Properties

Label 91.10.a.a.1.8
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(13.5332\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+11.5332 q^{2} -234.910 q^{3} -378.986 q^{4} -380.245 q^{5} -2709.25 q^{6} +2401.00 q^{7} -10275.9 q^{8} +35499.6 q^{9} +O(q^{10})\) \(q+11.5332 q^{2} -234.910 q^{3} -378.986 q^{4} -380.245 q^{5} -2709.25 q^{6} +2401.00 q^{7} -10275.9 q^{8} +35499.6 q^{9} -4385.43 q^{10} +9635.33 q^{11} +89027.6 q^{12} +28561.0 q^{13} +27691.1 q^{14} +89323.3 q^{15} +75527.8 q^{16} +464938. q^{17} +409422. q^{18} -346431. q^{19} +144108. q^{20} -564018. q^{21} +111126. q^{22} +1.34068e6 q^{23} +2.41390e6 q^{24} -1.80854e6 q^{25} +329398. q^{26} -3.71546e6 q^{27} -909946. q^{28} +3.44157e6 q^{29} +1.03018e6 q^{30} -555920. q^{31} +6.13232e6 q^{32} -2.26343e6 q^{33} +5.36220e6 q^{34} -912969. q^{35} -1.34539e7 q^{36} -2.25698e7 q^{37} -3.99544e6 q^{38} -6.70926e6 q^{39} +3.90736e6 q^{40} -1.05339e7 q^{41} -6.50491e6 q^{42} +1.25357e7 q^{43} -3.65166e6 q^{44} -1.34985e7 q^{45} +1.54622e7 q^{46} -5.29902e7 q^{47} -1.77422e7 q^{48} +5.76480e6 q^{49} -2.08581e7 q^{50} -1.09218e8 q^{51} -1.08242e7 q^{52} +2.37622e7 q^{53} -4.28510e7 q^{54} -3.66379e6 q^{55} -2.46724e7 q^{56} +8.13800e7 q^{57} +3.96921e7 q^{58} +1.37819e8 q^{59} -3.38523e7 q^{60} -1.62366e8 q^{61} -6.41151e6 q^{62} +8.52345e7 q^{63} +3.20548e7 q^{64} -1.08602e7 q^{65} -2.61045e7 q^{66} +1.66478e8 q^{67} -1.76205e8 q^{68} -3.14938e8 q^{69} -1.05294e7 q^{70} +1.90038e8 q^{71} -3.64789e8 q^{72} +2.33715e8 q^{73} -2.60301e8 q^{74} +4.24843e8 q^{75} +1.31293e8 q^{76} +2.31344e7 q^{77} -7.73789e7 q^{78} +2.38147e8 q^{79} -2.87191e7 q^{80} +1.74061e8 q^{81} -1.21489e8 q^{82} -5.70069e8 q^{83} +2.13755e8 q^{84} -1.76791e8 q^{85} +1.44576e8 q^{86} -8.08457e8 q^{87} -9.90115e7 q^{88} +3.40415e8 q^{89} -1.55681e8 q^{90} +6.85750e7 q^{91} -5.08098e8 q^{92} +1.30591e8 q^{93} -6.11144e8 q^{94} +1.31729e8 q^{95} -1.44054e9 q^{96} -3.80642e8 q^{97} +6.64863e7 q^{98} +3.42050e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.5332 0.509698 0.254849 0.966981i \(-0.417974\pi\)
0.254849 + 0.966981i \(0.417974\pi\)
\(3\) −234.910 −1.67438 −0.837192 0.546908i \(-0.815805\pi\)
−0.837192 + 0.546908i \(0.815805\pi\)
\(4\) −378.986 −0.740208
\(5\) −380.245 −0.272081 −0.136041 0.990703i \(-0.543438\pi\)
−0.136041 + 0.990703i \(0.543438\pi\)
\(6\) −2709.25 −0.853431
\(7\) 2401.00 0.377964
\(8\) −10275.9 −0.886981
\(9\) 35499.6 1.80356
\(10\) −4385.43 −0.138679
\(11\) 9635.33 0.198427 0.0992133 0.995066i \(-0.468367\pi\)
0.0992133 + 0.995066i \(0.468367\pi\)
\(12\) 89027.6 1.23939
\(13\) 28561.0 0.277350
\(14\) 27691.1 0.192648
\(15\) 89323.3 0.455569
\(16\) 75527.8 0.288116
\(17\) 464938. 1.35013 0.675064 0.737759i \(-0.264116\pi\)
0.675064 + 0.737759i \(0.264116\pi\)
\(18\) 409422. 0.919274
\(19\) −346431. −0.609853 −0.304927 0.952376i \(-0.598632\pi\)
−0.304927 + 0.952376i \(0.598632\pi\)
\(20\) 144108. 0.201397
\(21\) −564018. −0.632858
\(22\) 111126. 0.101138
\(23\) 1.34068e6 0.998961 0.499480 0.866325i \(-0.333524\pi\)
0.499480 + 0.866325i \(0.333524\pi\)
\(24\) 2.41390e6 1.48515
\(25\) −1.80854e6 −0.925972
\(26\) 329398. 0.141365
\(27\) −3.71546e6 −1.34548
\(28\) −909946. −0.279772
\(29\) 3.44157e6 0.903577 0.451788 0.892125i \(-0.350786\pi\)
0.451788 + 0.892125i \(0.350786\pi\)
\(30\) 1.03018e6 0.232203
\(31\) −555920. −0.108115 −0.0540573 0.998538i \(-0.517215\pi\)
−0.0540573 + 0.998538i \(0.517215\pi\)
\(32\) 6.13232e6 1.03383
\(33\) −2.26343e6 −0.332242
\(34\) 5.36220e6 0.688158
\(35\) −912969. −0.102837
\(36\) −1.34539e7 −1.33501
\(37\) −2.25698e7 −1.97980 −0.989898 0.141778i \(-0.954718\pi\)
−0.989898 + 0.141778i \(0.954718\pi\)
\(38\) −3.99544e6 −0.310841
\(39\) −6.70926e6 −0.464391
\(40\) 3.90736e6 0.241331
\(41\) −1.05339e7 −0.582184 −0.291092 0.956695i \(-0.594019\pi\)
−0.291092 + 0.956695i \(0.594019\pi\)
\(42\) −6.50491e6 −0.322567
\(43\) 1.25357e7 0.559166 0.279583 0.960122i \(-0.409804\pi\)
0.279583 + 0.960122i \(0.409804\pi\)
\(44\) −3.65166e6 −0.146877
\(45\) −1.34985e7 −0.490717
\(46\) 1.54622e7 0.509168
\(47\) −5.29902e7 −1.58400 −0.792001 0.610520i \(-0.790960\pi\)
−0.792001 + 0.610520i \(0.790960\pi\)
\(48\) −1.77422e7 −0.482416
\(49\) 5.76480e6 0.142857
\(50\) −2.08581e7 −0.471966
\(51\) −1.09218e8 −2.26063
\(52\) −1.08242e7 −0.205297
\(53\) 2.37622e7 0.413662 0.206831 0.978377i \(-0.433685\pi\)
0.206831 + 0.978377i \(0.433685\pi\)
\(54\) −4.28510e7 −0.685787
\(55\) −3.66379e6 −0.0539882
\(56\) −2.46724e7 −0.335247
\(57\) 8.13800e7 1.02113
\(58\) 3.96921e7 0.460551
\(59\) 1.37819e8 1.48073 0.740365 0.672205i \(-0.234652\pi\)
0.740365 + 0.672205i \(0.234652\pi\)
\(60\) −3.38523e7 −0.337216
\(61\) −1.62366e8 −1.50145 −0.750727 0.660612i \(-0.770297\pi\)
−0.750727 + 0.660612i \(0.770297\pi\)
\(62\) −6.41151e6 −0.0551058
\(63\) 8.52345e7 0.681683
\(64\) 3.20548e7 0.238827
\(65\) −1.08602e7 −0.0754618
\(66\) −2.61045e7 −0.169343
\(67\) 1.66478e8 1.00930 0.504650 0.863324i \(-0.331621\pi\)
0.504650 + 0.863324i \(0.331621\pi\)
\(68\) −1.76205e8 −0.999376
\(69\) −3.14938e8 −1.67264
\(70\) −1.05294e7 −0.0524159
\(71\) 1.90038e8 0.887521 0.443760 0.896145i \(-0.353644\pi\)
0.443760 + 0.896145i \(0.353644\pi\)
\(72\) −3.64789e8 −1.59973
\(73\) 2.33715e8 0.963239 0.481619 0.876380i \(-0.340049\pi\)
0.481619 + 0.876380i \(0.340049\pi\)
\(74\) −2.60301e8 −1.00910
\(75\) 4.24843e8 1.55043
\(76\) 1.31293e8 0.451418
\(77\) 2.31344e7 0.0749982
\(78\) −7.73789e7 −0.236699
\(79\) 2.38147e8 0.687897 0.343949 0.938988i \(-0.388235\pi\)
0.343949 + 0.938988i \(0.388235\pi\)
\(80\) −2.87191e7 −0.0783909
\(81\) 1.74061e8 0.449281
\(82\) −1.21489e8 −0.296738
\(83\) −5.70069e8 −1.31849 −0.659244 0.751929i \(-0.729123\pi\)
−0.659244 + 0.751929i \(0.729123\pi\)
\(84\) 2.13755e8 0.468446
\(85\) −1.76791e8 −0.367345
\(86\) 1.44576e8 0.285006
\(87\) −8.08457e8 −1.51294
\(88\) −9.90115e7 −0.176000
\(89\) 3.40415e8 0.575114 0.287557 0.957763i \(-0.407157\pi\)
0.287557 + 0.957763i \(0.407157\pi\)
\(90\) −1.55681e8 −0.250117
\(91\) 6.85750e7 0.104828
\(92\) −5.08098e8 −0.739439
\(93\) 1.30591e8 0.181026
\(94\) −6.11144e8 −0.807362
\(95\) 1.31729e8 0.165930
\(96\) −1.44054e9 −1.73103
\(97\) −3.80642e8 −0.436560 −0.218280 0.975886i \(-0.570045\pi\)
−0.218280 + 0.975886i \(0.570045\pi\)
\(98\) 6.64863e7 0.0728140
\(99\) 3.42050e8 0.357875
\(100\) 6.85412e8 0.685412
\(101\) −5.05267e8 −0.483142 −0.241571 0.970383i \(-0.577663\pi\)
−0.241571 + 0.970383i \(0.577663\pi\)
\(102\) −1.25963e9 −1.15224
\(103\) −2.10460e9 −1.84248 −0.921239 0.388998i \(-0.872821\pi\)
−0.921239 + 0.388998i \(0.872821\pi\)
\(104\) −2.93489e8 −0.246004
\(105\) 2.14465e8 0.172189
\(106\) 2.74054e8 0.210843
\(107\) −5.41253e8 −0.399184 −0.199592 0.979879i \(-0.563962\pi\)
−0.199592 + 0.979879i \(0.563962\pi\)
\(108\) 1.40811e9 0.995933
\(109\) −5.95006e8 −0.403740 −0.201870 0.979412i \(-0.564702\pi\)
−0.201870 + 0.979412i \(0.564702\pi\)
\(110\) −4.22551e7 −0.0275177
\(111\) 5.30187e9 3.31494
\(112\) 1.81342e8 0.108897
\(113\) −1.08103e9 −0.623711 −0.311855 0.950130i \(-0.600950\pi\)
−0.311855 + 0.950130i \(0.600950\pi\)
\(114\) 9.38567e8 0.520468
\(115\) −5.09786e8 −0.271799
\(116\) −1.30431e9 −0.668835
\(117\) 1.01390e9 0.500219
\(118\) 1.58949e9 0.754726
\(119\) 1.11632e9 0.510301
\(120\) −9.17876e8 −0.404081
\(121\) −2.26511e9 −0.960627
\(122\) −1.87260e9 −0.765288
\(123\) 2.47451e9 0.974801
\(124\) 2.10686e8 0.0800273
\(125\) 1.43036e9 0.524021
\(126\) 9.83022e8 0.347453
\(127\) 3.60564e9 1.22989 0.614943 0.788571i \(-0.289179\pi\)
0.614943 + 0.788571i \(0.289179\pi\)
\(128\) −2.77006e9 −0.912103
\(129\) −2.94476e9 −0.936259
\(130\) −1.25252e8 −0.0384627
\(131\) 4.25245e9 1.26159 0.630795 0.775949i \(-0.282729\pi\)
0.630795 + 0.775949i \(0.282729\pi\)
\(132\) 8.57811e8 0.245928
\(133\) −8.31780e8 −0.230503
\(134\) 1.92002e9 0.514439
\(135\) 1.41279e9 0.366079
\(136\) −4.77765e9 −1.19754
\(137\) 3.08378e9 0.747895 0.373948 0.927450i \(-0.378004\pi\)
0.373948 + 0.927450i \(0.378004\pi\)
\(138\) −3.63222e9 −0.852544
\(139\) −1.93335e9 −0.439283 −0.219641 0.975581i \(-0.570489\pi\)
−0.219641 + 0.975581i \(0.570489\pi\)
\(140\) 3.46003e8 0.0761208
\(141\) 1.24479e10 2.65223
\(142\) 2.19174e9 0.452368
\(143\) 2.75195e8 0.0550336
\(144\) 2.68120e9 0.519635
\(145\) −1.30864e9 −0.245846
\(146\) 2.69547e9 0.490961
\(147\) −1.35421e9 −0.239198
\(148\) 8.55366e9 1.46546
\(149\) −7.56328e8 −0.125711 −0.0628553 0.998023i \(-0.520021\pi\)
−0.0628553 + 0.998023i \(0.520021\pi\)
\(150\) 4.89978e9 0.790253
\(151\) 5.53056e9 0.865711 0.432856 0.901463i \(-0.357506\pi\)
0.432856 + 0.901463i \(0.357506\pi\)
\(152\) 3.55988e9 0.540928
\(153\) 1.65051e10 2.43504
\(154\) 2.66813e8 0.0382264
\(155\) 2.11386e8 0.0294160
\(156\) 2.54272e9 0.343746
\(157\) −9.78891e9 −1.28584 −0.642918 0.765935i \(-0.722276\pi\)
−0.642918 + 0.765935i \(0.722276\pi\)
\(158\) 2.74659e9 0.350620
\(159\) −5.58198e9 −0.692630
\(160\) −2.33179e9 −0.281287
\(161\) 3.21896e9 0.377572
\(162\) 2.00747e9 0.228998
\(163\) −6.91942e9 −0.767760 −0.383880 0.923383i \(-0.625412\pi\)
−0.383880 + 0.923383i \(0.625412\pi\)
\(164\) 3.99219e9 0.430938
\(165\) 8.60660e8 0.0903970
\(166\) −6.57469e9 −0.672031
\(167\) −1.62555e10 −1.61724 −0.808622 0.588328i \(-0.799786\pi\)
−0.808622 + 0.588328i \(0.799786\pi\)
\(168\) 5.79578e9 0.561333
\(169\) 8.15731e8 0.0769231
\(170\) −2.03895e9 −0.187235
\(171\) −1.22981e10 −1.09991
\(172\) −4.75086e9 −0.413899
\(173\) −6.18228e9 −0.524736 −0.262368 0.964968i \(-0.584503\pi\)
−0.262368 + 0.964968i \(0.584503\pi\)
\(174\) −9.32406e9 −0.771140
\(175\) −4.34230e9 −0.349984
\(176\) 7.27735e8 0.0571698
\(177\) −3.23751e10 −2.47931
\(178\) 3.92606e9 0.293135
\(179\) −1.46231e10 −1.06464 −0.532318 0.846545i \(-0.678679\pi\)
−0.532318 + 0.846545i \(0.678679\pi\)
\(180\) 5.11577e9 0.363232
\(181\) −1.24276e10 −0.860662 −0.430331 0.902671i \(-0.641603\pi\)
−0.430331 + 0.902671i \(0.641603\pi\)
\(182\) 7.90885e8 0.0534309
\(183\) 3.81415e10 2.51401
\(184\) −1.37766e10 −0.886059
\(185\) 8.58208e9 0.538666
\(186\) 1.50613e9 0.0922684
\(187\) 4.47983e9 0.267901
\(188\) 2.00826e10 1.17249
\(189\) −8.92083e9 −0.508542
\(190\) 1.51925e9 0.0845741
\(191\) 3.87058e9 0.210439 0.105219 0.994449i \(-0.466445\pi\)
0.105219 + 0.994449i \(0.466445\pi\)
\(192\) −7.52998e9 −0.399888
\(193\) 2.40332e9 0.124682 0.0623411 0.998055i \(-0.480143\pi\)
0.0623411 + 0.998055i \(0.480143\pi\)
\(194\) −4.39000e9 −0.222514
\(195\) 2.55116e9 0.126352
\(196\) −2.18478e9 −0.105744
\(197\) 9.77107e9 0.462215 0.231107 0.972928i \(-0.425765\pi\)
0.231107 + 0.972928i \(0.425765\pi\)
\(198\) 3.94492e9 0.182408
\(199\) −2.43479e10 −1.10058 −0.550292 0.834972i \(-0.685483\pi\)
−0.550292 + 0.834972i \(0.685483\pi\)
\(200\) 1.85843e10 0.821319
\(201\) −3.91073e10 −1.68996
\(202\) −5.82732e9 −0.246257
\(203\) 8.26320e9 0.341520
\(204\) 4.13923e10 1.67334
\(205\) 4.00546e9 0.158402
\(206\) −2.42727e10 −0.939107
\(207\) 4.75934e10 1.80169
\(208\) 2.15715e9 0.0799089
\(209\) −3.33798e9 −0.121011
\(210\) 2.47346e9 0.0877644
\(211\) −2.56316e10 −0.890235 −0.445117 0.895472i \(-0.646838\pi\)
−0.445117 + 0.895472i \(0.646838\pi\)
\(212\) −9.00557e9 −0.306196
\(213\) −4.46418e10 −1.48605
\(214\) −6.24235e9 −0.203463
\(215\) −4.76664e9 −0.152139
\(216\) 3.81797e10 1.19341
\(217\) −1.33476e9 −0.0408635
\(218\) −6.86229e9 −0.205786
\(219\) −5.49020e10 −1.61283
\(220\) 1.38853e9 0.0399625
\(221\) 1.32791e10 0.374458
\(222\) 6.11473e10 1.68962
\(223\) 2.99270e10 0.810384 0.405192 0.914232i \(-0.367205\pi\)
0.405192 + 0.914232i \(0.367205\pi\)
\(224\) 1.47237e10 0.390752
\(225\) −6.42023e10 −1.67005
\(226\) −1.24676e10 −0.317904
\(227\) −3.69918e10 −0.924675 −0.462338 0.886704i \(-0.652989\pi\)
−0.462338 + 0.886704i \(0.652989\pi\)
\(228\) −3.08419e10 −0.755848
\(229\) −5.13838e10 −1.23472 −0.617358 0.786683i \(-0.711797\pi\)
−0.617358 + 0.786683i \(0.711797\pi\)
\(230\) −5.87943e9 −0.138535
\(231\) −5.43450e9 −0.125576
\(232\) −3.53651e10 −0.801455
\(233\) 8.53089e10 1.89624 0.948119 0.317917i \(-0.102983\pi\)
0.948119 + 0.317917i \(0.102983\pi\)
\(234\) 1.16935e10 0.254961
\(235\) 2.01493e10 0.430977
\(236\) −5.22317e10 −1.09605
\(237\) −5.59431e10 −1.15181
\(238\) 1.28746e10 0.260099
\(239\) −3.61054e10 −0.715783 −0.357892 0.933763i \(-0.616504\pi\)
−0.357892 + 0.933763i \(0.616504\pi\)
\(240\) 6.74639e9 0.131257
\(241\) −6.77087e9 −0.129291 −0.0646454 0.997908i \(-0.520592\pi\)
−0.0646454 + 0.997908i \(0.520592\pi\)
\(242\) −2.61238e10 −0.489630
\(243\) 3.22429e10 0.593207
\(244\) 6.15347e10 1.11139
\(245\) −2.19204e9 −0.0388688
\(246\) 2.85389e10 0.496854
\(247\) −9.89441e9 −0.169143
\(248\) 5.71257e9 0.0958956
\(249\) 1.33915e11 2.20766
\(250\) 1.64965e10 0.267093
\(251\) −1.03841e11 −1.65134 −0.825671 0.564153i \(-0.809203\pi\)
−0.825671 + 0.564153i \(0.809203\pi\)
\(252\) −3.23027e10 −0.504587
\(253\) 1.29179e10 0.198220
\(254\) 4.15843e10 0.626871
\(255\) 4.15298e10 0.615077
\(256\) −4.83595e10 −0.703724
\(257\) −6.13242e10 −0.876865 −0.438432 0.898764i \(-0.644466\pi\)
−0.438432 + 0.898764i \(0.644466\pi\)
\(258\) −3.39623e10 −0.477209
\(259\) −5.41902e10 −0.748293
\(260\) 4.11586e9 0.0558574
\(261\) 1.22174e11 1.62966
\(262\) 4.90441e10 0.643030
\(263\) −2.47001e10 −0.318344 −0.159172 0.987251i \(-0.550882\pi\)
−0.159172 + 0.987251i \(0.550882\pi\)
\(264\) 2.32588e10 0.294693
\(265\) −9.03548e9 −0.112550
\(266\) −9.59305e9 −0.117487
\(267\) −7.99669e10 −0.962963
\(268\) −6.30930e10 −0.747092
\(269\) −1.31008e11 −1.52550 −0.762750 0.646694i \(-0.776151\pi\)
−0.762750 + 0.646694i \(0.776151\pi\)
\(270\) 1.62939e10 0.186590
\(271\) −7.02105e10 −0.790752 −0.395376 0.918519i \(-0.629386\pi\)
−0.395376 + 0.918519i \(0.629386\pi\)
\(272\) 3.51157e10 0.388993
\(273\) −1.61089e10 −0.175523
\(274\) 3.55657e10 0.381201
\(275\) −1.74259e10 −0.183737
\(276\) 1.19357e11 1.23810
\(277\) −6.84815e10 −0.698899 −0.349450 0.936955i \(-0.613631\pi\)
−0.349450 + 0.936955i \(0.613631\pi\)
\(278\) −2.22976e10 −0.223902
\(279\) −1.97349e10 −0.194992
\(280\) 9.38156e9 0.0912145
\(281\) 6.43019e10 0.615241 0.307620 0.951509i \(-0.400467\pi\)
0.307620 + 0.951509i \(0.400467\pi\)
\(282\) 1.43564e11 1.35184
\(283\) 1.73671e11 1.60949 0.804745 0.593621i \(-0.202302\pi\)
0.804745 + 0.593621i \(0.202302\pi\)
\(284\) −7.20219e10 −0.656950
\(285\) −3.09444e10 −0.277830
\(286\) 3.17386e9 0.0280505
\(287\) −2.52918e10 −0.220045
\(288\) 2.17695e11 1.86458
\(289\) 9.75796e10 0.822846
\(290\) −1.50927e10 −0.125307
\(291\) 8.94164e10 0.730969
\(292\) −8.85749e10 −0.712997
\(293\) −1.92848e10 −0.152866 −0.0764329 0.997075i \(-0.524353\pi\)
−0.0764329 + 0.997075i \(0.524353\pi\)
\(294\) −1.56183e10 −0.121919
\(295\) −5.24052e10 −0.402879
\(296\) 2.31925e11 1.75604
\(297\) −3.57997e10 −0.266978
\(298\) −8.72285e9 −0.0640745
\(299\) 3.82910e10 0.277062
\(300\) −1.61010e11 −1.14764
\(301\) 3.00982e10 0.211345
\(302\) 6.37848e10 0.441251
\(303\) 1.18692e11 0.808966
\(304\) −2.61652e10 −0.175708
\(305\) 6.17391e10 0.408518
\(306\) 1.90356e11 1.24114
\(307\) 1.16564e11 0.748932 0.374466 0.927241i \(-0.377826\pi\)
0.374466 + 0.927241i \(0.377826\pi\)
\(308\) −8.76764e9 −0.0555142
\(309\) 4.94391e11 3.08502
\(310\) 2.43795e9 0.0149933
\(311\) 2.58691e11 1.56805 0.784024 0.620731i \(-0.213164\pi\)
0.784024 + 0.620731i \(0.213164\pi\)
\(312\) 6.89435e10 0.411906
\(313\) −8.35496e10 −0.492033 −0.246017 0.969266i \(-0.579122\pi\)
−0.246017 + 0.969266i \(0.579122\pi\)
\(314\) −1.12897e11 −0.655388
\(315\) −3.24100e10 −0.185473
\(316\) −9.02546e10 −0.509187
\(317\) −2.59507e11 −1.44338 −0.721692 0.692215i \(-0.756635\pi\)
−0.721692 + 0.692215i \(0.756635\pi\)
\(318\) −6.43778e10 −0.353032
\(319\) 3.31606e10 0.179294
\(320\) −1.21887e10 −0.0649804
\(321\) 1.27146e11 0.668388
\(322\) 3.71248e10 0.192448
\(323\) −1.61069e11 −0.823380
\(324\) −6.59667e10 −0.332562
\(325\) −5.16537e10 −0.256818
\(326\) −7.98027e10 −0.391326
\(327\) 1.39773e11 0.676017
\(328\) 1.08245e11 0.516386
\(329\) −1.27230e11 −0.598696
\(330\) 9.92612e9 0.0460752
\(331\) 1.11042e11 0.508466 0.254233 0.967143i \(-0.418177\pi\)
0.254233 + 0.967143i \(0.418177\pi\)
\(332\) 2.16048e11 0.975955
\(333\) −8.01219e11 −3.57069
\(334\) −1.87477e11 −0.824307
\(335\) −6.33025e10 −0.274612
\(336\) −4.25990e10 −0.182336
\(337\) −1.56179e11 −0.659613 −0.329807 0.944049i \(-0.606984\pi\)
−0.329807 + 0.944049i \(0.606984\pi\)
\(338\) 9.40795e9 0.0392075
\(339\) 2.53943e11 1.04433
\(340\) 6.70012e10 0.271912
\(341\) −5.35647e9 −0.0214528
\(342\) −1.41836e11 −0.560622
\(343\) 1.38413e10 0.0539949
\(344\) −1.28815e11 −0.495969
\(345\) 1.19754e11 0.455096
\(346\) −7.13011e10 −0.267457
\(347\) 1.62017e11 0.599900 0.299950 0.953955i \(-0.403030\pi\)
0.299950 + 0.953955i \(0.403030\pi\)
\(348\) 3.06394e11 1.11989
\(349\) −5.00058e11 −1.80429 −0.902144 0.431435i \(-0.858007\pi\)
−0.902144 + 0.431435i \(0.858007\pi\)
\(350\) −5.00804e10 −0.178386
\(351\) −1.06117e11 −0.373168
\(352\) 5.90870e10 0.205140
\(353\) 1.88578e11 0.646406 0.323203 0.946330i \(-0.395240\pi\)
0.323203 + 0.946330i \(0.395240\pi\)
\(354\) −3.73387e11 −1.26370
\(355\) −7.22612e10 −0.241478
\(356\) −1.29013e11 −0.425704
\(357\) −2.62234e11 −0.854440
\(358\) −1.68650e11 −0.542643
\(359\) −1.00227e11 −0.318462 −0.159231 0.987241i \(-0.550901\pi\)
−0.159231 + 0.987241i \(0.550901\pi\)
\(360\) 1.38709e11 0.435256
\(361\) −2.02673e11 −0.628079
\(362\) −1.43329e11 −0.438678
\(363\) 5.32096e11 1.60846
\(364\) −2.59890e10 −0.0775949
\(365\) −8.88691e10 −0.262079
\(366\) 4.39891e11 1.28139
\(367\) 3.34161e11 0.961520 0.480760 0.876852i \(-0.340361\pi\)
0.480760 + 0.876852i \(0.340361\pi\)
\(368\) 1.01258e11 0.287816
\(369\) −3.73948e11 −1.05001
\(370\) 9.89784e10 0.274557
\(371\) 5.70531e10 0.156350
\(372\) −4.94922e10 −0.133997
\(373\) −5.45099e11 −1.45809 −0.729047 0.684464i \(-0.760036\pi\)
−0.729047 + 0.684464i \(0.760036\pi\)
\(374\) 5.16666e10 0.136549
\(375\) −3.36004e11 −0.877413
\(376\) 5.44521e11 1.40498
\(377\) 9.82946e10 0.250607
\(378\) −1.02885e11 −0.259203
\(379\) 1.20175e11 0.299184 0.149592 0.988748i \(-0.452204\pi\)
0.149592 + 0.988748i \(0.452204\pi\)
\(380\) −4.99234e10 −0.122823
\(381\) −8.46999e11 −2.05930
\(382\) 4.46400e10 0.107260
\(383\) 8.46764e10 0.201079 0.100540 0.994933i \(-0.467943\pi\)
0.100540 + 0.994933i \(0.467943\pi\)
\(384\) 6.50713e11 1.52721
\(385\) −8.79676e9 −0.0204056
\(386\) 2.77179e10 0.0635503
\(387\) 4.45012e11 1.00849
\(388\) 1.44258e11 0.323145
\(389\) −4.74270e11 −1.05015 −0.525077 0.851055i \(-0.675963\pi\)
−0.525077 + 0.851055i \(0.675963\pi\)
\(390\) 2.94230e10 0.0644014
\(391\) 6.23331e11 1.34872
\(392\) −5.92384e10 −0.126712
\(393\) −9.98942e11 −2.11239
\(394\) 1.12691e11 0.235590
\(395\) −9.05544e10 −0.187164
\(396\) −1.29632e11 −0.264902
\(397\) −3.13547e11 −0.633498 −0.316749 0.948509i \(-0.602591\pi\)
−0.316749 + 0.948509i \(0.602591\pi\)
\(398\) −2.80808e11 −0.560965
\(399\) 1.95393e11 0.385951
\(400\) −1.36595e11 −0.266787
\(401\) 5.34078e11 1.03147 0.515733 0.856749i \(-0.327519\pi\)
0.515733 + 0.856749i \(0.327519\pi\)
\(402\) −4.51031e11 −0.861368
\(403\) −1.58776e10 −0.0299856
\(404\) 1.91489e11 0.357625
\(405\) −6.61858e10 −0.122241
\(406\) 9.53007e10 0.174072
\(407\) −2.17468e11 −0.392844
\(408\) 1.12232e12 2.00514
\(409\) 6.14246e11 1.08539 0.542697 0.839929i \(-0.317403\pi\)
0.542697 + 0.839929i \(0.317403\pi\)
\(410\) 4.61955e10 0.0807370
\(411\) −7.24410e11 −1.25226
\(412\) 7.97615e11 1.36382
\(413\) 3.30904e11 0.559664
\(414\) 5.48902e11 0.918318
\(415\) 2.16766e11 0.358736
\(416\) 1.75145e11 0.286734
\(417\) 4.54163e11 0.735529
\(418\) −3.84974e10 −0.0616791
\(419\) 7.53195e11 1.19384 0.596918 0.802303i \(-0.296392\pi\)
0.596918 + 0.802303i \(0.296392\pi\)
\(420\) −8.12794e10 −0.127456
\(421\) 8.27712e11 1.28413 0.642066 0.766649i \(-0.278078\pi\)
0.642066 + 0.766649i \(0.278078\pi\)
\(422\) −2.95613e11 −0.453751
\(423\) −1.88113e12 −2.85685
\(424\) −2.44178e11 −0.366910
\(425\) −8.40859e11 −1.25018
\(426\) −5.14861e11 −0.757438
\(427\) −3.89842e11 −0.567496
\(428\) 2.05127e11 0.295479
\(429\) −6.46459e10 −0.0921474
\(430\) −5.49744e10 −0.0775448
\(431\) −3.37497e11 −0.471110 −0.235555 0.971861i \(-0.575691\pi\)
−0.235555 + 0.971861i \(0.575691\pi\)
\(432\) −2.80621e11 −0.387653
\(433\) −3.17581e11 −0.434169 −0.217085 0.976153i \(-0.569655\pi\)
−0.217085 + 0.976153i \(0.569655\pi\)
\(434\) −1.53940e10 −0.0208280
\(435\) 3.07412e11 0.411642
\(436\) 2.25499e11 0.298852
\(437\) −4.64451e11 −0.609219
\(438\) −6.33193e11 −0.822058
\(439\) −3.19897e11 −0.411074 −0.205537 0.978649i \(-0.565894\pi\)
−0.205537 + 0.978649i \(0.565894\pi\)
\(440\) 3.76487e10 0.0478865
\(441\) 2.04648e11 0.257652
\(442\) 1.53150e11 0.190861
\(443\) 1.22294e12 1.50865 0.754324 0.656502i \(-0.227965\pi\)
0.754324 + 0.656502i \(0.227965\pi\)
\(444\) −2.00934e12 −2.45375
\(445\) −1.29441e11 −0.156478
\(446\) 3.45152e11 0.413051
\(447\) 1.77669e11 0.210488
\(448\) 7.69636e10 0.0902681
\(449\) −9.72481e11 −1.12920 −0.564602 0.825363i \(-0.690970\pi\)
−0.564602 + 0.825363i \(0.690970\pi\)
\(450\) −7.40455e11 −0.851221
\(451\) −1.01497e11 −0.115521
\(452\) 4.09694e11 0.461675
\(453\) −1.29918e12 −1.44953
\(454\) −4.26632e11 −0.471305
\(455\) −2.60753e10 −0.0285219
\(456\) −8.36251e11 −0.905722
\(457\) −3.56791e10 −0.0382640 −0.0191320 0.999817i \(-0.506090\pi\)
−0.0191320 + 0.999817i \(0.506090\pi\)
\(458\) −5.92618e11 −0.629332
\(459\) −1.72746e12 −1.81657
\(460\) 1.93202e11 0.201188
\(461\) 4.05615e10 0.0418273 0.0209136 0.999781i \(-0.493342\pi\)
0.0209136 + 0.999781i \(0.493342\pi\)
\(462\) −6.26769e10 −0.0640057
\(463\) 9.38401e11 0.949017 0.474508 0.880251i \(-0.342626\pi\)
0.474508 + 0.880251i \(0.342626\pi\)
\(464\) 2.59934e11 0.260335
\(465\) −4.96566e10 −0.0492537
\(466\) 9.83880e11 0.966508
\(467\) −1.79799e11 −0.174928 −0.0874642 0.996168i \(-0.527876\pi\)
−0.0874642 + 0.996168i \(0.527876\pi\)
\(468\) −3.84256e11 −0.370266
\(469\) 3.99714e11 0.381480
\(470\) 2.32385e11 0.219668
\(471\) 2.29951e12 2.15299
\(472\) −1.41622e12 −1.31338
\(473\) 1.20786e11 0.110953
\(474\) −6.45200e11 −0.587073
\(475\) 6.26534e11 0.564707
\(476\) −4.23069e11 −0.377728
\(477\) 8.43549e11 0.746067
\(478\) −4.16409e11 −0.364833
\(479\) 2.16628e11 0.188021 0.0940103 0.995571i \(-0.470031\pi\)
0.0940103 + 0.995571i \(0.470031\pi\)
\(480\) 5.47760e11 0.470982
\(481\) −6.44617e11 −0.549097
\(482\) −7.80894e10 −0.0658992
\(483\) −7.56165e11 −0.632200
\(484\) 8.58445e11 0.711064
\(485\) 1.44737e11 0.118780
\(486\) 3.71863e11 0.302356
\(487\) 1.84903e12 1.48958 0.744790 0.667299i \(-0.232550\pi\)
0.744790 + 0.667299i \(0.232550\pi\)
\(488\) 1.66846e12 1.33176
\(489\) 1.62544e12 1.28553
\(490\) −2.52811e10 −0.0198113
\(491\) −4.15346e10 −0.0322510 −0.0161255 0.999870i \(-0.505133\pi\)
−0.0161255 + 0.999870i \(0.505133\pi\)
\(492\) −9.37805e11 −0.721555
\(493\) 1.60012e12 1.21994
\(494\) −1.14114e11 −0.0862118
\(495\) −1.30063e11 −0.0973712
\(496\) −4.19874e10 −0.0311495
\(497\) 4.56282e11 0.335451
\(498\) 1.54446e12 1.12524
\(499\) 1.80627e12 1.30416 0.652079 0.758151i \(-0.273897\pi\)
0.652079 + 0.758151i \(0.273897\pi\)
\(500\) −5.42085e11 −0.387885
\(501\) 3.81857e12 2.70789
\(502\) −1.19761e12 −0.841685
\(503\) 2.03742e12 1.41914 0.709568 0.704637i \(-0.248890\pi\)
0.709568 + 0.704637i \(0.248890\pi\)
\(504\) −8.75859e11 −0.604640
\(505\) 1.92125e11 0.131454
\(506\) 1.48984e11 0.101032
\(507\) −1.91623e11 −0.128799
\(508\) −1.36649e12 −0.910372
\(509\) −2.13776e12 −1.41166 −0.705829 0.708382i \(-0.749425\pi\)
−0.705829 + 0.708382i \(0.749425\pi\)
\(510\) 4.78970e11 0.313503
\(511\) 5.61150e11 0.364070
\(512\) 8.60531e11 0.553416
\(513\) 1.28715e12 0.820543
\(514\) −7.07261e11 −0.446936
\(515\) 8.00265e11 0.501304
\(516\) 1.11602e12 0.693026
\(517\) −5.10579e11 −0.314308
\(518\) −6.24983e11 −0.381403
\(519\) 1.45228e12 0.878610
\(520\) 1.11598e11 0.0669332
\(521\) −2.62432e12 −1.56044 −0.780220 0.625505i \(-0.784893\pi\)
−0.780220 + 0.625505i \(0.784893\pi\)
\(522\) 1.40905e12 0.830634
\(523\) −1.92752e12 −1.12653 −0.563263 0.826278i \(-0.690454\pi\)
−0.563263 + 0.826278i \(0.690454\pi\)
\(524\) −1.61162e12 −0.933839
\(525\) 1.02005e12 0.586009
\(526\) −2.84869e11 −0.162259
\(527\) −2.58468e11 −0.145969
\(528\) −1.70952e11 −0.0957242
\(529\) −3.74228e9 −0.00207772
\(530\) −1.04208e11 −0.0573664
\(531\) 4.89253e12 2.67059
\(532\) 3.15234e11 0.170620
\(533\) −3.00858e11 −0.161469
\(534\) −9.22270e11 −0.490820
\(535\) 2.05809e11 0.108611
\(536\) −1.71071e12 −0.895230
\(537\) 3.43511e12 1.78261
\(538\) −1.51093e12 −0.777544
\(539\) 5.55458e10 0.0283466
\(540\) −5.35428e11 −0.270975
\(541\) 3.85220e12 1.93340 0.966699 0.255918i \(-0.0823777\pi\)
0.966699 + 0.255918i \(0.0823777\pi\)
\(542\) −8.09749e11 −0.403045
\(543\) 2.91936e12 1.44108
\(544\) 2.85115e12 1.39581
\(545\) 2.26248e11 0.109850
\(546\) −1.85787e11 −0.0894639
\(547\) 6.08818e11 0.290767 0.145383 0.989375i \(-0.453558\pi\)
0.145383 + 0.989375i \(0.453558\pi\)
\(548\) −1.16871e12 −0.553598
\(549\) −5.76394e12 −2.70797
\(550\) −2.00975e11 −0.0936506
\(551\) −1.19226e12 −0.551049
\(552\) 3.23626e12 1.48360
\(553\) 5.71792e11 0.260001
\(554\) −7.89808e11 −0.356228
\(555\) −2.01601e12 −0.901934
\(556\) 7.32714e11 0.325161
\(557\) 1.06887e12 0.470518 0.235259 0.971933i \(-0.424406\pi\)
0.235259 + 0.971933i \(0.424406\pi\)
\(558\) −2.27606e11 −0.0993870
\(559\) 3.58032e11 0.155085
\(560\) −6.89545e10 −0.0296290
\(561\) −1.05236e12 −0.448570
\(562\) 7.41603e11 0.313587
\(563\) −8.63203e11 −0.362097 −0.181049 0.983474i \(-0.557949\pi\)
−0.181049 + 0.983474i \(0.557949\pi\)
\(564\) −4.71759e12 −1.96320
\(565\) 4.11055e11 0.169700
\(566\) 2.00297e12 0.820354
\(567\) 4.17920e11 0.169812
\(568\) −1.95281e12 −0.787214
\(569\) 7.35633e11 0.294209 0.147105 0.989121i \(-0.453005\pi\)
0.147105 + 0.989121i \(0.453005\pi\)
\(570\) −3.56886e11 −0.141610
\(571\) −3.20450e12 −1.26153 −0.630766 0.775973i \(-0.717259\pi\)
−0.630766 + 0.775973i \(0.717259\pi\)
\(572\) −1.04295e11 −0.0407363
\(573\) −9.09237e11 −0.352356
\(574\) −2.91694e11 −0.112157
\(575\) −2.42466e12 −0.925009
\(576\) 1.13793e12 0.430740
\(577\) −1.15728e12 −0.434658 −0.217329 0.976098i \(-0.569734\pi\)
−0.217329 + 0.976098i \(0.569734\pi\)
\(578\) 1.12540e12 0.419403
\(579\) −5.64564e11 −0.208766
\(580\) 4.95957e11 0.181977
\(581\) −1.36874e12 −0.498341
\(582\) 1.03125e12 0.372573
\(583\) 2.28957e11 0.0820816
\(584\) −2.40163e12 −0.854374
\(585\) −3.85532e11 −0.136100
\(586\) −2.22414e11 −0.0779154
\(587\) −1.58237e12 −0.550093 −0.275047 0.961431i \(-0.588693\pi\)
−0.275047 + 0.961431i \(0.588693\pi\)
\(588\) 5.13226e11 0.177056
\(589\) 1.92588e11 0.0659341
\(590\) −6.04397e11 −0.205347
\(591\) −2.29532e12 −0.773926
\(592\) −1.70465e12 −0.570410
\(593\) 5.05810e12 1.67974 0.839869 0.542789i \(-0.182632\pi\)
0.839869 + 0.542789i \(0.182632\pi\)
\(594\) −4.12884e11 −0.136078
\(595\) −4.24474e11 −0.138843
\(596\) 2.86638e11 0.0930520
\(597\) 5.71956e12 1.84280
\(598\) 4.41616e11 0.141218
\(599\) −4.38369e12 −1.39129 −0.695647 0.718384i \(-0.744882\pi\)
−0.695647 + 0.718384i \(0.744882\pi\)
\(600\) −4.36564e12 −1.37520
\(601\) −4.24767e12 −1.32806 −0.664028 0.747708i \(-0.731154\pi\)
−0.664028 + 0.747708i \(0.731154\pi\)
\(602\) 3.47127e11 0.107722
\(603\) 5.90990e12 1.82034
\(604\) −2.09601e12 −0.640806
\(605\) 8.61297e11 0.261369
\(606\) 1.36889e12 0.412328
\(607\) −2.38694e12 −0.713662 −0.356831 0.934169i \(-0.616143\pi\)
−0.356831 + 0.934169i \(0.616143\pi\)
\(608\) −2.12443e12 −0.630486
\(609\) −1.94111e12 −0.571836
\(610\) 7.12046e11 0.208221
\(611\) −1.51345e12 −0.439323
\(612\) −6.25521e12 −1.80244
\(613\) −1.38296e12 −0.395583 −0.197792 0.980244i \(-0.563377\pi\)
−0.197792 + 0.980244i \(0.563377\pi\)
\(614\) 1.34435e12 0.381729
\(615\) −9.40920e11 −0.265225
\(616\) −2.37727e11 −0.0665219
\(617\) −5.67379e12 −1.57612 −0.788061 0.615597i \(-0.788915\pi\)
−0.788061 + 0.615597i \(0.788915\pi\)
\(618\) 5.70189e12 1.57243
\(619\) −2.52272e12 −0.690654 −0.345327 0.938482i \(-0.612232\pi\)
−0.345327 + 0.938482i \(0.612232\pi\)
\(620\) −8.01124e10 −0.0217740
\(621\) −4.98123e12 −1.34408
\(622\) 2.98352e12 0.799231
\(623\) 8.17338e11 0.217373
\(624\) −5.06735e11 −0.133798
\(625\) 2.98842e12 0.783395
\(626\) −9.63590e11 −0.250789
\(627\) 7.84123e11 0.202619
\(628\) 3.70986e12 0.951786
\(629\) −1.04936e13 −2.67298
\(630\) −3.73790e11 −0.0945354
\(631\) −2.46725e12 −0.619556 −0.309778 0.950809i \(-0.600255\pi\)
−0.309778 + 0.950809i \(0.600255\pi\)
\(632\) −2.44717e12 −0.610152
\(633\) 6.02111e12 1.49060
\(634\) −2.99293e12 −0.735690
\(635\) −1.37103e12 −0.334629
\(636\) 2.11549e12 0.512690
\(637\) 1.64648e11 0.0396214
\(638\) 3.82447e11 0.0913856
\(639\) 6.74628e12 1.60070
\(640\) 1.05330e12 0.248166
\(641\) 1.29084e12 0.302004 0.151002 0.988533i \(-0.451750\pi\)
0.151002 + 0.988533i \(0.451750\pi\)
\(642\) 1.46639e12 0.340676
\(643\) 6.53843e12 1.50843 0.754213 0.656629i \(-0.228018\pi\)
0.754213 + 0.656629i \(0.228018\pi\)
\(644\) −1.21994e12 −0.279481
\(645\) 1.11973e12 0.254739
\(646\) −1.85763e12 −0.419675
\(647\) −4.04993e12 −0.908611 −0.454306 0.890846i \(-0.650113\pi\)
−0.454306 + 0.890846i \(0.650113\pi\)
\(648\) −1.78863e12 −0.398504
\(649\) 1.32794e12 0.293816
\(650\) −5.95730e11 −0.130900
\(651\) 3.13549e11 0.0684212
\(652\) 2.62237e12 0.568302
\(653\) −6.08688e12 −1.31004 −0.655021 0.755611i \(-0.727340\pi\)
−0.655021 + 0.755611i \(0.727340\pi\)
\(654\) 1.61202e12 0.344564
\(655\) −1.61697e12 −0.343255
\(656\) −7.95600e11 −0.167736
\(657\) 8.29679e12 1.73726
\(658\) −1.46736e12 −0.305154
\(659\) −7.83895e12 −1.61910 −0.809550 0.587051i \(-0.800289\pi\)
−0.809550 + 0.587051i \(0.800289\pi\)
\(660\) −3.26179e11 −0.0669126
\(661\) 7.69873e12 1.56860 0.784300 0.620382i \(-0.213022\pi\)
0.784300 + 0.620382i \(0.213022\pi\)
\(662\) 1.28067e12 0.259164
\(663\) −3.11939e12 −0.626987
\(664\) 5.85796e12 1.16947
\(665\) 3.16281e11 0.0627156
\(666\) −9.24058e12 −1.81997
\(667\) 4.61402e12 0.902638
\(668\) 6.16061e12 1.19710
\(669\) −7.03013e12 −1.35689
\(670\) −7.30078e11 −0.139969
\(671\) −1.56446e12 −0.297928
\(672\) −3.45874e12 −0.654269
\(673\) 7.27917e12 1.36777 0.683886 0.729589i \(-0.260288\pi\)
0.683886 + 0.729589i \(0.260288\pi\)
\(674\) −1.80124e12 −0.336204
\(675\) 6.71956e12 1.24587
\(676\) −3.09151e11 −0.0569391
\(677\) −5.97563e12 −1.09329 −0.546645 0.837365i \(-0.684095\pi\)
−0.546645 + 0.837365i \(0.684095\pi\)
\(678\) 2.92877e12 0.532294
\(679\) −9.13921e11 −0.165004
\(680\) 1.81668e12 0.325828
\(681\) 8.68973e12 1.54826
\(682\) −6.17770e10 −0.0109345
\(683\) −5.32965e12 −0.937142 −0.468571 0.883426i \(-0.655231\pi\)
−0.468571 + 0.883426i \(0.655231\pi\)
\(684\) 4.66083e12 0.814162
\(685\) −1.17259e12 −0.203488
\(686\) 1.59634e11 0.0275211
\(687\) 1.20706e13 2.06739
\(688\) 9.46794e11 0.161104
\(689\) 6.78673e11 0.114729
\(690\) 1.38114e12 0.231961
\(691\) −6.28767e12 −1.04915 −0.524576 0.851363i \(-0.675776\pi\)
−0.524576 + 0.851363i \(0.675776\pi\)
\(692\) 2.34300e12 0.388414
\(693\) 8.21262e11 0.135264
\(694\) 1.86857e12 0.305768
\(695\) 7.35148e11 0.119521
\(696\) 8.30761e12 1.34194
\(697\) −4.89760e12 −0.786024
\(698\) −5.76724e12 −0.919642
\(699\) −2.00399e13 −3.17503
\(700\) 1.64567e12 0.259061
\(701\) 1.64564e12 0.257398 0.128699 0.991684i \(-0.458920\pi\)
0.128699 + 0.991684i \(0.458920\pi\)
\(702\) −1.22387e12 −0.190203
\(703\) 7.81889e12 1.20739
\(704\) 3.08859e11 0.0473896
\(705\) −4.73326e12 −0.721622
\(706\) 2.17490e12 0.329472
\(707\) −1.21315e12 −0.182610
\(708\) 1.22697e13 1.83521
\(709\) −9.72322e12 −1.44511 −0.722557 0.691311i \(-0.757033\pi\)
−0.722557 + 0.691311i \(0.757033\pi\)
\(710\) −8.33399e11 −0.123081
\(711\) 8.45412e12 1.24067
\(712\) −3.49807e12 −0.510115
\(713\) −7.45308e11 −0.108002
\(714\) −3.02438e12 −0.435506
\(715\) −1.04642e11 −0.0149736
\(716\) 5.54196e12 0.788052
\(717\) 8.48151e12 1.19850
\(718\) −1.15593e12 −0.162320
\(719\) −5.77433e11 −0.0805790 −0.0402895 0.999188i \(-0.512828\pi\)
−0.0402895 + 0.999188i \(0.512828\pi\)
\(720\) −1.01952e12 −0.141383
\(721\) −5.05315e12 −0.696391
\(722\) −2.33746e12 −0.320131
\(723\) 1.59054e12 0.216482
\(724\) 4.70988e12 0.637069
\(725\) −6.22420e12 −0.836686
\(726\) 6.13674e12 0.819829
\(727\) 8.82668e12 1.17190 0.585952 0.810345i \(-0.300721\pi\)
0.585952 + 0.810345i \(0.300721\pi\)
\(728\) −7.04668e11 −0.0929808
\(729\) −1.10002e13 −1.44254
\(730\) −1.02494e12 −0.133581
\(731\) 5.82833e12 0.754946
\(732\) −1.44551e13 −1.86089
\(733\) 6.59469e12 0.843775 0.421887 0.906648i \(-0.361368\pi\)
0.421887 + 0.906648i \(0.361368\pi\)
\(734\) 3.85393e12 0.490085
\(735\) 5.14931e11 0.0650813
\(736\) 8.22146e12 1.03276
\(737\) 1.60407e12 0.200272
\(738\) −4.31280e12 −0.535187
\(739\) 1.52438e13 1.88016 0.940078 0.340960i \(-0.110752\pi\)
0.940078 + 0.340960i \(0.110752\pi\)
\(740\) −3.25249e12 −0.398725
\(741\) 2.32429e12 0.283210
\(742\) 6.58002e11 0.0796911
\(743\) −1.34907e13 −1.62400 −0.811999 0.583659i \(-0.801621\pi\)
−0.811999 + 0.583659i \(0.801621\pi\)
\(744\) −1.34194e12 −0.160566
\(745\) 2.87590e11 0.0342035
\(746\) −6.28671e12 −0.743188
\(747\) −2.02372e13 −2.37798
\(748\) −1.69780e12 −0.198303
\(749\) −1.29955e12 −0.150877
\(750\) −3.87519e12 −0.447216
\(751\) 1.18476e13 1.35909 0.679547 0.733632i \(-0.262176\pi\)
0.679547 + 0.733632i \(0.262176\pi\)
\(752\) −4.00223e12 −0.456375
\(753\) 2.43932e13 2.76498
\(754\) 1.13365e12 0.127734
\(755\) −2.10297e12 −0.235544
\(756\) 3.38087e12 0.376427
\(757\) −4.74039e12 −0.524665 −0.262333 0.964978i \(-0.584492\pi\)
−0.262333 + 0.964978i \(0.584492\pi\)
\(758\) 1.38600e12 0.152493
\(759\) −3.03453e12 −0.331897
\(760\) −1.35363e12 −0.147176
\(761\) 1.49535e11 0.0161626 0.00808130 0.999967i \(-0.497428\pi\)
0.00808130 + 0.999967i \(0.497428\pi\)
\(762\) −9.76857e12 −1.04962
\(763\) −1.42861e12 −0.152600
\(764\) −1.46690e12 −0.155769
\(765\) −6.27599e12 −0.662530
\(766\) 9.76585e11 0.102490
\(767\) 3.93626e12 0.410681
\(768\) 1.13601e13 1.17830
\(769\) −4.50550e12 −0.464595 −0.232298 0.972645i \(-0.574624\pi\)
−0.232298 + 0.972645i \(0.574624\pi\)
\(770\) −1.01454e11 −0.0104007
\(771\) 1.44056e13 1.46821
\(772\) −9.10827e11 −0.0922907
\(773\) −1.22201e12 −0.123103 −0.0615513 0.998104i \(-0.519605\pi\)
−0.0615513 + 0.998104i \(0.519605\pi\)
\(774\) 5.13239e12 0.514026
\(775\) 1.00540e12 0.100111
\(776\) 3.91143e12 0.387220
\(777\) 1.27298e13 1.25293
\(778\) −5.46983e12 −0.535261
\(779\) 3.64926e12 0.355047
\(780\) −9.66856e11 −0.0935268
\(781\) 1.83108e12 0.176108
\(782\) 7.18897e12 0.687443
\(783\) −1.27870e13 −1.21574
\(784\) 4.35403e11 0.0411594
\(785\) 3.72219e12 0.349852
\(786\) −1.15209e13 −1.07668
\(787\) 1.68549e13 1.56617 0.783086 0.621913i \(-0.213644\pi\)
0.783086 + 0.621913i \(0.213644\pi\)
\(788\) −3.70310e12 −0.342135
\(789\) 5.80228e12 0.533031
\(790\) −1.04438e12 −0.0953972
\(791\) −2.59554e12 −0.235740
\(792\) −3.51487e12 −0.317428
\(793\) −4.63735e12 −0.416428
\(794\) −3.61619e12 −0.322893
\(795\) 2.12252e12 0.188452
\(796\) 9.22753e12 0.814661
\(797\) −1.65939e13 −1.45675 −0.728377 0.685177i \(-0.759725\pi\)
−0.728377 + 0.685177i \(0.759725\pi\)
\(798\) 2.25350e12 0.196718
\(799\) −2.46372e13 −2.13860
\(800\) −1.10905e13 −0.957300
\(801\) 1.20846e13 1.03726
\(802\) 6.15961e12 0.525737
\(803\) 2.25192e12 0.191132
\(804\) 1.48211e13 1.25092
\(805\) −1.22400e12 −0.102730
\(806\) −1.83119e11 −0.0152836
\(807\) 3.07750e13 2.55427
\(808\) 5.19206e12 0.428538
\(809\) 2.62948e12 0.215825 0.107912 0.994160i \(-0.465583\pi\)
0.107912 + 0.994160i \(0.465583\pi\)
\(810\) −7.63331e11 −0.0623061
\(811\) −7.91920e12 −0.642817 −0.321409 0.946941i \(-0.604156\pi\)
−0.321409 + 0.946941i \(0.604156\pi\)
\(812\) −3.13164e12 −0.252796
\(813\) 1.64931e13 1.32402
\(814\) −2.50809e12 −0.200232
\(815\) 2.63108e12 0.208893
\(816\) −8.24903e12 −0.651324
\(817\) −4.34275e12 −0.341009
\(818\) 7.08419e12 0.553223
\(819\) 2.43438e12 0.189065
\(820\) −1.51801e12 −0.117250
\(821\) −8.59319e12 −0.660101 −0.330050 0.943963i \(-0.607066\pi\)
−0.330050 + 0.943963i \(0.607066\pi\)
\(822\) −8.35473e12 −0.638277
\(823\) −2.02642e13 −1.53968 −0.769840 0.638237i \(-0.779664\pi\)
−0.769840 + 0.638237i \(0.779664\pi\)
\(824\) 2.16266e13 1.63424
\(825\) 4.09351e12 0.307647
\(826\) 3.81637e12 0.285259
\(827\) −3.46046e11 −0.0257252 −0.0128626 0.999917i \(-0.504094\pi\)
−0.0128626 + 0.999917i \(0.504094\pi\)
\(828\) −1.80373e13 −1.33363
\(829\) 6.85280e12 0.503933 0.251966 0.967736i \(-0.418923\pi\)
0.251966 + 0.967736i \(0.418923\pi\)
\(830\) 2.50000e12 0.182847
\(831\) 1.60870e13 1.17023
\(832\) 9.15517e11 0.0662387
\(833\) 2.68028e12 0.192875
\(834\) 5.23793e12 0.374898
\(835\) 6.18107e12 0.440022
\(836\) 1.26505e12 0.0895733
\(837\) 2.06550e12 0.145466
\(838\) 8.68671e12 0.608496
\(839\) 2.18455e13 1.52206 0.761031 0.648715i \(-0.224693\pi\)
0.761031 + 0.648715i \(0.224693\pi\)
\(840\) −2.20382e12 −0.152728
\(841\) −2.66277e12 −0.183549
\(842\) 9.54612e12 0.654520
\(843\) −1.51051e13 −1.03015
\(844\) 9.71403e12 0.658959
\(845\) −3.10178e11 −0.0209293
\(846\) −2.16954e13 −1.45613
\(847\) −5.43852e12 −0.363083
\(848\) 1.79471e12 0.119183
\(849\) −4.07970e13 −2.69491
\(850\) −9.69775e12 −0.637215
\(851\) −3.02588e13 −1.97774
\(852\) 1.69187e13 1.09999
\(853\) 2.07810e13 1.34399 0.671993 0.740557i \(-0.265438\pi\)
0.671993 + 0.740557i \(0.265438\pi\)
\(854\) −4.49611e12 −0.289252
\(855\) 4.67631e12 0.299265
\(856\) 5.56185e12 0.354068
\(857\) −1.14535e13 −0.725310 −0.362655 0.931923i \(-0.618130\pi\)
−0.362655 + 0.931923i \(0.618130\pi\)
\(858\) −7.45571e11 −0.0469674
\(859\) −2.56086e13 −1.60478 −0.802392 0.596798i \(-0.796439\pi\)
−0.802392 + 0.596798i \(0.796439\pi\)
\(860\) 1.80649e12 0.112614
\(861\) 5.94129e12 0.368440
\(862\) −3.89240e12 −0.240124
\(863\) 2.66648e12 0.163640 0.0818199 0.996647i \(-0.473927\pi\)
0.0818199 + 0.996647i \(0.473927\pi\)
\(864\) −2.27844e13 −1.39100
\(865\) 2.35078e12 0.142771
\(866\) −3.66271e12 −0.221295
\(867\) −2.29224e13 −1.37776
\(868\) 5.05857e11 0.0302475
\(869\) 2.29463e12 0.136497
\(870\) 3.54543e12 0.209813
\(871\) 4.75478e12 0.279930
\(872\) 6.11421e12 0.358110
\(873\) −1.35126e13 −0.787364
\(874\) −5.35659e12 −0.310518
\(875\) 3.43428e12 0.198061
\(876\) 2.08071e13 1.19383
\(877\) −3.06283e13 −1.74833 −0.874166 0.485627i \(-0.838591\pi\)
−0.874166 + 0.485627i \(0.838591\pi\)
\(878\) −3.68942e12 −0.209524
\(879\) 4.53018e12 0.255956
\(880\) −2.76718e11 −0.0155548
\(881\) 1.45437e13 0.813364 0.406682 0.913570i \(-0.366686\pi\)
0.406682 + 0.913570i \(0.366686\pi\)
\(882\) 2.36024e12 0.131325
\(883\) −1.47021e13 −0.813873 −0.406937 0.913456i \(-0.633403\pi\)
−0.406937 + 0.913456i \(0.633403\pi\)
\(884\) −5.03260e12 −0.277177
\(885\) 1.23105e13 0.674575
\(886\) 1.41043e13 0.768955
\(887\) −5.08614e12 −0.275887 −0.137944 0.990440i \(-0.544049\pi\)
−0.137944 + 0.990440i \(0.544049\pi\)
\(888\) −5.44814e13 −2.94029
\(889\) 8.65713e12 0.464854
\(890\) −1.49287e12 −0.0797565
\(891\) 1.67713e12 0.0891493
\(892\) −1.13419e13 −0.599852
\(893\) 1.83575e13 0.966008
\(894\) 2.04908e12 0.107285
\(895\) 5.56037e12 0.289668
\(896\) −6.65091e12 −0.344743
\(897\) −8.99493e12 −0.463908
\(898\) −1.12158e13 −0.575554
\(899\) −1.91323e12 −0.0976899
\(900\) 2.43318e13 1.23618
\(901\) 1.10480e13 0.558497
\(902\) −1.17058e12 −0.0588807
\(903\) −7.07037e12 −0.353873
\(904\) 1.11085e13 0.553219
\(905\) 4.72553e12 0.234170
\(906\) −1.49837e13 −0.738825
\(907\) 3.47131e13 1.70318 0.851591 0.524207i \(-0.175638\pi\)
0.851591 + 0.524207i \(0.175638\pi\)
\(908\) 1.40194e13 0.684452
\(909\) −1.79368e13 −0.871378
\(910\) −3.00731e11 −0.0145376
\(911\) 3.18746e13 1.53325 0.766623 0.642098i \(-0.221936\pi\)
0.766623 + 0.642098i \(0.221936\pi\)
\(912\) 6.14645e12 0.294203
\(913\) −5.49281e12 −0.261623
\(914\) −4.11492e11 −0.0195031
\(915\) −1.45031e13 −0.684016
\(916\) 1.94738e13 0.913946
\(917\) 1.02101e13 0.476836
\(918\) −1.99231e13 −0.925900
\(919\) −3.81830e13 −1.76584 −0.882918 0.469528i \(-0.844424\pi\)
−0.882918 + 0.469528i \(0.844424\pi\)
\(920\) 5.23850e12 0.241080
\(921\) −2.73821e13 −1.25400
\(922\) 4.67802e11 0.0213193
\(923\) 5.42768e12 0.246154
\(924\) 2.05960e12 0.0929522
\(925\) 4.08184e13 1.83324
\(926\) 1.08227e13 0.483712
\(927\) −7.47124e13 −3.32303
\(928\) 2.11048e13 0.934147
\(929\) −1.69518e13 −0.746699 −0.373350 0.927691i \(-0.621791\pi\)
−0.373350 + 0.927691i \(0.621791\pi\)
\(930\) −5.72697e11 −0.0251045
\(931\) −1.99710e12 −0.0871219
\(932\) −3.23309e13 −1.40361
\(933\) −6.07690e13 −2.62552
\(934\) −2.07364e12 −0.0891606
\(935\) −1.70344e12 −0.0728910
\(936\) −1.04187e13 −0.443684
\(937\) −3.11021e13 −1.31814 −0.659069 0.752082i \(-0.729049\pi\)
−0.659069 + 0.752082i \(0.729049\pi\)
\(938\) 4.60996e12 0.194440
\(939\) 1.96266e13 0.823853
\(940\) −7.63631e12 −0.319013
\(941\) −7.48321e10 −0.00311125 −0.00155562 0.999999i \(-0.500495\pi\)
−0.00155562 + 0.999999i \(0.500495\pi\)
\(942\) 2.65206e13 1.09737
\(943\) −1.41225e13 −0.581579
\(944\) 1.04092e13 0.426622
\(945\) 3.39210e12 0.138365
\(946\) 1.39304e12 0.0565527
\(947\) −3.04222e13 −1.22918 −0.614590 0.788847i \(-0.710678\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(948\) 2.12017e13 0.852575
\(949\) 6.67514e12 0.267154
\(950\) 7.22591e12 0.287830
\(951\) 6.09606e13 2.41678
\(952\) −1.14711e13 −0.452627
\(953\) −1.20445e13 −0.473009 −0.236504 0.971630i \(-0.576002\pi\)
−0.236504 + 0.971630i \(0.576002\pi\)
\(954\) 9.72878e12 0.380269
\(955\) −1.47177e12 −0.0572565
\(956\) 1.36835e13 0.529829
\(957\) −7.78975e12 −0.300206
\(958\) 2.49841e12 0.0958337
\(959\) 7.40416e12 0.282678
\(960\) 2.86324e12 0.108802
\(961\) −2.61306e13 −0.988311
\(962\) −7.43447e12 −0.279874
\(963\) −1.92142e13 −0.719954
\(964\) 2.56607e12 0.0957020
\(965\) −9.13853e11 −0.0339237
\(966\) −8.72097e12 −0.322231
\(967\) 4.54088e12 0.167002 0.0835009 0.996508i \(-0.473390\pi\)
0.0835009 + 0.996508i \(0.473390\pi\)
\(968\) 2.32760e13 0.852057
\(969\) 3.78367e13 1.37866
\(970\) 1.66928e12 0.0605418
\(971\) −1.86719e13 −0.674064 −0.337032 0.941493i \(-0.609423\pi\)
−0.337032 + 0.941493i \(0.609423\pi\)
\(972\) −1.22196e13 −0.439097
\(973\) −4.64198e12 −0.166033
\(974\) 2.13251e13 0.759236
\(975\) 1.21339e13 0.430013
\(976\) −1.22632e13 −0.432592
\(977\) 3.14964e13 1.10595 0.552975 0.833198i \(-0.313493\pi\)
0.552975 + 0.833198i \(0.313493\pi\)
\(978\) 1.87464e13 0.655230
\(979\) 3.28002e12 0.114118
\(980\) 8.30753e11 0.0287710
\(981\) −2.11225e13 −0.728172
\(982\) −4.79025e11 −0.0164383
\(983\) 3.49452e12 0.119370 0.0596852 0.998217i \(-0.480990\pi\)
0.0596852 + 0.998217i \(0.480990\pi\)
\(984\) −2.54278e13 −0.864629
\(985\) −3.71540e12 −0.125760
\(986\) 1.84544e13 0.621803
\(987\) 2.98875e13 1.00245
\(988\) 3.74985e12 0.125201
\(989\) 1.68063e13 0.558585
\(990\) −1.50004e12 −0.0496299
\(991\) −2.75726e13 −0.908126 −0.454063 0.890970i \(-0.650026\pi\)
−0.454063 + 0.890970i \(0.650026\pi\)
\(992\) −3.40908e12 −0.111772
\(993\) −2.60849e13 −0.851367
\(994\) 5.26237e12 0.170979
\(995\) 9.25818e12 0.299448
\(996\) −5.07519e13 −1.63412
\(997\) 6.81005e12 0.218284 0.109142 0.994026i \(-0.465190\pi\)
0.109142 + 0.994026i \(0.465190\pi\)
\(998\) 2.08320e13 0.664727
\(999\) 8.38574e13 2.66377
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 91.10.a.a.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.8 12 1.1 even 1 trivial