09422e13 −1.84759
\(971\) −4.13526e12 −0.149285 −0.0746426 0.997210i \(-0.523782\pi\)
−0.0746426 + 0.997210i \(0.523782\pi\)
\(972\) −1.41977e13 −0.510177
\(973\) −1.33306e13 −0.476807
\(974\) 1.48061e13 0.527139
\(975\) 3.51761e13 1.24660
\(976\) −2.63956e12 −0.0931124
\(977\) 3.74336e13 1.31443 0.657213 0.753705i \(-0.271735\pi\)
0.657213 + 0.753705i \(0.271735\pi\)
\(978\) 3.06426e12 0.107103
\(979\) −4.13414e12 −0.143835
\(980\) 2.35281e13 0.814835
\(981\) 1.73510e12 0.0598155
\(982\) 1.48581e13 0.509871
\(983\) 1.37412e13 0.469390 0.234695 0.972069i \(-0.424591\pi\)
0.234695 + 0.972069i \(0.424591\pi\)
\(984\) −6.39307e12 −0.217386
\(985\) −4.88830e13 −1.65461
\(986\) −4.61491e12 −0.155495
\(987\) −2.33066e12 −0.0781721
\(988\) −4.52211e12 −0.150985
\(989\) 7.89307e13 2.62339
\(990\) 3.43116e12 0.113523
\(991\) −2.92261e13 −0.962584 −0.481292 0.876560i \(-0.659832\pi\)
−0.481292 + 0.876560i \(0.659832\pi\)
\(992\) 2.03587e11 0.00667494
\(993\) 8.52927e12 0.278381
\(994\) −4.49582e12 −0.146073
\(995\) 9.57564e13 3.09716
\(996\) 2.22374e12 0.0716006
\(997\) 1.27081e13 0.407335 0.203668 0.979040i \(-0.434714\pi\)
0.203668 + 0.979040i \(0.434714\pi\)
\(998\) −1.91375e13 −0.610658
\(999\) 2.20207e13 0.699499
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 38.10.a.d.1.2 4
3.2 odd 2 342.10.a.l.1.4 4
4.3 odd 2 304.10.a.e.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.10.a.d.1.2 4 1.1 even 1 trivial
304.10.a.e.1.3 4 4.3 odd 2
342.10.a.l.1.4 4 3.2 odd 2