Properties

Label 43.8.a.b.1.6
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-4.71679\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.71679 q^{2} -24.7485 q^{3} -114.185 q^{4} +385.350 q^{5} +91.9851 q^{6} -1546.77 q^{7} +900.153 q^{8} -1574.51 q^{9} +O(q^{10})\) \(q-3.71679 q^{2} -24.7485 q^{3} -114.185 q^{4} +385.350 q^{5} +91.9851 q^{6} -1546.77 q^{7} +900.153 q^{8} -1574.51 q^{9} -1432.27 q^{10} -1662.15 q^{11} +2825.92 q^{12} +12725.9 q^{13} +5749.00 q^{14} -9536.85 q^{15} +11270.1 q^{16} +23803.1 q^{17} +5852.12 q^{18} +30546.0 q^{19} -44001.4 q^{20} +38280.2 q^{21} +6177.85 q^{22} -63784.8 q^{23} -22277.4 q^{24} +70369.8 q^{25} -47299.5 q^{26} +93091.8 q^{27} +176618. q^{28} +190003. q^{29} +35446.5 q^{30} +143977. q^{31} -157108. q^{32} +41135.6 q^{33} -88471.1 q^{34} -596047. q^{35} +179786. q^{36} -53873.0 q^{37} -113533. q^{38} -314947. q^{39} +346874. q^{40} -355277. q^{41} -142279. q^{42} -79507.0 q^{43} +189793. q^{44} -606738. q^{45} +237075. q^{46} -39435.7 q^{47} -278917. q^{48} +1.56894e6 q^{49} -261550. q^{50} -589091. q^{51} -1.45311e6 q^{52} +1.01320e6 q^{53} -346003. q^{54} -640508. q^{55} -1.39233e6 q^{56} -755970. q^{57} -706202. q^{58} +1.24979e6 q^{59} +1.08897e6 q^{60} -751833. q^{61} -535133. q^{62} +2.43540e6 q^{63} -858631. q^{64} +4.90393e6 q^{65} -152893. q^{66} -3.22613e6 q^{67} -2.71797e6 q^{68} +1.57858e6 q^{69} +2.21538e6 q^{70} +1.10265e6 q^{71} -1.41730e6 q^{72} +4.53962e6 q^{73} +200235. q^{74} -1.74155e6 q^{75} -3.48791e6 q^{76} +2.57095e6 q^{77} +1.17059e6 q^{78} +3.63589e6 q^{79} +4.34292e6 q^{80} +1.13957e6 q^{81} +1.32049e6 q^{82} -149479. q^{83} -4.37104e6 q^{84} +9.17253e6 q^{85} +295511. q^{86} -4.70230e6 q^{87} -1.49618e6 q^{88} -8.46943e6 q^{89} +2.25512e6 q^{90} -1.96840e7 q^{91} +7.28329e6 q^{92} -3.56322e6 q^{93} +146574. q^{94} +1.17709e7 q^{95} +3.88819e6 q^{96} +5.99834e6 q^{97} -5.83143e6 q^{98} +2.61707e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.71679 −0.328521 −0.164260 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(3\) −24.7485 −0.529206 −0.264603 0.964357i \(-0.585241\pi\)
−0.264603 + 0.964357i \(0.585241\pi\)
\(4\) −114.185 −0.892074
\(5\) 385.350 1.37867 0.689336 0.724442i \(-0.257903\pi\)
0.689336 + 0.724442i \(0.257903\pi\)
\(6\) 91.9851 0.173855
\(7\) −1546.77 −1.70444 −0.852219 0.523185i \(-0.824744\pi\)
−0.852219 + 0.523185i \(0.824744\pi\)
\(8\) 900.153 0.621586
\(9\) −1574.51 −0.719941
\(10\) −1432.27 −0.452922
\(11\) −1662.15 −0.376526 −0.188263 0.982119i \(-0.560286\pi\)
−0.188263 + 0.982119i \(0.560286\pi\)
\(12\) 2825.92 0.472091
\(13\) 12725.9 1.60652 0.803261 0.595628i \(-0.203097\pi\)
0.803261 + 0.595628i \(0.203097\pi\)
\(14\) 5749.00 0.559944
\(15\) −9536.85 −0.729601
\(16\) 11270.1 0.687870
\(17\) 23803.1 1.17507 0.587533 0.809200i \(-0.300099\pi\)
0.587533 + 0.809200i \(0.300099\pi\)
\(18\) 5852.12 0.236516
\(19\) 30546.0 1.02169 0.510843 0.859674i \(-0.329333\pi\)
0.510843 + 0.859674i \(0.329333\pi\)
\(20\) −44001.4 −1.22988
\(21\) 38280.2 0.902000
\(22\) 6177.85 0.123697
\(23\) −63784.8 −1.09312 −0.546562 0.837419i \(-0.684064\pi\)
−0.546562 + 0.837419i \(0.684064\pi\)
\(24\) −22277.4 −0.328947
\(25\) 70369.8 0.900734
\(26\) −47299.5 −0.527776
\(27\) 93091.8 0.910203
\(28\) 176618. 1.52049
\(29\) 190003. 1.44666 0.723332 0.690500i \(-0.242609\pi\)
0.723332 + 0.690500i \(0.242609\pi\)
\(30\) 35446.5 0.239689
\(31\) 143977. 0.868015 0.434008 0.900909i \(-0.357099\pi\)
0.434008 + 0.900909i \(0.357099\pi\)
\(32\) −157108. −0.847566
\(33\) 41135.6 0.199260
\(34\) −88471.1 −0.386034
\(35\) −596047. −2.34986
\(36\) 179786. 0.642240
\(37\) −53873.0 −0.174850 −0.0874249 0.996171i \(-0.527864\pi\)
−0.0874249 + 0.996171i \(0.527864\pi\)
\(38\) −113533. −0.335645
\(39\) −314947. −0.850181
\(40\) 346874. 0.856963
\(41\) −355277. −0.805051 −0.402526 0.915409i \(-0.631868\pi\)
−0.402526 + 0.915409i \(0.631868\pi\)
\(42\) −142279. −0.296326
\(43\) −79507.0 −0.152499
\(44\) 189793. 0.335889
\(45\) −606738. −0.992561
\(46\) 237075. 0.359114
\(47\) −39435.7 −0.0554047 −0.0277024 0.999616i \(-0.508819\pi\)
−0.0277024 + 0.999616i \(0.508819\pi\)
\(48\) −278917. −0.364025
\(49\) 1.56894e6 1.90511
\(50\) −261550. −0.295910
\(51\) −589091. −0.621852
\(52\) −1.45311e6 −1.43314
\(53\) 1.01320e6 0.934823 0.467411 0.884040i \(-0.345187\pi\)
0.467411 + 0.884040i \(0.345187\pi\)
\(54\) −346003. −0.299021
\(55\) −640508. −0.519105
\(56\) −1.39233e6 −1.05946
\(57\) −755970. −0.540682
\(58\) −706202. −0.475260
\(59\) 1.24979e6 0.792236 0.396118 0.918200i \(-0.370357\pi\)
0.396118 + 0.918200i \(0.370357\pi\)
\(60\) 1.08897e6 0.650858
\(61\) −751833. −0.424098 −0.212049 0.977259i \(-0.568014\pi\)
−0.212049 + 0.977259i \(0.568014\pi\)
\(62\) −535133. −0.285161
\(63\) 2.43540e6 1.22709
\(64\) −858631. −0.409427
\(65\) 4.90393e6 2.21486
\(66\) −152893. −0.0654610
\(67\) −3.22613e6 −1.31045 −0.655224 0.755435i \(-0.727426\pi\)
−0.655224 + 0.755435i \(0.727426\pi\)
\(68\) −2.71797e6 −1.04825
\(69\) 1.57858e6 0.578488
\(70\) 2.21538e6 0.771978
\(71\) 1.10265e6 0.365623 0.182811 0.983148i \(-0.441480\pi\)
0.182811 + 0.983148i \(0.441480\pi\)
\(72\) −1.41730e6 −0.447505
\(73\) 4.53962e6 1.36581 0.682904 0.730508i \(-0.260717\pi\)
0.682904 + 0.730508i \(0.260717\pi\)
\(74\) 200235. 0.0574418
\(75\) −1.74155e6 −0.476674
\(76\) −3.48791e6 −0.911419
\(77\) 2.57095e6 0.641765
\(78\) 1.17059e6 0.279302
\(79\) 3.63589e6 0.829691 0.414846 0.909892i \(-0.363836\pi\)
0.414846 + 0.909892i \(0.363836\pi\)
\(80\) 4.34292e6 0.948347
\(81\) 1.13957e6 0.238255
\(82\) 1.32049e6 0.264476
\(83\) −149479. −0.0286951 −0.0143476 0.999897i \(-0.504567\pi\)
−0.0143476 + 0.999897i \(0.504567\pi\)
\(84\) −4.37104e6 −0.804650
\(85\) 9.17253e6 1.62003
\(86\) 295511. 0.0500990
\(87\) −4.70230e6 −0.765584
\(88\) −1.49618e6 −0.234043
\(89\) −8.46943e6 −1.27347 −0.636736 0.771082i \(-0.719716\pi\)
−0.636736 + 0.771082i \(0.719716\pi\)
\(90\) 2.25512e6 0.326077
\(91\) −1.96840e7 −2.73822
\(92\) 7.28329e6 0.975148
\(93\) −3.56322e6 −0.459359
\(94\) 146574. 0.0182016
\(95\) 1.17709e7 1.40857
\(96\) 3.88819e6 0.448537
\(97\) 5.99834e6 0.667313 0.333657 0.942695i \(-0.391717\pi\)
0.333657 + 0.942695i \(0.391717\pi\)
\(98\) −5.83143e6 −0.625869
\(99\) 2.61707e6 0.271076
\(100\) −8.03521e6 −0.803521
\(101\) −8.50960e6 −0.821834 −0.410917 0.911673i \(-0.634791\pi\)
−0.410917 + 0.911673i \(0.634791\pi\)
\(102\) 2.18953e6 0.204291
\(103\) 1.74827e6 0.157644 0.0788222 0.996889i \(-0.474884\pi\)
0.0788222 + 0.996889i \(0.474884\pi\)
\(104\) 1.14552e7 0.998591
\(105\) 1.47513e7 1.24356
\(106\) −3.76585e6 −0.307109
\(107\) 1.06723e7 0.842196 0.421098 0.907015i \(-0.361645\pi\)
0.421098 + 0.907015i \(0.361645\pi\)
\(108\) −1.06297e7 −0.811969
\(109\) 5.16184e6 0.381779 0.190889 0.981612i \(-0.438863\pi\)
0.190889 + 0.981612i \(0.438863\pi\)
\(110\) 2.38063e6 0.170537
\(111\) 1.33328e6 0.0925316
\(112\) −1.74321e7 −1.17243
\(113\) 1.76860e7 1.15307 0.576534 0.817073i \(-0.304405\pi\)
0.576534 + 0.817073i \(0.304405\pi\)
\(114\) 2.80978e6 0.177626
\(115\) −2.45795e7 −1.50706
\(116\) −2.16956e7 −1.29053
\(117\) −2.00370e7 −1.15660
\(118\) −4.64520e6 −0.260266
\(119\) −3.68178e7 −2.00283
\(120\) −8.58462e6 −0.453510
\(121\) −1.67244e7 −0.858228
\(122\) 2.79440e6 0.139325
\(123\) 8.79258e6 0.426038
\(124\) −1.64401e7 −0.774334
\(125\) −2.98845e6 −0.136855
\(126\) −9.05187e6 −0.403126
\(127\) 4.09391e7 1.77347 0.886737 0.462274i \(-0.152966\pi\)
0.886737 + 0.462274i \(0.152966\pi\)
\(128\) 2.33012e7 0.982071
\(129\) 1.96768e6 0.0807032
\(130\) −1.82269e7 −0.727629
\(131\) 4.15303e7 1.61404 0.807021 0.590522i \(-0.201078\pi\)
0.807021 + 0.590522i \(0.201078\pi\)
\(132\) −4.69709e6 −0.177754
\(133\) −4.72476e7 −1.74140
\(134\) 1.19908e7 0.430509
\(135\) 3.58730e7 1.25487
\(136\) 2.14264e7 0.730404
\(137\) −3.50608e7 −1.16493 −0.582466 0.812855i \(-0.697912\pi\)
−0.582466 + 0.812855i \(0.697912\pi\)
\(138\) −5.86725e6 −0.190045
\(139\) 5.57867e7 1.76189 0.880945 0.473219i \(-0.156908\pi\)
0.880945 + 0.473219i \(0.156908\pi\)
\(140\) 6.80599e7 2.09625
\(141\) 975974. 0.0293205
\(142\) −4.09831e6 −0.120115
\(143\) −2.11523e7 −0.604897
\(144\) −1.77448e7 −0.495226
\(145\) 7.32178e7 1.99447
\(146\) −1.68728e7 −0.448697
\(147\) −3.88290e7 −1.00820
\(148\) 6.15151e6 0.155979
\(149\) −1.78918e7 −0.443099 −0.221550 0.975149i \(-0.571112\pi\)
−0.221550 + 0.975149i \(0.571112\pi\)
\(150\) 6.47298e6 0.156597
\(151\) −2.85564e7 −0.674970 −0.337485 0.941331i \(-0.609576\pi\)
−0.337485 + 0.941331i \(0.609576\pi\)
\(152\) 2.74961e7 0.635065
\(153\) −3.74782e7 −0.845978
\(154\) −9.55568e6 −0.210833
\(155\) 5.54816e7 1.19671
\(156\) 3.59624e7 0.758424
\(157\) 6.75273e7 1.39261 0.696307 0.717744i \(-0.254825\pi\)
0.696307 + 0.717744i \(0.254825\pi\)
\(158\) −1.35139e7 −0.272571
\(159\) −2.50752e7 −0.494714
\(160\) −6.05416e7 −1.16851
\(161\) 9.86601e7 1.86316
\(162\) −4.23554e6 −0.0782719
\(163\) −7.20769e7 −1.30359 −0.651793 0.758397i \(-0.725983\pi\)
−0.651793 + 0.758397i \(0.725983\pi\)
\(164\) 4.05675e7 0.718165
\(165\) 1.58516e7 0.274714
\(166\) 555583. 0.00942694
\(167\) 2.87709e7 0.478019 0.239010 0.971017i \(-0.423177\pi\)
0.239010 + 0.971017i \(0.423177\pi\)
\(168\) 3.44580e7 0.560670
\(169\) 9.91998e7 1.58091
\(170\) −3.40924e7 −0.532213
\(171\) −4.80951e7 −0.735553
\(172\) 9.07854e6 0.136040
\(173\) −1.24594e8 −1.82951 −0.914755 0.404010i \(-0.867616\pi\)
−0.914755 + 0.404010i \(0.867616\pi\)
\(174\) 1.74775e7 0.251510
\(175\) −1.08846e8 −1.53525
\(176\) −1.87325e7 −0.259001
\(177\) −3.09304e7 −0.419256
\(178\) 3.14791e7 0.418362
\(179\) −1.45448e7 −0.189550 −0.0947749 0.995499i \(-0.530213\pi\)
−0.0947749 + 0.995499i \(0.530213\pi\)
\(180\) 6.92807e7 0.885438
\(181\) −4.71999e7 −0.591651 −0.295826 0.955242i \(-0.595595\pi\)
−0.295826 + 0.955242i \(0.595595\pi\)
\(182\) 7.31612e7 0.899562
\(183\) 1.86067e7 0.224436
\(184\) −5.74160e7 −0.679471
\(185\) −2.07600e7 −0.241060
\(186\) 1.32437e7 0.150909
\(187\) −3.95642e7 −0.442442
\(188\) 4.50298e6 0.0494251
\(189\) −1.43991e8 −1.55139
\(190\) −4.37501e7 −0.462744
\(191\) −5.66800e6 −0.0588590 −0.0294295 0.999567i \(-0.509369\pi\)
−0.0294295 + 0.999567i \(0.509369\pi\)
\(192\) 2.12498e7 0.216671
\(193\) 1.30205e8 1.30369 0.651847 0.758350i \(-0.273994\pi\)
0.651847 + 0.758350i \(0.273994\pi\)
\(194\) −2.22946e7 −0.219226
\(195\) −1.21365e8 −1.17212
\(196\) −1.79150e8 −1.69950
\(197\) 1.88254e8 1.75434 0.877169 0.480182i \(-0.159429\pi\)
0.877169 + 0.480182i \(0.159429\pi\)
\(198\) −9.72708e6 −0.0890542
\(199\) 1.14630e8 1.03113 0.515564 0.856851i \(-0.327583\pi\)
0.515564 + 0.856851i \(0.327583\pi\)
\(200\) 6.33436e7 0.559884
\(201\) 7.98419e7 0.693497
\(202\) 3.16284e7 0.269990
\(203\) −2.93890e8 −2.46575
\(204\) 6.72657e7 0.554738
\(205\) −1.36906e8 −1.10990
\(206\) −6.49796e6 −0.0517895
\(207\) 1.00430e8 0.786985
\(208\) 1.43422e8 1.10508
\(209\) −5.07720e7 −0.384691
\(210\) −5.48274e7 −0.408536
\(211\) 3.12083e7 0.228708 0.114354 0.993440i \(-0.463520\pi\)
0.114354 + 0.993440i \(0.463520\pi\)
\(212\) −1.15693e8 −0.833931
\(213\) −2.72889e7 −0.193490
\(214\) −3.96666e7 −0.276679
\(215\) −3.06380e7 −0.210245
\(216\) 8.37969e7 0.565770
\(217\) −2.22699e8 −1.47948
\(218\) −1.91855e7 −0.125422
\(219\) −1.12349e8 −0.722794
\(220\) 7.31367e7 0.463080
\(221\) 3.02916e8 1.88777
\(222\) −4.95551e6 −0.0303986
\(223\) −1.46395e8 −0.884015 −0.442007 0.897011i \(-0.645733\pi\)
−0.442007 + 0.897011i \(0.645733\pi\)
\(224\) 2.43009e8 1.44462
\(225\) −1.10798e8 −0.648475
\(226\) −6.57352e7 −0.378807
\(227\) −2.31019e7 −0.131086 −0.0655432 0.997850i \(-0.520878\pi\)
−0.0655432 + 0.997850i \(0.520878\pi\)
\(228\) 8.63207e7 0.482329
\(229\) −2.46033e8 −1.35385 −0.676924 0.736053i \(-0.736687\pi\)
−0.676924 + 0.736053i \(0.736687\pi\)
\(230\) 9.13568e7 0.495100
\(231\) −6.36272e7 −0.339626
\(232\) 1.71032e8 0.899226
\(233\) 2.56892e8 1.33047 0.665235 0.746634i \(-0.268331\pi\)
0.665235 + 0.746634i \(0.268331\pi\)
\(234\) 7.44735e7 0.379967
\(235\) −1.51965e7 −0.0763849
\(236\) −1.42708e8 −0.706733
\(237\) −8.99830e7 −0.439078
\(238\) 1.36844e8 0.657971
\(239\) 2.98201e8 1.41292 0.706458 0.707755i \(-0.250292\pi\)
0.706458 + 0.707755i \(0.250292\pi\)
\(240\) −1.07481e8 −0.501871
\(241\) −2.29672e8 −1.05694 −0.528468 0.848953i \(-0.677233\pi\)
−0.528468 + 0.848953i \(0.677233\pi\)
\(242\) 6.21612e7 0.281946
\(243\) −2.31794e8 −1.03629
\(244\) 8.58484e7 0.378327
\(245\) 6.04592e8 2.62652
\(246\) −3.26802e7 −0.139962
\(247\) 3.88726e8 1.64136
\(248\) 1.29601e8 0.539546
\(249\) 3.69939e6 0.0151856
\(250\) 1.11075e7 0.0449598
\(251\) 1.54752e8 0.617702 0.308851 0.951111i \(-0.400056\pi\)
0.308851 + 0.951111i \(0.400056\pi\)
\(252\) −2.78087e8 −1.09466
\(253\) 1.06020e8 0.411589
\(254\) −1.52162e8 −0.582623
\(255\) −2.27007e8 −0.857329
\(256\) 2.32992e7 0.0867961
\(257\) 2.73313e8 1.00437 0.502185 0.864760i \(-0.332530\pi\)
0.502185 + 0.864760i \(0.332530\pi\)
\(258\) −7.31346e6 −0.0265127
\(259\) 8.33289e7 0.298021
\(260\) −5.59957e8 −1.97582
\(261\) −2.99162e8 −1.04151
\(262\) −1.54359e8 −0.530247
\(263\) 2.20336e8 0.746863 0.373431 0.927658i \(-0.378181\pi\)
0.373431 + 0.927658i \(0.378181\pi\)
\(264\) 3.70284e7 0.123857
\(265\) 3.90437e8 1.28881
\(266\) 1.75609e8 0.572087
\(267\) 2.09606e8 0.673929
\(268\) 3.68377e8 1.16902
\(269\) −1.74740e8 −0.547342 −0.273671 0.961823i \(-0.588238\pi\)
−0.273671 + 0.961823i \(0.588238\pi\)
\(270\) −1.33332e8 −0.412251
\(271\) −5.89750e8 −1.80001 −0.900005 0.435879i \(-0.856438\pi\)
−0.900005 + 0.435879i \(0.856438\pi\)
\(272\) 2.68262e8 0.808292
\(273\) 4.87149e8 1.44908
\(274\) 1.30314e8 0.382704
\(275\) −1.16965e8 −0.339149
\(276\) −1.80251e8 −0.516054
\(277\) −3.46288e8 −0.978944 −0.489472 0.872019i \(-0.662810\pi\)
−0.489472 + 0.872019i \(0.662810\pi\)
\(278\) −2.07347e8 −0.578818
\(279\) −2.26693e8 −0.624920
\(280\) −5.36533e8 −1.46064
\(281\) −2.47819e8 −0.666289 −0.333145 0.942876i \(-0.608110\pi\)
−0.333145 + 0.942876i \(0.608110\pi\)
\(282\) −3.62749e6 −0.00963240
\(283\) 2.38221e8 0.624779 0.312390 0.949954i \(-0.398871\pi\)
0.312390 + 0.949954i \(0.398871\pi\)
\(284\) −1.25906e8 −0.326162
\(285\) −2.91313e8 −0.745423
\(286\) 7.86186e7 0.198721
\(287\) 5.49530e8 1.37216
\(288\) 2.47368e8 0.610197
\(289\) 1.56248e8 0.380779
\(290\) −2.72135e8 −0.655227
\(291\) −1.48450e8 −0.353146
\(292\) −5.18359e8 −1.21840
\(293\) 6.02688e8 1.39977 0.699884 0.714257i \(-0.253235\pi\)
0.699884 + 0.714257i \(0.253235\pi\)
\(294\) 1.44319e8 0.331214
\(295\) 4.81606e8 1.09223
\(296\) −4.84939e7 −0.108684
\(297\) −1.54732e8 −0.342715
\(298\) 6.64999e7 0.145567
\(299\) −8.11718e8 −1.75613
\(300\) 1.98860e8 0.425229
\(301\) 1.22979e8 0.259924
\(302\) 1.06138e8 0.221742
\(303\) 2.10600e8 0.434920
\(304\) 3.44256e8 0.702787
\(305\) −2.89719e8 −0.584692
\(306\) 1.39299e8 0.277921
\(307\) −5.50528e8 −1.08591 −0.542957 0.839761i \(-0.682695\pi\)
−0.542957 + 0.839761i \(0.682695\pi\)
\(308\) −2.93565e8 −0.572502
\(309\) −4.32672e7 −0.0834264
\(310\) −2.06213e8 −0.393143
\(311\) −8.54076e8 −1.61003 −0.805017 0.593251i \(-0.797844\pi\)
−0.805017 + 0.593251i \(0.797844\pi\)
\(312\) −2.83500e8 −0.528461
\(313\) 6.27627e8 1.15690 0.578451 0.815717i \(-0.303657\pi\)
0.578451 + 0.815717i \(0.303657\pi\)
\(314\) −2.50985e8 −0.457503
\(315\) 9.38482e8 1.69176
\(316\) −4.15166e8 −0.740146
\(317\) −8.96686e7 −0.158100 −0.0790502 0.996871i \(-0.525189\pi\)
−0.0790502 + 0.996871i \(0.525189\pi\)
\(318\) 9.31992e7 0.162524
\(319\) −3.15813e8 −0.544706
\(320\) −3.30874e8 −0.564465
\(321\) −2.64123e8 −0.445695
\(322\) −3.66699e8 −0.612088
\(323\) 7.27090e8 1.20055
\(324\) −1.30122e8 −0.212541
\(325\) 8.95519e8 1.44705
\(326\) 2.67895e8 0.428255
\(327\) −1.27748e8 −0.202040
\(328\) −3.19803e8 −0.500408
\(329\) 6.09977e7 0.0944339
\(330\) −5.89172e7 −0.0902492
\(331\) 2.00092e8 0.303271 0.151635 0.988436i \(-0.451546\pi\)
0.151635 + 0.988436i \(0.451546\pi\)
\(332\) 1.70684e7 0.0255982
\(333\) 8.48236e7 0.125881
\(334\) −1.06935e8 −0.157039
\(335\) −1.24319e9 −1.80668
\(336\) 4.31420e8 0.620459
\(337\) 4.67534e8 0.665439 0.332719 0.943026i \(-0.392034\pi\)
0.332719 + 0.943026i \(0.392034\pi\)
\(338\) −3.68705e8 −0.519362
\(339\) −4.37703e8 −0.610211
\(340\) −1.04737e9 −1.44519
\(341\) −2.39311e8 −0.326830
\(342\) 1.78759e8 0.241645
\(343\) −1.15296e9 −1.54271
\(344\) −7.15684e7 −0.0947910
\(345\) 6.08306e8 0.797545
\(346\) 4.63089e8 0.601032
\(347\) −1.38116e8 −0.177456 −0.0887279 0.996056i \(-0.528280\pi\)
−0.0887279 + 0.996056i \(0.528280\pi\)
\(348\) 5.36934e8 0.682958
\(349\) −6.09602e7 −0.0767639 −0.0383820 0.999263i \(-0.512220\pi\)
−0.0383820 + 0.999263i \(0.512220\pi\)
\(350\) 4.04556e8 0.504360
\(351\) 1.18468e9 1.46226
\(352\) 2.61136e8 0.319130
\(353\) −8.21666e7 −0.0994223 −0.0497111 0.998764i \(-0.515830\pi\)
−0.0497111 + 0.998764i \(0.515830\pi\)
\(354\) 1.14962e8 0.137734
\(355\) 4.24906e8 0.504073
\(356\) 9.67086e8 1.13603
\(357\) 9.11186e8 1.05991
\(358\) 5.40601e7 0.0622711
\(359\) −1.75626e8 −0.200336 −0.100168 0.994971i \(-0.531938\pi\)
−0.100168 + 0.994971i \(0.531938\pi\)
\(360\) −5.46157e8 −0.616962
\(361\) 3.91890e7 0.0438418
\(362\) 1.75432e8 0.194370
\(363\) 4.13905e8 0.454180
\(364\) 2.24762e9 2.44269
\(365\) 1.74934e9 1.88300
\(366\) −6.91574e7 −0.0737318
\(367\) 3.51295e7 0.0370972 0.0185486 0.999828i \(-0.494095\pi\)
0.0185486 + 0.999828i \(0.494095\pi\)
\(368\) −7.18858e8 −0.751927
\(369\) 5.59387e8 0.579589
\(370\) 7.71605e7 0.0791934
\(371\) −1.56718e9 −1.59335
\(372\) 4.06868e8 0.409782
\(373\) −1.67183e9 −1.66806 −0.834031 0.551718i \(-0.813972\pi\)
−0.834031 + 0.551718i \(0.813972\pi\)
\(374\) 1.47052e8 0.145352
\(375\) 7.39598e7 0.0724246
\(376\) −3.54981e7 −0.0344388
\(377\) 2.41796e9 2.32410
\(378\) 5.35185e8 0.509663
\(379\) −1.75757e9 −1.65835 −0.829175 0.558989i \(-0.811189\pi\)
−0.829175 + 0.558989i \(0.811189\pi\)
\(380\) −1.34407e9 −1.25655
\(381\) −1.01318e9 −0.938534
\(382\) 2.10668e7 0.0193364
\(383\) −2.17759e8 −0.198053 −0.0990263 0.995085i \(-0.531573\pi\)
−0.0990263 + 0.995085i \(0.531573\pi\)
\(384\) −5.76670e8 −0.519718
\(385\) 9.90716e8 0.884783
\(386\) −4.83943e8 −0.428291
\(387\) 1.25185e8 0.109790
\(388\) −6.84923e8 −0.595293
\(389\) 4.98336e8 0.429239 0.214619 0.976698i \(-0.431149\pi\)
0.214619 + 0.976698i \(0.431149\pi\)
\(390\) 4.51088e8 0.385066
\(391\) −1.51827e9 −1.28449
\(392\) 1.41229e9 1.18419
\(393\) −1.02781e9 −0.854162
\(394\) −6.99702e8 −0.576337
\(395\) 1.40109e9 1.14387
\(396\) −2.98831e8 −0.241820
\(397\) −9.41907e8 −0.755512 −0.377756 0.925905i \(-0.623304\pi\)
−0.377756 + 0.925905i \(0.623304\pi\)
\(398\) −4.26056e8 −0.338747
\(399\) 1.16931e9 0.921560
\(400\) 7.93073e8 0.619588
\(401\) −1.13370e9 −0.877994 −0.438997 0.898488i \(-0.644666\pi\)
−0.438997 + 0.898488i \(0.644666\pi\)
\(402\) −2.96756e8 −0.227828
\(403\) 1.83224e9 1.39449
\(404\) 9.71672e8 0.733137
\(405\) 4.39133e8 0.328476
\(406\) 1.09233e9 0.810051
\(407\) 8.95447e7 0.0658354
\(408\) −5.30272e8 −0.386534
\(409\) 2.01366e9 1.45531 0.727653 0.685946i \(-0.240611\pi\)
0.727653 + 0.685946i \(0.240611\pi\)
\(410\) 5.08851e8 0.364626
\(411\) 8.67704e8 0.616489
\(412\) −1.99627e8 −0.140631
\(413\) −1.93313e9 −1.35032
\(414\) −3.73276e8 −0.258541
\(415\) −5.76019e7 −0.0395611
\(416\) −1.99934e9 −1.36163
\(417\) −1.38064e9 −0.932403
\(418\) 1.88709e8 0.126379
\(419\) −2.42088e9 −1.60777 −0.803886 0.594784i \(-0.797238\pi\)
−0.803886 + 0.594784i \(0.797238\pi\)
\(420\) −1.68438e9 −1.10935
\(421\) 6.39130e8 0.417448 0.208724 0.977975i \(-0.433069\pi\)
0.208724 + 0.977975i \(0.433069\pi\)
\(422\) −1.15995e8 −0.0751355
\(423\) 6.20919e7 0.0398881
\(424\) 9.12034e8 0.581073
\(425\) 1.67502e9 1.05842
\(426\) 1.01427e8 0.0635654
\(427\) 1.16291e9 0.722850
\(428\) −1.21862e9 −0.751301
\(429\) 5.23488e8 0.320115
\(430\) 1.13875e8 0.0690700
\(431\) −2.22935e9 −1.34125 −0.670623 0.741798i \(-0.733973\pi\)
−0.670623 + 0.741798i \(0.733973\pi\)
\(432\) 1.04915e9 0.626102
\(433\) −6.57831e8 −0.389409 −0.194705 0.980862i \(-0.562375\pi\)
−0.194705 + 0.980862i \(0.562375\pi\)
\(434\) 8.27725e8 0.486040
\(435\) −1.81203e9 −1.05549
\(436\) −5.89407e8 −0.340575
\(437\) −1.94837e9 −1.11683
\(438\) 4.17578e8 0.237453
\(439\) −1.25699e9 −0.709097 −0.354549 0.935038i \(-0.615365\pi\)
−0.354549 + 0.935038i \(0.615365\pi\)
\(440\) −5.76555e8 −0.322668
\(441\) −2.47031e9 −1.37157
\(442\) −1.12587e9 −0.620171
\(443\) 6.21522e8 0.339659 0.169830 0.985473i \(-0.445678\pi\)
0.169830 + 0.985473i \(0.445678\pi\)
\(444\) −1.52241e8 −0.0825450
\(445\) −3.26370e9 −1.75570
\(446\) 5.44120e8 0.290417
\(447\) 4.42795e8 0.234491
\(448\) 1.32810e9 0.697843
\(449\) 3.37102e8 0.175752 0.0878758 0.996131i \(-0.471992\pi\)
0.0878758 + 0.996131i \(0.471992\pi\)
\(450\) 4.11813e8 0.213038
\(451\) 5.90522e8 0.303122
\(452\) −2.01948e9 −1.02862
\(453\) 7.06730e8 0.357199
\(454\) 8.58650e7 0.0430646
\(455\) −7.58522e9 −3.77510
\(456\) −6.80488e8 −0.336081
\(457\) 3.14261e9 1.54022 0.770112 0.637909i \(-0.220200\pi\)
0.770112 + 0.637909i \(0.220200\pi\)
\(458\) 9.14455e8 0.444767
\(459\) 2.21587e9 1.06955
\(460\) 2.80662e9 1.34441
\(461\) 6.04340e8 0.287295 0.143647 0.989629i \(-0.454117\pi\)
0.143647 + 0.989629i \(0.454117\pi\)
\(462\) 2.36489e8 0.111574
\(463\) −1.62669e8 −0.0761677 −0.0380839 0.999275i \(-0.512125\pi\)
−0.0380839 + 0.999275i \(0.512125\pi\)
\(464\) 2.14135e9 0.995117
\(465\) −1.37309e9 −0.633305
\(466\) −9.54814e8 −0.437087
\(467\) 1.96986e9 0.895005 0.447502 0.894283i \(-0.352314\pi\)
0.447502 + 0.894283i \(0.352314\pi\)
\(468\) 2.28794e9 1.03177
\(469\) 4.99006e9 2.23358
\(470\) 5.64824e7 0.0250940
\(471\) −1.67120e9 −0.736980
\(472\) 1.12500e9 0.492443
\(473\) 1.32152e8 0.0574196
\(474\) 3.34448e8 0.144246
\(475\) 2.14952e9 0.920267
\(476\) 4.20406e9 1.78667
\(477\) −1.59529e9 −0.673017
\(478\) −1.10835e9 −0.464173
\(479\) −2.56454e9 −1.06619 −0.533097 0.846054i \(-0.678972\pi\)
−0.533097 + 0.846054i \(0.678972\pi\)
\(480\) 1.49832e9 0.618385
\(481\) −6.85582e8 −0.280900
\(482\) 8.53643e8 0.347225
\(483\) −2.44169e9 −0.985998
\(484\) 1.90969e9 0.765603
\(485\) 2.31146e9 0.920005
\(486\) 8.61531e8 0.340443
\(487\) −2.97468e9 −1.16705 −0.583523 0.812096i \(-0.698326\pi\)
−0.583523 + 0.812096i \(0.698326\pi\)
\(488\) −6.76764e8 −0.263614
\(489\) 1.78380e9 0.689865
\(490\) −2.24714e9 −0.862868
\(491\) 1.23197e9 0.469696 0.234848 0.972032i \(-0.424541\pi\)
0.234848 + 0.972032i \(0.424541\pi\)
\(492\) −1.00398e9 −0.380058
\(493\) 4.52266e9 1.69993
\(494\) −1.44481e9 −0.539221
\(495\) 1.00849e9 0.373725
\(496\) 1.62263e9 0.597082
\(497\) −1.70554e9 −0.623181
\(498\) −1.37499e7 −0.00498880
\(499\) 2.78394e9 1.00301 0.501507 0.865153i \(-0.332779\pi\)
0.501507 + 0.865153i \(0.332779\pi\)
\(500\) 3.41238e8 0.122085
\(501\) −7.12037e8 −0.252971
\(502\) −5.75181e8 −0.202928
\(503\) 3.98467e8 0.139606 0.0698030 0.997561i \(-0.477763\pi\)
0.0698030 + 0.997561i \(0.477763\pi\)
\(504\) 2.19223e9 0.762745
\(505\) −3.27918e9 −1.13304
\(506\) −3.94052e8 −0.135216
\(507\) −2.45505e9 −0.836628
\(508\) −4.67465e9 −1.58207
\(509\) 2.69989e9 0.907473 0.453736 0.891136i \(-0.350091\pi\)
0.453736 + 0.891136i \(0.350091\pi\)
\(510\) 8.43736e8 0.281651
\(511\) −7.02173e9 −2.32794
\(512\) −3.06915e9 −1.01059
\(513\) 2.84359e9 0.929942
\(514\) −1.01585e9 −0.329956
\(515\) 6.73697e8 0.217340
\(516\) −2.24681e8 −0.0719932
\(517\) 6.55478e7 0.0208613
\(518\) −3.09716e8 −0.0979060
\(519\) 3.08351e9 0.968188
\(520\) 4.41428e9 1.37673
\(521\) 3.82871e8 0.118610 0.0593049 0.998240i \(-0.481112\pi\)
0.0593049 + 0.998240i \(0.481112\pi\)
\(522\) 1.11192e9 0.342159
\(523\) 2.76861e9 0.846264 0.423132 0.906068i \(-0.360931\pi\)
0.423132 + 0.906068i \(0.360931\pi\)
\(524\) −4.74215e9 −1.43985
\(525\) 2.69377e9 0.812462
\(526\) −8.18943e8 −0.245360
\(527\) 3.42710e9 1.01997
\(528\) 4.63601e8 0.137065
\(529\) 6.63671e8 0.194921
\(530\) −1.45117e9 −0.423402
\(531\) −1.96781e9 −0.570363
\(532\) 5.39499e9 1.55346
\(533\) −4.52122e9 −1.29333
\(534\) −7.79062e8 −0.221400
\(535\) 4.11256e9 1.16111
\(536\) −2.90401e9 −0.814556
\(537\) 3.59963e8 0.100311
\(538\) 6.49471e8 0.179813
\(539\) −2.60781e9 −0.717323
\(540\) −4.09617e9 −1.11944
\(541\) −3.30164e9 −0.896477 −0.448238 0.893914i \(-0.647948\pi\)
−0.448238 + 0.893914i \(0.647948\pi\)
\(542\) 2.19198e9 0.591341
\(543\) 1.16813e9 0.313105
\(544\) −3.73966e9 −0.995945
\(545\) 1.98912e9 0.526347
\(546\) −1.81063e9 −0.476054
\(547\) 6.45168e9 1.68546 0.842728 0.538340i \(-0.180948\pi\)
0.842728 + 0.538340i \(0.180948\pi\)
\(548\) 4.00344e9 1.03921
\(549\) 1.18377e9 0.305326
\(550\) 4.34734e8 0.111418
\(551\) 5.80385e9 1.47804
\(552\) 1.42096e9 0.359580
\(553\) −5.62388e9 −1.41416
\(554\) 1.28708e9 0.321604
\(555\) 5.13779e8 0.127571
\(556\) −6.37003e9 −1.57174
\(557\) 6.11372e9 1.49904 0.749519 0.661983i \(-0.230285\pi\)
0.749519 + 0.661983i \(0.230285\pi\)
\(558\) 8.42572e8 0.205299
\(559\) −1.01180e9 −0.244992
\(560\) −6.71748e9 −1.61640
\(561\) 9.79156e8 0.234143
\(562\) 9.21092e8 0.218890
\(563\) −4.59129e9 −1.08432 −0.542158 0.840277i \(-0.682392\pi\)
−0.542158 + 0.840277i \(0.682392\pi\)
\(564\) −1.11442e8 −0.0261561
\(565\) 6.81531e9 1.58970
\(566\) −8.85416e8 −0.205253
\(567\) −1.76265e9 −0.406092
\(568\) 9.92552e8 0.227266
\(569\) −4.85419e9 −1.10465 −0.552324 0.833630i \(-0.686259\pi\)
−0.552324 + 0.833630i \(0.686259\pi\)
\(570\) 1.08275e9 0.244887
\(571\) −3.97419e9 −0.893351 −0.446675 0.894696i \(-0.647392\pi\)
−0.446675 + 0.894696i \(0.647392\pi\)
\(572\) 2.41528e9 0.539612
\(573\) 1.40275e8 0.0311486
\(574\) −2.04249e9 −0.450783
\(575\) −4.48852e9 −0.984614
\(576\) 1.35192e9 0.294763
\(577\) 2.46945e9 0.535161 0.267581 0.963535i \(-0.413776\pi\)
0.267581 + 0.963535i \(0.413776\pi\)
\(578\) −5.80743e8 −0.125094
\(579\) −3.22237e9 −0.689923
\(580\) −8.36041e9 −1.77922
\(581\) 2.31209e8 0.0489090
\(582\) 5.51757e8 0.116016
\(583\) −1.68408e9 −0.351985
\(584\) 4.08635e9 0.848967
\(585\) −7.72128e9 −1.59457
\(586\) −2.24006e9 −0.459853
\(587\) 8.02057e9 1.63671 0.818355 0.574713i \(-0.194886\pi\)
0.818355 + 0.574713i \(0.194886\pi\)
\(588\) 4.43371e9 0.899386
\(589\) 4.39793e9 0.886839
\(590\) −1.79003e9 −0.358821
\(591\) −4.65902e9 −0.928407
\(592\) −6.07152e8 −0.120274
\(593\) 1.99128e9 0.392141 0.196070 0.980590i \(-0.437182\pi\)
0.196070 + 0.980590i \(0.437182\pi\)
\(594\) 5.75107e8 0.112589
\(595\) −1.41878e10 −2.76124
\(596\) 2.04298e9 0.395278
\(597\) −2.83692e9 −0.545679
\(598\) 3.01699e9 0.576925
\(599\) 4.99536e9 0.949670 0.474835 0.880075i \(-0.342508\pi\)
0.474835 + 0.880075i \(0.342508\pi\)
\(600\) −1.56766e9 −0.296294
\(601\) −4.19069e9 −0.787454 −0.393727 0.919227i \(-0.628814\pi\)
−0.393727 + 0.919227i \(0.628814\pi\)
\(602\) −4.57086e8 −0.0853906
\(603\) 5.07957e9 0.943445
\(604\) 3.26073e9 0.602124
\(605\) −6.44477e9 −1.18321
\(606\) −7.82756e8 −0.142880
\(607\) −2.91949e9 −0.529842 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(608\) −4.79903e9 −0.865946
\(609\) 7.27335e9 1.30489
\(610\) 1.07682e9 0.192084
\(611\) −5.01854e8 −0.0890088
\(612\) 4.27947e9 0.754675
\(613\) −3.83809e9 −0.672983 −0.336491 0.941687i \(-0.609240\pi\)
−0.336491 + 0.941687i \(0.609240\pi\)
\(614\) 2.04620e9 0.356745
\(615\) 3.38822e9 0.587366
\(616\) 2.31425e9 0.398912
\(617\) −5.20036e9 −0.891323 −0.445661 0.895202i \(-0.647031\pi\)
−0.445661 + 0.895202i \(0.647031\pi\)
\(618\) 1.60815e8 0.0274073
\(619\) −6.96863e9 −1.18095 −0.590473 0.807058i \(-0.701059\pi\)
−0.590473 + 0.807058i \(0.701059\pi\)
\(620\) −6.33519e9 −1.06755
\(621\) −5.93784e9 −0.994965
\(622\) 3.17442e9 0.528930
\(623\) 1.31002e10 2.17055
\(624\) −3.54947e9 −0.584814
\(625\) −6.64924e9 −1.08941
\(626\) −2.33276e9 −0.380067
\(627\) 1.25653e9 0.203581
\(628\) −7.71064e9 −1.24232
\(629\) −1.28234e9 −0.205460
\(630\) −3.48814e9 −0.555779
\(631\) 1.04787e10 1.66037 0.830183 0.557490i \(-0.188236\pi\)
0.830183 + 0.557490i \(0.188236\pi\)
\(632\) 3.27286e9 0.515724
\(633\) −7.72361e8 −0.121034
\(634\) 3.33279e8 0.0519393
\(635\) 1.57759e10 2.44504
\(636\) 2.86322e9 0.441322
\(637\) 1.99662e10 3.06060
\(638\) 1.17381e9 0.178947
\(639\) −1.73613e9 −0.263227
\(640\) 8.97911e9 1.35395
\(641\) −2.92164e9 −0.438151 −0.219076 0.975708i \(-0.570304\pi\)
−0.219076 + 0.975708i \(0.570304\pi\)
\(642\) 9.81689e8 0.146420
\(643\) 7.68989e9 1.14073 0.570364 0.821392i \(-0.306802\pi\)
0.570364 + 0.821392i \(0.306802\pi\)
\(644\) −1.12655e10 −1.66208
\(645\) 7.58246e8 0.111263
\(646\) −2.70244e9 −0.394405
\(647\) 1.38869e8 0.0201577 0.0100789 0.999949i \(-0.496792\pi\)
0.0100789 + 0.999949i \(0.496792\pi\)
\(648\) 1.02579e9 0.148096
\(649\) −2.07733e9 −0.298297
\(650\) −3.32846e9 −0.475386
\(651\) 5.51147e9 0.782950
\(652\) 8.23013e9 1.16289
\(653\) 4.91204e9 0.690344 0.345172 0.938539i \(-0.387821\pi\)
0.345172 + 0.938539i \(0.387821\pi\)
\(654\) 4.74812e8 0.0663742
\(655\) 1.60037e10 2.22523
\(656\) −4.00399e9 −0.553771
\(657\) −7.14768e9 −0.983301
\(658\) −2.26716e8 −0.0310235
\(659\) −1.44098e9 −0.196137 −0.0980686 0.995180i \(-0.531266\pi\)
−0.0980686 + 0.995180i \(0.531266\pi\)
\(660\) −1.81003e9 −0.245065
\(661\) 6.29813e9 0.848216 0.424108 0.905612i \(-0.360588\pi\)
0.424108 + 0.905612i \(0.360588\pi\)
\(662\) −7.43698e8 −0.0996309
\(663\) −7.49671e9 −0.999019
\(664\) −1.34554e8 −0.0178365
\(665\) −1.82069e10 −2.40082
\(666\) −3.15271e8 −0.0413547
\(667\) −1.21193e10 −1.58138
\(668\) −3.28522e9 −0.426429
\(669\) 3.62306e9 0.467826
\(670\) 4.62067e9 0.593531
\(671\) 1.24966e9 0.159684
\(672\) −6.01412e9 −0.764504
\(673\) −1.12010e10 −1.41646 −0.708229 0.705983i \(-0.750505\pi\)
−0.708229 + 0.705983i \(0.750505\pi\)
\(674\) −1.73772e9 −0.218611
\(675\) 6.55086e9 0.819851
\(676\) −1.13272e10 −1.41029
\(677\) 1.35096e10 1.67333 0.836666 0.547714i \(-0.184502\pi\)
0.836666 + 0.547714i \(0.184502\pi\)
\(678\) 1.62685e9 0.200467
\(679\) −9.27802e9 −1.13739
\(680\) 8.25667e9 1.00699
\(681\) 5.71739e8 0.0693718
\(682\) 8.89468e8 0.107371
\(683\) 7.42709e9 0.891962 0.445981 0.895042i \(-0.352855\pi\)
0.445981 + 0.895042i \(0.352855\pi\)
\(684\) 5.49176e9 0.656168
\(685\) −1.35107e10 −1.60606
\(686\) 4.28530e9 0.506812
\(687\) 6.08896e9 0.716465
\(688\) −8.96049e8 −0.104899
\(689\) 1.28939e10 1.50181
\(690\) −2.26095e9 −0.262010
\(691\) 6.66846e9 0.768869 0.384434 0.923152i \(-0.374397\pi\)
0.384434 + 0.923152i \(0.374397\pi\)
\(692\) 1.42268e10 1.63206
\(693\) −4.04799e9 −0.462033
\(694\) 5.13347e8 0.0582979
\(695\) 2.14974e10 2.42907
\(696\) −4.23279e9 −0.475876
\(697\) −8.45669e9 −0.945988
\(698\) 2.26576e8 0.0252185
\(699\) −6.35770e9 −0.704093
\(700\) 1.24286e10 1.36955
\(701\) −8.02719e9 −0.880137 −0.440069 0.897964i \(-0.645046\pi\)
−0.440069 + 0.897964i \(0.645046\pi\)
\(702\) −4.40319e9 −0.480383
\(703\) −1.64561e9 −0.178642
\(704\) 1.42717e9 0.154160
\(705\) 3.76092e8 0.0404233
\(706\) 3.05396e8 0.0326623
\(707\) 1.31624e10 1.40077
\(708\) 3.53181e9 0.374008
\(709\) 6.52080e9 0.687130 0.343565 0.939129i \(-0.388365\pi\)
0.343565 + 0.939129i \(0.388365\pi\)
\(710\) −1.57929e9 −0.165599
\(711\) −5.72475e9 −0.597328
\(712\) −7.62378e9 −0.791572
\(713\) −9.18354e9 −0.948849
\(714\) −3.38669e9 −0.348202
\(715\) −8.15104e9 −0.833953
\(716\) 1.66081e9 0.169092
\(717\) −7.38003e9 −0.747724
\(718\) 6.52765e8 0.0658145
\(719\) 1.19598e10 1.19997 0.599986 0.800010i \(-0.295173\pi\)
0.599986 + 0.800010i \(0.295173\pi\)
\(720\) −6.83798e9 −0.682753
\(721\) −2.70417e9 −0.268695
\(722\) −1.45657e8 −0.0144030
\(723\) 5.68404e9 0.559337
\(724\) 5.38954e9 0.527797
\(725\) 1.33705e10 1.30306
\(726\) −1.53840e9 −0.149208
\(727\) −3.08388e9 −0.297665 −0.148832 0.988862i \(-0.547551\pi\)
−0.148832 + 0.988862i \(0.547551\pi\)
\(728\) −1.77186e10 −1.70204
\(729\) 3.24434e9 0.310156
\(730\) −6.50195e9 −0.618605
\(731\) −1.89251e9 −0.179196
\(732\) −2.12462e9 −0.200213
\(733\) −1.53686e10 −1.44135 −0.720676 0.693272i \(-0.756168\pi\)
−0.720676 + 0.693272i \(0.756168\pi\)
\(734\) −1.30569e8 −0.0121872
\(735\) −1.49628e10 −1.38997
\(736\) 1.00211e10 0.926495
\(737\) 5.36229e9 0.493417
\(738\) −2.07912e9 −0.190407
\(739\) −8.64423e9 −0.787899 −0.393949 0.919132i \(-0.628892\pi\)
−0.393949 + 0.919132i \(0.628892\pi\)
\(740\) 2.37049e9 0.215044
\(741\) −9.62039e9 −0.868618
\(742\) 5.82489e9 0.523448
\(743\) 6.82591e9 0.610520 0.305260 0.952269i \(-0.401257\pi\)
0.305260 + 0.952269i \(0.401257\pi\)
\(744\) −3.20744e9 −0.285531
\(745\) −6.89460e9 −0.610888
\(746\) 6.21385e9 0.547993
\(747\) 2.35357e8 0.0206588
\(748\) 4.51766e9 0.394691
\(749\) −1.65075e10 −1.43547
\(750\) −2.74893e8 −0.0237930
\(751\) −1.00999e10 −0.870113 −0.435057 0.900403i \(-0.643272\pi\)
−0.435057 + 0.900403i \(0.643272\pi\)
\(752\) −4.44442e8 −0.0381112
\(753\) −3.82989e9 −0.326892
\(754\) −8.98705e9 −0.763515
\(755\) −1.10042e10 −0.930562
\(756\) 1.64417e10 1.38395
\(757\) −1.25390e10 −1.05057 −0.525287 0.850925i \(-0.676042\pi\)
−0.525287 + 0.850925i \(0.676042\pi\)
\(758\) 6.53253e9 0.544803
\(759\) −2.62383e9 −0.217816
\(760\) 1.05956e10 0.875546
\(761\) 1.68387e10 1.38504 0.692518 0.721400i \(-0.256501\pi\)
0.692518 + 0.721400i \(0.256501\pi\)
\(762\) 3.76578e9 0.308328
\(763\) −7.98415e9 −0.650718
\(764\) 6.47203e8 0.0525066
\(765\) −1.44422e10 −1.16632
\(766\) 8.09365e8 0.0650644
\(767\) 1.59047e10 1.27274
\(768\) −5.76620e8 −0.0459331
\(769\) 5.50473e9 0.436510 0.218255 0.975892i \(-0.429964\pi\)
0.218255 + 0.975892i \(0.429964\pi\)
\(770\) −3.68228e9 −0.290670
\(771\) −6.76408e9 −0.531519
\(772\) −1.48675e10 −1.16299
\(773\) −1.58271e9 −0.123246 −0.0616230 0.998099i \(-0.519628\pi\)
−0.0616230 + 0.998099i \(0.519628\pi\)
\(774\) −4.65285e8 −0.0360683
\(775\) 1.01316e10 0.781851
\(776\) 5.39942e9 0.414793
\(777\) −2.06227e9 −0.157714
\(778\) −1.85221e9 −0.141014
\(779\) −1.08523e10 −0.822509
\(780\) 1.38581e10 1.04562
\(781\) −1.83276e9 −0.137666
\(782\) 5.64311e9 0.421983
\(783\) 1.76877e10 1.31676
\(784\) 1.76821e10 1.31047
\(785\) 2.60217e10 1.91996
\(786\) 3.82016e9 0.280610
\(787\) −9.16094e9 −0.669929 −0.334964 0.942231i \(-0.608724\pi\)
−0.334964 + 0.942231i \(0.608724\pi\)
\(788\) −2.14959e10 −1.56500
\(789\) −5.45300e9 −0.395244
\(790\) −5.20757e9 −0.375786
\(791\) −2.73561e10 −1.96534
\(792\) 2.35576e9 0.168497
\(793\) −9.56774e9 −0.681323
\(794\) 3.50087e9 0.248202
\(795\) −9.66273e9 −0.682048
\(796\) −1.30891e10 −0.919842
\(797\) −1.42782e10 −0.999006 −0.499503 0.866312i \(-0.666484\pi\)
−0.499503 + 0.866312i \(0.666484\pi\)
\(798\) −4.34607e9 −0.302752
\(799\) −9.38691e8 −0.0651042
\(800\) −1.10557e10 −0.763431
\(801\) 1.33352e10 0.916824
\(802\) 4.21371e9 0.288440
\(803\) −7.54551e9 −0.514262
\(804\) −9.11678e9 −0.618651
\(805\) 3.80187e10 2.56869
\(806\) −6.81004e9 −0.458118
\(807\) 4.32455e9 0.289657
\(808\) −7.65994e9 −0.510841
\(809\) 8.21284e9 0.545348 0.272674 0.962106i \(-0.412092\pi\)
0.272674 + 0.962106i \(0.412092\pi\)
\(810\) −1.63217e9 −0.107911
\(811\) −1.87787e10 −1.23621 −0.618104 0.786096i \(-0.712099\pi\)
−0.618104 + 0.786096i \(0.712099\pi\)
\(812\) 3.35580e10 2.19963
\(813\) 1.45954e10 0.952577
\(814\) −3.32819e8 −0.0216283
\(815\) −2.77748e10 −1.79721
\(816\) −6.63910e9 −0.427753
\(817\) −2.42862e9 −0.155806
\(818\) −7.48434e9 −0.478098
\(819\) 3.09926e10 1.97135
\(820\) 1.56327e10 0.990114
\(821\) −1.50345e10 −0.948173 −0.474086 0.880478i \(-0.657222\pi\)
−0.474086 + 0.880478i \(0.657222\pi\)
\(822\) −3.22507e9 −0.202530
\(823\) 9.26651e9 0.579451 0.289726 0.957110i \(-0.406436\pi\)
0.289726 + 0.957110i \(0.406436\pi\)
\(824\) 1.57371e9 0.0979896
\(825\) 2.89471e9 0.179480
\(826\) 7.18504e9 0.443608
\(827\) −2.93861e10 −1.80665 −0.903323 0.428962i \(-0.858879\pi\)
−0.903323 + 0.428962i \(0.858879\pi\)
\(828\) −1.14676e10 −0.702049
\(829\) 3.66600e9 0.223487 0.111743 0.993737i \(-0.464357\pi\)
0.111743 + 0.993737i \(0.464357\pi\)
\(830\) 2.14094e8 0.0129967
\(831\) 8.57011e9 0.518063
\(832\) −1.09268e10 −0.657753
\(833\) 3.73457e10 2.23863
\(834\) 5.13154e9 0.306314
\(835\) 1.10869e10 0.659031
\(836\) 5.79742e9 0.343173
\(837\) 1.34031e10 0.790070
\(838\) 8.99791e9 0.528187
\(839\) −1.20233e9 −0.0702840 −0.0351420 0.999382i \(-0.511188\pi\)
−0.0351420 + 0.999382i \(0.511188\pi\)
\(840\) 1.32784e10 0.772980
\(841\) 1.88513e10 1.09284
\(842\) −2.37551e9 −0.137140
\(843\) 6.13316e9 0.352604
\(844\) −3.56354e9 −0.204025
\(845\) 3.82267e10 2.17956
\(846\) −2.30782e8 −0.0131041
\(847\) 2.58688e10 1.46280
\(848\) 1.14188e10 0.643037
\(849\) −5.89561e9 −0.330637
\(850\) −6.22570e9 −0.347714
\(851\) 3.43628e9 0.191133
\(852\) 3.11600e9 0.172607
\(853\) 2.76893e10 1.52753 0.763767 0.645491i \(-0.223347\pi\)
0.763767 + 0.645491i \(0.223347\pi\)
\(854\) −4.32229e9 −0.237471
\(855\) −1.85334e10 −1.01409
\(856\) 9.60666e9 0.523497
\(857\) −3.11448e10 −1.69025 −0.845127 0.534565i \(-0.820475\pi\)
−0.845127 + 0.534565i \(0.820475\pi\)
\(858\) −1.94569e9 −0.105164
\(859\) 1.06707e10 0.574405 0.287202 0.957870i \(-0.407275\pi\)
0.287202 + 0.957870i \(0.407275\pi\)
\(860\) 3.49842e9 0.187554
\(861\) −1.36001e10 −0.726156
\(862\) 8.28604e9 0.440627
\(863\) 2.01685e10 1.06816 0.534079 0.845435i \(-0.320659\pi\)
0.534079 + 0.845435i \(0.320659\pi\)
\(864\) −1.46255e10 −0.771457
\(865\) −4.80122e10 −2.52229
\(866\) 2.44502e9 0.127929
\(867\) −3.86692e9 −0.201511
\(868\) 2.54290e10 1.31980
\(869\) −6.04338e9 −0.312400
\(870\) 6.73494e9 0.346750
\(871\) −4.10553e10 −2.10526
\(872\) 4.64644e9 0.237308
\(873\) −9.44444e9 −0.480426
\(874\) 7.24169e9 0.366902
\(875\) 4.62244e9 0.233261
\(876\) 1.28286e10 0.644786
\(877\) −8.78810e9 −0.439943 −0.219971 0.975506i \(-0.570596\pi\)
−0.219971 + 0.975506i \(0.570596\pi\)
\(878\) 4.67197e9 0.232953
\(879\) −1.49156e10 −0.740765
\(880\) −7.21857e9 −0.357077
\(881\) −2.67005e10 −1.31554 −0.657769 0.753220i \(-0.728500\pi\)
−0.657769 + 0.753220i \(0.728500\pi\)
\(882\) 9.18164e9 0.450589
\(883\) 2.75848e10 1.34836 0.674181 0.738566i \(-0.264497\pi\)
0.674181 + 0.738566i \(0.264497\pi\)
\(884\) −3.45886e10 −1.68403
\(885\) −1.19190e10 −0.578016
\(886\) −2.31007e9 −0.111585
\(887\) 1.57163e10 0.756165 0.378083 0.925772i \(-0.376584\pi\)
0.378083 + 0.925772i \(0.376584\pi\)
\(888\) 1.20015e9 0.0575163
\(889\) −6.33231e10 −3.02278
\(890\) 1.21305e10 0.576784
\(891\) −1.89413e9 −0.0897093
\(892\) 1.67162e10 0.788606
\(893\) −1.20460e9 −0.0566062
\(894\) −1.64578e9 −0.0770352
\(895\) −5.60486e9 −0.261327
\(896\) −3.60415e10 −1.67388
\(897\) 2.00888e10 0.929354
\(898\) −1.25294e9 −0.0577381
\(899\) 2.73561e10 1.25573
\(900\) 1.26515e10 0.578488
\(901\) 2.41173e10 1.09848
\(902\) −2.19485e9 −0.0995821
\(903\) −3.04354e9 −0.137554
\(904\) 1.59201e10 0.716731
\(905\) −1.81885e10 −0.815692
\(906\) −2.62677e9 −0.117347
\(907\) −2.21424e10 −0.985369 −0.492685 0.870208i \(-0.663984\pi\)
−0.492685 + 0.870208i \(0.663984\pi\)
\(908\) 2.63790e9 0.116939
\(909\) 1.33984e10 0.591672
\(910\) 2.81927e10 1.24020
\(911\) 2.20181e10 0.964862 0.482431 0.875934i \(-0.339754\pi\)
0.482431 + 0.875934i \(0.339754\pi\)
\(912\) −8.51982e9 −0.371919
\(913\) 2.48456e8 0.0108044
\(914\) −1.16804e10 −0.505996
\(915\) 7.17012e9 0.309423
\(916\) 2.80934e10 1.20773
\(917\) −6.42376e10 −2.75104
\(918\) −8.23594e9 −0.351369
\(919\) 1.01632e10 0.431943 0.215972 0.976400i \(-0.430708\pi\)
0.215972 + 0.976400i \(0.430708\pi\)
\(920\) −2.21253e10 −0.936766
\(921\) 1.36248e10 0.574672
\(922\) −2.24620e9 −0.0943824
\(923\) 1.40322e10 0.587381
\(924\) 7.26530e9 0.302972
\(925\) −3.79103e9 −0.157493
\(926\) 6.04606e8 0.0250227
\(927\) −2.75267e9 −0.113495
\(928\) −2.98510e10 −1.22614
\(929\) −8.04262e9 −0.329111 −0.164556 0.986368i \(-0.552619\pi\)
−0.164556 + 0.986368i \(0.552619\pi\)
\(930\) 5.10348e9 0.208054
\(931\) 4.79249e10 1.94643
\(932\) −2.93333e10 −1.18688
\(933\) 2.11371e10 0.852040
\(934\) −7.32154e9 −0.294028
\(935\) −1.52461e10 −0.609983
\(936\) −1.80364e10 −0.718926
\(937\) −3.08374e10 −1.22459 −0.612293 0.790631i \(-0.709753\pi\)
−0.612293 + 0.790631i \(0.709753\pi\)
\(938\) −1.85470e10 −0.733777
\(939\) −1.55329e10 −0.612240
\(940\) 1.73522e9 0.0681409
\(941\) 8.87704e8 0.0347300 0.0173650 0.999849i \(-0.494472\pi\)
0.0173650 + 0.999849i \(0.494472\pi\)
\(942\) 6.21151e9 0.242113
\(943\) 2.26613e10 0.880021
\(944\) 1.40852e10 0.544955
\(945\) −5.54871e10 −2.13885
\(946\) −4.91182e8 −0.0188635
\(947\) −8.02900e9 −0.307211 −0.153605 0.988132i \(-0.549088\pi\)
−0.153605 + 0.988132i \(0.549088\pi\)
\(948\) 1.02748e10 0.391690
\(949\) 5.77707e10 2.19420
\(950\) −7.98931e9 −0.302327
\(951\) 2.21917e9 0.0836678
\(952\) −3.31416e10 −1.24493
\(953\) −6.27853e9 −0.234981 −0.117491 0.993074i \(-0.537485\pi\)
−0.117491 + 0.993074i \(0.537485\pi\)
\(954\) 5.92937e9 0.221100
\(955\) −2.18417e9 −0.0811472
\(956\) −3.40502e10 −1.26043
\(957\) 7.81590e9 0.288262
\(958\) 9.53187e9 0.350267
\(959\) 5.42309e10 1.98555
\(960\) 8.18863e9 0.298718
\(961\) −6.78322e9 −0.246549
\(962\) 2.54816e9 0.0922815
\(963\) −1.68036e10 −0.606331
\(964\) 2.62252e10 0.942865
\(965\) 5.01744e10 1.79737
\(966\) 9.07526e9 0.323921
\(967\) −1.81180e10 −0.644343 −0.322172 0.946681i \(-0.604413\pi\)
−0.322172 + 0.946681i \(0.604413\pi\)
\(968\) −1.50546e10 −0.533463
\(969\) −1.79944e10 −0.635337
\(970\) −8.59121e9 −0.302241
\(971\) −2.56716e10 −0.899883 −0.449942 0.893058i \(-0.648555\pi\)
−0.449942 + 0.893058i \(0.648555\pi\)
\(972\) 2.64676e10 0.924447
\(973\) −8.62889e10 −3.00303
\(974\) 1.10562e10 0.383399
\(975\) −2.21628e10 −0.765787
\(976\) −8.47320e9 −0.291725
\(977\) 2.60682e10 0.894295 0.447148 0.894460i \(-0.352440\pi\)
0.447148 + 0.894460i \(0.352440\pi\)
\(978\) −6.63000e9 −0.226635
\(979\) 1.40774e10 0.479495
\(980\) −6.90356e10 −2.34305
\(981\) −8.12737e9 −0.274858
\(982\) −4.57899e9 −0.154305
\(983\) 1.74429e10 0.585708 0.292854 0.956157i \(-0.405395\pi\)
0.292854 + 0.956157i \(0.405395\pi\)
\(984\) 7.91466e9 0.264819
\(985\) 7.25439e10 2.41866
\(986\) −1.68098e10 −0.558461
\(987\) −1.50960e9 −0.0499750
\(988\) −4.43868e10 −1.46421
\(989\) 5.07134e9 0.166700
\(990\) −3.74833e9 −0.122776
\(991\) 1.57746e10 0.514874 0.257437 0.966295i \(-0.417122\pi\)
0.257437 + 0.966295i \(0.417122\pi\)
\(992\) −2.26199e10 −0.735700
\(993\) −4.95197e9 −0.160493
\(994\) 6.33913e9 0.204728
\(995\) 4.41727e10 1.42159
\(996\) −4.22417e8 −0.0135467
\(997\) 2.18743e9 0.0699040 0.0349520 0.999389i \(-0.488872\pi\)
0.0349520 + 0.999389i \(0.488872\pi\)
\(998\) −1.03473e10 −0.329511
\(999\) −5.01514e9 −0.159149
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.b.1.6 13
3.2 odd 2 387.8.a.d.1.8 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.6 13 1.1 even 1 trivial
387.8.a.d.1.8 13 3.2 odd 2