Properties

Label 43.10.a.a.1.4
Level 43
Weight 10
Character 43.1
Self dual yes
Analytic conductor 22.147
Analytic rank 1
Dimension 15
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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 1.4
Root \(21.7142\) 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

\(f(q)\) \(=\) \(q-23.7142 q^{2} +2.00552 q^{3} +50.3616 q^{4} -764.097 q^{5} -47.5592 q^{6} +4079.63 q^{7} +10947.4 q^{8} -19679.0 q^{9} +O(q^{10})\) \(q-23.7142 q^{2} +2.00552 q^{3} +50.3616 q^{4} -764.097 q^{5} -47.5592 q^{6} +4079.63 q^{7} +10947.4 q^{8} -19679.0 q^{9} +18119.9 q^{10} +11874.8 q^{11} +101.001 q^{12} +108937. q^{13} -96745.1 q^{14} -1532.41 q^{15} -285393. q^{16} +190628. q^{17} +466671. q^{18} +764217. q^{19} -38481.2 q^{20} +8181.78 q^{21} -281600. q^{22} -923194. q^{23} +21955.2 q^{24} -1.36928e6 q^{25} -2.58336e6 q^{26} -78941.2 q^{27} +205457. q^{28} -3.16980e6 q^{29} +36339.9 q^{30} -1.00773e7 q^{31} +1.16280e6 q^{32} +23815.1 q^{33} -4.52058e6 q^{34} -3.11724e6 q^{35} -991065. q^{36} +6.60642e6 q^{37} -1.81228e7 q^{38} +218476. q^{39} -8.36486e6 q^{40} -2.24033e7 q^{41} -194024. q^{42} -3.41880e6 q^{43} +598032. q^{44} +1.50367e7 q^{45} +2.18928e7 q^{46} +1.96091e7 q^{47} -572361. q^{48} -2.37102e7 q^{49} +3.24713e7 q^{50} +382308. q^{51} +5.48626e6 q^{52} +2.18335e7 q^{53} +1.87203e6 q^{54} -9.07347e6 q^{55} +4.46612e7 q^{56} +1.53265e6 q^{57} +7.51692e7 q^{58} -1.05400e8 q^{59} -77174.8 q^{60} +6.39871e7 q^{61} +2.38974e8 q^{62} -8.02830e7 q^{63} +1.18546e8 q^{64} -8.32387e7 q^{65} -564754. q^{66} +6.09774e7 q^{67} +9.60032e6 q^{68} -1.85148e6 q^{69} +7.39227e7 q^{70} -3.75615e8 q^{71} -2.15433e8 q^{72} +3.38605e8 q^{73} -1.56666e8 q^{74} -2.74612e6 q^{75} +3.84872e7 q^{76} +4.84446e7 q^{77} -5.18097e6 q^{78} -6.12882e8 q^{79} +2.18068e8 q^{80} +3.87183e8 q^{81} +5.31277e8 q^{82} +3.98173e8 q^{83} +412048. q^{84} -1.45658e8 q^{85} +8.10740e7 q^{86} -6.35710e6 q^{87} +1.29997e8 q^{88} -8.15783e8 q^{89} -3.56582e8 q^{90} +4.44424e8 q^{91} -4.64936e7 q^{92} -2.02102e7 q^{93} -4.65014e8 q^{94} -5.83937e8 q^{95} +2.33202e6 q^{96} +6.93925e8 q^{97} +5.62268e8 q^{98} -2.33683e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.7142 −1.04803 −0.524014 0.851710i \(-0.675566\pi\)
−0.524014 + 0.851710i \(0.675566\pi\)
\(3\) 2.00552 0.0142949 0.00714745 0.999974i \(-0.497725\pi\)
0.00714745 + 0.999974i \(0.497725\pi\)
\(4\) 50.3616 0.0983625
\(5\) −764.097 −0.546744 −0.273372 0.961908i \(-0.588139\pi\)
−0.273372 + 0.961908i \(0.588139\pi\)
\(6\) −47.5592 −0.0149815
\(7\) 4079.63 0.642214 0.321107 0.947043i \(-0.395945\pi\)
0.321107 + 0.947043i \(0.395945\pi\)
\(8\) 10947.4 0.944941
\(9\) −19679.0 −0.999796
\(10\) 18119.9 0.573003
\(11\) 11874.8 0.244544 0.122272 0.992497i \(-0.460982\pi\)
0.122272 + 0.992497i \(0.460982\pi\)
\(12\) 101.001 0.00140608
\(13\) 108937. 1.05787 0.528934 0.848663i \(-0.322592\pi\)
0.528934 + 0.848663i \(0.322592\pi\)
\(14\) −96745.1 −0.673058
\(15\) −1532.41 −0.00781565
\(16\) −285393. −1.08869
\(17\) 190628. 0.553562 0.276781 0.960933i \(-0.410732\pi\)
0.276781 + 0.960933i \(0.410732\pi\)
\(18\) 466671. 1.04781
\(19\) 764217. 1.34532 0.672660 0.739951i \(-0.265152\pi\)
0.672660 + 0.739951i \(0.265152\pi\)
\(20\) −38481.2 −0.0537791
\(21\) 8181.78 0.00918039
\(22\) −281600. −0.256289
\(23\) −923194. −0.687888 −0.343944 0.938990i \(-0.611763\pi\)
−0.343944 + 0.938990i \(0.611763\pi\)
\(24\) 21955.2 0.0135078
\(25\) −1.36928e6 −0.701071
\(26\) −2.58336e6 −1.10868
\(27\) −78941.2 −0.0285869
\(28\) 205457. 0.0631698
\(29\) −3.16980e6 −0.832226 −0.416113 0.909313i \(-0.636608\pi\)
−0.416113 + 0.909313i \(0.636608\pi\)
\(30\) 36339.9 0.00819102
\(31\) −1.00773e7 −1.95982 −0.979909 0.199444i \(-0.936086\pi\)
−0.979909 + 0.199444i \(0.936086\pi\)
\(32\) 1.16280e6 0.196034
\(33\) 23815.1 0.00349574
\(34\) −4.52058e6 −0.580148
\(35\) −3.11724e6 −0.351126
\(36\) −991065. −0.0983424
\(37\) 6.60642e6 0.579506 0.289753 0.957101i \(-0.406427\pi\)
0.289753 + 0.957101i \(0.406427\pi\)
\(38\) −1.81228e7 −1.40993
\(39\) 218476. 0.0151221
\(40\) −8.36486e6 −0.516641
\(41\) −2.24033e7 −1.23818 −0.619092 0.785318i \(-0.712499\pi\)
−0.619092 + 0.785318i \(0.712499\pi\)
\(42\) −194024. −0.00962131
\(43\) −3.41880e6 −0.152499
\(44\) 598032. 0.0240540
\(45\) 1.50367e7 0.546632
\(46\) 2.18928e7 0.720926
\(47\) 1.96091e7 0.586162 0.293081 0.956088i \(-0.405319\pi\)
0.293081 + 0.956088i \(0.405319\pi\)
\(48\) −572361. −0.0155627
\(49\) −2.37102e7 −0.587561
\(50\) 3.24713e7 0.734742
\(51\) 382308. 0.00791311
\(52\) 5.48626e6 0.104055
\(53\) 2.18335e7 0.380086 0.190043 0.981776i \(-0.439137\pi\)
0.190043 + 0.981776i \(0.439137\pi\)
\(54\) 1.87203e6 0.0299599
\(55\) −9.07347e6 −0.133703
\(56\) 4.46612e7 0.606855
\(57\) 1.53265e6 0.0192312
\(58\) 7.51692e7 0.872196
\(59\) −1.05400e8 −1.13242 −0.566210 0.824261i \(-0.691591\pi\)
−0.566210 + 0.824261i \(0.691591\pi\)
\(60\) −77174.8 −0.000768767 0
\(61\) 6.39871e7 0.591709 0.295854 0.955233i \(-0.404396\pi\)
0.295854 + 0.955233i \(0.404396\pi\)
\(62\) 2.38974e8 2.05394
\(63\) −8.02830e7 −0.642083
\(64\) 1.18546e8 0.883239
\(65\) −8.32387e7 −0.578383
\(66\) −564754. −0.00366363
\(67\) 6.09774e7 0.369686 0.184843 0.982768i \(-0.440822\pi\)
0.184843 + 0.982768i \(0.440822\pi\)
\(68\) 9.60032e6 0.0544497
\(69\) −1.85148e6 −0.00983330
\(70\) 7.39227e7 0.367990
\(71\) −3.75615e8 −1.75420 −0.877102 0.480304i \(-0.840526\pi\)
−0.877102 + 0.480304i \(0.840526\pi\)
\(72\) −2.15433e8 −0.944748
\(73\) 3.38605e8 1.39553 0.697766 0.716325i \(-0.254177\pi\)
0.697766 + 0.716325i \(0.254177\pi\)
\(74\) −1.56666e8 −0.607339
\(75\) −2.74612e6 −0.0100218
\(76\) 3.84872e7 0.132329
\(77\) 4.84446e7 0.157050
\(78\) −5.18097e6 −0.0158484
\(79\) −6.12882e8 −1.77033 −0.885166 0.465275i \(-0.845955\pi\)
−0.885166 + 0.465275i \(0.845955\pi\)
\(80\) 2.18068e8 0.595233
\(81\) 3.87183e8 0.999387
\(82\) 5.31277e8 1.29765
\(83\) 3.98173e8 0.920916 0.460458 0.887682i \(-0.347685\pi\)
0.460458 + 0.887682i \(0.347685\pi\)
\(84\) 412048. 0.000903007 0
\(85\) −1.45658e8 −0.302656
\(86\) 8.10740e7 0.159823
\(87\) −6.35710e6 −0.0118966
\(88\) 1.29997e8 0.231080
\(89\) −8.15783e8 −1.37822 −0.689111 0.724655i \(-0.741999\pi\)
−0.689111 + 0.724655i \(0.741999\pi\)
\(90\) −3.56582e8 −0.572885
\(91\) 4.44424e8 0.679378
\(92\) −4.64936e7 −0.0676624
\(93\) −2.02102e7 −0.0280154
\(94\) −4.65014e8 −0.614314
\(95\) −5.83937e8 −0.735545
\(96\) 2.33202e6 0.00280228
\(97\) 6.93925e8 0.795866 0.397933 0.917415i \(-0.369728\pi\)
0.397933 + 0.917415i \(0.369728\pi\)
\(98\) 5.62268e8 0.615780
\(99\) −2.33683e8 −0.244494
\(100\) −6.89592e7 −0.0689592
\(101\) −1.18596e8 −0.113403 −0.0567014 0.998391i \(-0.518058\pi\)
−0.0567014 + 0.998391i \(0.518058\pi\)
\(102\) −9.06611e6 −0.00829316
\(103\) −1.30566e9 −1.14304 −0.571521 0.820587i \(-0.693647\pi\)
−0.571521 + 0.820587i \(0.693647\pi\)
\(104\) 1.19258e9 0.999623
\(105\) −6.25168e6 −0.00501932
\(106\) −5.17764e8 −0.398341
\(107\) −1.24496e9 −0.918182 −0.459091 0.888389i \(-0.651825\pi\)
−0.459091 + 0.888389i \(0.651825\pi\)
\(108\) −3.97561e6 −0.00281188
\(109\) −8.52511e8 −0.578470 −0.289235 0.957258i \(-0.593401\pi\)
−0.289235 + 0.957258i \(0.593401\pi\)
\(110\) 2.15170e8 0.140125
\(111\) 1.32493e7 0.00828399
\(112\) −1.16430e9 −0.699170
\(113\) −1.50939e9 −0.870858 −0.435429 0.900223i \(-0.643403\pi\)
−0.435429 + 0.900223i \(0.643403\pi\)
\(114\) −3.63456e7 −0.0201549
\(115\) 7.05410e8 0.376098
\(116\) −1.59636e8 −0.0818599
\(117\) −2.14377e9 −1.05765
\(118\) 2.49948e9 1.18681
\(119\) 7.77691e8 0.355505
\(120\) −1.67759e7 −0.00738533
\(121\) −2.21694e9 −0.940198
\(122\) −1.51740e9 −0.620127
\(123\) −4.49303e7 −0.0176997
\(124\) −5.07508e8 −0.192773
\(125\) 2.53864e9 0.930050
\(126\) 1.90384e9 0.672921
\(127\) 1.49537e9 0.510071 0.255036 0.966932i \(-0.417913\pi\)
0.255036 + 0.966932i \(0.417913\pi\)
\(128\) −3.40658e9 −1.12169
\(129\) −6.85647e6 −0.00217995
\(130\) 1.97394e9 0.606161
\(131\) −1.51968e8 −0.0450850 −0.0225425 0.999746i \(-0.507176\pi\)
−0.0225425 + 0.999746i \(0.507176\pi\)
\(132\) 1.19937e6 0.000343850 0
\(133\) 3.11773e9 0.863984
\(134\) −1.44603e9 −0.387441
\(135\) 6.03188e7 0.0156297
\(136\) 2.08687e9 0.523083
\(137\) −2.01944e9 −0.489765 −0.244883 0.969553i \(-0.578749\pi\)
−0.244883 + 0.969553i \(0.578749\pi\)
\(138\) 4.39064e7 0.0103056
\(139\) 2.32789e9 0.528927 0.264463 0.964396i \(-0.414805\pi\)
0.264463 + 0.964396i \(0.414805\pi\)
\(140\) −1.56989e8 −0.0345377
\(141\) 3.93265e7 0.00837913
\(142\) 8.90739e9 1.83845
\(143\) 1.29360e9 0.258696
\(144\) 5.61624e9 1.08846
\(145\) 2.42204e9 0.455014
\(146\) −8.02973e9 −1.46256
\(147\) −4.75513e7 −0.00839913
\(148\) 3.32710e8 0.0570017
\(149\) −1.41126e9 −0.234568 −0.117284 0.993098i \(-0.537419\pi\)
−0.117284 + 0.993098i \(0.537419\pi\)
\(150\) 6.51219e7 0.0105031
\(151\) 4.13179e9 0.646758 0.323379 0.946269i \(-0.395181\pi\)
0.323379 + 0.946269i \(0.395181\pi\)
\(152\) 8.36617e9 1.27125
\(153\) −3.75136e9 −0.553448
\(154\) −1.14882e9 −0.164593
\(155\) 7.70003e9 1.07152
\(156\) 1.10028e7 0.00148745
\(157\) 1.22045e10 1.60314 0.801568 0.597904i \(-0.204000\pi\)
0.801568 + 0.597904i \(0.204000\pi\)
\(158\) 1.45340e10 1.85536
\(159\) 4.37875e7 0.00543330
\(160\) −8.88493e8 −0.107180
\(161\) −3.76629e9 −0.441772
\(162\) −9.18172e9 −1.04739
\(163\) −1.00930e10 −1.11990 −0.559948 0.828528i \(-0.689179\pi\)
−0.559948 + 0.828528i \(0.689179\pi\)
\(164\) −1.12827e9 −0.121791
\(165\) −1.81970e7 −0.00191127
\(166\) −9.44233e9 −0.965145
\(167\) −1.39128e10 −1.38417 −0.692086 0.721815i \(-0.743308\pi\)
−0.692086 + 0.721815i \(0.743308\pi\)
\(168\) 8.95690e7 0.00867493
\(169\) 1.26283e9 0.119085
\(170\) 3.45416e9 0.317192
\(171\) −1.50390e10 −1.34505
\(172\) −1.72176e8 −0.0150001
\(173\) −2.03248e10 −1.72512 −0.862558 0.505958i \(-0.831139\pi\)
−0.862558 + 0.505958i \(0.831139\pi\)
\(174\) 1.50753e8 0.0124680
\(175\) −5.58616e9 −0.450238
\(176\) −3.38897e9 −0.266232
\(177\) −2.11382e8 −0.0161878
\(178\) 1.93456e10 1.44442
\(179\) −5.53330e9 −0.402852 −0.201426 0.979504i \(-0.564558\pi\)
−0.201426 + 0.979504i \(0.564558\pi\)
\(180\) 7.57270e8 0.0537681
\(181\) 2.15970e10 1.49569 0.747843 0.663875i \(-0.231089\pi\)
0.747843 + 0.663875i \(0.231089\pi\)
\(182\) −1.05391e10 −0.712007
\(183\) 1.28327e8 0.00845842
\(184\) −1.01065e10 −0.650014
\(185\) −5.04795e9 −0.316841
\(186\) 4.79268e8 0.0293609
\(187\) 2.26366e9 0.135370
\(188\) 9.87546e8 0.0576564
\(189\) −3.22051e8 −0.0183589
\(190\) 1.38476e10 0.770872
\(191\) −1.65024e10 −0.897217 −0.448608 0.893728i \(-0.648080\pi\)
−0.448608 + 0.893728i \(0.648080\pi\)
\(192\) 2.37747e8 0.0126258
\(193\) −3.40195e10 −1.76490 −0.882450 0.470407i \(-0.844107\pi\)
−0.882450 + 0.470407i \(0.844107\pi\)
\(194\) −1.64559e10 −0.834090
\(195\) −1.66937e8 −0.00826792
\(196\) −1.19408e9 −0.0577940
\(197\) −2.48463e10 −1.17534 −0.587670 0.809101i \(-0.699955\pi\)
−0.587670 + 0.809101i \(0.699955\pi\)
\(198\) 5.54160e9 0.256237
\(199\) 2.87975e10 1.30172 0.650858 0.759200i \(-0.274409\pi\)
0.650858 + 0.759200i \(0.274409\pi\)
\(200\) −1.49900e10 −0.662471
\(201\) 1.22291e8 0.00528462
\(202\) 2.81241e9 0.118849
\(203\) −1.29316e10 −0.534467
\(204\) 1.92536e7 0.000778354 0
\(205\) 1.71183e10 0.676970
\(206\) 3.09626e10 1.19794
\(207\) 1.81675e10 0.687748
\(208\) −3.10899e10 −1.15169
\(209\) 9.07490e9 0.328991
\(210\) 1.48253e8 0.00526039
\(211\) 3.08043e10 1.06989 0.534946 0.844886i \(-0.320332\pi\)
0.534946 + 0.844886i \(0.320332\pi\)
\(212\) 1.09957e9 0.0373863
\(213\) −7.53303e8 −0.0250762
\(214\) 2.95232e10 0.962281
\(215\) 2.61230e9 0.0833776
\(216\) −8.64199e8 −0.0270129
\(217\) −4.11116e10 −1.25862
\(218\) 2.02166e10 0.606253
\(219\) 6.79078e8 0.0199490
\(220\) −4.56955e8 −0.0131514
\(221\) 2.07665e10 0.585595
\(222\) −3.14196e8 −0.00868185
\(223\) 2.40525e10 0.651312 0.325656 0.945488i \(-0.394415\pi\)
0.325656 + 0.945488i \(0.394415\pi\)
\(224\) 4.74380e9 0.125895
\(225\) 2.69460e10 0.700928
\(226\) 3.57938e10 0.912683
\(227\) −5.03039e9 −0.125743 −0.0628717 0.998022i \(-0.520026\pi\)
−0.0628717 + 0.998022i \(0.520026\pi\)
\(228\) 7.71869e7 0.00189163
\(229\) −4.34034e9 −0.104295 −0.0521476 0.998639i \(-0.516607\pi\)
−0.0521476 + 0.998639i \(0.516607\pi\)
\(230\) −1.67282e10 −0.394162
\(231\) 9.71567e7 0.00224501
\(232\) −3.47010e10 −0.786405
\(233\) 8.25255e9 0.183437 0.0917184 0.995785i \(-0.470764\pi\)
0.0917184 + 0.995785i \(0.470764\pi\)
\(234\) 5.08378e10 1.10845
\(235\) −1.49833e10 −0.320480
\(236\) −5.30812e9 −0.111388
\(237\) −1.22915e9 −0.0253067
\(238\) −1.84423e10 −0.372579
\(239\) −3.54632e10 −0.703052 −0.351526 0.936178i \(-0.614337\pi\)
−0.351526 + 0.936178i \(0.614337\pi\)
\(240\) 4.37340e8 0.00850880
\(241\) −8.17008e9 −0.156009 −0.0780045 0.996953i \(-0.524855\pi\)
−0.0780045 + 0.996953i \(0.524855\pi\)
\(242\) 5.25728e10 0.985354
\(243\) 2.33030e9 0.0428730
\(244\) 3.22249e9 0.0582020
\(245\) 1.81169e10 0.321245
\(246\) 1.06549e9 0.0185498
\(247\) 8.32518e10 1.42317
\(248\) −1.10320e11 −1.85191
\(249\) 7.98543e8 0.0131644
\(250\) −6.02018e10 −0.974718
\(251\) 3.32427e10 0.528646 0.264323 0.964434i \(-0.414852\pi\)
0.264323 + 0.964434i \(0.414852\pi\)
\(252\) −4.04318e9 −0.0631569
\(253\) −1.09627e10 −0.168219
\(254\) −3.54614e10 −0.534569
\(255\) −2.92120e8 −0.00432644
\(256\) 2.00885e10 0.292326
\(257\) 5.99005e10 0.856508 0.428254 0.903658i \(-0.359129\pi\)
0.428254 + 0.903658i \(0.359129\pi\)
\(258\) 1.62596e8 0.00228465
\(259\) 2.69518e10 0.372167
\(260\) −4.19204e9 −0.0568912
\(261\) 6.23785e10 0.832056
\(262\) 3.60380e9 0.0472503
\(263\) −2.35411e10 −0.303407 −0.151703 0.988426i \(-0.548476\pi\)
−0.151703 + 0.988426i \(0.548476\pi\)
\(264\) 2.60712e8 0.00330327
\(265\) −1.66829e10 −0.207810
\(266\) −7.39343e10 −0.905479
\(267\) −1.63607e9 −0.0197016
\(268\) 3.07092e9 0.0363632
\(269\) −5.01380e10 −0.583824 −0.291912 0.956445i \(-0.594291\pi\)
−0.291912 + 0.956445i \(0.594291\pi\)
\(270\) −1.43041e9 −0.0163804
\(271\) −3.54796e7 −0.000399592 0 −0.000199796 1.00000i \(-0.500064\pi\)
−0.000199796 1.00000i \(0.500064\pi\)
\(272\) −5.44038e10 −0.602655
\(273\) 8.91301e8 0.00971164
\(274\) 4.78893e10 0.513288
\(275\) −1.62599e10 −0.171443
\(276\) −9.32438e7 −0.000967228 0
\(277\) 6.02234e10 0.614619 0.307310 0.951610i \(-0.400571\pi\)
0.307310 + 0.951610i \(0.400571\pi\)
\(278\) −5.52039e10 −0.554330
\(279\) 1.98311e11 1.95942
\(280\) −3.41255e10 −0.331794
\(281\) −4.41869e10 −0.422781 −0.211390 0.977402i \(-0.567799\pi\)
−0.211390 + 0.977402i \(0.567799\pi\)
\(282\) −9.32594e8 −0.00878156
\(283\) 4.63295e10 0.429357 0.214679 0.976685i \(-0.431130\pi\)
0.214679 + 0.976685i \(0.431130\pi\)
\(284\) −1.89166e10 −0.172548
\(285\) −1.17110e9 −0.0105146
\(286\) −3.06767e10 −0.271120
\(287\) −9.13974e10 −0.795180
\(288\) −2.28827e10 −0.195993
\(289\) −8.22489e10 −0.693570
\(290\) −5.74366e10 −0.476868
\(291\) 1.39168e9 0.0113768
\(292\) 1.70527e10 0.137268
\(293\) 1.36697e11 1.08356 0.541782 0.840519i \(-0.317750\pi\)
0.541782 + 0.840519i \(0.317750\pi\)
\(294\) 1.12764e9 0.00880252
\(295\) 8.05360e10 0.619143
\(296\) 7.23229e10 0.547599
\(297\) −9.37408e8 −0.00699076
\(298\) 3.34668e10 0.245833
\(299\) −1.00570e11 −0.727695
\(300\) −1.38299e8 −0.000985765 0
\(301\) −1.39475e10 −0.0979367
\(302\) −9.79819e10 −0.677821
\(303\) −2.37847e8 −0.00162108
\(304\) −2.18102e11 −1.46463
\(305\) −4.88924e10 −0.323513
\(306\) 8.89603e10 0.580029
\(307\) −1.70586e10 −0.109602 −0.0548012 0.998497i \(-0.517453\pi\)
−0.0548012 + 0.998497i \(0.517453\pi\)
\(308\) 2.43975e9 0.0154478
\(309\) −2.61853e9 −0.0163397
\(310\) −1.82600e11 −1.12298
\(311\) 1.68094e11 1.01890 0.509450 0.860500i \(-0.329849\pi\)
0.509450 + 0.860500i \(0.329849\pi\)
\(312\) 2.39174e9 0.0142895
\(313\) −1.12744e11 −0.663962 −0.331981 0.943286i \(-0.607717\pi\)
−0.331981 + 0.943286i \(0.607717\pi\)
\(314\) −2.89419e11 −1.68013
\(315\) 6.13440e10 0.351055
\(316\) −3.08657e10 −0.174134
\(317\) 9.37662e10 0.521531 0.260765 0.965402i \(-0.416025\pi\)
0.260765 + 0.965402i \(0.416025\pi\)
\(318\) −1.03839e9 −0.00569425
\(319\) −3.76406e10 −0.203516
\(320\) −9.05809e10 −0.482905
\(321\) −2.49680e9 −0.0131253
\(322\) 8.93145e10 0.462989
\(323\) 1.45681e11 0.744718
\(324\) 1.94992e10 0.0983022
\(325\) −1.49166e11 −0.741641
\(326\) 2.39348e11 1.17368
\(327\) −1.70973e9 −0.00826917
\(328\) −2.45258e11 −1.17001
\(329\) 7.99980e10 0.376441
\(330\) 4.31527e8 0.00200307
\(331\) −2.93005e11 −1.34168 −0.670840 0.741602i \(-0.734066\pi\)
−0.670840 + 0.741602i \(0.734066\pi\)
\(332\) 2.00526e10 0.0905836
\(333\) −1.30008e11 −0.579388
\(334\) 3.29930e11 1.45065
\(335\) −4.65927e10 −0.202123
\(336\) −2.33502e9 −0.00999458
\(337\) 3.82436e11 1.61519 0.807596 0.589736i \(-0.200768\pi\)
0.807596 + 0.589736i \(0.200768\pi\)
\(338\) −2.99470e10 −0.124804
\(339\) −3.02710e9 −0.0124488
\(340\) −7.33558e9 −0.0297700
\(341\) −1.19665e11 −0.479263
\(342\) 3.56638e11 1.40965
\(343\) −2.61357e11 −1.01955
\(344\) −3.74269e10 −0.144102
\(345\) 1.41471e9 0.00537629
\(346\) 4.81985e11 1.80797
\(347\) −9.85113e10 −0.364757 −0.182378 0.983228i \(-0.558380\pi\)
−0.182378 + 0.983228i \(0.558380\pi\)
\(348\) −3.20154e8 −0.00117018
\(349\) −2.38389e11 −0.860145 −0.430073 0.902794i \(-0.641512\pi\)
−0.430073 + 0.902794i \(0.641512\pi\)
\(350\) 1.32471e11 0.471862
\(351\) −8.59964e9 −0.0302412
\(352\) 1.38080e10 0.0479389
\(353\) 1.97866e10 0.0678243 0.0339122 0.999425i \(-0.489203\pi\)
0.0339122 + 0.999425i \(0.489203\pi\)
\(354\) 5.01275e9 0.0169653
\(355\) 2.87006e11 0.959100
\(356\) −4.10841e10 −0.135566
\(357\) 1.55967e9 0.00508191
\(358\) 1.31217e11 0.422200
\(359\) 5.77059e10 0.183356 0.0916781 0.995789i \(-0.470777\pi\)
0.0916781 + 0.995789i \(0.470777\pi\)
\(360\) 1.64612e11 0.516535
\(361\) 2.61341e11 0.809887
\(362\) −5.12156e11 −1.56752
\(363\) −4.44611e9 −0.0134400
\(364\) 2.23819e10 0.0668253
\(365\) −2.58727e11 −0.762999
\(366\) −3.04318e9 −0.00886466
\(367\) 3.72822e11 1.07276 0.536382 0.843975i \(-0.319791\pi\)
0.536382 + 0.843975i \(0.319791\pi\)
\(368\) 2.63473e11 0.748895
\(369\) 4.40875e11 1.23793
\(370\) 1.19708e11 0.332059
\(371\) 8.90727e10 0.244097
\(372\) −1.01782e9 −0.00275567
\(373\) −1.51504e11 −0.405260 −0.202630 0.979255i \(-0.564949\pi\)
−0.202630 + 0.979255i \(0.564949\pi\)
\(374\) −5.36808e10 −0.141872
\(375\) 5.09129e9 0.0132950
\(376\) 2.14668e11 0.553888
\(377\) −3.45310e11 −0.880385
\(378\) 7.63718e9 0.0192406
\(379\) 3.48542e11 0.867719 0.433860 0.900981i \(-0.357151\pi\)
0.433860 + 0.900981i \(0.357151\pi\)
\(380\) −2.94080e10 −0.0723501
\(381\) 2.99899e9 0.00729142
\(382\) 3.91341e11 0.940308
\(383\) −1.32499e11 −0.314643 −0.157321 0.987547i \(-0.550286\pi\)
−0.157321 + 0.987547i \(0.550286\pi\)
\(384\) −6.83196e9 −0.0160345
\(385\) −3.70164e10 −0.0858660
\(386\) 8.06744e11 1.84966
\(387\) 6.72785e10 0.152467
\(388\) 3.49472e10 0.0782834
\(389\) −7.39994e11 −1.63853 −0.819266 0.573413i \(-0.805619\pi\)
−0.819266 + 0.573413i \(0.805619\pi\)
\(390\) 3.95877e9 0.00866501
\(391\) −1.75986e11 −0.380789
\(392\) −2.59564e11 −0.555211
\(393\) −3.04775e8 −0.000644486 0
\(394\) 5.89209e11 1.23179
\(395\) 4.68301e11 0.967918
\(396\) −1.17687e10 −0.0240491
\(397\) −5.46776e10 −0.110472 −0.0552360 0.998473i \(-0.517591\pi\)
−0.0552360 + 0.998473i \(0.517591\pi\)
\(398\) −6.82909e11 −1.36423
\(399\) 6.25266e9 0.0123506
\(400\) 3.90783e11 0.763248
\(401\) −3.83788e11 −0.741212 −0.370606 0.928790i \(-0.620850\pi\)
−0.370606 + 0.928790i \(0.620850\pi\)
\(402\) −2.90004e9 −0.00553843
\(403\) −1.09779e12 −2.07323
\(404\) −5.97269e9 −0.0111546
\(405\) −2.95846e11 −0.546408
\(406\) 3.06663e11 0.560137
\(407\) 7.84496e10 0.141715
\(408\) 4.18526e9 0.00747742
\(409\) −7.70773e11 −1.36198 −0.680992 0.732291i \(-0.738451\pi\)
−0.680992 + 0.732291i \(0.738451\pi\)
\(410\) −4.05947e11 −0.709483
\(411\) −4.05002e9 −0.00700115
\(412\) −6.57551e10 −0.112433
\(413\) −4.29994e11 −0.727256
\(414\) −4.30828e11 −0.720779
\(415\) −3.04243e11 −0.503505
\(416\) 1.26672e11 0.207378
\(417\) 4.66863e9 0.00756096
\(418\) −2.15204e11 −0.344791
\(419\) 1.70733e11 0.270617 0.135309 0.990804i \(-0.456797\pi\)
0.135309 + 0.990804i \(0.456797\pi\)
\(420\) −3.14845e8 −0.000493713 0
\(421\) −1.17127e12 −1.81714 −0.908572 0.417728i \(-0.862827\pi\)
−0.908572 + 0.417728i \(0.862827\pi\)
\(422\) −7.30498e11 −1.12128
\(423\) −3.85887e11 −0.586042
\(424\) 2.39020e11 0.359159
\(425\) −2.61023e11 −0.388086
\(426\) 1.78639e10 0.0262805
\(427\) 2.61044e11 0.380004
\(428\) −6.26983e10 −0.0903148
\(429\) 2.59435e9 0.00369803
\(430\) −6.19484e10 −0.0873821
\(431\) 1.01493e12 1.41673 0.708364 0.705847i \(-0.249434\pi\)
0.708364 + 0.705847i \(0.249434\pi\)
\(432\) 2.25293e10 0.0311222
\(433\) 4.90479e11 0.670540 0.335270 0.942122i \(-0.391172\pi\)
0.335270 + 0.942122i \(0.391172\pi\)
\(434\) 9.74928e11 1.31907
\(435\) 4.85745e9 0.00650439
\(436\) −4.29338e10 −0.0568998
\(437\) −7.05521e11 −0.925430
\(438\) −1.61038e10 −0.0209071
\(439\) 7.89685e11 1.01476 0.507380 0.861722i \(-0.330614\pi\)
0.507380 + 0.861722i \(0.330614\pi\)
\(440\) −9.93306e10 −0.126342
\(441\) 4.66593e11 0.587441
\(442\) −4.92459e11 −0.613720
\(443\) −1.09410e12 −1.34971 −0.674857 0.737949i \(-0.735795\pi\)
−0.674857 + 0.737949i \(0.735795\pi\)
\(444\) 6.67256e8 0.000814834 0
\(445\) 6.23337e11 0.753535
\(446\) −5.70386e11 −0.682593
\(447\) −2.83031e9 −0.00335312
\(448\) 4.83625e11 0.567228
\(449\) −5.93385e11 −0.689015 −0.344507 0.938784i \(-0.611954\pi\)
−0.344507 + 0.938784i \(0.611954\pi\)
\(450\) −6.39003e11 −0.734592
\(451\) −2.66034e11 −0.302791
\(452\) −7.60151e10 −0.0856598
\(453\) 8.28639e9 0.00924535
\(454\) 1.19291e11 0.131783
\(455\) −3.39583e11 −0.371445
\(456\) 1.67785e10 0.0181724
\(457\) 4.56802e11 0.489898 0.244949 0.969536i \(-0.421229\pi\)
0.244949 + 0.969536i \(0.421229\pi\)
\(458\) 1.02928e11 0.109304
\(459\) −1.50484e10 −0.0158246
\(460\) 3.55256e10 0.0369940
\(461\) 1.69775e12 1.75073 0.875367 0.483460i \(-0.160620\pi\)
0.875367 + 0.483460i \(0.160620\pi\)
\(462\) −2.30399e9 −0.00235284
\(463\) 3.87467e11 0.391851 0.195925 0.980619i \(-0.437229\pi\)
0.195925 + 0.980619i \(0.437229\pi\)
\(464\) 9.04639e11 0.906034
\(465\) 1.54426e10 0.0153173
\(466\) −1.95702e11 −0.192247
\(467\) 1.51160e12 1.47066 0.735328 0.677711i \(-0.237028\pi\)
0.735328 + 0.677711i \(0.237028\pi\)
\(468\) −1.07964e11 −0.104033
\(469\) 2.48766e11 0.237417
\(470\) 3.55316e11 0.335872
\(471\) 2.44763e10 0.0229167
\(472\) −1.15385e12 −1.07007
\(473\) −4.05974e10 −0.0372927
\(474\) 2.91482e10 0.0265222
\(475\) −1.04643e12 −0.943166
\(476\) 3.91658e10 0.0349684
\(477\) −4.29661e11 −0.380009
\(478\) 8.40980e11 0.736818
\(479\) 1.14506e12 0.993847 0.496924 0.867794i \(-0.334463\pi\)
0.496924 + 0.867794i \(0.334463\pi\)
\(480\) −1.78189e9 −0.00153213
\(481\) 7.19685e11 0.613041
\(482\) 1.93747e11 0.163502
\(483\) −7.55338e9 −0.00631508
\(484\) −1.11649e11 −0.0924803
\(485\) −5.30226e11 −0.435135
\(486\) −5.52612e10 −0.0449321
\(487\) 1.23558e12 0.995382 0.497691 0.867354i \(-0.334181\pi\)
0.497691 + 0.867354i \(0.334181\pi\)
\(488\) 7.00490e11 0.559130
\(489\) −2.02418e10 −0.0160088
\(490\) −4.29627e11 −0.336674
\(491\) 1.82342e12 1.41586 0.707931 0.706281i \(-0.249629\pi\)
0.707931 + 0.706281i \(0.249629\pi\)
\(492\) −2.26277e9 −0.00174099
\(493\) −6.04252e11 −0.460688
\(494\) −1.97425e12 −1.49152
\(495\) 1.78557e11 0.133676
\(496\) 2.87598e12 2.13363
\(497\) −1.53237e12 −1.12657
\(498\) −1.89368e10 −0.0137967
\(499\) 1.34781e12 0.973145 0.486573 0.873640i \(-0.338247\pi\)
0.486573 + 0.873640i \(0.338247\pi\)
\(500\) 1.27850e11 0.0914821
\(501\) −2.79024e10 −0.0197866
\(502\) −7.88323e11 −0.554036
\(503\) 1.07497e12 0.748757 0.374378 0.927276i \(-0.377856\pi\)
0.374378 + 0.927276i \(0.377856\pi\)
\(504\) −8.78888e11 −0.606731
\(505\) 9.06189e10 0.0620023
\(506\) 2.59972e11 0.176298
\(507\) 2.53264e9 0.00170230
\(508\) 7.53091e10 0.0501719
\(509\) −8.76635e11 −0.578881 −0.289440 0.957196i \(-0.593469\pi\)
−0.289440 + 0.957196i \(0.593469\pi\)
\(510\) 6.92739e9 0.00453423
\(511\) 1.38138e12 0.896231
\(512\) 1.26779e12 0.815326
\(513\) −6.03283e10 −0.0384585
\(514\) −1.42049e12 −0.897644
\(515\) 9.97651e11 0.624951
\(516\) −3.45303e8 −0.000214426 0
\(517\) 2.32853e11 0.143343
\(518\) −6.39138e11 −0.390042
\(519\) −4.07617e10 −0.0246604
\(520\) −9.11245e11 −0.546537
\(521\) 1.32443e11 0.0787517 0.0393758 0.999224i \(-0.487463\pi\)
0.0393758 + 0.999224i \(0.487463\pi\)
\(522\) −1.47925e12 −0.872018
\(523\) −3.27846e12 −1.91607 −0.958037 0.286643i \(-0.907461\pi\)
−0.958037 + 0.286643i \(0.907461\pi\)
\(524\) −7.65337e9 −0.00443467
\(525\) −1.12032e10 −0.00643611
\(526\) 5.58257e11 0.317979
\(527\) −1.92101e12 −1.08488
\(528\) −6.79665e9 −0.00380577
\(529\) −9.48865e11 −0.526810
\(530\) 3.95622e11 0.217790
\(531\) 2.07417e12 1.13219
\(532\) 1.57014e11 0.0849836
\(533\) −2.44056e12 −1.30984
\(534\) 3.87980e10 0.0206478
\(535\) 9.51272e11 0.502010
\(536\) 6.67542e11 0.349331
\(537\) −1.10971e10 −0.00575873
\(538\) 1.18898e12 0.611864
\(539\) −2.81553e11 −0.143685
\(540\) 3.03775e9 0.00153738
\(541\) 1.03137e12 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(542\) 8.41368e8 0.000418783 0
\(543\) 4.33133e10 0.0213807
\(544\) 2.21662e11 0.108517
\(545\) 6.51402e11 0.316275
\(546\) −2.11365e10 −0.0101781
\(547\) 1.11547e12 0.532741 0.266371 0.963871i \(-0.414175\pi\)
0.266371 + 0.963871i \(0.414175\pi\)
\(548\) −1.01702e11 −0.0481746
\(549\) −1.25920e12 −0.591588
\(550\) 3.85589e11 0.179677
\(551\) −2.42242e12 −1.11961
\(552\) −2.02689e10 −0.00929189
\(553\) −2.50033e12 −1.13693
\(554\) −1.42815e12 −0.644138
\(555\) −1.01238e10 −0.00452922
\(556\) 1.17236e11 0.0520266
\(557\) 2.62897e12 1.15728 0.578638 0.815585i \(-0.303584\pi\)
0.578638 + 0.815585i \(0.303584\pi\)
\(558\) −4.70277e12 −2.05352
\(559\) −3.72435e11 −0.161323
\(560\) 8.89637e11 0.382267
\(561\) 4.53981e9 0.00193511
\(562\) 1.04786e12 0.443086
\(563\) 3.88359e12 1.62909 0.814546 0.580099i \(-0.196986\pi\)
0.814546 + 0.580099i \(0.196986\pi\)
\(564\) 1.98054e9 0.000824192 0
\(565\) 1.15332e12 0.476136
\(566\) −1.09867e12 −0.449979
\(567\) 1.57956e12 0.641820
\(568\) −4.11199e12 −1.65762
\(569\) −3.46357e12 −1.38522 −0.692609 0.721313i \(-0.743539\pi\)
−0.692609 + 0.721313i \(0.743539\pi\)
\(570\) 2.77716e10 0.0110195
\(571\) 6.88790e11 0.271159 0.135580 0.990766i \(-0.456710\pi\)
0.135580 + 0.990766i \(0.456710\pi\)
\(572\) 6.51480e10 0.0254460
\(573\) −3.30959e10 −0.0128256
\(574\) 2.16741e12 0.833371
\(575\) 1.26411e12 0.482259
\(576\) −2.33287e12 −0.883058
\(577\) −1.20927e12 −0.454184 −0.227092 0.973873i \(-0.572922\pi\)
−0.227092 + 0.973873i \(0.572922\pi\)
\(578\) 1.95047e12 0.726880
\(579\) −6.82268e10 −0.0252291
\(580\) 1.21978e11 0.0447564
\(581\) 1.62440e12 0.591425
\(582\) −3.30025e10 −0.0119232
\(583\) 2.59268e11 0.0929480
\(584\) 3.70683e12 1.31870
\(585\) 1.63805e12 0.578264
\(586\) −3.24166e12 −1.13561
\(587\) 3.86701e12 1.34432 0.672162 0.740404i \(-0.265366\pi\)
0.672162 + 0.740404i \(0.265366\pi\)
\(588\) −2.39476e9 −0.000826160 0
\(589\) −7.70124e12 −2.63658
\(590\) −1.90984e12 −0.648879
\(591\) −4.98297e10 −0.0168014
\(592\) −1.88542e12 −0.630901
\(593\) −1.67086e10 −0.00554872 −0.00277436 0.999996i \(-0.500883\pi\)
−0.00277436 + 0.999996i \(0.500883\pi\)
\(594\) 2.22298e10 0.00732652
\(595\) −5.94232e11 −0.194370
\(596\) −7.10732e10 −0.0230727
\(597\) 5.77540e10 0.0186079
\(598\) 2.38494e12 0.762645
\(599\) 7.82498e11 0.248349 0.124174 0.992260i \(-0.460372\pi\)
0.124174 + 0.992260i \(0.460372\pi\)
\(600\) −3.00628e10 −0.00946997
\(601\) 4.64544e12 1.45242 0.726210 0.687473i \(-0.241280\pi\)
0.726210 + 0.687473i \(0.241280\pi\)
\(602\) 3.30752e11 0.102640
\(603\) −1.19997e12 −0.369610
\(604\) 2.08084e11 0.0636168
\(605\) 1.69396e12 0.514047
\(606\) 5.64034e9 0.00169894
\(607\) −3.78070e12 −1.13038 −0.565188 0.824962i \(-0.691197\pi\)
−0.565188 + 0.824962i \(0.691197\pi\)
\(608\) 8.88632e11 0.263728
\(609\) −2.59346e10 −0.00764016
\(610\) 1.15944e12 0.339051
\(611\) 2.13616e12 0.620082
\(612\) −1.88925e11 −0.0544386
\(613\) −4.92457e12 −1.40863 −0.704314 0.709889i \(-0.748745\pi\)
−0.704314 + 0.709889i \(0.748745\pi\)
\(614\) 4.04530e11 0.114866
\(615\) 3.43312e10 0.00967722
\(616\) 5.30341e11 0.148403
\(617\) 1.68798e12 0.468904 0.234452 0.972128i \(-0.424670\pi\)
0.234452 + 0.972128i \(0.424670\pi\)
\(618\) 6.20962e10 0.0171245
\(619\) −1.50175e12 −0.411141 −0.205570 0.978642i \(-0.565905\pi\)
−0.205570 + 0.978642i \(0.565905\pi\)
\(620\) 3.87786e11 0.105397
\(621\) 7.28781e10 0.0196646
\(622\) −3.98622e12 −1.06784
\(623\) −3.32809e12 −0.885114
\(624\) −6.23515e10 −0.0164633
\(625\) 7.34606e11 0.192573
\(626\) 2.67363e12 0.695851
\(627\) 1.81999e10 0.00470289
\(628\) 6.14637e11 0.157688
\(629\) 1.25937e12 0.320792
\(630\) −1.45472e12 −0.367915
\(631\) −6.62550e12 −1.66374 −0.831872 0.554968i \(-0.812731\pi\)
−0.831872 + 0.554968i \(0.812731\pi\)
\(632\) −6.70944e12 −1.67286
\(633\) 6.17786e10 0.0152940
\(634\) −2.22359e12 −0.546579
\(635\) −1.14261e12 −0.278878
\(636\) 2.20521e9 0.000534433 0
\(637\) −2.58293e12 −0.621562
\(638\) 8.92617e11 0.213291
\(639\) 7.39171e12 1.75385
\(640\) 2.60296e12 0.613278
\(641\) 7.46690e12 1.74694 0.873472 0.486875i \(-0.161863\pi\)
0.873472 + 0.486875i \(0.161863\pi\)
\(642\) 5.92094e10 0.0137557
\(643\) 4.63171e12 1.06854 0.534271 0.845313i \(-0.320586\pi\)
0.534271 + 0.845313i \(0.320586\pi\)
\(644\) −1.89677e11 −0.0434538
\(645\) 5.23901e9 0.00119188
\(646\) −3.45470e12 −0.780485
\(647\) −2.14353e12 −0.480906 −0.240453 0.970661i \(-0.577296\pi\)
−0.240453 + 0.970661i \(0.577296\pi\)
\(648\) 4.23864e12 0.944362
\(649\) −1.25160e12 −0.276927
\(650\) 3.53734e12 0.777261
\(651\) −8.24502e10 −0.0179919
\(652\) −5.08302e11 −0.110156
\(653\) 8.17170e12 1.75874 0.879372 0.476135i \(-0.157962\pi\)
0.879372 + 0.476135i \(0.157962\pi\)
\(654\) 4.05448e10 0.00866633
\(655\) 1.16119e11 0.0246499
\(656\) 6.39375e12 1.34800
\(657\) −6.66339e12 −1.39525
\(658\) −1.89708e12 −0.394521
\(659\) 3.65558e12 0.755044 0.377522 0.926001i \(-0.376776\pi\)
0.377522 + 0.926001i \(0.376776\pi\)
\(660\) −9.16432e8 −0.000187998 0
\(661\) −3.52150e12 −0.717499 −0.358750 0.933434i \(-0.616797\pi\)
−0.358750 + 0.933434i \(0.616797\pi\)
\(662\) 6.94837e12 1.40612
\(663\) 4.16476e10 0.00837103
\(664\) 4.35894e12 0.870211
\(665\) −2.38225e12 −0.472378
\(666\) 3.08302e12 0.607215
\(667\) 2.92634e12 0.572479
\(668\) −7.00671e11 −0.136151
\(669\) 4.82379e10 0.00931045
\(670\) 1.10491e12 0.211831
\(671\) 7.59831e11 0.144699
\(672\) 9.51378e9 0.00179966
\(673\) −5.32239e12 −1.00009 −0.500045 0.866000i \(-0.666683\pi\)
−0.500045 + 0.866000i \(0.666683\pi\)
\(674\) −9.06915e12 −1.69277
\(675\) 1.08093e11 0.0200415
\(676\) 6.35983e10 0.0117135
\(677\) −9.87601e11 −0.180689 −0.0903447 0.995911i \(-0.528797\pi\)
−0.0903447 + 0.995911i \(0.528797\pi\)
\(678\) 7.17852e10 0.0130467
\(679\) 2.83096e12 0.511116
\(680\) −1.59457e12 −0.285992
\(681\) −1.00885e10 −0.00179749
\(682\) 2.83776e12 0.502281
\(683\) 8.97820e11 0.157869 0.0789343 0.996880i \(-0.474848\pi\)
0.0789343 + 0.996880i \(0.474848\pi\)
\(684\) −7.57389e11 −0.132302
\(685\) 1.54305e12 0.267776
\(686\) 6.19786e12 1.06852
\(687\) −8.70463e9 −0.00149089
\(688\) 9.75701e11 0.166023
\(689\) 2.37848e12 0.402081
\(690\) −3.35488e10 −0.00563450
\(691\) 3.95252e12 0.659512 0.329756 0.944066i \(-0.393034\pi\)
0.329756 + 0.944066i \(0.393034\pi\)
\(692\) −1.02359e12 −0.169687
\(693\) −9.53341e11 −0.157018
\(694\) 2.33611e12 0.382275
\(695\) −1.77873e12 −0.289187
\(696\) −6.95936e10 −0.0112416
\(697\) −4.27070e12 −0.685412
\(698\) 5.65320e12 0.901456
\(699\) 1.65506e10 0.00262221
\(700\) −2.81328e11 −0.0442866
\(701\) −1.15110e13 −1.80046 −0.900230 0.435415i \(-0.856601\pi\)
−0.900230 + 0.435415i \(0.856601\pi\)
\(702\) 2.03933e11 0.0316936
\(703\) 5.04874e12 0.779622
\(704\) 1.40771e12 0.215991
\(705\) −3.00492e10 −0.00458123
\(706\) −4.69223e11 −0.0710818
\(707\) −4.83828e11 −0.0728289
\(708\) −1.06455e10 −0.00159228
\(709\) 1.05617e12 0.156974 0.0784869 0.996915i \(-0.474991\pi\)
0.0784869 + 0.996915i \(0.474991\pi\)
\(710\) −6.80611e12 −1.00516
\(711\) 1.20609e13 1.76997
\(712\) −8.93067e12 −1.30234
\(713\) 9.30329e12 1.34814
\(714\) −3.69864e10 −0.00532599
\(715\) −9.88439e11 −0.141440
\(716\) −2.78666e11 −0.0396255
\(717\) −7.11222e10 −0.0100501
\(718\) −1.36845e12 −0.192162
\(719\) 8.18317e12 1.14194 0.570968 0.820973i \(-0.306568\pi\)
0.570968 + 0.820973i \(0.306568\pi\)
\(720\) −4.29135e12 −0.595111
\(721\) −5.32661e12 −0.734078
\(722\) −6.19748e12 −0.848784
\(723\) −1.63853e10 −0.00223013
\(724\) 1.08766e12 0.147120
\(725\) 4.34035e12 0.583450
\(726\) 1.05436e11 0.0140855
\(727\) −9.27117e12 −1.23092 −0.615460 0.788168i \(-0.711030\pi\)
−0.615460 + 0.788168i \(0.711030\pi\)
\(728\) 4.86527e12 0.641972
\(729\) −7.61625e12 −0.998774
\(730\) 6.13549e12 0.799644
\(731\) −6.51718e11 −0.0844173
\(732\) 6.46277e9 0.000831992 0
\(733\) −4.24321e12 −0.542909 −0.271454 0.962451i \(-0.587505\pi\)
−0.271454 + 0.962451i \(0.587505\pi\)
\(734\) −8.84116e12 −1.12429
\(735\) 3.63338e10 0.00459217
\(736\) −1.07349e12 −0.134849
\(737\) 7.24092e11 0.0904045
\(738\) −1.04550e13 −1.29739
\(739\) 2.15623e12 0.265947 0.132974 0.991120i \(-0.457547\pi\)
0.132974 + 0.991120i \(0.457547\pi\)
\(740\) −2.54223e11 −0.0311653
\(741\) 1.66963e11 0.0203441
\(742\) −2.11229e12 −0.255820
\(743\) −8.84199e12 −1.06439 −0.532194 0.846622i \(-0.678632\pi\)
−0.532194 + 0.846622i \(0.678632\pi\)
\(744\) −2.21248e11 −0.0264729
\(745\) 1.07834e12 0.128248
\(746\) 3.59278e12 0.424724
\(747\) −7.83563e12 −0.920728
\(748\) 1.14001e11 0.0133154
\(749\) −5.07899e12 −0.589670
\(750\) −1.20736e11 −0.0139335
\(751\) −2.48740e12 −0.285342 −0.142671 0.989770i \(-0.545569\pi\)
−0.142671 + 0.989770i \(0.545569\pi\)
\(752\) −5.59630e12 −0.638147
\(753\) 6.66689e10 0.00755694
\(754\) 8.18873e12 0.922668
\(755\) −3.15709e12 −0.353611
\(756\) −1.62190e10 −0.00180583
\(757\) 4.59030e12 0.508054 0.254027 0.967197i \(-0.418245\pi\)
0.254027 + 0.967197i \(0.418245\pi\)
\(758\) −8.26539e12 −0.909394
\(759\) −2.19859e10 −0.00240468
\(760\) −6.39257e12 −0.695047
\(761\) −1.26409e13 −1.36631 −0.683153 0.730275i \(-0.739392\pi\)
−0.683153 + 0.730275i \(0.739392\pi\)
\(762\) −7.11185e10 −0.00764161
\(763\) −3.47793e12 −0.371502
\(764\) −8.31089e11 −0.0882525
\(765\) 2.86640e12 0.302594
\(766\) 3.14210e12 0.329755
\(767\) −1.14820e13 −1.19795
\(768\) 4.02879e10 0.00417878
\(769\) 7.97884e11 0.0822757 0.0411378 0.999153i \(-0.486902\pi\)
0.0411378 + 0.999153i \(0.486902\pi\)
\(770\) 8.77814e11 0.0899900
\(771\) 1.20132e11 0.0122437
\(772\) −1.71328e12 −0.173600
\(773\) 1.55667e13 1.56816 0.784079 0.620662i \(-0.213136\pi\)
0.784079 + 0.620662i \(0.213136\pi\)
\(774\) −1.59545e12 −0.159790
\(775\) 1.37986e13 1.37397
\(776\) 7.59665e12 0.752046
\(777\) 5.40523e10 0.00532010
\(778\) 1.75483e13 1.71723
\(779\) −1.71210e13 −1.66576
\(780\) −8.40721e9 −0.000813254 0
\(781\) −4.46033e12 −0.428981
\(782\) 4.17337e12 0.399077
\(783\) 2.50228e11 0.0237908
\(784\) 6.76672e12 0.639670
\(785\) −9.32540e12 −0.876504
\(786\) 7.22749e9 0.000675439 0
\(787\) −1.64141e13 −1.52521 −0.762607 0.646862i \(-0.776081\pi\)
−0.762607 + 0.646862i \(0.776081\pi\)
\(788\) −1.25130e12 −0.115609
\(789\) −4.72121e10 −0.00433717
\(790\) −1.11054e13 −1.01440
\(791\) −6.15774e12 −0.559277
\(792\) −2.55822e12 −0.231033
\(793\) 6.97058e12 0.625950
\(794\) 1.29663e12 0.115778
\(795\) −3.34579e10 −0.00297062
\(796\) 1.45029e12 0.128040
\(797\) −2.56157e12 −0.224877 −0.112438 0.993659i \(-0.535866\pi\)
−0.112438 + 0.993659i \(0.535866\pi\)
\(798\) −1.48277e11 −0.0129437
\(799\) 3.73804e12 0.324477
\(800\) −1.59220e12 −0.137433
\(801\) 1.60538e13 1.37794
\(802\) 9.10122e12 0.776810
\(803\) 4.02085e12 0.341270
\(804\) 6.15880e9 0.000519809 0
\(805\) 2.87782e12 0.241536
\(806\) 2.60332e13 2.17280
\(807\) −1.00553e11 −0.00834571
\(808\) −1.29831e12 −0.107159
\(809\) −2.05067e13 −1.68317 −0.841584 0.540126i \(-0.818377\pi\)
−0.841584 + 0.540126i \(0.818377\pi\)
\(810\) 7.01573e12 0.572651
\(811\) 2.03393e13 1.65098 0.825491 0.564415i \(-0.190898\pi\)
0.825491 + 0.564415i \(0.190898\pi\)
\(812\) −6.51258e11 −0.0525716
\(813\) −7.11550e7 −5.71213e−6 0
\(814\) −1.86037e12 −0.148521
\(815\) 7.71207e12 0.612296
\(816\) −1.09108e11 −0.00861490
\(817\) −2.61271e12 −0.205159
\(818\) 1.82782e13 1.42740
\(819\) −8.74581e12 −0.679239
\(820\) 8.62107e11 0.0665885
\(821\) 7.12024e12 0.546954 0.273477 0.961879i \(-0.411826\pi\)
0.273477 + 0.961879i \(0.411826\pi\)
\(822\) 9.60429e10 0.00733740
\(823\) 1.59854e13 1.21458 0.607289 0.794481i \(-0.292257\pi\)
0.607289 + 0.794481i \(0.292257\pi\)
\(824\) −1.42935e13 −1.08011
\(825\) −3.26095e10 −0.00245076
\(826\) 1.01969e13 0.762184
\(827\) −9.14084e11 −0.0679534 −0.0339767 0.999423i \(-0.510817\pi\)
−0.0339767 + 0.999423i \(0.510817\pi\)
\(828\) 9.14946e11 0.0676486
\(829\) −1.85974e13 −1.36759 −0.683797 0.729672i \(-0.739673\pi\)
−0.683797 + 0.729672i \(0.739673\pi\)
\(830\) 7.21486e12 0.527687
\(831\) 1.20779e11 0.00878592
\(832\) 1.29141e13 0.934350
\(833\) −4.51982e12 −0.325251
\(834\) −1.10713e11 −0.00792409
\(835\) 1.06307e13 0.756787
\(836\) 4.57027e11 0.0323603
\(837\) 7.95513e11 0.0560251
\(838\) −4.04880e12 −0.283614
\(839\) 1.49886e13 1.04432 0.522159 0.852848i \(-0.325127\pi\)
0.522159 + 0.852848i \(0.325127\pi\)
\(840\) −6.84394e10 −0.00474296
\(841\) −4.45949e12 −0.307400
\(842\) 2.77758e13 1.90442
\(843\) −8.86177e10 −0.00604361
\(844\) 1.55135e12 0.105237
\(845\) −9.64927e11 −0.0651088
\(846\) 9.15099e12 0.614188
\(847\) −9.04429e12 −0.603808
\(848\) −6.23113e12 −0.413795
\(849\) 9.29148e10 0.00613762
\(850\) 6.18994e12 0.406725
\(851\) −6.09901e12 −0.398636
\(852\) −3.79375e10 −0.00246656
\(853\) 2.59861e13 1.68062 0.840310 0.542106i \(-0.182373\pi\)
0.840310 + 0.542106i \(0.182373\pi\)
\(854\) −6.19044e12 −0.398255
\(855\) 1.14913e13 0.735395
\(856\) −1.36291e13 −0.867628
\(857\) 1.59617e13 1.01080 0.505400 0.862885i \(-0.331345\pi\)
0.505400 + 0.862885i \(0.331345\pi\)
\(858\) −6.15228e10 −0.00387564
\(859\) −2.03909e13 −1.27781 −0.638905 0.769286i \(-0.720612\pi\)
−0.638905 + 0.769286i \(0.720612\pi\)
\(860\) 1.31560e11 0.00820123
\(861\) −1.83299e11 −0.0113670
\(862\) −2.40681e13 −1.48477
\(863\) −3.06757e13 −1.88255 −0.941273 0.337646i \(-0.890369\pi\)
−0.941273 + 0.337646i \(0.890369\pi\)
\(864\) −9.17929e10 −0.00560399
\(865\) 1.55301e13 0.943196
\(866\) −1.16313e13 −0.702745
\(867\) −1.64952e11 −0.00991451
\(868\) −2.07045e12 −0.123801
\(869\) −7.27782e12 −0.432925
\(870\) −1.15190e11 −0.00681678
\(871\) 6.64272e12 0.391079
\(872\) −9.33275e12 −0.546620
\(873\) −1.36557e13 −0.795703
\(874\) 1.67308e13 0.969877
\(875\) 1.03567e13 0.597291
\(876\) 3.41995e10 0.00196224
\(877\) −1.24011e13 −0.707885 −0.353943 0.935267i \(-0.615159\pi\)
−0.353943 + 0.935267i \(0.615159\pi\)
\(878\) −1.87267e13 −1.06350
\(879\) 2.74149e11 0.0154895
\(880\) 2.58950e12 0.145561
\(881\) −2.11899e13 −1.18505 −0.592526 0.805552i \(-0.701869\pi\)
−0.592526 + 0.805552i \(0.701869\pi\)
\(882\) −1.10649e13 −0.615654
\(883\) 3.32814e13 1.84238 0.921188 0.389118i \(-0.127220\pi\)
0.921188 + 0.389118i \(0.127220\pi\)
\(884\) 1.04583e12 0.0576006
\(885\) 1.61516e11 0.00885059
\(886\) 2.59458e13 1.41454
\(887\) −2.83029e13 −1.53523 −0.767616 0.640910i \(-0.778557\pi\)
−0.767616 + 0.640910i \(0.778557\pi\)
\(888\) 1.45045e11 0.00782788
\(889\) 6.10055e12 0.327575
\(890\) −1.47819e13 −0.789725
\(891\) 4.59770e12 0.244394
\(892\) 1.21133e12 0.0640647
\(893\) 1.49856e13 0.788575
\(894\) 6.71183e10 0.00351417
\(895\) 4.22798e12 0.220257
\(896\) −1.38976e13 −0.720367
\(897\) −2.01696e11 −0.0104023
\(898\) 1.40716e13 0.722106
\(899\) 3.19430e13 1.63101
\(900\) 1.35705e12 0.0689451
\(901\) 4.16207e12 0.210401
\(902\) 6.30878e12 0.317334
\(903\) −2.79719e10 −0.00140000
\(904\) −1.65238e13 −0.822910
\(905\) −1.65022e13 −0.817757
\(906\) −1.96505e11 −0.00968938
\(907\) −1.55925e13 −0.765039 −0.382519 0.923947i \(-0.624943\pi\)
−0.382519 + 0.923947i \(0.624943\pi\)
\(908\) −2.53338e11 −0.0123684
\(909\) 2.33385e12 0.113380
\(910\) 8.05293e12 0.389285
\(911\) −8.70057e12 −0.418519 −0.209259 0.977860i \(-0.567105\pi\)
−0.209259 + 0.977860i \(0.567105\pi\)
\(912\) −4.37408e11 −0.0209368
\(913\) 4.72820e12 0.225205
\(914\) −1.08327e13 −0.513426
\(915\) −9.80546e10 −0.00462459
\(916\) −2.18586e11 −0.0102587
\(917\) −6.19975e11 −0.0289542
\(918\) 3.56860e11 0.0165846
\(919\) −3.60733e13 −1.66827 −0.834135 0.551560i \(-0.814033\pi\)
−0.834135 + 0.551560i \(0.814033\pi\)
\(920\) 7.72239e12 0.355391
\(921\) −3.42113e10 −0.00156676
\(922\) −4.02608e13 −1.83482
\(923\) −4.09184e13 −1.85572
\(924\) 4.89297e9 0.000220825 0
\(925\) −9.04604e12 −0.406275
\(926\) −9.18847e12 −0.410671
\(927\) 2.56941e13 1.14281
\(928\) −3.68585e12 −0.163144
\(929\) 1.24731e13 0.549418 0.274709 0.961527i \(-0.411418\pi\)
0.274709 + 0.961527i \(0.411418\pi\)
\(930\) −3.66207e11 −0.0160529
\(931\) −1.81198e13 −0.790458
\(932\) 4.15612e11 0.0180433
\(933\) 3.37117e11 0.0145651
\(934\) −3.58464e13 −1.54129
\(935\) −1.72966e12 −0.0740129
\(936\) −2.34687e13 −0.999419
\(937\) 2.42282e13 1.02682 0.513408 0.858145i \(-0.328383\pi\)
0.513408 + 0.858145i \(0.328383\pi\)
\(938\) −5.89927e12 −0.248820
\(939\) −2.26110e11 −0.00949128
\(940\) −7.54582e11 −0.0315232
\(941\) −1.17034e13 −0.486586 −0.243293 0.969953i \(-0.578228\pi\)
−0.243293 + 0.969953i \(0.578228\pi\)
\(942\) −5.80435e11 −0.0240173
\(943\) 2.06826e13 0.851733
\(944\) 3.00805e13 1.23285
\(945\) 2.46078e11 0.0100376
\(946\) 9.62734e11 0.0390838
\(947\) −3.01213e13 −1.21702 −0.608512 0.793545i \(-0.708233\pi\)
−0.608512 + 0.793545i \(0.708233\pi\)
\(948\) −6.19018e10 −0.00248923
\(949\) 3.68867e13 1.47629
\(950\) 2.48152e13 0.988464
\(951\) 1.88050e11 0.00745523
\(952\) 8.51367e12 0.335931
\(953\) −1.01806e13 −0.399813 −0.199906 0.979815i \(-0.564064\pi\)
−0.199906 + 0.979815i \(0.564064\pi\)
\(954\) 1.01891e13 0.398260
\(955\) 1.26095e13 0.490548
\(956\) −1.78598e12 −0.0691540
\(957\) −7.54891e10 −0.00290925
\(958\) −2.71542e13 −1.04158
\(959\) −8.23857e12 −0.314534
\(960\) −1.81662e11 −0.00690308
\(961\) 7.51120e13 2.84089
\(962\) −1.70667e13 −0.642484
\(963\) 2.44996e13 0.917995
\(964\) −4.11458e11 −0.0153454
\(965\) 2.59942e13 0.964947
\(966\) 1.79122e11 0.00661838
\(967\) 9.47914e11 0.0348618 0.0174309 0.999848i \(-0.494451\pi\)
0.0174309 + 0.999848i \(0.494451\pi\)
\(968\) −2.42696e13 −0.888432
\(969\) 2.92166e11 0.0106457
\(970\) 1.25739e13 0.456033
\(971\) 2.66136e13 0.960764 0.480382 0.877059i \(-0.340498\pi\)
0.480382 + 0.877059i \(0.340498\pi\)
\(972\) 1.17358e11 0.00421710
\(973\) 9.49693e12 0.339684
\(974\) −2.93007e13 −1.04319
\(975\) −2.99155e11 −0.0106017
\(976\) −1.82615e13 −0.644186
\(977\) −2.65754e13 −0.933155 −0.466578 0.884480i \(-0.654513\pi\)
−0.466578 + 0.884480i \(0.654513\pi\)
\(978\) 4.80017e11 0.0167777
\(979\) −9.68722e12 −0.337037
\(980\) 9.12397e11 0.0315985
\(981\) 1.67765e13 0.578352
\(982\) −4.32410e13 −1.48386
\(983\) 2.71576e12 0.0927687 0.0463843 0.998924i \(-0.485230\pi\)
0.0463843 + 0.998924i \(0.485230\pi\)
\(984\) −4.91869e11 −0.0167252
\(985\) 1.89850e13 0.642609
\(986\) 1.43293e13 0.482814
\(987\) 1.60437e11 0.00538119
\(988\) 4.19269e12 0.139987
\(989\) 3.15622e12 0.104902
\(990\) −4.23432e12 −0.140096
\(991\) 6.39401e12 0.210592 0.105296 0.994441i \(-0.466421\pi\)
0.105296 + 0.994441i \(0.466421\pi\)
\(992\) −1.17179e13 −0.384190
\(993\) −5.87627e11 −0.0191792
\(994\) 3.63389e13 1.18068
\(995\) −2.20041e13 −0.711705
\(996\) 4.02159e10 0.00129488
\(997\) 1.78534e13 0.572259 0.286130 0.958191i \(-0.407631\pi\)
0.286130 + 0.958191i \(0.407631\pi\)
\(998\) −3.19623e13 −1.01988
\(999\) −5.21519e11 −0.0165663
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.1.4 15
3.2 odd 2 387.10.a.c.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.4 15 1.1 even 1 trivial
387.10.a.c.1.12 15 3.2 odd 2