Properties

Label 91.8.a.c.1.4
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-9.51136\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-11.5114 q^{2} +23.9058 q^{3} +4.51150 q^{4} -301.396 q^{5} -275.188 q^{6} +343.000 q^{7} +1421.52 q^{8} -1615.51 q^{9} +O(q^{10})\) \(q-11.5114 q^{2} +23.9058 q^{3} +4.51150 q^{4} -301.396 q^{5} -275.188 q^{6} +343.000 q^{7} +1421.52 q^{8} -1615.51 q^{9} +3469.48 q^{10} +8165.11 q^{11} +107.851 q^{12} -2197.00 q^{13} -3948.40 q^{14} -7205.13 q^{15} -16941.1 q^{16} -18445.8 q^{17} +18596.7 q^{18} +25451.2 q^{19} -1359.75 q^{20} +8199.69 q^{21} -93991.6 q^{22} +100507. q^{23} +33982.6 q^{24} +12714.8 q^{25} +25290.5 q^{26} -90902.1 q^{27} +1547.44 q^{28} +1639.36 q^{29} +82940.8 q^{30} -183210. q^{31} +13060.7 q^{32} +195194. q^{33} +212337. q^{34} -103379. q^{35} -7288.38 q^{36} +564340. q^{37} -292978. q^{38} -52521.1 q^{39} -428441. q^{40} -738171. q^{41} -94389.6 q^{42} -759297. q^{43} +36836.9 q^{44} +486910. q^{45} -1.15698e6 q^{46} -1.10773e6 q^{47} -404991. q^{48} +117649. q^{49} -146365. q^{50} -440963. q^{51} -9911.76 q^{52} -822620. q^{53} +1.04641e6 q^{54} -2.46094e6 q^{55} +487582. q^{56} +608432. q^{57} -18871.3 q^{58} -768137. q^{59} -32505.9 q^{60} -1.66731e6 q^{61} +2.10900e6 q^{62} -554121. q^{63} +2.01812e6 q^{64} +662168. q^{65} -2.24694e6 q^{66} +1.69335e6 q^{67} -83218.4 q^{68} +2.40271e6 q^{69} +1.19003e6 q^{70} -3.19149e6 q^{71} -2.29648e6 q^{72} -1.83029e6 q^{73} -6.49632e6 q^{74} +303958. q^{75} +114823. q^{76} +2.80063e6 q^{77} +604589. q^{78} +4.85216e6 q^{79} +5.10599e6 q^{80} +1.36004e6 q^{81} +8.49735e6 q^{82} -3.84497e6 q^{83} +36992.9 q^{84} +5.55951e6 q^{85} +8.74055e6 q^{86} +39190.2 q^{87} +1.16069e7 q^{88} -2.37126e6 q^{89} -5.60499e6 q^{90} -753571. q^{91} +453439. q^{92} -4.37979e6 q^{93} +1.27515e7 q^{94} -7.67091e6 q^{95} +312226. q^{96} -4.74338e6 q^{97} -1.35430e6 q^{98} -1.31908e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.5114 −1.01747 −0.508735 0.860923i \(-0.669887\pi\)
−0.508735 + 0.860923i \(0.669887\pi\)
\(3\) 23.9058 0.511186 0.255593 0.966784i \(-0.417729\pi\)
0.255593 + 0.966784i \(0.417729\pi\)
\(4\) 4.51150 0.0352461
\(5\) −301.396 −1.07831 −0.539154 0.842207i \(-0.681256\pi\)
−0.539154 + 0.842207i \(0.681256\pi\)
\(6\) −275.188 −0.520117
\(7\) 343.000 0.377964
\(8\) 1421.52 0.981609
\(9\) −1615.51 −0.738689
\(10\) 3469.48 1.09715
\(11\) 8165.11 1.84964 0.924821 0.380403i \(-0.124215\pi\)
0.924821 + 0.380403i \(0.124215\pi\)
\(12\) 107.851 0.0180173
\(13\) −2197.00 −0.277350
\(14\) −3948.40 −0.384568
\(15\) −7205.13 −0.551217
\(16\) −16941.1 −1.03400
\(17\) −18445.8 −0.910599 −0.455299 0.890338i \(-0.650468\pi\)
−0.455299 + 0.890338i \(0.650468\pi\)
\(18\) 18596.7 0.751594
\(19\) 25451.2 0.851278 0.425639 0.904893i \(-0.360049\pi\)
0.425639 + 0.904893i \(0.360049\pi\)
\(20\) −1359.75 −0.0380062
\(21\) 8199.69 0.193210
\(22\) −93991.6 −1.88196
\(23\) 100507. 1.72246 0.861232 0.508212i \(-0.169693\pi\)
0.861232 + 0.508212i \(0.169693\pi\)
\(24\) 33982.6 0.501785
\(25\) 12714.8 0.162750
\(26\) 25290.5 0.282196
\(27\) −90902.1 −0.888794
\(28\) 1547.44 0.0133218
\(29\) 1639.36 0.0124819 0.00624096 0.999981i \(-0.498013\pi\)
0.00624096 + 0.999981i \(0.498013\pi\)
\(30\) 82940.8 0.560847
\(31\) −183210. −1.10455 −0.552274 0.833663i \(-0.686240\pi\)
−0.552274 + 0.833663i \(0.686240\pi\)
\(32\) 13060.7 0.0704597
\(33\) 195194. 0.945511
\(34\) 212337. 0.926508
\(35\) −103379. −0.407562
\(36\) −7288.38 −0.0260359
\(37\) 564340. 1.83162 0.915808 0.401615i \(-0.131551\pi\)
0.915808 + 0.401615i \(0.131551\pi\)
\(38\) −292978. −0.866150
\(39\) −52521.1 −0.141778
\(40\) −428441. −1.05848
\(41\) −738171. −1.67268 −0.836341 0.548209i \(-0.815310\pi\)
−0.836341 + 0.548209i \(0.815310\pi\)
\(42\) −94389.6 −0.196586
\(43\) −759297. −1.45637 −0.728186 0.685380i \(-0.759636\pi\)
−0.728186 + 0.685380i \(0.759636\pi\)
\(44\) 36836.9 0.0651926
\(45\) 486910. 0.796535
\(46\) −1.15698e6 −1.75256
\(47\) −1.10773e6 −1.55630 −0.778150 0.628078i \(-0.783842\pi\)
−0.778150 + 0.628078i \(0.783842\pi\)
\(48\) −404991. −0.528568
\(49\) 117649. 0.142857
\(50\) −146365. −0.165593
\(51\) −440963. −0.465486
\(52\) −9911.76 −0.00977551
\(53\) −822620. −0.758986 −0.379493 0.925195i \(-0.623901\pi\)
−0.379493 + 0.925195i \(0.623901\pi\)
\(54\) 1.04641e6 0.904321
\(55\) −2.46094e6 −1.99449
\(56\) 487582. 0.371013
\(57\) 608432. 0.435161
\(58\) −18871.3 −0.0127000
\(59\) −768137. −0.486919 −0.243459 0.969911i \(-0.578282\pi\)
−0.243459 + 0.969911i \(0.578282\pi\)
\(60\) −32505.9 −0.0194282
\(61\) −1.66731e6 −0.940508 −0.470254 0.882531i \(-0.655838\pi\)
−0.470254 + 0.882531i \(0.655838\pi\)
\(62\) 2.10900e6 1.12384
\(63\) −554121. −0.279198
\(64\) 2.01812e6 0.962313
\(65\) 662168. 0.299069
\(66\) −2.24694e6 −0.962030
\(67\) 1.69335e6 0.687835 0.343917 0.939000i \(-0.388246\pi\)
0.343917 + 0.939000i \(0.388246\pi\)
\(68\) −83218.4 −0.0320951
\(69\) 2.40271e6 0.880500
\(70\) 1.19003e6 0.414683
\(71\) −3.19149e6 −1.05825 −0.529127 0.848543i \(-0.677480\pi\)
−0.529127 + 0.848543i \(0.677480\pi\)
\(72\) −2.29648e6 −0.725103
\(73\) −1.83029e6 −0.550668 −0.275334 0.961349i \(-0.588789\pi\)
−0.275334 + 0.961349i \(0.588789\pi\)
\(74\) −6.49632e6 −1.86362
\(75\) 303958. 0.0831955
\(76\) 114823. 0.0300042
\(77\) 2.80063e6 0.699099
\(78\) 604589. 0.144254
\(79\) 4.85216e6 1.10724 0.553618 0.832771i \(-0.313247\pi\)
0.553618 + 0.832771i \(0.313247\pi\)
\(80\) 5.10599e6 1.11498
\(81\) 1.36004e6 0.284350
\(82\) 8.49735e6 1.70190
\(83\) −3.84497e6 −0.738107 −0.369053 0.929408i \(-0.620318\pi\)
−0.369053 + 0.929408i \(0.620318\pi\)
\(84\) 36992.9 0.00680990
\(85\) 5.55951e6 0.981907
\(86\) 8.74055e6 1.48182
\(87\) 39190.2 0.00638058
\(88\) 1.16069e7 1.81562
\(89\) −2.37126e6 −0.356545 −0.178272 0.983981i \(-0.557051\pi\)
−0.178272 + 0.983981i \(0.557051\pi\)
\(90\) −5.60499e6 −0.810450
\(91\) −753571. −0.104828
\(92\) 453439. 0.0607101
\(93\) −4.37979e6 −0.564629
\(94\) 1.27515e7 1.58349
\(95\) −7.67091e6 −0.917940
\(96\) 312226. 0.0360180
\(97\) −4.74338e6 −0.527700 −0.263850 0.964564i \(-0.584992\pi\)
−0.263850 + 0.964564i \(0.584992\pi\)
\(98\) −1.35430e6 −0.145353
\(99\) −1.31908e7 −1.36631
\(100\) 57363.0 0.00573630
\(101\) 8.69794e6 0.840024 0.420012 0.907519i \(-0.362026\pi\)
0.420012 + 0.907519i \(0.362026\pi\)
\(102\) 5.07608e6 0.473618
\(103\) 5.63467e6 0.508087 0.254043 0.967193i \(-0.418239\pi\)
0.254043 + 0.967193i \(0.418239\pi\)
\(104\) −3.12308e6 −0.272249
\(105\) −2.47136e6 −0.208340
\(106\) 9.46947e6 0.772245
\(107\) −3.33224e6 −0.262962 −0.131481 0.991319i \(-0.541973\pi\)
−0.131481 + 0.991319i \(0.541973\pi\)
\(108\) −410105. −0.0313265
\(109\) 181248. 0.0134054 0.00670270 0.999978i \(-0.497866\pi\)
0.00670270 + 0.999978i \(0.497866\pi\)
\(110\) 2.83287e7 2.02933
\(111\) 1.34910e7 0.936297
\(112\) −5.81080e6 −0.390817
\(113\) −1.78210e7 −1.16187 −0.580935 0.813950i \(-0.697313\pi\)
−0.580935 + 0.813950i \(0.697313\pi\)
\(114\) −7.00389e6 −0.442764
\(115\) −3.02925e7 −1.85735
\(116\) 7395.97 0.000439939 0
\(117\) 3.54928e6 0.204875
\(118\) 8.84231e6 0.495426
\(119\) −6.32692e6 −0.344174
\(120\) −1.02422e7 −0.541079
\(121\) 4.71819e7 2.42118
\(122\) 1.91930e7 0.956939
\(123\) −1.76466e7 −0.855052
\(124\) −826554. −0.0389310
\(125\) 1.97144e7 0.902814
\(126\) 6.37869e6 0.284076
\(127\) −4.53740e7 −1.96560 −0.982798 0.184685i \(-0.940874\pi\)
−0.982798 + 0.184685i \(0.940874\pi\)
\(128\) −2.49030e7 −1.04958
\(129\) −1.81516e7 −0.744477
\(130\) −7.62246e6 −0.304294
\(131\) −4.50612e6 −0.175127 −0.0875636 0.996159i \(-0.527908\pi\)
−0.0875636 + 0.996159i \(0.527908\pi\)
\(132\) 880616. 0.0333256
\(133\) 8.72977e6 0.321753
\(134\) −1.94927e7 −0.699852
\(135\) 2.73976e7 0.958394
\(136\) −2.62211e7 −0.893852
\(137\) −2.00557e7 −0.666372 −0.333186 0.942861i \(-0.608124\pi\)
−0.333186 + 0.942861i \(0.608124\pi\)
\(138\) −2.76585e7 −0.895883
\(139\) 5.63022e7 1.77817 0.889085 0.457742i \(-0.151342\pi\)
0.889085 + 0.457742i \(0.151342\pi\)
\(140\) −466394. −0.0143650
\(141\) −2.64813e7 −0.795559
\(142\) 3.67384e7 1.07674
\(143\) −1.79387e7 −0.512998
\(144\) 2.73686e7 0.763807
\(145\) −494097. −0.0134594
\(146\) 2.10691e7 0.560289
\(147\) 2.81249e6 0.0730266
\(148\) 2.54602e6 0.0645573
\(149\) −1.35095e7 −0.334571 −0.167285 0.985909i \(-0.553500\pi\)
−0.167285 + 0.985909i \(0.553500\pi\)
\(150\) −3.49898e6 −0.0846489
\(151\) −4.15725e7 −0.982624 −0.491312 0.870984i \(-0.663482\pi\)
−0.491312 + 0.870984i \(0.663482\pi\)
\(152\) 3.61795e7 0.835621
\(153\) 2.97995e7 0.672649
\(154\) −3.22391e7 −0.711313
\(155\) 5.52190e7 1.19104
\(156\) −236949. −0.00499710
\(157\) 5.56692e7 1.14806 0.574032 0.818833i \(-0.305378\pi\)
0.574032 + 0.818833i \(0.305378\pi\)
\(158\) −5.58550e7 −1.12658
\(159\) −1.96654e7 −0.387983
\(160\) −3.93644e6 −0.0759773
\(161\) 3.44740e7 0.651030
\(162\) −1.56559e7 −0.289317
\(163\) −1.25415e7 −0.226826 −0.113413 0.993548i \(-0.536178\pi\)
−0.113413 + 0.993548i \(0.536178\pi\)
\(164\) −3.33026e6 −0.0589555
\(165\) −5.88307e7 −1.01955
\(166\) 4.42608e7 0.751002
\(167\) 4.76687e7 0.792001 0.396000 0.918250i \(-0.370398\pi\)
0.396000 + 0.918250i \(0.370398\pi\)
\(168\) 1.16560e7 0.189657
\(169\) 4.82681e6 0.0769231
\(170\) −6.39975e7 −0.999061
\(171\) −4.11168e7 −0.628829
\(172\) −3.42557e6 −0.0513314
\(173\) −9.74049e7 −1.43028 −0.715138 0.698984i \(-0.753636\pi\)
−0.715138 + 0.698984i \(0.753636\pi\)
\(174\) −451133. −0.00649206
\(175\) 4.36119e6 0.0615137
\(176\) −1.38326e8 −1.91254
\(177\) −1.83629e7 −0.248906
\(178\) 2.72964e7 0.362774
\(179\) 1.24140e8 1.61781 0.808905 0.587939i \(-0.200061\pi\)
0.808905 + 0.587939i \(0.200061\pi\)
\(180\) 2.19669e6 0.0280747
\(181\) −5.09706e7 −0.638918 −0.319459 0.947600i \(-0.603501\pi\)
−0.319459 + 0.947600i \(0.603501\pi\)
\(182\) 8.67463e6 0.106660
\(183\) −3.98585e7 −0.480775
\(184\) 1.42873e8 1.69079
\(185\) −1.70090e8 −1.97505
\(186\) 5.04174e7 0.574494
\(187\) −1.50612e8 −1.68428
\(188\) −4.99755e6 −0.0548535
\(189\) −3.11794e7 −0.335932
\(190\) 8.83027e7 0.933977
\(191\) −8.85715e7 −0.919766 −0.459883 0.887979i \(-0.652109\pi\)
−0.459883 + 0.887979i \(0.652109\pi\)
\(192\) 4.82447e7 0.491921
\(193\) 1.60849e7 0.161053 0.0805264 0.996752i \(-0.474340\pi\)
0.0805264 + 0.996752i \(0.474340\pi\)
\(194\) 5.46028e7 0.536919
\(195\) 1.58297e7 0.152880
\(196\) 530773. 0.00503516
\(197\) −7.18559e7 −0.669624 −0.334812 0.942285i \(-0.608673\pi\)
−0.334812 + 0.942285i \(0.608673\pi\)
\(198\) 1.51845e8 1.39018
\(199\) −1.69138e8 −1.52144 −0.760721 0.649079i \(-0.775154\pi\)
−0.760721 + 0.649079i \(0.775154\pi\)
\(200\) 1.80744e7 0.159757
\(201\) 4.04808e7 0.351612
\(202\) −1.00125e8 −0.854699
\(203\) 562300. 0.00471772
\(204\) −1.98940e6 −0.0164065
\(205\) 2.22482e8 1.80367
\(206\) −6.48627e7 −0.516963
\(207\) −1.62371e8 −1.27236
\(208\) 3.72196e7 0.286781
\(209\) 2.07812e8 1.57456
\(210\) 2.84487e7 0.211980
\(211\) −2.36714e7 −0.173474 −0.0867372 0.996231i \(-0.527644\pi\)
−0.0867372 + 0.996231i \(0.527644\pi\)
\(212\) −3.71125e6 −0.0267513
\(213\) −7.62952e7 −0.540965
\(214\) 3.83587e7 0.267556
\(215\) 2.28850e8 1.57042
\(216\) −1.29219e8 −0.872447
\(217\) −6.28412e7 −0.417480
\(218\) −2.08641e6 −0.0136396
\(219\) −4.37546e7 −0.281494
\(220\) −1.11025e7 −0.0702978
\(221\) 4.05255e7 0.252555
\(222\) −1.55300e8 −0.952655
\(223\) −2.27490e8 −1.37371 −0.686855 0.726795i \(-0.741009\pi\)
−0.686855 + 0.726795i \(0.741009\pi\)
\(224\) 4.47981e6 0.0266313
\(225\) −2.05410e7 −0.120221
\(226\) 2.05144e8 1.18217
\(227\) 1.57710e8 0.894886 0.447443 0.894313i \(-0.352335\pi\)
0.447443 + 0.894313i \(0.352335\pi\)
\(228\) 2.74494e6 0.0153377
\(229\) 1.97476e8 1.08665 0.543325 0.839522i \(-0.317165\pi\)
0.543325 + 0.839522i \(0.317165\pi\)
\(230\) 3.48709e8 1.88980
\(231\) 6.69514e7 0.357370
\(232\) 2.33038e6 0.0122524
\(233\) −8.59413e7 −0.445099 −0.222549 0.974921i \(-0.571438\pi\)
−0.222549 + 0.974921i \(0.571438\pi\)
\(234\) −4.08571e7 −0.208455
\(235\) 3.33867e8 1.67817
\(236\) −3.46545e6 −0.0171620
\(237\) 1.15995e8 0.566004
\(238\) 7.28315e7 0.350187
\(239\) −2.62303e8 −1.24283 −0.621413 0.783483i \(-0.713441\pi\)
−0.621413 + 0.783483i \(0.713441\pi\)
\(240\) 1.22063e8 0.569960
\(241\) −1.78206e7 −0.0820090 −0.0410045 0.999159i \(-0.513056\pi\)
−0.0410045 + 0.999159i \(0.513056\pi\)
\(242\) −5.43128e8 −2.46347
\(243\) 2.31316e8 1.03415
\(244\) −7.52208e6 −0.0331492
\(245\) −3.54590e7 −0.154044
\(246\) 2.03136e8 0.869990
\(247\) −5.59164e7 −0.236102
\(248\) −2.60438e8 −1.08423
\(249\) −9.19170e7 −0.377310
\(250\) −2.26940e8 −0.918587
\(251\) −3.88360e8 −1.55016 −0.775079 0.631864i \(-0.782290\pi\)
−0.775079 + 0.631864i \(0.782290\pi\)
\(252\) −2.49992e6 −0.00984064
\(253\) 8.20653e8 3.18594
\(254\) 5.22317e8 1.99994
\(255\) 1.32905e8 0.501937
\(256\) 2.83491e7 0.105608
\(257\) −1.76175e8 −0.647410 −0.323705 0.946158i \(-0.604929\pi\)
−0.323705 + 0.946158i \(0.604929\pi\)
\(258\) 2.08950e8 0.757484
\(259\) 1.93569e8 0.692286
\(260\) 2.98737e6 0.0105410
\(261\) −2.64841e6 −0.00922025
\(262\) 5.18716e7 0.178187
\(263\) −2.45839e8 −0.833309 −0.416654 0.909065i \(-0.636797\pi\)
−0.416654 + 0.909065i \(0.636797\pi\)
\(264\) 2.77472e8 0.928122
\(265\) 2.47935e8 0.818421
\(266\) −1.00492e8 −0.327374
\(267\) −5.66868e7 −0.182261
\(268\) 7.63953e6 0.0242435
\(269\) −3.98799e8 −1.24917 −0.624584 0.780957i \(-0.714732\pi\)
−0.624584 + 0.780957i \(0.714732\pi\)
\(270\) −3.15384e8 −0.975137
\(271\) −2.09254e8 −0.638678 −0.319339 0.947641i \(-0.603461\pi\)
−0.319339 + 0.947641i \(0.603461\pi\)
\(272\) 3.12493e8 0.941563
\(273\) −1.80147e7 −0.0535869
\(274\) 2.30869e8 0.678013
\(275\) 1.03818e8 0.301029
\(276\) 1.08398e7 0.0310342
\(277\) 1.29079e8 0.364902 0.182451 0.983215i \(-0.441597\pi\)
0.182451 + 0.983215i \(0.441597\pi\)
\(278\) −6.48115e8 −1.80924
\(279\) 2.95979e8 0.815917
\(280\) −1.46955e8 −0.400067
\(281\) −2.90525e8 −0.781109 −0.390555 0.920580i \(-0.627717\pi\)
−0.390555 + 0.920580i \(0.627717\pi\)
\(282\) 3.04836e8 0.809458
\(283\) 2.86057e8 0.750241 0.375120 0.926976i \(-0.377601\pi\)
0.375120 + 0.926976i \(0.377601\pi\)
\(284\) −1.43984e7 −0.0372993
\(285\) −1.83379e8 −0.469238
\(286\) 2.06499e8 0.521961
\(287\) −2.53193e8 −0.632214
\(288\) −2.10997e7 −0.0520478
\(289\) −7.00897e7 −0.170809
\(290\) 5.68773e6 0.0136945
\(291\) −1.13394e8 −0.269753
\(292\) −8.25736e6 −0.0194089
\(293\) 5.16778e8 1.20024 0.600118 0.799911i \(-0.295120\pi\)
0.600118 + 0.799911i \(0.295120\pi\)
\(294\) −3.23756e7 −0.0743024
\(295\) 2.31514e8 0.525049
\(296\) 8.02221e8 1.79793
\(297\) −7.42226e8 −1.64395
\(298\) 1.55513e8 0.340416
\(299\) −2.20815e8 −0.477726
\(300\) 1.37131e6 0.00293231
\(301\) −2.60439e8 −0.550457
\(302\) 4.78557e8 0.999791
\(303\) 2.07931e8 0.429408
\(304\) −4.31172e8 −0.880224
\(305\) 5.02522e8 1.01416
\(306\) −3.43033e8 −0.684401
\(307\) 1.57322e8 0.310317 0.155159 0.987890i \(-0.450411\pi\)
0.155159 + 0.987890i \(0.450411\pi\)
\(308\) 1.26351e7 0.0246405
\(309\) 1.34701e8 0.259727
\(310\) −6.35646e8 −1.21185
\(311\) −1.02817e9 −1.93823 −0.969113 0.246619i \(-0.920680\pi\)
−0.969113 + 0.246619i \(0.920680\pi\)
\(312\) −7.46598e7 −0.139170
\(313\) −5.94891e7 −0.109656 −0.0548280 0.998496i \(-0.517461\pi\)
−0.0548280 + 0.998496i \(0.517461\pi\)
\(314\) −6.40829e8 −1.16812
\(315\) 1.67010e8 0.301062
\(316\) 2.18905e7 0.0390257
\(317\) −6.27861e8 −1.10702 −0.553511 0.832842i \(-0.686712\pi\)
−0.553511 + 0.832842i \(0.686712\pi\)
\(318\) 2.26375e8 0.394761
\(319\) 1.33856e7 0.0230871
\(320\) −6.08253e8 −1.03767
\(321\) −7.96600e7 −0.134423
\(322\) −3.96843e8 −0.662404
\(323\) −4.69469e8 −0.775173
\(324\) 6.13580e6 0.0100222
\(325\) −2.79345e7 −0.0451387
\(326\) 1.44370e8 0.230789
\(327\) 4.33288e6 0.00685266
\(328\) −1.04933e9 −1.64192
\(329\) −3.79953e8 −0.588226
\(330\) 6.77221e8 1.03737
\(331\) 5.86045e8 0.888245 0.444123 0.895966i \(-0.353515\pi\)
0.444123 + 0.895966i \(0.353515\pi\)
\(332\) −1.73466e7 −0.0260154
\(333\) −9.11698e8 −1.35299
\(334\) −5.48732e8 −0.805837
\(335\) −5.10369e8 −0.741698
\(336\) −1.38912e8 −0.199780
\(337\) 1.27872e9 1.82000 0.910002 0.414605i \(-0.136080\pi\)
0.910002 + 0.414605i \(0.136080\pi\)
\(338\) −5.55632e7 −0.0782670
\(339\) −4.26026e8 −0.593932
\(340\) 2.50817e7 0.0346084
\(341\) −1.49593e9 −2.04302
\(342\) 4.73310e8 0.639815
\(343\) 4.03536e7 0.0539949
\(344\) −1.07936e9 −1.42959
\(345\) −7.24168e8 −0.949451
\(346\) 1.12126e9 1.45526
\(347\) 1.13865e9 1.46298 0.731490 0.681852i \(-0.238825\pi\)
0.731490 + 0.681852i \(0.238825\pi\)
\(348\) 176807. 0.000224891 0
\(349\) −9.59006e8 −1.20763 −0.603813 0.797126i \(-0.706353\pi\)
−0.603813 + 0.797126i \(0.706353\pi\)
\(350\) −5.02032e7 −0.0625883
\(351\) 1.99712e8 0.246507
\(352\) 1.06642e8 0.130325
\(353\) 1.12980e8 0.136707 0.0683536 0.997661i \(-0.478225\pi\)
0.0683536 + 0.997661i \(0.478225\pi\)
\(354\) 2.11382e8 0.253255
\(355\) 9.61905e8 1.14112
\(356\) −1.06979e7 −0.0125668
\(357\) −1.51250e8 −0.175937
\(358\) −1.42903e9 −1.64607
\(359\) −2.35730e8 −0.268896 −0.134448 0.990921i \(-0.542926\pi\)
−0.134448 + 0.990921i \(0.542926\pi\)
\(360\) 6.92152e8 0.781885
\(361\) −2.46107e8 −0.275326
\(362\) 5.86742e8 0.650080
\(363\) 1.12792e9 1.23767
\(364\) −3.39974e6 −0.00369479
\(365\) 5.51643e8 0.593791
\(366\) 4.58825e8 0.489174
\(367\) 1.21921e9 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(368\) −1.70271e9 −1.78103
\(369\) 1.19252e9 1.23559
\(370\) 1.95797e9 2.00955
\(371\) −2.82159e8 −0.286870
\(372\) −1.97594e7 −0.0199010
\(373\) 1.62860e9 1.62492 0.812462 0.583013i \(-0.198127\pi\)
0.812462 + 0.583013i \(0.198127\pi\)
\(374\) 1.73375e9 1.71371
\(375\) 4.71289e8 0.461506
\(376\) −1.57467e9 −1.52768
\(377\) −3.60167e6 −0.00346186
\(378\) 3.58918e8 0.341801
\(379\) −7.52232e8 −0.709765 −0.354882 0.934911i \(-0.615479\pi\)
−0.354882 + 0.934911i \(0.615479\pi\)
\(380\) −3.46073e7 −0.0323538
\(381\) −1.08470e9 −1.00479
\(382\) 1.01958e9 0.935835
\(383\) −7.70245e7 −0.0700540 −0.0350270 0.999386i \(-0.511152\pi\)
−0.0350270 + 0.999386i \(0.511152\pi\)
\(384\) −5.95327e8 −0.536533
\(385\) −8.44101e8 −0.753845
\(386\) −1.85159e8 −0.163866
\(387\) 1.22665e9 1.07581
\(388\) −2.13998e7 −0.0185993
\(389\) −6.01354e8 −0.517973 −0.258986 0.965881i \(-0.583388\pi\)
−0.258986 + 0.965881i \(0.583388\pi\)
\(390\) −1.82221e8 −0.155551
\(391\) −1.85394e9 −1.56847
\(392\) 1.67241e8 0.140230
\(393\) −1.07722e8 −0.0895226
\(394\) 8.27160e8 0.681322
\(395\) −1.46242e9 −1.19394
\(396\) −5.95104e7 −0.0481571
\(397\) −1.83806e9 −1.47433 −0.737163 0.675715i \(-0.763835\pi\)
−0.737163 + 0.675715i \(0.763835\pi\)
\(398\) 1.94701e9 1.54802
\(399\) 2.08692e8 0.164476
\(400\) −2.15403e8 −0.168284
\(401\) 1.40955e9 1.09163 0.545817 0.837905i \(-0.316220\pi\)
0.545817 + 0.837905i \(0.316220\pi\)
\(402\) −4.65990e8 −0.357754
\(403\) 4.02513e8 0.306346
\(404\) 3.92407e7 0.0296075
\(405\) −4.09910e8 −0.306617
\(406\) −6.47285e6 −0.00480014
\(407\) 4.60790e9 3.38784
\(408\) −6.26838e8 −0.456925
\(409\) 2.23639e9 1.61628 0.808140 0.588990i \(-0.200474\pi\)
0.808140 + 0.588990i \(0.200474\pi\)
\(410\) −2.56107e9 −1.83518
\(411\) −4.79448e8 −0.340640
\(412\) 2.54208e7 0.0179081
\(413\) −2.63471e8 −0.184038
\(414\) 1.86911e9 1.29459
\(415\) 1.15886e9 0.795907
\(416\) −2.86943e7 −0.0195420
\(417\) 1.34595e9 0.908976
\(418\) −2.39220e9 −1.60207
\(419\) 1.46487e9 0.972859 0.486429 0.873720i \(-0.338299\pi\)
0.486429 + 0.873720i \(0.338299\pi\)
\(420\) −1.11495e7 −0.00734318
\(421\) 1.43831e9 0.939430 0.469715 0.882818i \(-0.344357\pi\)
0.469715 + 0.882818i \(0.344357\pi\)
\(422\) 2.72490e8 0.176505
\(423\) 1.78956e9 1.14962
\(424\) −1.16937e9 −0.745027
\(425\) −2.34536e8 −0.148200
\(426\) 8.78262e8 0.550415
\(427\) −5.71888e8 −0.355479
\(428\) −1.50334e7 −0.00926839
\(429\) −4.28840e8 −0.262238
\(430\) −2.63437e9 −1.59785
\(431\) −7.35475e8 −0.442484 −0.221242 0.975219i \(-0.571011\pi\)
−0.221242 + 0.975219i \(0.571011\pi\)
\(432\) 1.53998e9 0.919016
\(433\) 1.13801e9 0.673658 0.336829 0.941566i \(-0.390646\pi\)
0.336829 + 0.941566i \(0.390646\pi\)
\(434\) 7.23388e8 0.424773
\(435\) −1.18118e7 −0.00688024
\(436\) 817699. 0.000472488 0
\(437\) 2.55803e9 1.46630
\(438\) 5.03675e8 0.286412
\(439\) 2.06154e9 1.16297 0.581483 0.813559i \(-0.302473\pi\)
0.581483 + 0.813559i \(0.302473\pi\)
\(440\) −3.49827e9 −1.95780
\(441\) −1.90063e8 −0.105527
\(442\) −4.66504e8 −0.256967
\(443\) 1.06794e9 0.583624 0.291812 0.956476i \(-0.405742\pi\)
0.291812 + 0.956476i \(0.405742\pi\)
\(444\) 6.08646e7 0.0330008
\(445\) 7.14689e8 0.384465
\(446\) 2.61872e9 1.39771
\(447\) −3.22956e8 −0.171028
\(448\) 6.92214e8 0.363720
\(449\) −5.31935e7 −0.0277330 −0.0138665 0.999904i \(-0.504414\pi\)
−0.0138665 + 0.999904i \(0.504414\pi\)
\(450\) 2.36455e8 0.122322
\(451\) −6.02725e9 −3.09386
\(452\) −8.03995e7 −0.0409514
\(453\) −9.93825e8 −0.502304
\(454\) −1.81545e9 −0.910520
\(455\) 2.27124e8 0.113037
\(456\) 8.64899e8 0.427158
\(457\) −4.22053e8 −0.206852 −0.103426 0.994637i \(-0.532981\pi\)
−0.103426 + 0.994637i \(0.532981\pi\)
\(458\) −2.27322e9 −1.10563
\(459\) 1.67677e9 0.809335
\(460\) −1.36665e8 −0.0654643
\(461\) 8.15164e8 0.387518 0.193759 0.981049i \(-0.437932\pi\)
0.193759 + 0.981049i \(0.437932\pi\)
\(462\) −7.70702e8 −0.363613
\(463\) −8.00723e8 −0.374929 −0.187464 0.982271i \(-0.560027\pi\)
−0.187464 + 0.982271i \(0.560027\pi\)
\(464\) −2.77726e7 −0.0129064
\(465\) 1.32005e9 0.608845
\(466\) 9.89302e8 0.452875
\(467\) −1.74070e9 −0.790887 −0.395444 0.918490i \(-0.629409\pi\)
−0.395444 + 0.918490i \(0.629409\pi\)
\(468\) 1.60126e7 0.00722106
\(469\) 5.80818e8 0.259977
\(470\) −3.84327e9 −1.70749
\(471\) 1.33082e9 0.586875
\(472\) −1.09192e9 −0.477964
\(473\) −6.19975e9 −2.69377
\(474\) −1.33526e9 −0.575892
\(475\) 3.23608e8 0.138545
\(476\) −2.85439e7 −0.0121308
\(477\) 1.32895e9 0.560654
\(478\) 3.01946e9 1.26454
\(479\) −2.49945e9 −1.03913 −0.519565 0.854431i \(-0.673906\pi\)
−0.519565 + 0.854431i \(0.673906\pi\)
\(480\) −9.41039e7 −0.0388386
\(481\) −1.23985e9 −0.507999
\(482\) 2.05139e8 0.0834417
\(483\) 8.24129e8 0.332798
\(484\) 2.12861e8 0.0853370
\(485\) 1.42964e9 0.569023
\(486\) −2.66276e9 −1.05222
\(487\) −3.88076e8 −0.152253 −0.0761265 0.997098i \(-0.524255\pi\)
−0.0761265 + 0.997098i \(0.524255\pi\)
\(488\) −2.37012e9 −0.923211
\(489\) −2.99815e8 −0.115950
\(490\) 4.08181e8 0.156735
\(491\) −1.73317e9 −0.660778 −0.330389 0.943845i \(-0.607180\pi\)
−0.330389 + 0.943845i \(0.607180\pi\)
\(492\) −7.96125e7 −0.0301372
\(493\) −3.02394e7 −0.0113660
\(494\) 6.43674e8 0.240227
\(495\) 3.97567e9 1.47330
\(496\) 3.10379e9 1.14211
\(497\) −1.09468e9 −0.399982
\(498\) 1.05809e9 0.383902
\(499\) 2.45395e9 0.884127 0.442063 0.896984i \(-0.354247\pi\)
0.442063 + 0.896984i \(0.354247\pi\)
\(500\) 8.89415e7 0.0318207
\(501\) 1.13956e9 0.404860
\(502\) 4.47055e9 1.57724
\(503\) 2.23819e9 0.784169 0.392084 0.919929i \(-0.371754\pi\)
0.392084 + 0.919929i \(0.371754\pi\)
\(504\) −7.87694e8 −0.274063
\(505\) −2.62153e9 −0.905805
\(506\) −9.44684e9 −3.24160
\(507\) 1.15389e8 0.0393220
\(508\) −2.04705e8 −0.0692796
\(509\) 1.62022e9 0.544580 0.272290 0.962215i \(-0.412219\pi\)
0.272290 + 0.962215i \(0.412219\pi\)
\(510\) −1.52991e9 −0.510706
\(511\) −6.27790e8 −0.208133
\(512\) 2.86125e9 0.942131
\(513\) −2.31357e9 −0.756610
\(514\) 2.02802e9 0.658720
\(515\) −1.69827e9 −0.547874
\(516\) −8.18910e7 −0.0262399
\(517\) −9.04478e9 −2.87860
\(518\) −2.22824e9 −0.704381
\(519\) −2.32854e9 −0.731137
\(520\) 9.41286e8 0.293569
\(521\) 1.97614e9 0.612189 0.306095 0.952001i \(-0.400978\pi\)
0.306095 + 0.952001i \(0.400978\pi\)
\(522\) 3.04868e7 0.00938133
\(523\) −5.67139e8 −0.173354 −0.0866769 0.996236i \(-0.527625\pi\)
−0.0866769 + 0.996236i \(0.527625\pi\)
\(524\) −2.03294e7 −0.00617255
\(525\) 1.04258e8 0.0314449
\(526\) 2.82994e9 0.847867
\(527\) 3.37947e9 1.00580
\(528\) −3.30680e9 −0.977662
\(529\) 6.69689e9 1.96688
\(530\) −2.85407e9 −0.832719
\(531\) 1.24093e9 0.359682
\(532\) 3.93844e7 0.0113405
\(533\) 1.62176e9 0.463919
\(534\) 6.52543e8 0.185445
\(535\) 1.00433e9 0.283555
\(536\) 2.40713e9 0.675184
\(537\) 2.96768e9 0.827002
\(538\) 4.59072e9 1.27099
\(539\) 9.60617e8 0.264235
\(540\) 1.23604e8 0.0337796
\(541\) −1.24208e9 −0.337256 −0.168628 0.985680i \(-0.553934\pi\)
−0.168628 + 0.985680i \(0.553934\pi\)
\(542\) 2.40880e9 0.649836
\(543\) −1.21849e9 −0.326606
\(544\) −2.40915e8 −0.0641605
\(545\) −5.46274e7 −0.0144552
\(546\) 2.07374e8 0.0545231
\(547\) −6.42413e9 −1.67826 −0.839129 0.543932i \(-0.816935\pi\)
−0.839129 + 0.543932i \(0.816935\pi\)
\(548\) −9.04814e7 −0.0234870
\(549\) 2.69356e9 0.694743
\(550\) −1.19509e9 −0.306288
\(551\) 4.17237e7 0.0106256
\(552\) 3.41550e9 0.864306
\(553\) 1.66429e9 0.418496
\(554\) −1.48587e9 −0.371277
\(555\) −4.06614e9 −1.00962
\(556\) 2.54007e8 0.0626735
\(557\) −2.72192e9 −0.667395 −0.333697 0.942680i \(-0.608296\pi\)
−0.333697 + 0.942680i \(0.608296\pi\)
\(558\) −3.40712e9 −0.830171
\(559\) 1.66818e9 0.403925
\(560\) 1.75136e9 0.421421
\(561\) −3.60051e9 −0.860982
\(562\) 3.34434e9 0.794756
\(563\) 4.65154e9 1.09854 0.549272 0.835644i \(-0.314905\pi\)
0.549272 + 0.835644i \(0.314905\pi\)
\(564\) −1.19470e8 −0.0280403
\(565\) 5.37119e9 1.25286
\(566\) −3.29291e9 −0.763348
\(567\) 4.66492e8 0.107474
\(568\) −4.53678e9 −1.03879
\(569\) −1.93553e9 −0.440461 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(570\) 2.11095e9 0.477436
\(571\) 1.43480e9 0.322526 0.161263 0.986911i \(-0.448443\pi\)
0.161263 + 0.986911i \(0.448443\pi\)
\(572\) −8.09306e7 −0.0180812
\(573\) −2.11737e9 −0.470172
\(574\) 2.91459e9 0.643260
\(575\) 1.27793e9 0.280331
\(576\) −3.26029e9 −0.710850
\(577\) −4.15398e9 −0.900222 −0.450111 0.892973i \(-0.648616\pi\)
−0.450111 + 0.892973i \(0.648616\pi\)
\(578\) 8.06829e8 0.173794
\(579\) 3.84523e8 0.0823279
\(580\) −2.22912e6 −0.000474390 0
\(581\) −1.31882e9 −0.278978
\(582\) 1.30532e9 0.274465
\(583\) −6.71678e9 −1.40385
\(584\) −2.60180e9 −0.540541
\(585\) −1.06974e9 −0.220919
\(586\) −5.94881e9 −1.22121
\(587\) 3.56244e9 0.726967 0.363483 0.931601i \(-0.381587\pi\)
0.363483 + 0.931601i \(0.381587\pi\)
\(588\) 1.26886e7 0.00257390
\(589\) −4.66293e9 −0.940277
\(590\) −2.66504e9 −0.534222
\(591\) −1.71777e9 −0.342302
\(592\) −9.56055e9 −1.89390
\(593\) 5.37876e9 1.05923 0.529615 0.848238i \(-0.322336\pi\)
0.529615 + 0.848238i \(0.322336\pi\)
\(594\) 8.54403e9 1.67267
\(595\) 1.90691e9 0.371126
\(596\) −6.09482e7 −0.0117923
\(597\) −4.04338e9 −0.777740
\(598\) 2.54188e9 0.486072
\(599\) 6.26420e9 1.19089 0.595446 0.803396i \(-0.296976\pi\)
0.595446 + 0.803396i \(0.296976\pi\)
\(600\) 4.32083e8 0.0816654
\(601\) 4.78903e8 0.0899885 0.0449942 0.998987i \(-0.485673\pi\)
0.0449942 + 0.998987i \(0.485673\pi\)
\(602\) 2.99801e9 0.560074
\(603\) −2.73562e9 −0.508096
\(604\) −1.87554e8 −0.0346336
\(605\) −1.42204e10 −2.61077
\(606\) −2.39357e9 −0.436910
\(607\) 5.46225e9 0.991313 0.495657 0.868519i \(-0.334927\pi\)
0.495657 + 0.868519i \(0.334927\pi\)
\(608\) 3.32410e8 0.0599808
\(609\) 1.34422e7 0.00241163
\(610\) −5.78471e9 −1.03188
\(611\) 2.43369e9 0.431640
\(612\) 1.34440e8 0.0237083
\(613\) 3.36384e9 0.589826 0.294913 0.955524i \(-0.404709\pi\)
0.294913 + 0.955524i \(0.404709\pi\)
\(614\) −1.81099e9 −0.315739
\(615\) 5.31862e9 0.922010
\(616\) 3.98116e9 0.686242
\(617\) −7.89816e8 −0.135372 −0.0676858 0.997707i \(-0.521562\pi\)
−0.0676858 + 0.997707i \(0.521562\pi\)
\(618\) −1.55060e9 −0.264264
\(619\) 5.58082e9 0.945760 0.472880 0.881127i \(-0.343214\pi\)
0.472880 + 0.881127i \(0.343214\pi\)
\(620\) 2.49120e8 0.0419796
\(621\) −9.13633e9 −1.53092
\(622\) 1.18357e10 1.97209
\(623\) −8.13342e8 −0.134761
\(624\) 8.89766e8 0.146599
\(625\) −6.93519e9 −1.13626
\(626\) 6.84801e8 0.111572
\(627\) 4.96792e9 0.804893
\(628\) 2.51152e8 0.0404648
\(629\) −1.04097e10 −1.66787
\(630\) −1.92251e9 −0.306321
\(631\) −7.62952e9 −1.20891 −0.604455 0.796639i \(-0.706609\pi\)
−0.604455 + 0.796639i \(0.706609\pi\)
\(632\) 6.89745e9 1.08687
\(633\) −5.65884e8 −0.0886777
\(634\) 7.22754e9 1.12636
\(635\) 1.36756e10 2.11952
\(636\) −8.87204e7 −0.0136749
\(637\) −2.58475e8 −0.0396214
\(638\) −1.54086e8 −0.0234904
\(639\) 5.15590e9 0.781720
\(640\) 7.50569e9 1.13178
\(641\) 7.41046e8 0.111133 0.0555663 0.998455i \(-0.482304\pi\)
0.0555663 + 0.998455i \(0.482304\pi\)
\(642\) 9.16995e8 0.136771
\(643\) −2.66540e9 −0.395388 −0.197694 0.980264i \(-0.563345\pi\)
−0.197694 + 0.980264i \(0.563345\pi\)
\(644\) 1.55529e8 0.0229463
\(645\) 5.47083e9 0.802776
\(646\) 5.40423e9 0.788715
\(647\) 2.53599e9 0.368114 0.184057 0.982916i \(-0.441077\pi\)
0.184057 + 0.982916i \(0.441077\pi\)
\(648\) 1.93332e9 0.279120
\(649\) −6.27192e9 −0.900626
\(650\) 3.21564e8 0.0459273
\(651\) −1.50227e9 −0.213410
\(652\) −5.65809e7 −0.00799472
\(653\) 3.39054e9 0.476511 0.238256 0.971202i \(-0.423424\pi\)
0.238256 + 0.971202i \(0.423424\pi\)
\(654\) −4.98773e7 −0.00697238
\(655\) 1.35813e9 0.188841
\(656\) 1.25054e10 1.72956
\(657\) 2.95686e9 0.406773
\(658\) 4.37378e9 0.598503
\(659\) −8.84416e9 −1.20381 −0.601904 0.798568i \(-0.705591\pi\)
−0.601904 + 0.798568i \(0.705591\pi\)
\(660\) −2.65414e8 −0.0359353
\(661\) −6.20676e9 −0.835911 −0.417955 0.908468i \(-0.637253\pi\)
−0.417955 + 0.908468i \(0.637253\pi\)
\(662\) −6.74618e9 −0.903763
\(663\) 9.68795e8 0.129102
\(664\) −5.46570e9 −0.724532
\(665\) −2.63112e9 −0.346949
\(666\) 1.04949e10 1.37663
\(667\) 1.64768e8 0.0214997
\(668\) 2.15057e8 0.0279149
\(669\) −5.43833e9 −0.702221
\(670\) 5.87504e9 0.754656
\(671\) −1.36138e10 −1.73960
\(672\) 1.07094e8 0.0136135
\(673\) 1.76630e9 0.223363 0.111682 0.993744i \(-0.464376\pi\)
0.111682 + 0.993744i \(0.464376\pi\)
\(674\) −1.47199e10 −1.85180
\(675\) −1.15581e9 −0.144651
\(676\) 2.17761e7 0.00271124
\(677\) −5.83167e9 −0.722325 −0.361162 0.932503i \(-0.617620\pi\)
−0.361162 + 0.932503i \(0.617620\pi\)
\(678\) 4.90414e9 0.604309
\(679\) −1.62698e9 −0.199452
\(680\) 7.90296e9 0.963848
\(681\) 3.77018e9 0.457453
\(682\) 1.72202e10 2.07871
\(683\) 4.08372e9 0.490437 0.245219 0.969468i \(-0.421140\pi\)
0.245219 + 0.969468i \(0.421140\pi\)
\(684\) −1.85498e8 −0.0221638
\(685\) 6.04473e9 0.718554
\(686\) −4.64525e8 −0.0549382
\(687\) 4.72082e9 0.555481
\(688\) 1.28633e10 1.50589
\(689\) 1.80730e9 0.210505
\(690\) 8.33616e9 0.966038
\(691\) 1.04320e9 0.120280 0.0601402 0.998190i \(-0.480845\pi\)
0.0601402 + 0.998190i \(0.480845\pi\)
\(692\) −4.39442e8 −0.0504116
\(693\) −4.52446e9 −0.516417
\(694\) −1.31075e10 −1.48854
\(695\) −1.69693e10 −1.91742
\(696\) 5.57097e7 0.00626324
\(697\) 1.36162e10 1.52314
\(698\) 1.10395e10 1.22872
\(699\) −2.05450e9 −0.227528
\(700\) 1.96755e7 0.00216812
\(701\) 5.53065e9 0.606406 0.303203 0.952926i \(-0.401944\pi\)
0.303203 + 0.952926i \(0.401944\pi\)
\(702\) −2.29896e9 −0.250814
\(703\) 1.43631e10 1.55921
\(704\) 1.64781e10 1.77993
\(705\) 7.98137e9 0.857858
\(706\) −1.30056e9 −0.139096
\(707\) 2.98339e9 0.317499
\(708\) −8.28444e7 −0.00877297
\(709\) −1.54751e10 −1.63069 −0.815344 0.578977i \(-0.803452\pi\)
−0.815344 + 0.578977i \(0.803452\pi\)
\(710\) −1.10728e10 −1.16106
\(711\) −7.83872e9 −0.817903
\(712\) −3.37079e9 −0.349987
\(713\) −1.84140e10 −1.90254
\(714\) 1.74110e9 0.179011
\(715\) 5.40668e9 0.553171
\(716\) 5.60059e8 0.0570215
\(717\) −6.27056e9 −0.635316
\(718\) 2.71358e9 0.273594
\(719\) −1.67550e9 −0.168110 −0.0840551 0.996461i \(-0.526787\pi\)
−0.0840551 + 0.996461i \(0.526787\pi\)
\(720\) −8.24879e9 −0.823620
\(721\) 1.93269e9 0.192039
\(722\) 2.83302e9 0.280136
\(723\) −4.26015e8 −0.0419219
\(724\) −2.29954e8 −0.0225194
\(725\) 2.08442e7 0.00203143
\(726\) −1.29839e10 −1.25929
\(727\) −1.85212e10 −1.78771 −0.893857 0.448352i \(-0.852011\pi\)
−0.893857 + 0.448352i \(0.852011\pi\)
\(728\) −1.07122e9 −0.102901
\(729\) 2.55539e9 0.244293
\(730\) −6.35017e9 −0.604164
\(731\) 1.40059e10 1.32617
\(732\) −1.79821e8 −0.0169454
\(733\) 6.30733e9 0.591537 0.295768 0.955260i \(-0.404424\pi\)
0.295768 + 0.955260i \(0.404424\pi\)
\(734\) −1.40347e10 −1.30999
\(735\) −8.47676e8 −0.0787452
\(736\) 1.31269e9 0.121364
\(737\) 1.38264e10 1.27225
\(738\) −1.37276e10 −1.25718
\(739\) −4.89349e9 −0.446029 −0.223015 0.974815i \(-0.571590\pi\)
−0.223015 + 0.974815i \(0.571590\pi\)
\(740\) −7.67361e8 −0.0696127
\(741\) −1.33673e9 −0.120692
\(742\) 3.24803e9 0.291881
\(743\) −5.97118e9 −0.534071 −0.267035 0.963687i \(-0.586044\pi\)
−0.267035 + 0.963687i \(0.586044\pi\)
\(744\) −6.22597e9 −0.554245
\(745\) 4.07172e9 0.360771
\(746\) −1.87474e10 −1.65331
\(747\) 6.21159e9 0.545231
\(748\) −6.79487e8 −0.0593644
\(749\) −1.14296e9 −0.0993904
\(750\) −5.42517e9 −0.469569
\(751\) 3.33059e9 0.286934 0.143467 0.989655i \(-0.454175\pi\)
0.143467 + 0.989655i \(0.454175\pi\)
\(752\) 1.87663e10 1.60922
\(753\) −9.28405e9 −0.792419
\(754\) 4.14602e7 0.00352234
\(755\) 1.25298e10 1.05957
\(756\) −1.40666e8 −0.0118403
\(757\) 1.37350e10 1.15078 0.575392 0.817878i \(-0.304850\pi\)
0.575392 + 0.817878i \(0.304850\pi\)
\(758\) 8.65921e9 0.722165
\(759\) 1.96184e10 1.62861
\(760\) −1.09044e10 −0.901058
\(761\) −1.96227e9 −0.161404 −0.0807018 0.996738i \(-0.525716\pi\)
−0.0807018 + 0.996738i \(0.525716\pi\)
\(762\) 1.24864e10 1.02234
\(763\) 6.21680e7 0.00506677
\(764\) −3.99590e8 −0.0324182
\(765\) −8.98146e9 −0.725324
\(766\) 8.86657e8 0.0712779
\(767\) 1.68760e9 0.135047
\(768\) 6.77707e8 0.0539856
\(769\) −1.39126e10 −1.10323 −0.551616 0.834098i \(-0.685989\pi\)
−0.551616 + 0.834098i \(0.685989\pi\)
\(770\) 9.71675e9 0.767015
\(771\) −4.21162e9 −0.330947
\(772\) 7.25671e7 0.00567648
\(773\) −2.28851e10 −1.78207 −0.891035 0.453935i \(-0.850020\pi\)
−0.891035 + 0.453935i \(0.850020\pi\)
\(774\) −1.41205e10 −1.09460
\(775\) −2.32949e9 −0.179765
\(776\) −6.74281e9 −0.517994
\(777\) 4.62741e9 0.353887
\(778\) 6.92241e9 0.527022
\(779\) −1.87874e10 −1.42392
\(780\) 7.14155e7 0.00538842
\(781\) −2.60589e10 −1.95739
\(782\) 2.13414e10 1.59588
\(783\) −1.49021e8 −0.0110938
\(784\) −1.99311e9 −0.147715
\(785\) −1.67785e10 −1.23797
\(786\) 1.24003e9 0.0910866
\(787\) −2.79341e9 −0.204279 −0.102139 0.994770i \(-0.532569\pi\)
−0.102139 + 0.994770i \(0.532569\pi\)
\(788\) −3.24178e8 −0.0236016
\(789\) −5.87699e9 −0.425976
\(790\) 1.68345e10 1.21480
\(791\) −6.11261e9 −0.439146
\(792\) −1.87511e10 −1.34118
\(793\) 3.66309e9 0.260850
\(794\) 2.11586e10 1.50008
\(795\) 5.92708e9 0.418365
\(796\) −7.63066e8 −0.0536249
\(797\) −3.24938e9 −0.227351 −0.113675 0.993518i \(-0.536262\pi\)
−0.113675 + 0.993518i \(0.536262\pi\)
\(798\) −2.40233e9 −0.167349
\(799\) 2.04331e10 1.41717
\(800\) 1.66064e8 0.0114673
\(801\) 3.83080e9 0.263375
\(802\) −1.62259e10 −1.11070
\(803\) −1.49445e10 −1.01854
\(804\) 1.82629e8 0.0123929
\(805\) −1.03903e10 −0.702012
\(806\) −4.63348e9 −0.311698
\(807\) −9.53361e9 −0.638558
\(808\) 1.23643e10 0.824574
\(809\) 8.68513e9 0.576709 0.288354 0.957524i \(-0.406892\pi\)
0.288354 + 0.957524i \(0.406892\pi\)
\(810\) 4.71862e9 0.311974
\(811\) −2.04298e10 −1.34491 −0.672453 0.740140i \(-0.734759\pi\)
−0.672453 + 0.740140i \(0.734759\pi\)
\(812\) 2.53682e6 0.000166281 0
\(813\) −5.00240e9 −0.326483
\(814\) −5.30432e10 −3.44702
\(815\) 3.77996e9 0.244588
\(816\) 7.47040e9 0.481314
\(817\) −1.93251e10 −1.23978
\(818\) −2.57439e10 −1.64452
\(819\) 1.21740e9 0.0774356
\(820\) 1.00373e9 0.0635722
\(821\) −3.30893e9 −0.208682 −0.104341 0.994542i \(-0.533273\pi\)
−0.104341 + 0.994542i \(0.533273\pi\)
\(822\) 5.51911e9 0.346591
\(823\) 2.44310e10 1.52772 0.763858 0.645385i \(-0.223303\pi\)
0.763858 + 0.645385i \(0.223303\pi\)
\(824\) 8.00980e9 0.498742
\(825\) 2.48185e9 0.153882
\(826\) 3.03291e9 0.187253
\(827\) 1.79678e10 1.10465 0.552326 0.833628i \(-0.313740\pi\)
0.552326 + 0.833628i \(0.313740\pi\)
\(828\) −7.32536e8 −0.0448459
\(829\) 5.72608e9 0.349073 0.174537 0.984651i \(-0.444157\pi\)
0.174537 + 0.984651i \(0.444157\pi\)
\(830\) −1.33400e10 −0.809812
\(831\) 3.08574e9 0.186533
\(832\) −4.43380e9 −0.266898
\(833\) −2.17013e9 −0.130086
\(834\) −1.54937e10 −0.924856
\(835\) −1.43672e10 −0.854021
\(836\) 9.37544e8 0.0554970
\(837\) 1.66542e10 0.981715
\(838\) −1.68626e10 −0.989855
\(839\) 2.01482e10 1.17779 0.588897 0.808208i \(-0.299562\pi\)
0.588897 + 0.808208i \(0.299562\pi\)
\(840\) −3.51309e9 −0.204509
\(841\) −1.72472e10 −0.999844
\(842\) −1.65569e10 −0.955842
\(843\) −6.94524e9 −0.399292
\(844\) −1.06793e8 −0.00611429
\(845\) −1.45478e9 −0.0829468
\(846\) −2.06003e10 −1.16971
\(847\) 1.61834e10 0.915118
\(848\) 1.39361e10 0.784794
\(849\) 6.83843e9 0.383513
\(850\) 2.69983e9 0.150789
\(851\) 5.67203e10 3.15489
\(852\) −3.44206e8 −0.0190669
\(853\) 2.16479e10 1.19425 0.597124 0.802149i \(-0.296310\pi\)
0.597124 + 0.802149i \(0.296310\pi\)
\(854\) 6.58321e9 0.361689
\(855\) 1.23925e10 0.678072
\(856\) −4.73685e9 −0.258126
\(857\) 2.26970e10 1.23179 0.615895 0.787829i \(-0.288795\pi\)
0.615895 + 0.787829i \(0.288795\pi\)
\(858\) 4.93654e9 0.266819
\(859\) 3.63609e9 0.195731 0.0978654 0.995200i \(-0.468799\pi\)
0.0978654 + 0.995200i \(0.468799\pi\)
\(860\) 1.03245e9 0.0553511
\(861\) −6.05278e9 −0.323179
\(862\) 8.46631e9 0.450214
\(863\) 3.06452e10 1.62302 0.811512 0.584336i \(-0.198645\pi\)
0.811512 + 0.584336i \(0.198645\pi\)
\(864\) −1.18724e9 −0.0626241
\(865\) 2.93575e10 1.54228
\(866\) −1.31001e10 −0.685427
\(867\) −1.67555e9 −0.0873154
\(868\) −2.83508e8 −0.0147145
\(869\) 3.96184e10 2.04799
\(870\) 1.35970e8 0.00700044
\(871\) −3.72028e9 −0.190771
\(872\) 2.57648e8 0.0131589
\(873\) 7.66299e9 0.389806
\(874\) −2.94465e10 −1.49191
\(875\) 6.76204e9 0.341232
\(876\) −1.97399e8 −0.00992157
\(877\) −5.57383e9 −0.279033 −0.139516 0.990220i \(-0.544555\pi\)
−0.139516 + 0.990220i \(0.544555\pi\)
\(878\) −2.37312e10 −1.18328
\(879\) 1.23540e10 0.613544
\(880\) 4.16910e10 2.06231
\(881\) 1.74820e10 0.861340 0.430670 0.902509i \(-0.358277\pi\)
0.430670 + 0.902509i \(0.358277\pi\)
\(882\) 2.18789e9 0.107371
\(883\) 1.37811e10 0.673631 0.336815 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(884\) 1.82831e8 0.00890157
\(885\) 5.53452e9 0.268398
\(886\) −1.22934e10 −0.593820
\(887\) −2.27458e10 −1.09438 −0.547190 0.837009i \(-0.684302\pi\)
−0.547190 + 0.837009i \(0.684302\pi\)
\(888\) 1.91777e10 0.919077
\(889\) −1.55633e10 −0.742925
\(890\) −8.22704e9 −0.391182
\(891\) 1.11048e10 0.525945
\(892\) −1.02632e9 −0.0484179
\(893\) −2.81932e10 −1.32484
\(894\) 3.71767e9 0.174016
\(895\) −3.74155e10 −1.74450
\(896\) −8.54174e9 −0.396706
\(897\) −5.27875e9 −0.244207
\(898\) 6.12329e8 0.0282175
\(899\) −3.00348e8 −0.0137869
\(900\) −9.26706e7 −0.00423734
\(901\) 1.51739e10 0.691132
\(902\) 6.93818e10 3.14791
\(903\) −6.22601e9 −0.281386
\(904\) −2.53329e10 −1.14050
\(905\) 1.53624e10 0.688951
\(906\) 1.14403e10 0.511079
\(907\) 2.46167e10 1.09548 0.547740 0.836649i \(-0.315488\pi\)
0.547740 + 0.836649i \(0.315488\pi\)
\(908\) 7.11507e8 0.0315412
\(909\) −1.40516e10 −0.620516
\(910\) −2.61450e9 −0.115012
\(911\) 1.80092e10 0.789187 0.394594 0.918856i \(-0.370885\pi\)
0.394594 + 0.918856i \(0.370885\pi\)
\(912\) −1.03075e10 −0.449958
\(913\) −3.13946e10 −1.36523
\(914\) 4.85840e9 0.210466
\(915\) 1.20132e10 0.518424
\(916\) 8.90912e8 0.0383002
\(917\) −1.54560e9 −0.0661918
\(918\) −1.93019e10 −0.823474
\(919\) 2.01851e10 0.857881 0.428940 0.903333i \(-0.358887\pi\)
0.428940 + 0.903333i \(0.358887\pi\)
\(920\) −4.30615e10 −1.82319
\(921\) 3.76092e9 0.158630
\(922\) −9.38364e9 −0.394288
\(923\) 7.01171e9 0.293507
\(924\) 3.02051e8 0.0125959
\(925\) 7.17549e9 0.298095
\(926\) 9.21742e9 0.381479
\(927\) −9.10287e9 −0.375318
\(928\) 2.14112e7 0.000879472 0
\(929\) −1.17886e10 −0.482398 −0.241199 0.970476i \(-0.577541\pi\)
−0.241199 + 0.970476i \(0.577541\pi\)
\(930\) −1.51956e10 −0.619482
\(931\) 2.99431e9 0.121611
\(932\) −3.87724e8 −0.0156880
\(933\) −2.45793e10 −0.990794
\(934\) 2.00378e10 0.804705
\(935\) 4.53940e10 1.81618
\(936\) 5.04538e9 0.201107
\(937\) 3.55634e10 1.41226 0.706130 0.708082i \(-0.250439\pi\)
0.706130 + 0.708082i \(0.250439\pi\)
\(938\) −6.68601e9 −0.264519
\(939\) −1.42214e9 −0.0560546
\(940\) 1.50624e9 0.0591490
\(941\) 1.29049e10 0.504882 0.252441 0.967612i \(-0.418767\pi\)
0.252441 + 0.967612i \(0.418767\pi\)
\(942\) −1.53195e10 −0.597128
\(943\) −7.41916e10 −2.88114
\(944\) 1.30131e10 0.503476
\(945\) 9.39737e9 0.362239
\(946\) 7.13675e10 2.74083
\(947\) 3.57581e10 1.36820 0.684099 0.729389i \(-0.260195\pi\)
0.684099 + 0.729389i \(0.260195\pi\)
\(948\) 5.23310e8 0.0199494
\(949\) 4.02115e9 0.152728
\(950\) −3.72517e9 −0.140966
\(951\) −1.50095e10 −0.565894
\(952\) −8.99385e9 −0.337844
\(953\) −1.55603e10 −0.582363 −0.291181 0.956668i \(-0.594048\pi\)
−0.291181 + 0.956668i \(0.594048\pi\)
\(954\) −1.52981e10 −0.570449
\(955\) 2.66951e10 0.991792
\(956\) −1.18338e9 −0.0438048
\(957\) 3.19993e8 0.0118018
\(958\) 2.87721e10 1.05728
\(959\) −6.87911e9 −0.251865
\(960\) −1.45408e10 −0.530443
\(961\) 6.05347e9 0.220025
\(962\) 1.42724e10 0.516874
\(963\) 5.38328e9 0.194247
\(964\) −8.03974e7 −0.00289050
\(965\) −4.84794e9 −0.173665
\(966\) −9.48685e9 −0.338612
\(967\) 1.36197e9 0.0484368 0.0242184 0.999707i \(-0.492290\pi\)
0.0242184 + 0.999707i \(0.492290\pi\)
\(968\) 6.70700e10 2.37665
\(969\) −1.12230e10 −0.396257
\(970\) −1.64571e10 −0.578964
\(971\) −4.41825e10 −1.54875 −0.774377 0.632724i \(-0.781937\pi\)
−0.774377 + 0.632724i \(0.781937\pi\)
\(972\) 1.04358e9 0.0364497
\(973\) 1.93116e10 0.672085
\(974\) 4.46729e9 0.154913
\(975\) −6.67797e8 −0.0230743
\(976\) 2.82461e10 0.972489
\(977\) −1.47857e10 −0.507238 −0.253619 0.967304i \(-0.581621\pi\)
−0.253619 + 0.967304i \(0.581621\pi\)
\(978\) 3.45127e9 0.117976
\(979\) −1.93616e10 −0.659480
\(980\) −1.59973e8 −0.00542945
\(981\) −2.92808e8 −0.00990242
\(982\) 1.99511e10 0.672322
\(983\) −1.09926e10 −0.369117 −0.184559 0.982822i \(-0.559086\pi\)
−0.184559 + 0.982822i \(0.559086\pi\)
\(984\) −2.50850e10 −0.839326
\(985\) 2.16571e10 0.722061
\(986\) 3.48096e8 0.0115646
\(987\) −9.08309e9 −0.300693
\(988\) −2.52267e8 −0.00832167
\(989\) −7.63149e10 −2.50855
\(990\) −4.57654e10 −1.49904
\(991\) −2.24046e10 −0.731272 −0.365636 0.930758i \(-0.619149\pi\)
−0.365636 + 0.930758i \(0.619149\pi\)
\(992\) −2.39285e9 −0.0778261
\(993\) 1.40099e10 0.454059
\(994\) 1.26013e10 0.406970
\(995\) 5.09776e10 1.64058
\(996\) −4.14684e8 −0.0132987
\(997\) 1.69909e10 0.542979 0.271489 0.962441i \(-0.412484\pi\)
0.271489 + 0.962441i \(0.412484\pi\)
\(998\) −2.82484e10 −0.899573
\(999\) −5.12997e10 −1.62793
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 91.8.a.c.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.c.1.4 10 1.1 even 1 trivial