Properties

Label 43.8.a.a.1.5
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(6.31419\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.31419 q^{2} +11.6940 q^{3} -58.8742 q^{4} +148.086 q^{5} -97.2261 q^{6} +122.189 q^{7} +1553.71 q^{8} -2050.25 q^{9} +O(q^{10})\) \(q-8.31419 q^{2} +11.6940 q^{3} -58.8742 q^{4} +148.086 q^{5} -97.2261 q^{6} +122.189 q^{7} +1553.71 q^{8} -2050.25 q^{9} -1231.21 q^{10} +5634.96 q^{11} -688.475 q^{12} -7738.55 q^{13} -1015.90 q^{14} +1731.71 q^{15} -5381.93 q^{16} +1712.43 q^{17} +17046.2 q^{18} -51155.1 q^{19} -8718.44 q^{20} +1428.88 q^{21} -46850.2 q^{22} -77057.3 q^{23} +18169.1 q^{24} -56195.6 q^{25} +64339.8 q^{26} -49550.4 q^{27} -7193.77 q^{28} -48088.8 q^{29} -14397.8 q^{30} -192432. q^{31} -154128. q^{32} +65895.2 q^{33} -14237.5 q^{34} +18094.4 q^{35} +120707. q^{36} +461302. q^{37} +425313. q^{38} -90494.6 q^{39} +230082. q^{40} +688771. q^{41} -11879.9 q^{42} +79507.0 q^{43} -331754. q^{44} -303613. q^{45} +640669. q^{46} -619513. q^{47} -62936.2 q^{48} -808613. q^{49} +467221. q^{50} +20025.1 q^{51} +455601. q^{52} +574333. q^{53} +411971. q^{54} +834458. q^{55} +189846. q^{56} -598208. q^{57} +399820. q^{58} -1.01363e6 q^{59} -101953. q^{60} -3.23249e6 q^{61} +1.59991e6 q^{62} -250518. q^{63} +1.97034e6 q^{64} -1.14597e6 q^{65} -547866. q^{66} +3.15895e6 q^{67} -100818. q^{68} -901108. q^{69} -150441. q^{70} +2.46302e6 q^{71} -3.18549e6 q^{72} -4.02731e6 q^{73} -3.83535e6 q^{74} -657151. q^{75} +3.01172e6 q^{76} +688530. q^{77} +752389. q^{78} -463495. q^{79} -796987. q^{80} +3.90446e6 q^{81} -5.72657e6 q^{82} -5.49654e6 q^{83} -84123.9 q^{84} +253587. q^{85} -661036. q^{86} -562351. q^{87} +8.75509e6 q^{88} -8.18997e6 q^{89} +2.52430e6 q^{90} -945564. q^{91} +4.53669e6 q^{92} -2.25030e6 q^{93} +5.15075e6 q^{94} -7.57535e6 q^{95} -1.80237e6 q^{96} +1.20009e7 q^{97} +6.72296e6 q^{98} -1.15531e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.31419 −0.734878 −0.367439 0.930048i \(-0.619765\pi\)
−0.367439 + 0.930048i \(0.619765\pi\)
\(3\) 11.6940 0.250057 0.125028 0.992153i \(-0.460098\pi\)
0.125028 + 0.992153i \(0.460098\pi\)
\(4\) −58.8742 −0.459955
\(5\) 148.086 0.529808 0.264904 0.964275i \(-0.414660\pi\)
0.264904 + 0.964275i \(0.414660\pi\)
\(6\) −97.2261 −0.183761
\(7\) 122.189 0.134644 0.0673222 0.997731i \(-0.478554\pi\)
0.0673222 + 0.997731i \(0.478554\pi\)
\(8\) 1553.71 1.07289
\(9\) −2050.25 −0.937472
\(10\) −1231.21 −0.389344
\(11\) 5634.96 1.27649 0.638244 0.769834i \(-0.279661\pi\)
0.638244 + 0.769834i \(0.279661\pi\)
\(12\) −688.475 −0.115015
\(13\) −7738.55 −0.976917 −0.488459 0.872587i \(-0.662441\pi\)
−0.488459 + 0.872587i \(0.662441\pi\)
\(14\) −1015.90 −0.0989472
\(15\) 1731.71 0.132482
\(16\) −5381.93 −0.328487
\(17\) 1712.43 0.0845360 0.0422680 0.999106i \(-0.486542\pi\)
0.0422680 + 0.999106i \(0.486542\pi\)
\(18\) 17046.2 0.688927
\(19\) −51155.1 −1.71101 −0.855503 0.517798i \(-0.826752\pi\)
−0.855503 + 0.517798i \(0.826752\pi\)
\(20\) −8718.44 −0.243688
\(21\) 1428.88 0.0336687
\(22\) −46850.2 −0.938062
\(23\) −77057.3 −1.32058 −0.660292 0.751009i \(-0.729568\pi\)
−0.660292 + 0.751009i \(0.729568\pi\)
\(24\) 18169.1 0.268283
\(25\) −56195.6 −0.719304
\(26\) 64339.8 0.717915
\(27\) −49550.4 −0.484478
\(28\) −7193.77 −0.0619304
\(29\) −48088.8 −0.366143 −0.183072 0.983100i \(-0.558604\pi\)
−0.183072 + 0.983100i \(0.558604\pi\)
\(30\) −14397.8 −0.0973581
\(31\) −192432. −1.16014 −0.580070 0.814566i \(-0.696975\pi\)
−0.580070 + 0.814566i \(0.696975\pi\)
\(32\) −154128. −0.831491
\(33\) 65895.2 0.319194
\(34\) −14237.5 −0.0621236
\(35\) 18094.4 0.0713357
\(36\) 120707. 0.431195
\(37\) 461302. 1.49720 0.748599 0.663023i \(-0.230727\pi\)
0.748599 + 0.663023i \(0.230727\pi\)
\(38\) 425313. 1.25738
\(39\) −90494.6 −0.244285
\(40\) 230082. 0.568425
\(41\) 688771. 1.56074 0.780371 0.625316i \(-0.215030\pi\)
0.780371 + 0.625316i \(0.215030\pi\)
\(42\) −11879.9 −0.0247424
\(43\) 79507.0 0.152499
\(44\) −331754. −0.587127
\(45\) −303613. −0.496680
\(46\) 640669. 0.970468
\(47\) −619513. −0.870379 −0.435189 0.900339i \(-0.643319\pi\)
−0.435189 + 0.900339i \(0.643319\pi\)
\(48\) −62936.2 −0.0821403
\(49\) −808613. −0.981871
\(50\) 467221. 0.528600
\(51\) 20025.1 0.0211388
\(52\) 455601. 0.449338
\(53\) 574333. 0.529905 0.264952 0.964261i \(-0.414644\pi\)
0.264952 + 0.964261i \(0.414644\pi\)
\(54\) 411971. 0.356032
\(55\) 834458. 0.676293
\(56\) 189846. 0.144458
\(57\) −598208. −0.427848
\(58\) 399820. 0.269071
\(59\) −1.01363e6 −0.642539 −0.321269 0.946988i \(-0.604109\pi\)
−0.321269 + 0.946988i \(0.604109\pi\)
\(60\) −101953. −0.0609357
\(61\) −3.23249e6 −1.82340 −0.911702 0.410852i \(-0.865231\pi\)
−0.911702 + 0.410852i \(0.865231\pi\)
\(62\) 1.59991e6 0.852561
\(63\) −250518. −0.126225
\(64\) 1.97034e6 0.939531
\(65\) −1.14597e6 −0.517579
\(66\) −547866. −0.234569
\(67\) 3.15895e6 1.28316 0.641580 0.767056i \(-0.278279\pi\)
0.641580 + 0.767056i \(0.278279\pi\)
\(68\) −100818. −0.0388827
\(69\) −901108. −0.330221
\(70\) −150441. −0.0524230
\(71\) 2.46302e6 0.816701 0.408350 0.912825i \(-0.366104\pi\)
0.408350 + 0.912825i \(0.366104\pi\)
\(72\) −3.18549e6 −1.00580
\(73\) −4.02731e6 −1.21167 −0.605836 0.795590i \(-0.707161\pi\)
−0.605836 + 0.795590i \(0.707161\pi\)
\(74\) −3.83535e6 −1.10026
\(75\) −657151. −0.179867
\(76\) 3.01172e6 0.786985
\(77\) 688530. 0.171872
\(78\) 752389. 0.179519
\(79\) −463495. −0.105767 −0.0528835 0.998601i \(-0.516841\pi\)
−0.0528835 + 0.998601i \(0.516841\pi\)
\(80\) −796987. −0.174035
\(81\) 3.90446e6 0.816325
\(82\) −5.72657e6 −1.14696
\(83\) −5.49654e6 −1.05516 −0.527578 0.849507i \(-0.676900\pi\)
−0.527578 + 0.849507i \(0.676900\pi\)
\(84\) −84123.9 −0.0154861
\(85\) 253587. 0.0447878
\(86\) −661036. −0.112068
\(87\) −562351. −0.0915566
\(88\) 8.75509e6 1.36953
\(89\) −8.18997e6 −1.23145 −0.615726 0.787960i \(-0.711137\pi\)
−0.615726 + 0.787960i \(0.711137\pi\)
\(90\) 2.52430e6 0.364999
\(91\) −945564. −0.131536
\(92\) 4.53669e6 0.607409
\(93\) −2.25030e6 −0.290101
\(94\) 5.15075e6 0.639622
\(95\) −7.57535e6 −0.906505
\(96\) −1.80237e6 −0.207920
\(97\) 1.20009e7 1.33510 0.667548 0.744567i \(-0.267344\pi\)
0.667548 + 0.744567i \(0.267344\pi\)
\(98\) 6.72296e6 0.721555
\(99\) −1.15531e7 −1.19667
\(100\) 3.30847e6 0.330847
\(101\) −1.45741e7 −1.40752 −0.703762 0.710436i \(-0.748498\pi\)
−0.703762 + 0.710436i \(0.748498\pi\)
\(102\) −166493. −0.0155344
\(103\) 3.29091e6 0.296747 0.148373 0.988931i \(-0.452596\pi\)
0.148373 + 0.988931i \(0.452596\pi\)
\(104\) −1.20234e7 −1.04812
\(105\) 211596. 0.0178380
\(106\) −4.77511e6 −0.389415
\(107\) 2.34488e7 1.85045 0.925226 0.379416i \(-0.123875\pi\)
0.925226 + 0.379416i \(0.123875\pi\)
\(108\) 2.91724e6 0.222838
\(109\) 2.05051e7 1.51659 0.758296 0.651910i \(-0.226032\pi\)
0.758296 + 0.651910i \(0.226032\pi\)
\(110\) −6.93785e6 −0.496993
\(111\) 5.39446e6 0.374384
\(112\) −657611. −0.0442289
\(113\) −4.02775e6 −0.262596 −0.131298 0.991343i \(-0.541914\pi\)
−0.131298 + 0.991343i \(0.541914\pi\)
\(114\) 4.97361e6 0.314416
\(115\) −1.14111e7 −0.699656
\(116\) 2.83119e6 0.168409
\(117\) 1.58660e7 0.915832
\(118\) 8.42755e6 0.472187
\(119\) 209240. 0.0113823
\(120\) 2.69058e6 0.142138
\(121\) 1.22656e7 0.629421
\(122\) 2.68755e7 1.33998
\(123\) 8.05448e6 0.390274
\(124\) 1.13293e7 0.533612
\(125\) −1.98910e7 −0.910901
\(126\) 2.08285e6 0.0927602
\(127\) −1.44813e7 −0.627328 −0.313664 0.949534i \(-0.601557\pi\)
−0.313664 + 0.949534i \(0.601557\pi\)
\(128\) 3.34664e6 0.141050
\(129\) 929754. 0.0381333
\(130\) 9.52781e6 0.380357
\(131\) −1.55714e7 −0.605170 −0.302585 0.953122i \(-0.597850\pi\)
−0.302585 + 0.953122i \(0.597850\pi\)
\(132\) −3.87953e6 −0.146815
\(133\) −6.25058e6 −0.230377
\(134\) −2.62641e7 −0.942966
\(135\) −7.33771e6 −0.256680
\(136\) 2.66062e6 0.0906976
\(137\) −2.42862e7 −0.806932 −0.403466 0.914995i \(-0.632195\pi\)
−0.403466 + 0.914995i \(0.632195\pi\)
\(138\) 7.49198e6 0.242672
\(139\) 5.15045e7 1.62665 0.813323 0.581812i \(-0.197656\pi\)
0.813323 + 0.581812i \(0.197656\pi\)
\(140\) −1.06530e6 −0.0328112
\(141\) −7.24459e6 −0.217644
\(142\) −2.04780e7 −0.600175
\(143\) −4.36064e7 −1.24702
\(144\) 1.10343e7 0.307947
\(145\) −7.12128e6 −0.193986
\(146\) 3.34838e7 0.890431
\(147\) −9.45591e6 −0.245523
\(148\) −2.71588e7 −0.688644
\(149\) 2.03855e7 0.504858 0.252429 0.967615i \(-0.418771\pi\)
0.252429 + 0.967615i \(0.418771\pi\)
\(150\) 5.46368e6 0.132180
\(151\) −1.46726e7 −0.346806 −0.173403 0.984851i \(-0.555476\pi\)
−0.173403 + 0.984851i \(0.555476\pi\)
\(152\) −7.94801e7 −1.83572
\(153\) −3.51091e6 −0.0792501
\(154\) −5.72457e6 −0.126305
\(155\) −2.84964e7 −0.614652
\(156\) 5.32780e6 0.112360
\(157\) −1.07322e7 −0.221329 −0.110665 0.993858i \(-0.535298\pi\)
−0.110665 + 0.993858i \(0.535298\pi\)
\(158\) 3.85358e6 0.0777257
\(159\) 6.71624e6 0.132506
\(160\) −2.28242e7 −0.440530
\(161\) −9.41554e6 −0.177809
\(162\) −3.24624e7 −0.599899
\(163\) −3.08777e7 −0.558454 −0.279227 0.960225i \(-0.590078\pi\)
−0.279227 + 0.960225i \(0.590078\pi\)
\(164\) −4.05509e7 −0.717871
\(165\) 9.75815e6 0.169112
\(166\) 4.56993e7 0.775410
\(167\) −9.20853e6 −0.152997 −0.0764985 0.997070i \(-0.524374\pi\)
−0.0764985 + 0.997070i \(0.524374\pi\)
\(168\) 2.22006e6 0.0361228
\(169\) −2.86337e6 −0.0456325
\(170\) −2.10837e6 −0.0329136
\(171\) 1.04881e8 1.60402
\(172\) −4.68091e6 −0.0701425
\(173\) 1.49747e7 0.219885 0.109943 0.993938i \(-0.464933\pi\)
0.109943 + 0.993938i \(0.464933\pi\)
\(174\) 4.67549e6 0.0672829
\(175\) −6.86647e6 −0.0968502
\(176\) −3.03270e7 −0.419309
\(177\) −1.18534e7 −0.160671
\(178\) 6.80930e7 0.904967
\(179\) 8.89995e7 1.15985 0.579925 0.814670i \(-0.303082\pi\)
0.579925 + 0.814670i \(0.303082\pi\)
\(180\) 1.78750e7 0.228450
\(181\) 5.85628e7 0.734086 0.367043 0.930204i \(-0.380370\pi\)
0.367043 + 0.930204i \(0.380370\pi\)
\(182\) 7.86160e6 0.0966632
\(183\) −3.78007e7 −0.455954
\(184\) −1.19725e8 −1.41684
\(185\) 6.83123e7 0.793228
\(186\) 1.87094e7 0.213189
\(187\) 9.64948e6 0.107909
\(188\) 3.64734e7 0.400335
\(189\) −6.05450e6 −0.0652322
\(190\) 6.29829e7 0.666170
\(191\) 2.81969e7 0.292809 0.146404 0.989225i \(-0.453230\pi\)
0.146404 + 0.989225i \(0.453230\pi\)
\(192\) 2.30411e7 0.234936
\(193\) −1.42485e8 −1.42665 −0.713325 0.700833i \(-0.752812\pi\)
−0.713325 + 0.700833i \(0.752812\pi\)
\(194\) −9.97778e7 −0.981133
\(195\) −1.34010e7 −0.129424
\(196\) 4.76064e7 0.451616
\(197\) 1.07088e8 0.997951 0.498976 0.866616i \(-0.333710\pi\)
0.498976 + 0.866616i \(0.333710\pi\)
\(198\) 9.60546e7 0.879407
\(199\) 1.37599e8 1.23774 0.618871 0.785493i \(-0.287590\pi\)
0.618871 + 0.785493i \(0.287590\pi\)
\(200\) −8.73115e7 −0.771732
\(201\) 3.69407e7 0.320863
\(202\) 1.21171e8 1.03436
\(203\) −5.87592e6 −0.0492992
\(204\) −1.17896e6 −0.00972288
\(205\) 1.01997e8 0.826894
\(206\) −2.73613e7 −0.218073
\(207\) 1.57987e8 1.23801
\(208\) 4.16483e7 0.320904
\(209\) −2.88257e8 −2.18408
\(210\) −1.75925e6 −0.0131087
\(211\) 1.52091e8 1.11459 0.557295 0.830314i \(-0.311839\pi\)
0.557295 + 0.830314i \(0.311839\pi\)
\(212\) −3.38134e7 −0.243732
\(213\) 2.88025e7 0.204222
\(214\) −1.94958e8 −1.35986
\(215\) 1.17739e7 0.0807950
\(216\) −7.69868e7 −0.519790
\(217\) −2.35130e7 −0.156206
\(218\) −1.70483e8 −1.11451
\(219\) −4.70953e7 −0.302987
\(220\) −4.91281e7 −0.311064
\(221\) −1.32517e7 −0.0825846
\(222\) −4.48506e7 −0.275127
\(223\) −1.58102e8 −0.954709 −0.477354 0.878711i \(-0.658404\pi\)
−0.477354 + 0.878711i \(0.658404\pi\)
\(224\) −1.88328e7 −0.111956
\(225\) 1.15215e8 0.674327
\(226\) 3.34875e7 0.192976
\(227\) 1.42553e8 0.808886 0.404443 0.914563i \(-0.367465\pi\)
0.404443 + 0.914563i \(0.367465\pi\)
\(228\) 3.52190e7 0.196791
\(229\) −1.08019e7 −0.0594395 −0.0297198 0.999558i \(-0.509461\pi\)
−0.0297198 + 0.999558i \(0.509461\pi\)
\(230\) 9.48740e7 0.514162
\(231\) 8.05166e6 0.0429777
\(232\) −7.47160e7 −0.392831
\(233\) 4.31324e7 0.223387 0.111694 0.993743i \(-0.464373\pi\)
0.111694 + 0.993743i \(0.464373\pi\)
\(234\) −1.31913e8 −0.673025
\(235\) −9.17412e7 −0.461134
\(236\) 5.96769e7 0.295539
\(237\) −5.42010e6 −0.0264477
\(238\) −1.73966e6 −0.00836459
\(239\) −1.51247e8 −0.716631 −0.358315 0.933601i \(-0.616649\pi\)
−0.358315 + 0.933601i \(0.616649\pi\)
\(240\) −9.31996e6 −0.0435186
\(241\) −1.16354e7 −0.0535453 −0.0267726 0.999642i \(-0.508523\pi\)
−0.0267726 + 0.999642i \(0.508523\pi\)
\(242\) −1.01979e8 −0.462548
\(243\) 1.54025e8 0.688605
\(244\) 1.90310e8 0.838683
\(245\) −1.19744e8 −0.520203
\(246\) −6.69665e7 −0.286804
\(247\) 3.95866e8 1.67151
\(248\) −2.98983e8 −1.24470
\(249\) −6.42766e7 −0.263849
\(250\) 1.65377e8 0.669401
\(251\) −1.76350e8 −0.703910 −0.351955 0.936017i \(-0.614483\pi\)
−0.351955 + 0.936017i \(0.614483\pi\)
\(252\) 1.47490e7 0.0580580
\(253\) −4.34215e8 −1.68571
\(254\) 1.20400e8 0.461009
\(255\) 2.96544e6 0.0111995
\(256\) −2.80028e8 −1.04319
\(257\) −3.46637e8 −1.27382 −0.636912 0.770937i \(-0.719789\pi\)
−0.636912 + 0.770937i \(0.719789\pi\)
\(258\) −7.73016e6 −0.0280233
\(259\) 5.63660e7 0.201589
\(260\) 6.74681e7 0.238063
\(261\) 9.85942e7 0.343249
\(262\) 1.29463e8 0.444726
\(263\) 3.71975e8 1.26087 0.630433 0.776244i \(-0.282878\pi\)
0.630433 + 0.776244i \(0.282878\pi\)
\(264\) 1.02382e8 0.342460
\(265\) 8.50505e7 0.280748
\(266\) 5.19686e7 0.169299
\(267\) −9.57735e7 −0.307933
\(268\) −1.85981e8 −0.590196
\(269\) 3.76594e8 1.17962 0.589808 0.807544i \(-0.299203\pi\)
0.589808 + 0.807544i \(0.299203\pi\)
\(270\) 6.10071e7 0.188629
\(271\) 1.87035e8 0.570862 0.285431 0.958399i \(-0.407863\pi\)
0.285431 + 0.958399i \(0.407863\pi\)
\(272\) −9.21617e6 −0.0277689
\(273\) −1.10574e7 −0.0328916
\(274\) 2.01920e8 0.592996
\(275\) −3.16660e8 −0.918182
\(276\) 5.30520e7 0.151887
\(277\) −2.58454e8 −0.730642 −0.365321 0.930882i \(-0.619041\pi\)
−0.365321 + 0.930882i \(0.619041\pi\)
\(278\) −4.28218e8 −1.19539
\(279\) 3.94533e8 1.08760
\(280\) 2.81135e7 0.0765352
\(281\) −3.07159e8 −0.825831 −0.412916 0.910769i \(-0.635490\pi\)
−0.412916 + 0.910769i \(0.635490\pi\)
\(282\) 6.02329e7 0.159942
\(283\) −6.04080e8 −1.58432 −0.792159 0.610315i \(-0.791043\pi\)
−0.792159 + 0.610315i \(0.791043\pi\)
\(284\) −1.45008e8 −0.375646
\(285\) −8.85861e7 −0.226678
\(286\) 3.62552e8 0.916409
\(287\) 8.41601e7 0.210145
\(288\) 3.16002e8 0.779499
\(289\) −4.07406e8 −0.992854
\(290\) 5.92077e7 0.142556
\(291\) 1.40338e8 0.333850
\(292\) 2.37105e8 0.557314
\(293\) −1.79377e8 −0.416610 −0.208305 0.978064i \(-0.566795\pi\)
−0.208305 + 0.978064i \(0.566795\pi\)
\(294\) 7.86183e7 0.180430
\(295\) −1.50105e8 −0.340422
\(296\) 7.16729e8 1.60633
\(297\) −2.79215e8 −0.618430
\(298\) −1.69489e8 −0.371009
\(299\) 5.96312e8 1.29010
\(300\) 3.86892e7 0.0827305
\(301\) 9.71487e6 0.0205331
\(302\) 1.21991e8 0.254860
\(303\) −1.70429e8 −0.351961
\(304\) 2.75313e8 0.562043
\(305\) −4.78686e8 −0.966054
\(306\) 2.91904e7 0.0582391
\(307\) −1.45485e8 −0.286969 −0.143484 0.989653i \(-0.545831\pi\)
−0.143484 + 0.989653i \(0.545831\pi\)
\(308\) −4.05366e7 −0.0790533
\(309\) 3.84839e7 0.0742036
\(310\) 2.36925e8 0.451694
\(311\) 3.56832e8 0.672670 0.336335 0.941742i \(-0.390813\pi\)
0.336335 + 0.941742i \(0.390813\pi\)
\(312\) −1.40602e8 −0.262090
\(313\) 6.50132e8 1.19838 0.599192 0.800605i \(-0.295489\pi\)
0.599192 + 0.800605i \(0.295489\pi\)
\(314\) 8.92293e7 0.162650
\(315\) −3.70981e7 −0.0668752
\(316\) 2.72879e7 0.0486480
\(317\) −1.34462e8 −0.237078 −0.118539 0.992949i \(-0.537821\pi\)
−0.118539 + 0.992949i \(0.537821\pi\)
\(318\) −5.58401e7 −0.0973759
\(319\) −2.70979e8 −0.467378
\(320\) 2.91779e8 0.497771
\(321\) 2.74210e8 0.462718
\(322\) 7.82826e7 0.130668
\(323\) −8.75995e7 −0.144642
\(324\) −2.29872e8 −0.375472
\(325\) 4.34872e8 0.702700
\(326\) 2.56723e8 0.410396
\(327\) 2.39786e8 0.379234
\(328\) 1.07015e9 1.67450
\(329\) −7.56976e7 −0.117192
\(330\) −8.11311e7 −0.124276
\(331\) 6.61036e8 1.00191 0.500953 0.865474i \(-0.332983\pi\)
0.500953 + 0.865474i \(0.332983\pi\)
\(332\) 3.23605e8 0.485324
\(333\) −9.45785e8 −1.40358
\(334\) 7.65615e7 0.112434
\(335\) 4.67796e8 0.679828
\(336\) −7.69010e6 −0.0110597
\(337\) −2.64722e8 −0.376778 −0.188389 0.982094i \(-0.560327\pi\)
−0.188389 + 0.982094i \(0.560327\pi\)
\(338\) 2.38066e7 0.0335343
\(339\) −4.71005e7 −0.0656638
\(340\) −1.49297e7 −0.0206004
\(341\) −1.08435e9 −1.48091
\(342\) −8.71999e8 −1.17876
\(343\) −1.99431e8 −0.266848
\(344\) 1.23531e8 0.163614
\(345\) −1.33441e8 −0.174954
\(346\) −1.24502e8 −0.161589
\(347\) −3.92072e8 −0.503748 −0.251874 0.967760i \(-0.581047\pi\)
−0.251874 + 0.967760i \(0.581047\pi\)
\(348\) 3.31080e7 0.0421119
\(349\) −2.58464e8 −0.325470 −0.162735 0.986670i \(-0.552032\pi\)
−0.162735 + 0.986670i \(0.552032\pi\)
\(350\) 5.70892e7 0.0711731
\(351\) 3.83448e8 0.473295
\(352\) −8.68507e8 −1.06139
\(353\) −2.36908e8 −0.286661 −0.143331 0.989675i \(-0.545781\pi\)
−0.143331 + 0.989675i \(0.545781\pi\)
\(354\) 9.85517e7 0.118074
\(355\) 3.64738e8 0.432695
\(356\) 4.82178e8 0.566412
\(357\) 2.44685e6 0.00284622
\(358\) −7.39959e8 −0.852348
\(359\) −1.22528e9 −1.39767 −0.698836 0.715282i \(-0.746298\pi\)
−0.698836 + 0.715282i \(0.746298\pi\)
\(360\) −4.71726e8 −0.532882
\(361\) 1.72297e9 1.92754
\(362\) −4.86903e8 −0.539464
\(363\) 1.43434e8 0.157391
\(364\) 5.56694e7 0.0605008
\(365\) −5.96388e8 −0.641953
\(366\) 3.14283e8 0.335071
\(367\) −4.80402e8 −0.507311 −0.253655 0.967295i \(-0.581633\pi\)
−0.253655 + 0.967295i \(0.581633\pi\)
\(368\) 4.14717e8 0.433795
\(369\) −1.41215e9 −1.46315
\(370\) −5.67962e8 −0.582925
\(371\) 7.01770e7 0.0713488
\(372\) 1.32484e8 0.133433
\(373\) −1.06518e9 −1.06278 −0.531389 0.847128i \(-0.678330\pi\)
−0.531389 + 0.847128i \(0.678330\pi\)
\(374\) −8.02276e7 −0.0793000
\(375\) −2.32605e8 −0.227777
\(376\) −9.62543e8 −0.933819
\(377\) 3.72138e8 0.357692
\(378\) 5.03383e7 0.0479377
\(379\) 1.57397e9 1.48511 0.742554 0.669786i \(-0.233614\pi\)
0.742554 + 0.669786i \(0.233614\pi\)
\(380\) 4.45993e8 0.416951
\(381\) −1.69344e8 −0.156867
\(382\) −2.34434e8 −0.215179
\(383\) −8.92980e8 −0.812168 −0.406084 0.913836i \(-0.633106\pi\)
−0.406084 + 0.913836i \(0.633106\pi\)
\(384\) 3.91356e7 0.0352706
\(385\) 1.01961e8 0.0910591
\(386\) 1.18464e9 1.04841
\(387\) −1.63009e8 −0.142963
\(388\) −7.06543e8 −0.614084
\(389\) 2.17007e9 1.86917 0.934586 0.355737i \(-0.115770\pi\)
0.934586 + 0.355737i \(0.115770\pi\)
\(390\) 1.11418e8 0.0951108
\(391\) −1.31955e8 −0.111637
\(392\) −1.25635e9 −1.05344
\(393\) −1.82092e8 −0.151327
\(394\) −8.90351e8 −0.733372
\(395\) −6.86370e7 −0.0560362
\(396\) 6.80179e8 0.550415
\(397\) −1.33189e9 −1.06832 −0.534162 0.845382i \(-0.679373\pi\)
−0.534162 + 0.845382i \(0.679373\pi\)
\(398\) −1.14403e9 −0.909589
\(399\) −7.30943e7 −0.0576074
\(400\) 3.02441e8 0.236282
\(401\) 3.57769e8 0.277075 0.138538 0.990357i \(-0.455760\pi\)
0.138538 + 0.990357i \(0.455760\pi\)
\(402\) −3.07132e8 −0.235795
\(403\) 1.48914e9 1.13336
\(404\) 8.58036e8 0.647397
\(405\) 5.78195e8 0.432495
\(406\) 4.88535e7 0.0362289
\(407\) 2.59942e9 1.91116
\(408\) 3.11132e7 0.0226795
\(409\) 7.47840e8 0.540477 0.270238 0.962793i \(-0.412897\pi\)
0.270238 + 0.962793i \(0.412897\pi\)
\(410\) −8.48024e8 −0.607666
\(411\) −2.84002e8 −0.201779
\(412\) −1.93750e8 −0.136490
\(413\) −1.23855e8 −0.0865143
\(414\) −1.31353e9 −0.909787
\(415\) −8.13960e8 −0.559030
\(416\) 1.19273e9 0.812298
\(417\) 6.02293e8 0.406754
\(418\) 2.39663e9 1.60503
\(419\) 9.00703e8 0.598181 0.299091 0.954225i \(-0.403317\pi\)
0.299091 + 0.954225i \(0.403317\pi\)
\(420\) −1.24576e7 −0.00820466
\(421\) −1.70587e9 −1.11419 −0.557094 0.830449i \(-0.688084\pi\)
−0.557094 + 0.830449i \(0.688084\pi\)
\(422\) −1.26451e9 −0.819088
\(423\) 1.27016e9 0.815955
\(424\) 8.92345e8 0.568529
\(425\) −9.62310e7 −0.0608070
\(426\) −2.39470e8 −0.150078
\(427\) −3.94974e8 −0.245511
\(428\) −1.38053e9 −0.851124
\(429\) −5.09933e8 −0.311826
\(430\) −9.78901e7 −0.0593744
\(431\) 1.88962e9 1.13685 0.568425 0.822735i \(-0.307553\pi\)
0.568425 + 0.822735i \(0.307553\pi\)
\(432\) 2.66677e8 0.159145
\(433\) 7.46212e7 0.0441728 0.0220864 0.999756i \(-0.492969\pi\)
0.0220864 + 0.999756i \(0.492969\pi\)
\(434\) 1.95492e8 0.114793
\(435\) −8.32762e7 −0.0485074
\(436\) −1.20722e9 −0.697564
\(437\) 3.94188e9 2.25953
\(438\) 3.91560e8 0.222658
\(439\) −1.72954e9 −0.975677 −0.487838 0.872934i \(-0.662214\pi\)
−0.487838 + 0.872934i \(0.662214\pi\)
\(440\) 1.29650e9 0.725587
\(441\) 1.65786e9 0.920476
\(442\) 1.10177e8 0.0606896
\(443\) 3.58748e9 1.96054 0.980270 0.197661i \(-0.0633346\pi\)
0.980270 + 0.197661i \(0.0633346\pi\)
\(444\) −3.17595e8 −0.172200
\(445\) −1.21282e9 −0.652433
\(446\) 1.31449e9 0.701594
\(447\) 2.38388e8 0.126243
\(448\) 2.40753e8 0.126503
\(449\) −3.27797e9 −1.70900 −0.854502 0.519447i \(-0.826138\pi\)
−0.854502 + 0.519447i \(0.826138\pi\)
\(450\) −9.57920e8 −0.495548
\(451\) 3.88120e9 1.99227
\(452\) 2.37130e8 0.120782
\(453\) −1.71581e8 −0.0867213
\(454\) −1.18522e9 −0.594432
\(455\) −1.40025e8 −0.0696891
\(456\) −9.29440e8 −0.459034
\(457\) −1.77071e9 −0.867842 −0.433921 0.900951i \(-0.642870\pi\)
−0.433921 + 0.900951i \(0.642870\pi\)
\(458\) 8.98089e7 0.0436808
\(459\) −8.48515e7 −0.0409558
\(460\) 6.71819e8 0.321810
\(461\) 2.35457e9 1.11933 0.559664 0.828719i \(-0.310930\pi\)
0.559664 + 0.828719i \(0.310930\pi\)
\(462\) −6.69431e7 −0.0315834
\(463\) −4.94391e8 −0.231493 −0.115746 0.993279i \(-0.536926\pi\)
−0.115746 + 0.993279i \(0.536926\pi\)
\(464\) 2.58811e8 0.120273
\(465\) −3.33237e8 −0.153698
\(466\) −3.58611e8 −0.164162
\(467\) −3.45960e9 −1.57187 −0.785935 0.618309i \(-0.787818\pi\)
−0.785935 + 0.618309i \(0.787818\pi\)
\(468\) −9.34096e8 −0.421241
\(469\) 3.85988e8 0.172770
\(470\) 7.62754e8 0.338877
\(471\) −1.25502e8 −0.0553449
\(472\) −1.57489e9 −0.689372
\(473\) 4.48019e8 0.194663
\(474\) 4.50638e7 0.0194358
\(475\) 2.87469e9 1.23073
\(476\) −1.23188e7 −0.00523534
\(477\) −1.17753e9 −0.496771
\(478\) 1.25750e9 0.526636
\(479\) −2.26891e9 −0.943284 −0.471642 0.881790i \(-0.656338\pi\)
−0.471642 + 0.881790i \(0.656338\pi\)
\(480\) −2.66906e8 −0.110158
\(481\) −3.56981e9 −1.46264
\(482\) 9.67389e7 0.0393492
\(483\) −1.10105e8 −0.0444624
\(484\) −7.22130e8 −0.289505
\(485\) 1.77716e9 0.707345
\(486\) −1.28060e9 −0.506041
\(487\) −2.51843e9 −0.988051 −0.494025 0.869447i \(-0.664475\pi\)
−0.494025 + 0.869447i \(0.664475\pi\)
\(488\) −5.02235e9 −1.95631
\(489\) −3.61083e8 −0.139645
\(490\) 9.95576e8 0.382286
\(491\) 7.20363e8 0.274642 0.137321 0.990527i \(-0.456151\pi\)
0.137321 + 0.990527i \(0.456151\pi\)
\(492\) −4.74201e8 −0.179508
\(493\) −8.23487e7 −0.0309523
\(494\) −3.29131e9 −1.22836
\(495\) −1.71085e9 −0.634006
\(496\) 1.03565e9 0.381091
\(497\) 3.00953e8 0.109964
\(498\) 5.34408e8 0.193896
\(499\) −2.15427e9 −0.776154 −0.388077 0.921627i \(-0.626861\pi\)
−0.388077 + 0.921627i \(0.626861\pi\)
\(500\) 1.17107e9 0.418973
\(501\) −1.07684e8 −0.0382579
\(502\) 1.46621e9 0.517288
\(503\) 2.80736e9 0.983579 0.491790 0.870714i \(-0.336343\pi\)
0.491790 + 0.870714i \(0.336343\pi\)
\(504\) −3.89231e8 −0.135426
\(505\) −2.15821e9 −0.745717
\(506\) 3.61015e9 1.23879
\(507\) −3.34842e7 −0.0114107
\(508\) 8.52575e8 0.288542
\(509\) −2.99574e9 −1.00691 −0.503457 0.864020i \(-0.667939\pi\)
−0.503457 + 0.864020i \(0.667939\pi\)
\(510\) −2.46552e7 −0.00823026
\(511\) −4.92092e8 −0.163145
\(512\) 1.89984e9 0.625563
\(513\) 2.53476e9 0.828944
\(514\) 2.88201e9 0.936105
\(515\) 4.87338e8 0.157219
\(516\) −5.47386e7 −0.0175396
\(517\) −3.49094e9 −1.11103
\(518\) −4.68637e8 −0.148144
\(519\) 1.75114e8 0.0549838
\(520\) −1.78050e9 −0.555304
\(521\) 2.58005e9 0.799275 0.399637 0.916673i \(-0.369136\pi\)
0.399637 + 0.916673i \(0.369136\pi\)
\(522\) −8.19731e8 −0.252246
\(523\) 1.70818e9 0.522129 0.261064 0.965321i \(-0.415926\pi\)
0.261064 + 0.965321i \(0.415926\pi\)
\(524\) 9.16753e8 0.278351
\(525\) −8.02965e7 −0.0242180
\(526\) −3.09267e9 −0.926582
\(527\) −3.29526e8 −0.0980736
\(528\) −3.54643e8 −0.104851
\(529\) 2.53300e9 0.743944
\(530\) −7.07126e8 −0.206315
\(531\) 2.07820e9 0.602362
\(532\) 3.67998e8 0.105963
\(533\) −5.33009e9 −1.52472
\(534\) 7.96279e8 0.226293
\(535\) 3.47244e9 0.980384
\(536\) 4.90809e9 1.37669
\(537\) 1.04076e9 0.290028
\(538\) −3.13108e9 −0.866873
\(539\) −4.55650e9 −1.25335
\(540\) 4.32002e8 0.118061
\(541\) −4.88301e9 −1.32586 −0.662929 0.748682i \(-0.730687\pi\)
−0.662929 + 0.748682i \(0.730687\pi\)
\(542\) −1.55505e9 −0.419514
\(543\) 6.84834e8 0.183563
\(544\) −2.63934e8 −0.0702909
\(545\) 3.03651e9 0.803503
\(546\) 9.19335e7 0.0241713
\(547\) −6.00910e9 −1.56984 −0.784918 0.619600i \(-0.787295\pi\)
−0.784918 + 0.619600i \(0.787295\pi\)
\(548\) 1.42983e9 0.371152
\(549\) 6.62742e9 1.70939
\(550\) 2.63277e9 0.674752
\(551\) 2.45999e9 0.626474
\(552\) −1.40006e9 −0.354290
\(553\) −5.66339e7 −0.0142409
\(554\) 2.14884e9 0.536932
\(555\) 7.98844e8 0.198352
\(556\) −3.03229e9 −0.748184
\(557\) −3.64266e9 −0.893152 −0.446576 0.894746i \(-0.647357\pi\)
−0.446576 + 0.894746i \(0.647357\pi\)
\(558\) −3.28022e9 −0.799252
\(559\) −6.15269e8 −0.148979
\(560\) −9.73829e7 −0.0234328
\(561\) 1.12841e8 0.0269834
\(562\) 2.55378e9 0.606885
\(563\) 9.78473e8 0.231084 0.115542 0.993303i \(-0.463140\pi\)
0.115542 + 0.993303i \(0.463140\pi\)
\(564\) 4.26519e8 0.100106
\(565\) −5.96452e8 −0.139125
\(566\) 5.02244e9 1.16428
\(567\) 4.77081e8 0.109914
\(568\) 3.82681e9 0.876229
\(569\) −2.98433e9 −0.679131 −0.339566 0.940582i \(-0.610280\pi\)
−0.339566 + 0.940582i \(0.610280\pi\)
\(570\) 7.36522e8 0.166580
\(571\) 4.09176e9 0.919779 0.459890 0.887976i \(-0.347889\pi\)
0.459890 + 0.887976i \(0.347889\pi\)
\(572\) 2.56729e9 0.573574
\(573\) 3.29734e8 0.0732188
\(574\) −6.99723e8 −0.154431
\(575\) 4.33028e9 0.949901
\(576\) −4.03969e9 −0.880783
\(577\) −3.21664e9 −0.697087 −0.348544 0.937293i \(-0.613324\pi\)
−0.348544 + 0.937293i \(0.613324\pi\)
\(578\) 3.38725e9 0.729626
\(579\) −1.66621e9 −0.356743
\(580\) 4.19260e8 0.0892247
\(581\) −6.71616e8 −0.142071
\(582\) −1.16680e9 −0.245339
\(583\) 3.23634e9 0.676417
\(584\) −6.25726e9 −1.29999
\(585\) 2.34952e9 0.485215
\(586\) 1.49137e9 0.306157
\(587\) −3.18322e9 −0.649582 −0.324791 0.945786i \(-0.605294\pi\)
−0.324791 + 0.945786i \(0.605294\pi\)
\(588\) 5.56710e8 0.112930
\(589\) 9.84387e9 1.98501
\(590\) 1.24800e9 0.250169
\(591\) 1.25229e9 0.249544
\(592\) −2.48269e9 −0.491810
\(593\) −1.62368e9 −0.319749 −0.159875 0.987137i \(-0.551109\pi\)
−0.159875 + 0.987137i \(0.551109\pi\)
\(594\) 2.32144e9 0.454470
\(595\) 3.09854e7 0.00603043
\(596\) −1.20018e9 −0.232212
\(597\) 1.60908e9 0.309506
\(598\) −4.95785e9 −0.948067
\(599\) −3.68608e9 −0.700763 −0.350382 0.936607i \(-0.613948\pi\)
−0.350382 + 0.936607i \(0.613948\pi\)
\(600\) −1.02102e9 −0.192977
\(601\) −2.91768e9 −0.548248 −0.274124 0.961694i \(-0.588388\pi\)
−0.274124 + 0.961694i \(0.588388\pi\)
\(602\) −8.07713e7 −0.0150893
\(603\) −6.47664e9 −1.20293
\(604\) 8.63837e8 0.159515
\(605\) 1.81637e9 0.333472
\(606\) 1.41698e9 0.258648
\(607\) −3.35727e9 −0.609292 −0.304646 0.952466i \(-0.598538\pi\)
−0.304646 + 0.952466i \(0.598538\pi\)
\(608\) 7.88445e9 1.42269
\(609\) −6.87130e7 −0.0123276
\(610\) 3.97989e9 0.709931
\(611\) 4.79414e9 0.850288
\(612\) 2.06702e8 0.0364514
\(613\) 1.51150e9 0.265030 0.132515 0.991181i \(-0.457695\pi\)
0.132515 + 0.991181i \(0.457695\pi\)
\(614\) 1.20959e9 0.210887
\(615\) 1.19275e9 0.206770
\(616\) 1.06977e9 0.184399
\(617\) −5.64788e9 −0.968026 −0.484013 0.875061i \(-0.660821\pi\)
−0.484013 + 0.875061i \(0.660821\pi\)
\(618\) −3.19963e8 −0.0545305
\(619\) −2.42139e9 −0.410343 −0.205172 0.978726i \(-0.565775\pi\)
−0.205172 + 0.978726i \(0.565775\pi\)
\(620\) 1.67770e9 0.282712
\(621\) 3.81822e9 0.639794
\(622\) −2.96677e9 −0.494330
\(623\) −1.00072e9 −0.165808
\(624\) 4.87035e8 0.0802443
\(625\) 1.44471e9 0.236701
\(626\) −5.40532e9 −0.880666
\(627\) −3.37088e9 −0.546143
\(628\) 6.31848e8 0.101801
\(629\) 7.89947e8 0.126567
\(630\) 3.08441e8 0.0491451
\(631\) 7.99860e9 1.26739 0.633696 0.773582i \(-0.281537\pi\)
0.633696 + 0.773582i \(0.281537\pi\)
\(632\) −7.20135e8 −0.113476
\(633\) 1.77855e9 0.278711
\(634\) 1.11794e9 0.174223
\(635\) −2.14448e9 −0.332363
\(636\) −3.95414e8 −0.0609469
\(637\) 6.25749e9 0.959207
\(638\) 2.25297e9 0.343465
\(639\) −5.04980e9 −0.765634
\(640\) 4.95591e8 0.0747297
\(641\) 9.69674e8 0.145419 0.0727097 0.997353i \(-0.476835\pi\)
0.0727097 + 0.997353i \(0.476835\pi\)
\(642\) −2.27984e9 −0.340041
\(643\) 3.12786e9 0.463990 0.231995 0.972717i \(-0.425475\pi\)
0.231995 + 0.972717i \(0.425475\pi\)
\(644\) 5.54333e8 0.0817843
\(645\) 1.37683e8 0.0202033
\(646\) 7.28319e8 0.106294
\(647\) −1.05397e10 −1.52990 −0.764949 0.644091i \(-0.777236\pi\)
−0.764949 + 0.644091i \(0.777236\pi\)
\(648\) 6.06638e9 0.875825
\(649\) −5.71179e9 −0.820193
\(650\) −3.61561e9 −0.516399
\(651\) −2.74961e8 −0.0390605
\(652\) 1.81790e9 0.256864
\(653\) −1.16184e10 −1.63286 −0.816430 0.577444i \(-0.804050\pi\)
−0.816430 + 0.577444i \(0.804050\pi\)
\(654\) −1.99363e9 −0.278691
\(655\) −2.30590e9 −0.320624
\(656\) −3.70692e9 −0.512683
\(657\) 8.25699e9 1.13591
\(658\) 6.29365e8 0.0861215
\(659\) −5.81607e9 −0.791645 −0.395822 0.918327i \(-0.629540\pi\)
−0.395822 + 0.918327i \(0.629540\pi\)
\(660\) −5.74503e8 −0.0777837
\(661\) 8.71459e9 1.17366 0.586830 0.809710i \(-0.300376\pi\)
0.586830 + 0.809710i \(0.300376\pi\)
\(662\) −5.49598e9 −0.736278
\(663\) −1.54966e8 −0.0206508
\(664\) −8.54003e9 −1.13206
\(665\) −9.25623e8 −0.122056
\(666\) 7.86344e9 1.03146
\(667\) 3.70560e9 0.483524
\(668\) 5.42145e8 0.0703717
\(669\) −1.84885e9 −0.238731
\(670\) −3.88934e9 −0.499591
\(671\) −1.82150e10 −2.32755
\(672\) −2.20230e8 −0.0279952
\(673\) 1.52007e10 1.92226 0.961129 0.276098i \(-0.0890415\pi\)
0.961129 + 0.276098i \(0.0890415\pi\)
\(674\) 2.20095e9 0.276886
\(675\) 2.78451e9 0.348487
\(676\) 1.68579e8 0.0209889
\(677\) −7.66446e9 −0.949339 −0.474670 0.880164i \(-0.657432\pi\)
−0.474670 + 0.880164i \(0.657432\pi\)
\(678\) 3.91602e8 0.0482549
\(679\) 1.46638e9 0.179763
\(680\) 3.93999e8 0.0480523
\(681\) 1.66702e9 0.202267
\(682\) 9.01546e9 1.08828
\(683\) −5.86341e9 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(684\) −6.17478e9 −0.737776
\(685\) −3.59644e9 −0.427519
\(686\) 1.65811e9 0.196101
\(687\) −1.26317e8 −0.0148632
\(688\) −4.27901e8 −0.0500938
\(689\) −4.44450e9 −0.517673
\(690\) 1.10946e9 0.128570
\(691\) −2.93527e9 −0.338435 −0.169218 0.985579i \(-0.554124\pi\)
−0.169218 + 0.985579i \(0.554124\pi\)
\(692\) −8.81623e8 −0.101137
\(693\) −1.41166e9 −0.161125
\(694\) 3.25976e9 0.370193
\(695\) 7.62708e9 0.861810
\(696\) −8.73729e8 −0.0982300
\(697\) 1.17947e9 0.131939
\(698\) 2.14892e9 0.239181
\(699\) 5.04390e8 0.0558594
\(700\) 4.04258e8 0.0445467
\(701\) −1.59500e10 −1.74883 −0.874415 0.485178i \(-0.838755\pi\)
−0.874415 + 0.485178i \(0.838755\pi\)
\(702\) −3.18806e9 −0.347814
\(703\) −2.35980e10 −2.56172
\(704\) 1.11028e10 1.19930
\(705\) −1.07282e9 −0.115310
\(706\) 1.96970e9 0.210661
\(707\) −1.78079e9 −0.189515
\(708\) 6.97862e8 0.0739015
\(709\) −5.26013e9 −0.554287 −0.277143 0.960829i \(-0.589388\pi\)
−0.277143 + 0.960829i \(0.589388\pi\)
\(710\) −3.03250e9 −0.317978
\(711\) 9.50280e8 0.0991535
\(712\) −1.27248e10 −1.32121
\(713\) 1.48283e10 1.53206
\(714\) −2.03436e7 −0.00209162
\(715\) −6.45750e9 −0.660683
\(716\) −5.23978e9 −0.533479
\(717\) −1.76869e9 −0.179198
\(718\) 1.01872e10 1.02712
\(719\) 1.60266e10 1.60802 0.804008 0.594618i \(-0.202697\pi\)
0.804008 + 0.594618i \(0.202697\pi\)
\(720\) 1.63402e9 0.163153
\(721\) 4.02113e8 0.0399553
\(722\) −1.43251e10 −1.41651
\(723\) −1.36064e8 −0.0133894
\(724\) −3.44784e9 −0.337647
\(725\) 2.70238e9 0.263368
\(726\) −1.19254e9 −0.115663
\(727\) 1.07033e10 1.03311 0.516556 0.856253i \(-0.327214\pi\)
0.516556 + 0.856253i \(0.327214\pi\)
\(728\) −1.46913e9 −0.141124
\(729\) −6.73787e9 −0.644134
\(730\) 4.95848e9 0.471757
\(731\) 1.36150e8 0.0128916
\(732\) 2.22549e9 0.209718
\(733\) 1.43369e10 1.34460 0.672298 0.740280i \(-0.265307\pi\)
0.672298 + 0.740280i \(0.265307\pi\)
\(734\) 3.99416e9 0.372811
\(735\) −1.40029e9 −0.130080
\(736\) 1.18767e10 1.09805
\(737\) 1.78006e10 1.63794
\(738\) 1.17409e10 1.07524
\(739\) 2.07369e10 1.89012 0.945060 0.326898i \(-0.106003\pi\)
0.945060 + 0.326898i \(0.106003\pi\)
\(740\) −4.02183e9 −0.364849
\(741\) 4.62926e9 0.417973
\(742\) −5.83465e8 −0.0524326
\(743\) −5.43590e9 −0.486196 −0.243098 0.970002i \(-0.578164\pi\)
−0.243098 + 0.970002i \(0.578164\pi\)
\(744\) −3.49630e9 −0.311246
\(745\) 3.01880e9 0.267478
\(746\) 8.85612e9 0.781012
\(747\) 1.12693e10 0.989178
\(748\) −5.68105e8 −0.0496333
\(749\) 2.86518e9 0.249153
\(750\) 1.93392e9 0.167388
\(751\) −3.49854e9 −0.301403 −0.150701 0.988579i \(-0.548153\pi\)
−0.150701 + 0.988579i \(0.548153\pi\)
\(752\) 3.33418e9 0.285908
\(753\) −2.06223e9 −0.176017
\(754\) −3.09403e9 −0.262860
\(755\) −2.17280e9 −0.183741
\(756\) 3.56454e8 0.0300039
\(757\) −2.02699e10 −1.69831 −0.849153 0.528148i \(-0.822887\pi\)
−0.849153 + 0.528148i \(0.822887\pi\)
\(758\) −1.30862e10 −1.09137
\(759\) −5.07771e9 −0.421523
\(760\) −1.17699e10 −0.972578
\(761\) 1.16941e10 0.961876 0.480938 0.876755i \(-0.340296\pi\)
0.480938 + 0.876755i \(0.340296\pi\)
\(762\) 1.40796e9 0.115278
\(763\) 2.50549e9 0.204201
\(764\) −1.66007e9 −0.134679
\(765\) −5.19916e8 −0.0419873
\(766\) 7.42441e9 0.596844
\(767\) 7.84406e9 0.627707
\(768\) −3.27465e9 −0.260856
\(769\) 1.64924e10 1.30780 0.653899 0.756582i \(-0.273132\pi\)
0.653899 + 0.756582i \(0.273132\pi\)
\(770\) −8.47727e8 −0.0669173
\(771\) −4.05358e9 −0.318528
\(772\) 8.38867e9 0.656195
\(773\) −4.04094e9 −0.314669 −0.157334 0.987545i \(-0.550290\pi\)
−0.157334 + 0.987545i \(0.550290\pi\)
\(774\) 1.35529e9 0.105060
\(775\) 1.08138e10 0.834493
\(776\) 1.86459e10 1.43241
\(777\) 6.59143e8 0.0504088
\(778\) −1.80423e10 −1.37361
\(779\) −3.52342e10 −2.67044
\(780\) 7.88971e8 0.0595292
\(781\) 1.38790e10 1.04251
\(782\) 1.09710e9 0.0820395
\(783\) 2.38282e9 0.177388
\(784\) 4.35190e9 0.322532
\(785\) −1.58928e9 −0.117262
\(786\) 1.51394e9 0.111207
\(787\) 2.46052e10 1.79935 0.899675 0.436560i \(-0.143803\pi\)
0.899675 + 0.436560i \(0.143803\pi\)
\(788\) −6.30473e9 −0.459012
\(789\) 4.34987e9 0.315288
\(790\) 5.70661e8 0.0411797
\(791\) −4.92146e8 −0.0353571
\(792\) −1.79501e10 −1.28389
\(793\) 2.50148e10 1.78131
\(794\) 1.10736e10 0.785087
\(795\) 9.94580e8 0.0702029
\(796\) −8.10105e9 −0.569305
\(797\) −1.84665e10 −1.29205 −0.646026 0.763316i \(-0.723570\pi\)
−0.646026 + 0.763316i \(0.723570\pi\)
\(798\) 6.07720e8 0.0423344
\(799\) −1.06087e9 −0.0735783
\(800\) 8.66133e9 0.598094
\(801\) 1.67915e10 1.15445
\(802\) −2.97456e9 −0.203616
\(803\) −2.26937e10 −1.54668
\(804\) −2.17486e9 −0.147582
\(805\) −1.39431e9 −0.0942048
\(806\) −1.23810e10 −0.832882
\(807\) 4.40389e9 0.294971
\(808\) −2.26438e10 −1.51012
\(809\) 1.80517e10 1.19867 0.599333 0.800500i \(-0.295433\pi\)
0.599333 + 0.800500i \(0.295433\pi\)
\(810\) −4.80722e9 −0.317831
\(811\) 1.23449e10 0.812673 0.406337 0.913723i \(-0.366806\pi\)
0.406337 + 0.913723i \(0.366806\pi\)
\(812\) 3.45940e8 0.0226754
\(813\) 2.18719e9 0.142748
\(814\) −2.16121e10 −1.40447
\(815\) −4.57254e9 −0.295874
\(816\) −1.07774e8 −0.00694381
\(817\) −4.06719e9 −0.260926
\(818\) −6.21768e9 −0.397184
\(819\) 1.93864e9 0.123312
\(820\) −6.00501e9 −0.380334
\(821\) −1.36091e10 −0.858276 −0.429138 0.903239i \(-0.641183\pi\)
−0.429138 + 0.903239i \(0.641183\pi\)
\(822\) 2.36125e9 0.148283
\(823\) −1.55999e10 −0.975490 −0.487745 0.872986i \(-0.662180\pi\)
−0.487745 + 0.872986i \(0.662180\pi\)
\(824\) 5.11312e9 0.318376
\(825\) −3.70302e9 −0.229598
\(826\) 1.02975e9 0.0635774
\(827\) −1.54413e10 −0.949325 −0.474662 0.880168i \(-0.657430\pi\)
−0.474662 + 0.880168i \(0.657430\pi\)
\(828\) −9.30135e9 −0.569429
\(829\) −5.48114e9 −0.334141 −0.167071 0.985945i \(-0.553431\pi\)
−0.167071 + 0.985945i \(0.553431\pi\)
\(830\) 6.76742e9 0.410818
\(831\) −3.02236e9 −0.182702
\(832\) −1.52476e10 −0.917844
\(833\) −1.38469e9 −0.0830034
\(834\) −5.00758e9 −0.298914
\(835\) −1.36365e9 −0.0810590
\(836\) 1.69709e10 1.00458
\(837\) 9.53506e9 0.562062
\(838\) −7.48862e9 −0.439590
\(839\) −9.70181e9 −0.567134 −0.283567 0.958952i \(-0.591518\pi\)
−0.283567 + 0.958952i \(0.591518\pi\)
\(840\) 3.28759e8 0.0191381
\(841\) −1.49373e10 −0.865939
\(842\) 1.41829e10 0.818792
\(843\) −3.59192e9 −0.206505
\(844\) −8.95425e9 −0.512661
\(845\) −4.24025e8 −0.0241764
\(846\) −1.05603e10 −0.599627
\(847\) 1.49872e9 0.0847481
\(848\) −3.09102e9 −0.174067
\(849\) −7.06411e9 −0.396169
\(850\) 8.00083e8 0.0446857
\(851\) −3.55467e10 −1.97718
\(852\) −1.69572e9 −0.0939327
\(853\) 7.25754e9 0.400376 0.200188 0.979758i \(-0.435845\pi\)
0.200188 + 0.979758i \(0.435845\pi\)
\(854\) 3.28389e9 0.180421
\(855\) 1.55314e10 0.849822
\(856\) 3.64326e10 1.98533
\(857\) 2.87404e10 1.55977 0.779884 0.625923i \(-0.215278\pi\)
0.779884 + 0.625923i \(0.215278\pi\)
\(858\) 4.23968e9 0.229154
\(859\) −7.30578e9 −0.393270 −0.196635 0.980477i \(-0.563001\pi\)
−0.196635 + 0.980477i \(0.563001\pi\)
\(860\) −6.93177e8 −0.0371620
\(861\) 9.84168e8 0.0525482
\(862\) −1.57106e10 −0.835446
\(863\) 3.12604e10 1.65560 0.827802 0.561021i \(-0.189591\pi\)
0.827802 + 0.561021i \(0.189591\pi\)
\(864\) 7.63711e9 0.402839
\(865\) 2.21754e9 0.116497
\(866\) −6.20415e8 −0.0324616
\(867\) −4.76421e9 −0.248270
\(868\) 1.38431e9 0.0718479
\(869\) −2.61177e9 −0.135010
\(870\) 6.92374e8 0.0356470
\(871\) −2.44457e10 −1.25354
\(872\) 3.18589e10 1.62713
\(873\) −2.46048e10 −1.25162
\(874\) −3.27735e10 −1.66048
\(875\) −2.43046e9 −0.122648
\(876\) 2.77270e9 0.139360
\(877\) −1.16138e10 −0.581403 −0.290702 0.956814i \(-0.593889\pi\)
−0.290702 + 0.956814i \(0.593889\pi\)
\(878\) 1.43798e10 0.717003
\(879\) −2.09763e9 −0.104176
\(880\) −4.49099e9 −0.222153
\(881\) −5.59437e9 −0.275636 −0.137818 0.990458i \(-0.544009\pi\)
−0.137818 + 0.990458i \(0.544009\pi\)
\(882\) −1.37838e10 −0.676437
\(883\) 2.50609e10 1.22500 0.612498 0.790472i \(-0.290165\pi\)
0.612498 + 0.790472i \(0.290165\pi\)
\(884\) 7.80185e8 0.0379852
\(885\) −1.75533e9 −0.0851248
\(886\) −2.98270e10 −1.44076
\(887\) −1.82957e10 −0.880271 −0.440135 0.897931i \(-0.645070\pi\)
−0.440135 + 0.897931i \(0.645070\pi\)
\(888\) 8.38142e9 0.401673
\(889\) −1.76945e9 −0.0844662
\(890\) 1.00836e10 0.479459
\(891\) 2.20015e10 1.04203
\(892\) 9.30815e9 0.439123
\(893\) 3.16913e10 1.48922
\(894\) −1.98200e9 −0.0927733
\(895\) 1.31796e10 0.614498
\(896\) 4.08923e8 0.0189917
\(897\) 6.97327e9 0.322599
\(898\) 2.72537e10 1.25591
\(899\) 9.25382e9 0.424778
\(900\) −6.78319e9 −0.310160
\(901\) 9.83504e8 0.0447960
\(902\) −3.22690e10 −1.46407
\(903\) 1.13606e8 0.00513444
\(904\) −6.25794e9 −0.281736
\(905\) 8.67233e9 0.388925
\(906\) 1.42656e9 0.0637295
\(907\) −7.13828e9 −0.317664 −0.158832 0.987306i \(-0.550773\pi\)
−0.158832 + 0.987306i \(0.550773\pi\)
\(908\) −8.39272e9 −0.372051
\(909\) 2.98805e10 1.31951
\(910\) 1.16419e9 0.0512129
\(911\) 3.00662e10 1.31754 0.658770 0.752344i \(-0.271077\pi\)
0.658770 + 0.752344i \(0.271077\pi\)
\(912\) 3.21951e9 0.140543
\(913\) −3.09728e10 −1.34689
\(914\) 1.47220e10 0.637758
\(915\) −5.59775e9 −0.241568
\(916\) 6.35952e8 0.0273395
\(917\) −1.90265e9 −0.0814828
\(918\) 7.05472e8 0.0300975
\(919\) 1.16122e10 0.493524 0.246762 0.969076i \(-0.420633\pi\)
0.246762 + 0.969076i \(0.420633\pi\)
\(920\) −1.77295e10 −0.750653
\(921\) −1.70130e9 −0.0717585
\(922\) −1.95763e10 −0.822570
\(923\) −1.90602e10 −0.797849
\(924\) −4.74035e8 −0.0197678
\(925\) −2.59231e10 −1.07694
\(926\) 4.11046e9 0.170119
\(927\) −6.74720e9 −0.278192
\(928\) 7.41185e9 0.304445
\(929\) 9.40477e8 0.0384851 0.0192426 0.999815i \(-0.493875\pi\)
0.0192426 + 0.999815i \(0.493875\pi\)
\(930\) 2.77059e9 0.112949
\(931\) 4.13647e10 1.67999
\(932\) −2.53939e9 −0.102748
\(933\) 4.17279e9 0.168206
\(934\) 2.87638e10 1.15513
\(935\) 1.42895e9 0.0571711
\(936\) 2.46511e10 0.982586
\(937\) 4.42595e9 0.175759 0.0878796 0.996131i \(-0.471991\pi\)
0.0878796 + 0.996131i \(0.471991\pi\)
\(938\) −3.20918e9 −0.126965
\(939\) 7.60264e9 0.299664
\(940\) 5.40119e9 0.212101
\(941\) 1.67372e10 0.654814 0.327407 0.944883i \(-0.393825\pi\)
0.327407 + 0.944883i \(0.393825\pi\)
\(942\) 1.04345e9 0.0406717
\(943\) −5.30748e10 −2.06109
\(944\) 5.45531e9 0.211065
\(945\) −8.96586e8 −0.0345606
\(946\) −3.72492e9 −0.143053
\(947\) 2.33893e10 0.894935 0.447468 0.894300i \(-0.352326\pi\)
0.447468 + 0.894300i \(0.352326\pi\)
\(948\) 3.19104e8 0.0121648
\(949\) 3.11655e10 1.18370
\(950\) −2.39007e10 −0.904438
\(951\) −1.57239e9 −0.0592829
\(952\) 3.25098e8 0.0122119
\(953\) 1.32227e10 0.494875 0.247437 0.968904i \(-0.420412\pi\)
0.247437 + 0.968904i \(0.420412\pi\)
\(954\) 9.79018e9 0.365066
\(955\) 4.17556e9 0.155132
\(956\) 8.90457e9 0.329618
\(957\) −3.16882e9 −0.116871
\(958\) 1.88641e10 0.693198
\(959\) −2.96750e9 −0.108649
\(960\) 3.41206e9 0.124471
\(961\) 9.51734e9 0.345926
\(962\) 2.96801e10 1.07486
\(963\) −4.80760e10 −1.73475
\(964\) 6.85024e8 0.0246284
\(965\) −2.11000e10 −0.755851
\(966\) 9.15436e8 0.0326744
\(967\) −1.96959e10 −0.700459 −0.350229 0.936664i \(-0.613896\pi\)
−0.350229 + 0.936664i \(0.613896\pi\)
\(968\) 1.90572e10 0.675299
\(969\) −1.02439e9 −0.0361686
\(970\) −1.47757e10 −0.519812
\(971\) −4.75127e10 −1.66549 −0.832746 0.553655i \(-0.813232\pi\)
−0.832746 + 0.553655i \(0.813232\pi\)
\(972\) −9.06812e9 −0.316727
\(973\) 6.29327e9 0.219019
\(974\) 2.09387e10 0.726097
\(975\) 5.08539e9 0.175715
\(976\) 1.73970e10 0.598964
\(977\) 4.37334e10 1.50031 0.750157 0.661260i \(-0.229978\pi\)
0.750157 + 0.661260i \(0.229978\pi\)
\(978\) 3.00211e9 0.102622
\(979\) −4.61502e10 −1.57193
\(980\) 7.04984e9 0.239270
\(981\) −4.20406e10 −1.42176
\(982\) −5.98924e9 −0.201828
\(983\) 1.27037e10 0.426572 0.213286 0.976990i \(-0.431583\pi\)
0.213286 + 0.976990i \(0.431583\pi\)
\(984\) 1.25143e10 0.418721
\(985\) 1.58582e10 0.528722
\(986\) 6.84663e8 0.0227461
\(987\) −8.85208e8 −0.0293046
\(988\) −2.33063e10 −0.768820
\(989\) −6.12659e9 −0.201387
\(990\) 1.42243e10 0.465917
\(991\) −2.85362e10 −0.931404 −0.465702 0.884941i \(-0.654198\pi\)
−0.465702 + 0.884941i \(0.654198\pi\)
\(992\) 2.96592e10 0.964646
\(993\) 7.73015e9 0.250533
\(994\) −2.50218e9 −0.0808103
\(995\) 2.03765e10 0.655766
\(996\) 3.78423e9 0.121358
\(997\) 3.65067e10 1.16665 0.583323 0.812240i \(-0.301752\pi\)
0.583323 + 0.812240i \(0.301752\pi\)
\(998\) 1.79110e10 0.570378
\(999\) −2.28577e10 −0.725359
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 43.8.a.a.1.5 11
3.2 odd 2 387.8.a.b.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.5 11 1.1 even 1 trivial
387.8.a.b.1.7 11 3.2 odd 2