Properties

Label 7.10.a.a.1.1
Level $7$
Weight $10$
Character 7.1
Self dual yes
Analytic conductor $3.605$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(3.60525085315\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 7.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.8924 q^{2} +109.817 q^{3} -226.645 q^{4} -2438.78 q^{5} -1855.08 q^{6} -2401.00 q^{7} +12477.5 q^{8} -7623.25 q^{9} +O(q^{10})\) \(q-16.8924 q^{2} +109.817 q^{3} -226.645 q^{4} -2438.78 q^{5} -1855.08 q^{6} -2401.00 q^{7} +12477.5 q^{8} -7623.25 q^{9} +41197.0 q^{10} -28548.3 q^{11} -24889.5 q^{12} +138149. q^{13} +40558.8 q^{14} -267819. q^{15} -94733.5 q^{16} -101010. q^{17} +128775. q^{18} -488928. q^{19} +552739. q^{20} -263670. q^{21} +482250. q^{22} -140071. q^{23} +1.37024e6 q^{24} +3.99453e6 q^{25} -2.33367e6 q^{26} -2.99869e6 q^{27} +544175. q^{28} -6.31716e6 q^{29} +4.52413e6 q^{30} -1.00903e6 q^{31} -4.78821e6 q^{32} -3.13508e6 q^{33} +1.70630e6 q^{34} +5.85552e6 q^{35} +1.72777e6 q^{36} +1.19206e7 q^{37} +8.25919e6 q^{38} +1.51711e7 q^{39} -3.04300e7 q^{40} -2.15106e7 q^{41} +4.45404e6 q^{42} +1.65957e7 q^{43} +6.47033e6 q^{44} +1.85915e7 q^{45} +2.36615e6 q^{46} -2.67441e7 q^{47} -1.04033e7 q^{48} +5.76480e6 q^{49} -6.74774e7 q^{50} -1.10926e7 q^{51} -3.13108e7 q^{52} +3.74991e7 q^{53} +5.06552e7 q^{54} +6.96230e7 q^{55} -2.99585e7 q^{56} -5.36926e7 q^{57} +1.06712e8 q^{58} +1.81907e7 q^{59} +6.07000e7 q^{60} -2.50111e7 q^{61} +1.70449e7 q^{62} +1.83034e7 q^{63} +1.29388e8 q^{64} -3.36915e8 q^{65} +5.29592e7 q^{66} -2.18572e8 q^{67} +2.28934e7 q^{68} -1.53822e7 q^{69} -9.89140e7 q^{70} +3.12688e8 q^{71} -9.51193e7 q^{72} -2.89038e8 q^{73} -2.01369e8 q^{74} +4.38667e8 q^{75} +1.10813e8 q^{76} +6.85444e7 q^{77} -2.56276e8 q^{78} +4.68685e8 q^{79} +2.31034e8 q^{80} -1.79258e8 q^{81} +3.63366e8 q^{82} -7.75407e7 q^{83} +5.97597e7 q^{84} +2.46341e8 q^{85} -2.80342e8 q^{86} -6.93730e8 q^{87} -3.56212e8 q^{88} +3.37680e8 q^{89} -3.14055e8 q^{90} -3.31695e8 q^{91} +3.17465e7 q^{92} -1.10808e8 q^{93} +4.51773e8 q^{94} +1.19239e9 q^{95} -5.25827e8 q^{96} -7.36733e8 q^{97} -9.73816e7 q^{98} +2.17631e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + 43384 q^{10} + 35316 q^{11} + 52136 q^{12} - 26530 q^{13} + 14406 q^{14} - 307136 q^{15} - 752 q^{16} - 463920 q^{17} + 332042 q^{18} - 925426 q^{19} + 473760 q^{20} + 206486 q^{21} + 1177888 q^{22} + 778128 q^{23} + 3301296 q^{24} + 2081722 q^{25} - 4127424 q^{26} - 2798612 q^{27} + 1488620 q^{28} - 10003584 q^{29} + 4095872 q^{30} + 2467260 q^{31} + 1284576 q^{32} - 15640784 q^{33} - 2246676 q^{34} + 5373438 q^{35} - 5612716 q^{36} + 30735552 q^{37} + 3504660 q^{38} + 47417944 q^{39} - 32409984 q^{40} - 19103448 q^{41} + 9575188 q^{42} + 4065100 q^{43} - 18650976 q^{44} + 22338298 q^{45} + 12367584 q^{46} - 82195020 q^{47} - 28806496 q^{48} + 11529602 q^{49} - 88312626 q^{50} + 59971356 q^{51} + 33466384 q^{52} - 55189812 q^{53} + 52834472 q^{54} + 82445816 q^{55} - 6281016 q^{56} + 31781116 q^{57} + 66558004 q^{58} - 7069218 q^{59} + 76165376 q^{60} + 44316386 q^{61} + 54910200 q^{62} - 26502238 q^{63} + 147417152 q^{64} - 369979260 q^{65} - 83258464 q^{66} - 241921336 q^{67} + 165645816 q^{68} - 195181152 q^{69} - 104164984 q^{70} + 206493816 q^{71} - 279147720 q^{72} - 499153188 q^{73} + 3571524 q^{74} + 813228014 q^{75} + 282511768 q^{76} - 84793716 q^{77} + 94970960 q^{78} + 468535096 q^{79} + 249904128 q^{80} - 585745634 q^{81} + 389586092 q^{82} + 444023958 q^{83} - 125178536 q^{84} + 173475060 q^{85} - 416830608 q^{86} + 28134340 q^{87} - 986010816 q^{88} + 636267396 q^{89} - 273242728 q^{90} + 63698530 q^{91} - 329431488 q^{92} - 791523960 q^{93} - 152223192 q^{94} + 1104747984 q^{95} - 1714981184 q^{96} - 1632716064 q^{97} - 34588806 q^{98} + 1409417860 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.8924 −0.746548 −0.373274 0.927721i \(-0.621765\pi\)
−0.373274 + 0.927721i \(0.621765\pi\)
\(3\) 109.817 0.782751 0.391375 0.920231i \(-0.371999\pi\)
0.391375 + 0.920231i \(0.371999\pi\)
\(4\) −226.645 −0.442667
\(5\) −2438.78 −1.74505 −0.872525 0.488569i \(-0.837519\pi\)
−0.872525 + 0.488569i \(0.837519\pi\)
\(6\) −1855.08 −0.584361
\(7\) −2401.00 −0.377964
\(8\) 12477.5 1.07702
\(9\) −7623.25 −0.387301
\(10\) 41197.0 1.30276
\(11\) −28548.3 −0.587912 −0.293956 0.955819i \(-0.594972\pi\)
−0.293956 + 0.955819i \(0.594972\pi\)
\(12\) −24889.5 −0.346498
\(13\) 138149. 1.34153 0.670767 0.741668i \(-0.265965\pi\)
0.670767 + 0.741668i \(0.265965\pi\)
\(14\) 40558.8 0.282168
\(15\) −267819. −1.36594
\(16\) −94733.5 −0.361380
\(17\) −101010. −0.293321 −0.146661 0.989187i \(-0.546853\pi\)
−0.146661 + 0.989187i \(0.546853\pi\)
\(18\) 128775. 0.289139
\(19\) −488928. −0.860704 −0.430352 0.902661i \(-0.641611\pi\)
−0.430352 + 0.902661i \(0.641611\pi\)
\(20\) 552739. 0.772476
\(21\) −263670. −0.295852
\(22\) 482250. 0.438905
\(23\) −140071. −0.104370 −0.0521848 0.998637i \(-0.516618\pi\)
−0.0521848 + 0.998637i \(0.516618\pi\)
\(24\) 1.37024e6 0.843038
\(25\) 3.99453e6 2.04520
\(26\) −2.33367e6 −1.00152
\(27\) −2.99869e6 −1.08591
\(28\) 544175. 0.167312
\(29\) −6.31716e6 −1.65856 −0.829279 0.558835i \(-0.811249\pi\)
−0.829279 + 0.558835i \(0.811249\pi\)
\(30\) 4.52413e6 1.01974
\(31\) −1.00903e6 −0.196234 −0.0981172 0.995175i \(-0.531282\pi\)
−0.0981172 + 0.995175i \(0.531282\pi\)
\(32\) −4.78821e6 −0.807232
\(33\) −3.13508e6 −0.460189
\(34\) 1.70630e6 0.218978
\(35\) 5.85552e6 0.659567
\(36\) 1.72777e6 0.171445
\(37\) 1.19206e7 1.04566 0.522832 0.852436i \(-0.324876\pi\)
0.522832 + 0.852436i \(0.324876\pi\)
\(38\) 8.25919e6 0.642556
\(39\) 1.51711e7 1.05009
\(40\) −3.04300e7 −1.87945
\(41\) −2.15106e7 −1.18884 −0.594422 0.804153i \(-0.702619\pi\)
−0.594422 + 0.804153i \(0.702619\pi\)
\(42\) 4.45404e6 0.220868
\(43\) 1.65957e7 0.740265 0.370133 0.928979i \(-0.379312\pi\)
0.370133 + 0.928979i \(0.379312\pi\)
\(44\) 6.47033e6 0.260249
\(45\) 1.85915e7 0.675860
\(46\) 2.36615e6 0.0779169
\(47\) −2.67441e7 −0.799443 −0.399721 0.916637i \(-0.630893\pi\)
−0.399721 + 0.916637i \(0.630893\pi\)
\(48\) −1.04033e7 −0.282870
\(49\) 5.76480e6 0.142857
\(50\) −6.74774e7 −1.52684
\(51\) −1.10926e7 −0.229597
\(52\) −3.13108e7 −0.593853
\(53\) 3.74991e7 0.652799 0.326399 0.945232i \(-0.394165\pi\)
0.326399 + 0.945232i \(0.394165\pi\)
\(54\) 5.06552e7 0.810684
\(55\) 6.96230e7 1.02594
\(56\) −2.99585e7 −0.407075
\(57\) −5.36926e7 −0.673717
\(58\) 1.06712e8 1.23819
\(59\) 1.81907e7 0.195441 0.0977207 0.995214i \(-0.468845\pi\)
0.0977207 + 0.995214i \(0.468845\pi\)
\(60\) 6.07000e7 0.604656
\(61\) −2.50111e7 −0.231285 −0.115643 0.993291i \(-0.536893\pi\)
−0.115643 + 0.993291i \(0.536893\pi\)
\(62\) 1.70449e7 0.146498
\(63\) 1.83034e7 0.146386
\(64\) 1.29388e8 0.964017
\(65\) −3.36915e8 −2.34105
\(66\) 5.29592e7 0.343553
\(67\) −2.18572e8 −1.32513 −0.662564 0.749005i \(-0.730532\pi\)
−0.662564 + 0.749005i \(0.730532\pi\)
\(68\) 2.28934e7 0.129844
\(69\) −1.53822e7 −0.0816954
\(70\) −9.89140e7 −0.492398
\(71\) 3.12688e8 1.46032 0.730161 0.683275i \(-0.239445\pi\)
0.730161 + 0.683275i \(0.239445\pi\)
\(72\) −9.51193e7 −0.417131
\(73\) −2.89038e8 −1.19125 −0.595624 0.803264i \(-0.703095\pi\)
−0.595624 + 0.803264i \(0.703095\pi\)
\(74\) −2.01369e8 −0.780638
\(75\) 4.38667e8 1.60088
\(76\) 1.10813e8 0.381005
\(77\) 6.85444e7 0.222210
\(78\) −2.56276e8 −0.783940
\(79\) 4.68685e8 1.35381 0.676907 0.736069i \(-0.263320\pi\)
0.676907 + 0.736069i \(0.263320\pi\)
\(80\) 2.31034e8 0.630626
\(81\) −1.79258e8 −0.462696
\(82\) 3.63366e8 0.887529
\(83\) −7.75407e7 −0.179341 −0.0896703 0.995972i \(-0.528581\pi\)
−0.0896703 + 0.995972i \(0.528581\pi\)
\(84\) 5.97597e7 0.130964
\(85\) 2.46341e8 0.511860
\(86\) −2.80342e8 −0.552643
\(87\) −6.93730e8 −1.29824
\(88\) −3.56212e8 −0.633193
\(89\) 3.37680e8 0.570493 0.285246 0.958454i \(-0.407925\pi\)
0.285246 + 0.958454i \(0.407925\pi\)
\(90\) −3.14055e8 −0.504562
\(91\) −3.31695e8 −0.507052
\(92\) 3.17465e7 0.0462010
\(93\) −1.10808e8 −0.153603
\(94\) 4.51773e8 0.596822
\(95\) 1.19239e9 1.50197
\(96\) −5.25827e8 −0.631862
\(97\) −7.36733e8 −0.844962 −0.422481 0.906372i \(-0.638841\pi\)
−0.422481 + 0.906372i \(0.638841\pi\)
\(98\) −9.73816e7 −0.106650
\(99\) 2.17631e8 0.227699
\(100\) −9.05342e8 −0.905342
\(101\) 1.97428e8 0.188783 0.0943914 0.995535i \(-0.469909\pi\)
0.0943914 + 0.995535i \(0.469909\pi\)
\(102\) 1.87381e8 0.171405
\(103\) −1.27449e9 −1.11576 −0.557879 0.829922i \(-0.688385\pi\)
−0.557879 + 0.829922i \(0.688385\pi\)
\(104\) 1.72375e9 1.44486
\(105\) 6.43035e8 0.516277
\(106\) −6.33451e8 −0.487345
\(107\) 1.81937e9 1.34182 0.670908 0.741541i \(-0.265905\pi\)
0.670908 + 0.741541i \(0.265905\pi\)
\(108\) 6.79639e8 0.480697
\(109\) −2.31425e8 −0.157033 −0.0785167 0.996913i \(-0.525018\pi\)
−0.0785167 + 0.996913i \(0.525018\pi\)
\(110\) −1.17610e9 −0.765911
\(111\) 1.30909e9 0.818494
\(112\) 2.27455e8 0.136589
\(113\) −1.51983e9 −0.876885 −0.438443 0.898759i \(-0.644470\pi\)
−0.438443 + 0.898759i \(0.644470\pi\)
\(114\) 9.06998e8 0.502961
\(115\) 3.41604e8 0.182130
\(116\) 1.43175e9 0.734188
\(117\) −1.05314e9 −0.519578
\(118\) −3.07286e8 −0.145906
\(119\) 2.42525e8 0.110865
\(120\) −3.34172e9 −1.47114
\(121\) −1.54294e9 −0.654359
\(122\) 4.22498e8 0.172666
\(123\) −2.36223e9 −0.930569
\(124\) 2.28691e8 0.0868665
\(125\) −4.97855e9 −1.82393
\(126\) −3.09190e8 −0.109284
\(127\) 4.20951e9 1.43587 0.717934 0.696111i \(-0.245088\pi\)
0.717934 + 0.696111i \(0.245088\pi\)
\(128\) 2.65883e8 0.0875478
\(129\) 1.82249e9 0.579443
\(130\) 5.69131e9 1.74770
\(131\) −4.12131e9 −1.22268 −0.611342 0.791367i \(-0.709370\pi\)
−0.611342 + 0.791367i \(0.709370\pi\)
\(132\) 7.10552e8 0.203710
\(133\) 1.17392e9 0.325315
\(134\) 3.69221e9 0.989271
\(135\) 7.31315e9 1.89497
\(136\) −1.26035e9 −0.315913
\(137\) −1.27942e9 −0.310292 −0.155146 0.987892i \(-0.549585\pi\)
−0.155146 + 0.987892i \(0.549585\pi\)
\(138\) 2.59843e8 0.0609895
\(139\) −4.02340e9 −0.914170 −0.457085 0.889423i \(-0.651107\pi\)
−0.457085 + 0.889423i \(0.651107\pi\)
\(140\) −1.32713e9 −0.291968
\(141\) −2.93695e9 −0.625764
\(142\) −5.28206e9 −1.09020
\(143\) −3.94391e9 −0.788705
\(144\) 7.22177e8 0.139963
\(145\) 1.54062e10 2.89427
\(146\) 4.88256e9 0.889323
\(147\) 6.33072e8 0.111822
\(148\) −2.70176e9 −0.462880
\(149\) −7.67785e8 −0.127615 −0.0638075 0.997962i \(-0.520324\pi\)
−0.0638075 + 0.997962i \(0.520324\pi\)
\(150\) −7.41016e9 −1.19514
\(151\) −9.84752e9 −1.54145 −0.770727 0.637165i \(-0.780107\pi\)
−0.770727 + 0.637165i \(0.780107\pi\)
\(152\) −6.10061e9 −0.926995
\(153\) 7.70023e8 0.113604
\(154\) −1.15788e9 −0.165890
\(155\) 2.46080e9 0.342439
\(156\) −3.43845e9 −0.464839
\(157\) 8.04096e9 1.05623 0.528116 0.849172i \(-0.322898\pi\)
0.528116 + 0.849172i \(0.322898\pi\)
\(158\) −7.91723e9 −1.01069
\(159\) 4.11803e9 0.510979
\(160\) 1.16774e10 1.40866
\(161\) 3.36311e8 0.0394480
\(162\) 3.02811e9 0.345425
\(163\) −9.07348e9 −1.00677 −0.503384 0.864063i \(-0.667912\pi\)
−0.503384 + 0.864063i \(0.667912\pi\)
\(164\) 4.87527e9 0.526262
\(165\) 7.64578e9 0.803053
\(166\) 1.30985e9 0.133886
\(167\) −8.83471e9 −0.878958 −0.439479 0.898253i \(-0.644837\pi\)
−0.439479 + 0.898253i \(0.644837\pi\)
\(168\) −3.28995e9 −0.318638
\(169\) 8.48058e9 0.799715
\(170\) −4.16130e9 −0.382128
\(171\) 3.72722e9 0.333352
\(172\) −3.76134e9 −0.327691
\(173\) 7.89871e9 0.670423 0.335211 0.942143i \(-0.391192\pi\)
0.335211 + 0.942143i \(0.391192\pi\)
\(174\) 1.17188e10 0.969196
\(175\) −9.59087e9 −0.773013
\(176\) 2.70448e9 0.212460
\(177\) 1.99765e9 0.152982
\(178\) −5.70424e9 −0.425900
\(179\) 4.59871e9 0.334810 0.167405 0.985888i \(-0.446461\pi\)
0.167405 + 0.985888i \(0.446461\pi\)
\(180\) −4.21367e9 −0.299181
\(181\) −6.70993e8 −0.0464691 −0.0232346 0.999730i \(-0.507396\pi\)
−0.0232346 + 0.999730i \(0.507396\pi\)
\(182\) 5.60314e9 0.378539
\(183\) −2.74664e9 −0.181039
\(184\) −1.74774e9 −0.112408
\(185\) −2.90719e10 −1.82474
\(186\) 1.87182e9 0.114672
\(187\) 2.88366e9 0.172447
\(188\) 6.06142e9 0.353887
\(189\) 7.19985e9 0.410436
\(190\) −2.01424e10 −1.12129
\(191\) 1.00686e10 0.547416 0.273708 0.961813i \(-0.411750\pi\)
0.273708 + 0.961813i \(0.411750\pi\)
\(192\) 1.42090e10 0.754585
\(193\) −1.97876e10 −1.02656 −0.513281 0.858221i \(-0.671570\pi\)
−0.513281 + 0.858221i \(0.671570\pi\)
\(194\) 1.24452e10 0.630805
\(195\) −3.69989e10 −1.83246
\(196\) −1.30657e9 −0.0632381
\(197\) −1.07508e10 −0.508560 −0.254280 0.967131i \(-0.581839\pi\)
−0.254280 + 0.967131i \(0.581839\pi\)
\(198\) −3.67631e9 −0.169988
\(199\) −1.58060e9 −0.0714468 −0.0357234 0.999362i \(-0.511374\pi\)
−0.0357234 + 0.999362i \(0.511374\pi\)
\(200\) 4.98419e10 2.20272
\(201\) −2.40029e10 −1.03724
\(202\) −3.33504e9 −0.140935
\(203\) 1.51675e10 0.626876
\(204\) 2.51408e9 0.101635
\(205\) 5.24596e10 2.07459
\(206\) 2.15293e10 0.832967
\(207\) 1.06780e9 0.0404225
\(208\) −1.30873e10 −0.484803
\(209\) 1.39580e10 0.506019
\(210\) −1.08624e10 −0.385425
\(211\) 4.44247e10 1.54295 0.771477 0.636257i \(-0.219518\pi\)
0.771477 + 0.636257i \(0.219518\pi\)
\(212\) −8.49899e9 −0.288972
\(213\) 3.43384e10 1.14307
\(214\) −3.07335e10 −1.00173
\(215\) −4.04733e10 −1.29180
\(216\) −3.74162e10 −1.16955
\(217\) 2.42267e9 0.0741697
\(218\) 3.90934e9 0.117233
\(219\) −3.17412e10 −0.932450
\(220\) −1.57797e10 −0.454148
\(221\) −1.39544e10 −0.393501
\(222\) −2.21137e10 −0.611045
\(223\) 2.40745e10 0.651907 0.325954 0.945386i \(-0.394315\pi\)
0.325954 + 0.945386i \(0.394315\pi\)
\(224\) 1.14965e10 0.305105
\(225\) −3.04513e10 −0.792109
\(226\) 2.56737e10 0.654636
\(227\) −1.71588e8 −0.00428915 −0.00214458 0.999998i \(-0.500683\pi\)
−0.00214458 + 0.999998i \(0.500683\pi\)
\(228\) 1.21692e10 0.298232
\(229\) −1.11777e10 −0.268592 −0.134296 0.990941i \(-0.542877\pi\)
−0.134296 + 0.990941i \(0.542877\pi\)
\(230\) −5.77052e9 −0.135969
\(231\) 7.52733e9 0.173935
\(232\) −7.88225e10 −1.78630
\(233\) 6.54487e9 0.145479 0.0727393 0.997351i \(-0.476826\pi\)
0.0727393 + 0.997351i \(0.476826\pi\)
\(234\) 1.77902e10 0.387890
\(235\) 6.52230e10 1.39507
\(236\) −4.12285e9 −0.0865154
\(237\) 5.14695e10 1.05970
\(238\) −4.09683e9 −0.0827660
\(239\) −9.08610e9 −0.180130 −0.0900651 0.995936i \(-0.528708\pi\)
−0.0900651 + 0.995936i \(0.528708\pi\)
\(240\) 2.53715e10 0.493623
\(241\) −5.36400e10 −1.02426 −0.512132 0.858906i \(-0.671144\pi\)
−0.512132 + 0.858906i \(0.671144\pi\)
\(242\) 2.60641e10 0.488510
\(243\) 3.93376e10 0.723735
\(244\) 5.66865e9 0.102382
\(245\) −1.40591e10 −0.249293
\(246\) 3.99038e10 0.694714
\(247\) −6.75448e10 −1.15466
\(248\) −1.25902e10 −0.211348
\(249\) −8.51528e9 −0.140379
\(250\) 8.40999e10 1.36165
\(251\) 3.92651e10 0.624418 0.312209 0.950013i \(-0.398931\pi\)
0.312209 + 0.950013i \(0.398931\pi\)
\(252\) −4.14839e9 −0.0648003
\(253\) 3.99880e9 0.0613602
\(254\) −7.11089e10 −1.07194
\(255\) 2.70524e10 0.400659
\(256\) −7.07381e10 −1.02938
\(257\) −7.70786e9 −0.110213 −0.0551067 0.998480i \(-0.517550\pi\)
−0.0551067 + 0.998480i \(0.517550\pi\)
\(258\) −3.07863e10 −0.432582
\(259\) −2.86215e10 −0.395224
\(260\) 7.63601e10 1.03630
\(261\) 4.81573e10 0.642362
\(262\) 6.96189e10 0.912791
\(263\) 3.65878e10 0.471559 0.235779 0.971807i \(-0.424236\pi\)
0.235779 + 0.971807i \(0.424236\pi\)
\(264\) −3.91181e10 −0.495632
\(265\) −9.14521e10 −1.13917
\(266\) −1.98303e10 −0.242863
\(267\) 3.70830e10 0.446554
\(268\) 4.95383e10 0.586590
\(269\) −1.65478e11 −1.92688 −0.963442 0.267916i \(-0.913665\pi\)
−0.963442 + 0.267916i \(0.913665\pi\)
\(270\) −1.23537e11 −1.41469
\(271\) 5.47248e10 0.616343 0.308171 0.951331i \(-0.400283\pi\)
0.308171 + 0.951331i \(0.400283\pi\)
\(272\) 9.56901e9 0.106000
\(273\) −3.64257e10 −0.396896
\(274\) 2.16125e10 0.231648
\(275\) −1.14037e11 −1.20240
\(276\) 3.48631e9 0.0361638
\(277\) −2.55019e10 −0.260264 −0.130132 0.991497i \(-0.541540\pi\)
−0.130132 + 0.991497i \(0.541540\pi\)
\(278\) 6.79651e10 0.682472
\(279\) 7.69207e9 0.0760019
\(280\) 7.30623e10 0.710367
\(281\) −2.44664e10 −0.234095 −0.117047 0.993126i \(-0.537343\pi\)
−0.117047 + 0.993126i \(0.537343\pi\)
\(282\) 4.96123e10 0.467163
\(283\) 1.83938e11 1.70464 0.852318 0.523024i \(-0.175196\pi\)
0.852318 + 0.523024i \(0.175196\pi\)
\(284\) −7.08693e10 −0.646436
\(285\) 1.30944e11 1.17567
\(286\) 6.66222e10 0.588806
\(287\) 5.16469e10 0.449341
\(288\) 3.65018e10 0.312642
\(289\) −1.08385e11 −0.913963
\(290\) −2.60248e11 −2.16071
\(291\) −8.09057e10 −0.661395
\(292\) 6.55091e10 0.527325
\(293\) 1.39840e11 1.10848 0.554241 0.832356i \(-0.313009\pi\)
0.554241 + 0.832356i \(0.313009\pi\)
\(294\) −1.06941e10 −0.0834801
\(295\) −4.43633e10 −0.341055
\(296\) 1.48740e11 1.12620
\(297\) 8.56073e10 0.638421
\(298\) 1.29698e10 0.0952706
\(299\) −1.93507e10 −0.140015
\(300\) −9.94219e10 −0.708657
\(301\) −3.98463e10 −0.279794
\(302\) 1.66349e11 1.15077
\(303\) 2.16809e10 0.147770
\(304\) 4.63179e10 0.311041
\(305\) 6.09966e10 0.403605
\(306\) −1.30076e10 −0.0848106
\(307\) −1.22633e11 −0.787922 −0.393961 0.919127i \(-0.628895\pi\)
−0.393961 + 0.919127i \(0.628895\pi\)
\(308\) −1.55353e10 −0.0983650
\(309\) −1.39961e11 −0.873361
\(310\) −4.15689e10 −0.255647
\(311\) −1.44117e11 −0.873564 −0.436782 0.899567i \(-0.643882\pi\)
−0.436782 + 0.899567i \(0.643882\pi\)
\(312\) 1.89297e11 1.13096
\(313\) −2.74753e11 −1.61805 −0.809027 0.587771i \(-0.800006\pi\)
−0.809027 + 0.587771i \(0.800006\pi\)
\(314\) −1.35832e11 −0.788528
\(315\) −4.46381e10 −0.255451
\(316\) −1.06225e11 −0.599288
\(317\) −1.23139e11 −0.684904 −0.342452 0.939535i \(-0.611257\pi\)
−0.342452 + 0.939535i \(0.611257\pi\)
\(318\) −6.95636e10 −0.381470
\(319\) 1.80344e11 0.975087
\(320\) −3.15550e11 −1.68226
\(321\) 1.99797e11 1.05031
\(322\) −5.68112e9 −0.0294498
\(323\) 4.93865e10 0.252463
\(324\) 4.06280e10 0.204820
\(325\) 5.51840e11 2.74371
\(326\) 1.53273e11 0.751601
\(327\) −2.54144e10 −0.122918
\(328\) −2.68399e11 −1.28041
\(329\) 6.42126e10 0.302161
\(330\) −1.29156e11 −0.599517
\(331\) 3.37245e11 1.54426 0.772129 0.635466i \(-0.219192\pi\)
0.772129 + 0.635466i \(0.219192\pi\)
\(332\) 1.75742e10 0.0793881
\(333\) −9.08741e10 −0.404987
\(334\) 1.49240e11 0.656184
\(335\) 5.33049e11 2.31242
\(336\) 2.49784e10 0.106915
\(337\) −2.84144e11 −1.20006 −0.600032 0.799976i \(-0.704846\pi\)
−0.600032 + 0.799976i \(0.704846\pi\)
\(338\) −1.43258e11 −0.597025
\(339\) −1.66903e11 −0.686382
\(340\) −5.58320e10 −0.226584
\(341\) 2.88060e10 0.115369
\(342\) −6.29619e10 −0.248863
\(343\) −1.38413e10 −0.0539949
\(344\) 2.07073e11 0.797280
\(345\) 3.75138e10 0.142563
\(346\) −1.33429e11 −0.500503
\(347\) 1.38776e11 0.513846 0.256923 0.966432i \(-0.417291\pi\)
0.256923 + 0.966432i \(0.417291\pi\)
\(348\) 1.57231e11 0.574686
\(349\) −1.52561e11 −0.550463 −0.275231 0.961378i \(-0.588754\pi\)
−0.275231 + 0.961378i \(0.588754\pi\)
\(350\) 1.62013e11 0.577091
\(351\) −4.14265e11 −1.45679
\(352\) 1.36695e11 0.474582
\(353\) 1.24628e11 0.427199 0.213599 0.976921i \(-0.431481\pi\)
0.213599 + 0.976921i \(0.431481\pi\)
\(354\) −3.37452e10 −0.114208
\(355\) −7.62578e11 −2.54834
\(356\) −7.65336e10 −0.252538
\(357\) 2.66333e10 0.0867797
\(358\) −7.76835e10 −0.249951
\(359\) −7.69379e10 −0.244464 −0.122232 0.992502i \(-0.539005\pi\)
−0.122232 + 0.992502i \(0.539005\pi\)
\(360\) 2.31975e11 0.727915
\(361\) −8.36371e10 −0.259189
\(362\) 1.13347e10 0.0346914
\(363\) −1.69441e11 −0.512200
\(364\) 7.51772e10 0.224455
\(365\) 7.04900e11 2.07879
\(366\) 4.63975e10 0.135154
\(367\) −4.36607e11 −1.25630 −0.628150 0.778092i \(-0.716187\pi\)
−0.628150 + 0.778092i \(0.716187\pi\)
\(368\) 1.32695e10 0.0377171
\(369\) 1.63981e11 0.460441
\(370\) 4.91095e11 1.36225
\(371\) −9.00353e10 −0.246735
\(372\) 2.51142e10 0.0679948
\(373\) 1.66545e11 0.445495 0.222747 0.974876i \(-0.428497\pi\)
0.222747 + 0.974876i \(0.428497\pi\)
\(374\) −4.87120e10 −0.128740
\(375\) −5.46729e11 −1.42768
\(376\) −3.33700e11 −0.861015
\(377\) −8.72707e11 −2.22501
\(378\) −1.21623e11 −0.306410
\(379\) 4.10213e11 1.02125 0.510627 0.859803i \(-0.329413\pi\)
0.510627 + 0.859803i \(0.329413\pi\)
\(380\) −2.70249e11 −0.664873
\(381\) 4.62275e11 1.12393
\(382\) −1.70083e11 −0.408672
\(383\) 7.01166e11 1.66505 0.832523 0.553990i \(-0.186895\pi\)
0.832523 + 0.553990i \(0.186895\pi\)
\(384\) 2.91984e10 0.0685281
\(385\) −1.67165e11 −0.387768
\(386\) 3.34261e11 0.766377
\(387\) −1.26513e11 −0.286706
\(388\) 1.66977e11 0.374037
\(389\) 8.46752e11 1.87492 0.937461 0.348090i \(-0.113170\pi\)
0.937461 + 0.348090i \(0.113170\pi\)
\(390\) 6.25002e11 1.36801
\(391\) 1.41486e10 0.0306138
\(392\) 7.19304e10 0.153860
\(393\) −4.52589e11 −0.957056
\(394\) 1.81607e11 0.379665
\(395\) −1.14302e12 −2.36247
\(396\) −4.93250e10 −0.100795
\(397\) 8.75287e11 1.76845 0.884226 0.467059i \(-0.154687\pi\)
0.884226 + 0.467059i \(0.154687\pi\)
\(398\) 2.67002e10 0.0533385
\(399\) 1.28916e11 0.254641
\(400\) −3.78416e11 −0.739094
\(401\) −2.47491e11 −0.477980 −0.238990 0.971022i \(-0.576816\pi\)
−0.238990 + 0.971022i \(0.576816\pi\)
\(402\) 4.05468e11 0.774353
\(403\) −1.39396e11 −0.263255
\(404\) −4.47461e10 −0.0835679
\(405\) 4.37171e11 0.807428
\(406\) −2.56216e11 −0.467993
\(407\) −3.40314e11 −0.614759
\(408\) −1.38408e11 −0.247281
\(409\) −9.48426e10 −0.167590 −0.0837951 0.996483i \(-0.526704\pi\)
−0.0837951 + 0.996483i \(0.526704\pi\)
\(410\) −8.86172e11 −1.54878
\(411\) −1.40502e11 −0.242881
\(412\) 2.88858e11 0.493909
\(413\) −4.36760e10 −0.0738699
\(414\) −1.80377e10 −0.0301773
\(415\) 1.89105e11 0.312958
\(416\) −6.61486e11 −1.08293
\(417\) −4.41838e11 −0.715567
\(418\) −2.35786e11 −0.377767
\(419\) 9.93237e11 1.57431 0.787154 0.616756i \(-0.211554\pi\)
0.787154 + 0.616756i \(0.211554\pi\)
\(420\) −1.45741e11 −0.228538
\(421\) 3.88328e11 0.602461 0.301230 0.953551i \(-0.402603\pi\)
0.301230 + 0.953551i \(0.402603\pi\)
\(422\) −7.50441e11 −1.15189
\(423\) 2.03877e11 0.309625
\(424\) 4.67896e11 0.703077
\(425\) −4.03487e11 −0.599901
\(426\) −5.80060e11 −0.853355
\(427\) 6.00516e10 0.0874177
\(428\) −4.12351e11 −0.593977
\(429\) −4.33108e11 −0.617359
\(430\) 6.83693e11 0.964390
\(431\) 2.08495e11 0.291036 0.145518 0.989356i \(-0.453515\pi\)
0.145518 + 0.989356i \(0.453515\pi\)
\(432\) 2.84076e11 0.392426
\(433\) −3.43270e11 −0.469289 −0.234645 0.972081i \(-0.575393\pi\)
−0.234645 + 0.972081i \(0.575393\pi\)
\(434\) −4.09249e10 −0.0553712
\(435\) 1.69186e12 2.26549
\(436\) 5.24515e10 0.0695134
\(437\) 6.84848e10 0.0898314
\(438\) 5.36187e11 0.696118
\(439\) 5.98857e11 0.769543 0.384771 0.923012i \(-0.374280\pi\)
0.384771 + 0.923012i \(0.374280\pi\)
\(440\) 8.68723e11 1.10495
\(441\) −4.39465e10 −0.0553288
\(442\) 2.35724e11 0.293767
\(443\) −3.05169e11 −0.376464 −0.188232 0.982125i \(-0.560276\pi\)
−0.188232 + 0.982125i \(0.560276\pi\)
\(444\) −2.96699e11 −0.362320
\(445\) −8.23528e11 −0.995539
\(446\) −4.06678e11 −0.486680
\(447\) −8.43158e10 −0.0998907
\(448\) −3.10661e11 −0.364364
\(449\) 2.40802e11 0.279610 0.139805 0.990179i \(-0.455352\pi\)
0.139805 + 0.990179i \(0.455352\pi\)
\(450\) 5.14397e11 0.591347
\(451\) 6.14090e11 0.698936
\(452\) 3.44463e11 0.388168
\(453\) −1.08142e12 −1.20657
\(454\) 2.89855e9 0.00320206
\(455\) 8.08932e11 0.884832
\(456\) −6.69950e11 −0.725606
\(457\) 2.36226e11 0.253341 0.126671 0.991945i \(-0.459571\pi\)
0.126671 + 0.991945i \(0.459571\pi\)
\(458\) 1.88819e11 0.200517
\(459\) 3.02897e11 0.318521
\(460\) −7.74229e10 −0.0806230
\(461\) −1.17120e12 −1.20775 −0.603873 0.797081i \(-0.706376\pi\)
−0.603873 + 0.797081i \(0.706376\pi\)
\(462\) −1.27155e11 −0.129851
\(463\) −1.78934e12 −1.80958 −0.904791 0.425856i \(-0.859973\pi\)
−0.904791 + 0.425856i \(0.859973\pi\)
\(464\) 5.98446e11 0.599369
\(465\) 2.70237e11 0.268044
\(466\) −1.10559e11 −0.108607
\(467\) −3.81202e11 −0.370877 −0.185438 0.982656i \(-0.559370\pi\)
−0.185438 + 0.982656i \(0.559370\pi\)
\(468\) 2.38690e11 0.230000
\(469\) 5.24791e11 0.500851
\(470\) −1.10178e12 −1.04148
\(471\) 8.83034e11 0.826767
\(472\) 2.26975e11 0.210494
\(473\) −4.73778e11 −0.435211
\(474\) −8.69446e11 −0.791115
\(475\) −1.95304e12 −1.76031
\(476\) −5.49671e10 −0.0490762
\(477\) −2.85865e11 −0.252830
\(478\) 1.53486e11 0.134476
\(479\) −1.71146e12 −1.48545 −0.742725 0.669597i \(-0.766467\pi\)
−0.742725 + 0.669597i \(0.766467\pi\)
\(480\) 1.28238e12 1.10263
\(481\) 1.64682e12 1.40279
\(482\) 9.06111e11 0.764663
\(483\) 3.69327e10 0.0308780
\(484\) 3.49701e11 0.289663
\(485\) 1.79673e12 1.47450
\(486\) −6.64508e11 −0.540303
\(487\) 7.38713e11 0.595108 0.297554 0.954705i \(-0.403829\pi\)
0.297554 + 0.954705i \(0.403829\pi\)
\(488\) −3.12076e11 −0.249099
\(489\) −9.96422e11 −0.788049
\(490\) 2.37492e11 0.186109
\(491\) 1.52117e12 1.18116 0.590582 0.806978i \(-0.298898\pi\)
0.590582 + 0.806978i \(0.298898\pi\)
\(492\) 5.35387e11 0.411932
\(493\) 6.38095e11 0.486490
\(494\) 1.14100e12 0.862012
\(495\) −5.30754e11 −0.397347
\(496\) 9.55887e10 0.0709151
\(497\) −7.50764e11 −0.551950
\(498\) 1.43844e11 0.104800
\(499\) −1.67720e12 −1.21097 −0.605484 0.795857i \(-0.707021\pi\)
−0.605484 + 0.795857i \(0.707021\pi\)
\(500\) 1.12837e12 0.807392
\(501\) −9.70200e11 −0.688005
\(502\) −6.63284e11 −0.466158
\(503\) 2.93328e11 0.204314 0.102157 0.994768i \(-0.467426\pi\)
0.102157 + 0.994768i \(0.467426\pi\)
\(504\) 2.28381e11 0.157661
\(505\) −4.81484e11 −0.329436
\(506\) −6.75494e10 −0.0458083
\(507\) 9.31311e11 0.625977
\(508\) −9.54065e11 −0.635611
\(509\) −2.87498e12 −1.89848 −0.949238 0.314559i \(-0.898143\pi\)
−0.949238 + 0.314559i \(0.898143\pi\)
\(510\) −4.56981e11 −0.299111
\(511\) 6.93980e11 0.450249
\(512\) 1.05881e12 0.680930
\(513\) 1.46614e12 0.934648
\(514\) 1.30205e11 0.0822796
\(515\) 3.10821e12 1.94706
\(516\) −4.13058e11 −0.256500
\(517\) 7.63497e11 0.470002
\(518\) 4.83487e11 0.295053
\(519\) 8.67412e11 0.524774
\(520\) −4.20386e12 −2.52135
\(521\) 1.69333e12 1.00687 0.503434 0.864034i \(-0.332070\pi\)
0.503434 + 0.864034i \(0.332070\pi\)
\(522\) −8.13494e11 −0.479554
\(523\) 9.48465e11 0.554324 0.277162 0.960823i \(-0.410606\pi\)
0.277162 + 0.960823i \(0.410606\pi\)
\(524\) 9.34075e11 0.541241
\(525\) −1.05324e12 −0.605077
\(526\) −6.18058e11 −0.352041
\(527\) 1.01922e11 0.0575597
\(528\) 2.96997e11 0.166303
\(529\) −1.78153e12 −0.989107
\(530\) 1.54485e12 0.850442
\(531\) −1.38673e11 −0.0756947
\(532\) −2.66063e11 −0.144006
\(533\) −2.97166e12 −1.59488
\(534\) −6.26422e11 −0.333374
\(535\) −4.43704e12 −2.34154
\(536\) −2.72724e12 −1.42719
\(537\) 5.05017e11 0.262072
\(538\) 2.79533e12 1.43851
\(539\) −1.64575e11 −0.0839875
\(540\) −1.65749e12 −0.838840
\(541\) 2.60488e12 1.30738 0.653688 0.756764i \(-0.273221\pi\)
0.653688 + 0.756764i \(0.273221\pi\)
\(542\) −9.24436e11 −0.460129
\(543\) −7.36864e10 −0.0363738
\(544\) 4.83657e11 0.236778
\(545\) 5.64396e11 0.274031
\(546\) 6.15320e11 0.296301
\(547\) 2.01576e12 0.962711 0.481356 0.876525i \(-0.340145\pi\)
0.481356 + 0.876525i \(0.340145\pi\)
\(548\) 2.89974e11 0.137356
\(549\) 1.90666e11 0.0895772
\(550\) 1.92636e12 0.897648
\(551\) 3.08863e12 1.42753
\(552\) −1.91932e11 −0.0879875
\(553\) −1.12531e12 −0.511693
\(554\) 4.30790e11 0.194300
\(555\) −3.19258e12 −1.42831
\(556\) 9.11886e11 0.404673
\(557\) −3.04194e12 −1.33907 −0.669534 0.742782i \(-0.733506\pi\)
−0.669534 + 0.742782i \(0.733506\pi\)
\(558\) −1.29938e11 −0.0567390
\(559\) 2.29267e12 0.993091
\(560\) −5.54713e11 −0.238354
\(561\) 3.16674e11 0.134983
\(562\) 4.13297e11 0.174763
\(563\) 4.23912e12 1.77823 0.889115 0.457684i \(-0.151321\pi\)
0.889115 + 0.457684i \(0.151321\pi\)
\(564\) 6.65647e11 0.277005
\(565\) 3.70654e12 1.53021
\(566\) −3.10716e12 −1.27259
\(567\) 4.30399e11 0.174883
\(568\) 3.90157e12 1.57280
\(569\) 1.34119e11 0.0536397 0.0268199 0.999640i \(-0.491462\pi\)
0.0268199 + 0.999640i \(0.491462\pi\)
\(570\) −2.21197e12 −0.877693
\(571\) −1.51210e11 −0.0595277 −0.0297638 0.999557i \(-0.509476\pi\)
−0.0297638 + 0.999557i \(0.509476\pi\)
\(572\) 8.93868e11 0.349133
\(573\) 1.10570e12 0.428491
\(574\) −8.72443e11 −0.335454
\(575\) −5.59520e11 −0.213457
\(576\) −9.86359e11 −0.373365
\(577\) −1.84681e12 −0.693633 −0.346817 0.937933i \(-0.612737\pi\)
−0.346817 + 0.937933i \(0.612737\pi\)
\(578\) 1.83089e12 0.682317
\(579\) −2.17301e12 −0.803541
\(580\) −3.49174e12 −1.28120
\(581\) 1.86175e11 0.0677843
\(582\) 1.36670e12 0.493763
\(583\) −1.07053e12 −0.383788
\(584\) −3.60648e12 −1.28300
\(585\) 2.56839e12 0.906690
\(586\) −2.36225e12 −0.827535
\(587\) −2.97730e12 −1.03503 −0.517513 0.855675i \(-0.673142\pi\)
−0.517513 + 0.855675i \(0.673142\pi\)
\(588\) −1.43483e11 −0.0494997
\(589\) 4.93342e11 0.168900
\(590\) 7.49404e11 0.254614
\(591\) −1.18062e12 −0.398076
\(592\) −1.12928e12 −0.377881
\(593\) 1.06093e12 0.352322 0.176161 0.984361i \(-0.443632\pi\)
0.176161 + 0.984361i \(0.443632\pi\)
\(594\) −1.44612e12 −0.476611
\(595\) −5.91465e11 −0.193465
\(596\) 1.74015e11 0.0564909
\(597\) −1.73576e11 −0.0559250
\(598\) 3.26880e11 0.104528
\(599\) −4.53108e12 −1.43807 −0.719037 0.694972i \(-0.755417\pi\)
−0.719037 + 0.694972i \(0.755417\pi\)
\(600\) 5.47348e12 1.72418
\(601\) −1.05588e12 −0.330127 −0.165063 0.986283i \(-0.552783\pi\)
−0.165063 + 0.986283i \(0.552783\pi\)
\(602\) 6.73101e11 0.208880
\(603\) 1.66623e12 0.513224
\(604\) 2.23190e12 0.682351
\(605\) 3.76290e12 1.14189
\(606\) −3.66244e11 −0.110317
\(607\) −3.25042e12 −0.971831 −0.485916 0.874006i \(-0.661514\pi\)
−0.485916 + 0.874006i \(0.661514\pi\)
\(608\) 2.34109e12 0.694788
\(609\) 1.66565e12 0.490688
\(610\) −1.03038e12 −0.301310
\(611\) −3.69466e12 −1.07248
\(612\) −1.74522e11 −0.0502886
\(613\) −2.48670e12 −0.711299 −0.355649 0.934619i \(-0.615740\pi\)
−0.355649 + 0.934619i \(0.615740\pi\)
\(614\) 2.07156e12 0.588221
\(615\) 5.76095e12 1.62389
\(616\) 8.55264e11 0.239325
\(617\) −4.06622e12 −1.12956 −0.564778 0.825243i \(-0.691038\pi\)
−0.564778 + 0.825243i \(0.691038\pi\)
\(618\) 2.36428e12 0.652006
\(619\) −4.24730e12 −1.16280 −0.581400 0.813618i \(-0.697495\pi\)
−0.581400 + 0.813618i \(0.697495\pi\)
\(620\) −5.57728e11 −0.151586
\(621\) 4.20030e11 0.113336
\(622\) 2.43450e12 0.652157
\(623\) −8.10769e11 −0.215626
\(624\) −1.43721e12 −0.379480
\(625\) 4.33978e12 1.13765
\(626\) 4.64125e12 1.20796
\(627\) 1.53283e12 0.396086
\(628\) −1.82245e12 −0.467559
\(629\) −1.20410e12 −0.306715
\(630\) 7.54046e11 0.190706
\(631\) 1.44130e12 0.361929 0.180965 0.983490i \(-0.442078\pi\)
0.180965 + 0.983490i \(0.442078\pi\)
\(632\) 5.84803e12 1.45808
\(633\) 4.87858e12 1.20775
\(634\) 2.08012e12 0.511314
\(635\) −1.02661e13 −2.50566
\(636\) −9.33333e11 −0.226193
\(637\) 7.96400e11 0.191648
\(638\) −3.04645e12 −0.727949
\(639\) −2.38370e12 −0.565585
\(640\) −6.48430e11 −0.152775
\(641\) 6.00980e12 1.40604 0.703022 0.711168i \(-0.251833\pi\)
0.703022 + 0.711168i \(0.251833\pi\)
\(642\) −3.37506e12 −0.784105
\(643\) 2.09069e12 0.482327 0.241163 0.970485i \(-0.422471\pi\)
0.241163 + 0.970485i \(0.422471\pi\)
\(644\) −7.62234e10 −0.0174623
\(645\) −4.44465e12 −1.01116
\(646\) −8.34259e11 −0.188475
\(647\) −1.34894e12 −0.302638 −0.151319 0.988485i \(-0.548352\pi\)
−0.151319 + 0.988485i \(0.548352\pi\)
\(648\) −2.23670e12 −0.498333
\(649\) −5.19314e11 −0.114902
\(650\) −9.32192e12 −2.04831
\(651\) 2.66051e11 0.0580563
\(652\) 2.05646e12 0.445663
\(653\) −7.38295e12 −1.58899 −0.794494 0.607272i \(-0.792264\pi\)
−0.794494 + 0.607272i \(0.792264\pi\)
\(654\) 4.29312e11 0.0917641
\(655\) 1.00510e13 2.13364
\(656\) 2.03777e12 0.429624
\(657\) 2.20341e12 0.461372
\(658\) −1.08471e12 −0.225578
\(659\) 3.34345e12 0.690574 0.345287 0.938497i \(-0.387782\pi\)
0.345287 + 0.938497i \(0.387782\pi\)
\(660\) −1.73288e12 −0.355485
\(661\) 8.14808e12 1.66016 0.830078 0.557648i \(-0.188296\pi\)
0.830078 + 0.557648i \(0.188296\pi\)
\(662\) −5.69690e12 −1.15286
\(663\) −1.53243e12 −0.308013
\(664\) −9.67516e11 −0.193153
\(665\) −2.86293e12 −0.567692
\(666\) 1.53509e12 0.302342
\(667\) 8.84853e11 0.173103
\(668\) 2.00235e12 0.389085
\(669\) 2.64379e12 0.510281
\(670\) −9.00451e12 −1.72633
\(671\) 7.14023e11 0.135976
\(672\) 1.26251e12 0.238821
\(673\) −6.60403e12 −1.24091 −0.620457 0.784241i \(-0.713053\pi\)
−0.620457 + 0.784241i \(0.713053\pi\)
\(674\) 4.79989e12 0.895905
\(675\) −1.19784e13 −2.22091
\(676\) −1.92208e12 −0.354007
\(677\) 3.16321e12 0.578735 0.289367 0.957218i \(-0.406555\pi\)
0.289367 + 0.957218i \(0.406555\pi\)
\(678\) 2.81940e12 0.512417
\(679\) 1.76890e12 0.319366
\(680\) 3.07373e12 0.551283
\(681\) −1.88433e10 −0.00335734
\(682\) −4.86603e11 −0.0861282
\(683\) 7.85876e12 1.38185 0.690925 0.722927i \(-0.257204\pi\)
0.690925 + 0.722927i \(0.257204\pi\)
\(684\) −8.44757e11 −0.147564
\(685\) 3.12022e12 0.541475
\(686\) 2.33813e11 0.0403098
\(687\) −1.22750e12 −0.210241
\(688\) −1.57217e12 −0.267517
\(689\) 5.18045e12 0.875752
\(690\) −6.33701e11 −0.106430
\(691\) 3.21525e12 0.536492 0.268246 0.963350i \(-0.413556\pi\)
0.268246 + 0.963350i \(0.413556\pi\)
\(692\) −1.79021e12 −0.296774
\(693\) −5.22531e11 −0.0860622
\(694\) −2.34427e12 −0.383611
\(695\) 9.81221e12 1.59527
\(696\) −8.65604e12 −1.39823
\(697\) 2.17278e12 0.348713
\(698\) 2.57712e12 0.410947
\(699\) 7.18737e11 0.113874
\(700\) 2.17373e12 0.342187
\(701\) −6.07789e12 −0.950652 −0.475326 0.879810i \(-0.657670\pi\)
−0.475326 + 0.879810i \(0.657670\pi\)
\(702\) 6.99795e12 1.08756
\(703\) −5.82834e12 −0.900007
\(704\) −3.69381e12 −0.566758
\(705\) 7.16259e12 1.09199
\(706\) −2.10528e12 −0.318924
\(707\) −4.74025e11 −0.0713532
\(708\) −4.52758e11 −0.0677200
\(709\) −6.79921e12 −1.01053 −0.505266 0.862963i \(-0.668606\pi\)
−0.505266 + 0.862963i \(0.668606\pi\)
\(710\) 1.28818e13 1.90245
\(711\) −3.57290e12 −0.524334
\(712\) 4.21341e12 0.614432
\(713\) 1.41336e11 0.0204809
\(714\) −4.49901e11 −0.0647851
\(715\) 9.61833e12 1.37633
\(716\) −1.04228e12 −0.148209
\(717\) −9.97807e11 −0.140997
\(718\) 1.29967e12 0.182504
\(719\) 8.63238e11 0.120462 0.0602311 0.998184i \(-0.480816\pi\)
0.0602311 + 0.998184i \(0.480816\pi\)
\(720\) −1.76123e12 −0.244242
\(721\) 3.06006e12 0.421717
\(722\) 1.41283e12 0.193497
\(723\) −5.89058e12 −0.801744
\(724\) 1.52078e11 0.0205703
\(725\) −2.52341e13 −3.39208
\(726\) 2.86228e12 0.382382
\(727\) −9.34730e12 −1.24103 −0.620514 0.784195i \(-0.713076\pi\)
−0.620514 + 0.784195i \(0.713076\pi\)
\(728\) −4.13873e12 −0.546105
\(729\) 7.84827e12 1.02920
\(730\) −1.19075e13 −1.55191
\(731\) −1.67633e12 −0.217135
\(732\) 6.22513e11 0.0801399
\(733\) −1.12045e13 −1.43359 −0.716793 0.697286i \(-0.754391\pi\)
−0.716793 + 0.697286i \(0.754391\pi\)
\(734\) 7.37536e12 0.937887
\(735\) −1.54393e12 −0.195134
\(736\) 6.70692e11 0.0842506
\(737\) 6.23985e12 0.779059
\(738\) −2.77003e12 −0.343741
\(739\) −4.55769e12 −0.562140 −0.281070 0.959687i \(-0.590689\pi\)
−0.281070 + 0.959687i \(0.590689\pi\)
\(740\) 6.58900e12 0.807750
\(741\) −7.41756e12 −0.903814
\(742\) 1.52092e12 0.184199
\(743\) 3.57016e12 0.429772 0.214886 0.976639i \(-0.431062\pi\)
0.214886 + 0.976639i \(0.431062\pi\)
\(744\) −1.38261e12 −0.165433
\(745\) 1.87246e12 0.222695
\(746\) −2.81336e12 −0.332583
\(747\) 5.91112e11 0.0694588
\(748\) −6.53567e11 −0.0763366
\(749\) −4.36830e12 −0.507159
\(750\) 9.23559e12 1.06583
\(751\) −2.38313e12 −0.273380 −0.136690 0.990614i \(-0.543646\pi\)
−0.136690 + 0.990614i \(0.543646\pi\)
\(752\) 2.53356e12 0.288902
\(753\) 4.31198e12 0.488764
\(754\) 1.47422e13 1.66108
\(755\) 2.40160e13 2.68992
\(756\) −1.63181e12 −0.181686
\(757\) 4.96674e12 0.549718 0.274859 0.961484i \(-0.411369\pi\)
0.274859 + 0.961484i \(0.411369\pi\)
\(758\) −6.92951e12 −0.762414
\(759\) 4.39135e11 0.0480298
\(760\) 1.48781e13 1.61765
\(761\) 1.01660e13 1.09880 0.549401 0.835559i \(-0.314856\pi\)
0.549401 + 0.835559i \(0.314856\pi\)
\(762\) −7.80896e12 −0.839065
\(763\) 5.55652e11 0.0593530
\(764\) −2.28200e12 −0.242323
\(765\) −1.87792e12 −0.198244
\(766\) −1.18444e13 −1.24304
\(767\) 2.51303e12 0.262191
\(768\) −7.76824e12 −0.805744
\(769\) −4.53519e12 −0.467656 −0.233828 0.972278i \(-0.575125\pi\)
−0.233828 + 0.972278i \(0.575125\pi\)
\(770\) 2.82382e12 0.289487
\(771\) −8.46453e11 −0.0862697
\(772\) 4.48476e12 0.454424
\(773\) −3.64760e12 −0.367451 −0.183725 0.982978i \(-0.558816\pi\)
−0.183725 + 0.982978i \(0.558816\pi\)
\(774\) 2.13712e12 0.214039
\(775\) −4.03059e12 −0.401339
\(776\) −9.19261e12 −0.910041
\(777\) −3.14312e12 −0.309362
\(778\) −1.43037e13 −1.39972
\(779\) 1.05171e13 1.02324
\(780\) 8.38563e12 0.811167
\(781\) −8.92670e12 −0.858542
\(782\) −2.39004e11 −0.0228547
\(783\) 1.89432e13 1.80105
\(784\) −5.46120e11 −0.0516256
\(785\) −1.96102e13 −1.84318
\(786\) 7.64533e12 0.714488
\(787\) −2.68084e12 −0.249106 −0.124553 0.992213i \(-0.539750\pi\)
−0.124553 + 0.992213i \(0.539750\pi\)
\(788\) 2.43662e12 0.225123
\(789\) 4.01796e12 0.369113
\(790\) 1.93084e13 1.76370
\(791\) 3.64912e12 0.331431
\(792\) 2.71549e12 0.245237
\(793\) −3.45525e12 −0.310277
\(794\) −1.47857e13 −1.32023
\(795\) −1.00430e13 −0.891683
\(796\) 3.58235e11 0.0316271
\(797\) −9.88359e12 −0.867665 −0.433833 0.900994i \(-0.642839\pi\)
−0.433833 + 0.900994i \(0.642839\pi\)
\(798\) −2.17770e12 −0.190102
\(799\) 2.70142e12 0.234494
\(800\) −1.91267e13 −1.65095
\(801\) −2.57422e12 −0.220953
\(802\) 4.18073e12 0.356835
\(803\) 8.25153e12 0.700349
\(804\) 5.44014e12 0.459154
\(805\) −8.20190e11 −0.0688388
\(806\) 2.35474e12 0.196533
\(807\) −1.81723e13 −1.50827
\(808\) 2.46341e12 0.203323
\(809\) 1.62405e13 1.33300 0.666502 0.745503i \(-0.267791\pi\)
0.666502 + 0.745503i \(0.267791\pi\)
\(810\) −7.38489e12 −0.602784
\(811\) −2.28548e13 −1.85517 −0.927584 0.373614i \(-0.878118\pi\)
−0.927584 + 0.373614i \(0.878118\pi\)
\(812\) −3.43764e12 −0.277497
\(813\) 6.00971e12 0.482443
\(814\) 5.74873e12 0.458947
\(815\) 2.21282e13 1.75686
\(816\) 1.05084e12 0.0829718
\(817\) −8.11410e12 −0.637149
\(818\) 1.60212e12 0.125114
\(819\) 2.52860e12 0.196382
\(820\) −1.18897e13 −0.918353
\(821\) 1.65900e13 1.27439 0.637193 0.770704i \(-0.280095\pi\)
0.637193 + 0.770704i \(0.280095\pi\)
\(822\) 2.37342e12 0.181322
\(823\) 9.87616e12 0.750393 0.375197 0.926945i \(-0.377575\pi\)
0.375197 + 0.926945i \(0.377575\pi\)
\(824\) −1.59025e13 −1.20169
\(825\) −1.25232e13 −0.941179
\(826\) 7.37794e11 0.0551474
\(827\) −3.15367e12 −0.234445 −0.117223 0.993106i \(-0.537399\pi\)
−0.117223 + 0.993106i \(0.537399\pi\)
\(828\) −2.42012e11 −0.0178937
\(829\) 2.02328e13 1.48785 0.743927 0.668261i \(-0.232961\pi\)
0.743927 + 0.668261i \(0.232961\pi\)
\(830\) −3.19444e12 −0.233638
\(831\) −2.80054e12 −0.203722
\(832\) 1.78748e13 1.29326
\(833\) −5.82302e11 −0.0419030
\(834\) 7.46372e12 0.534205
\(835\) 2.15459e13 1.53383
\(836\) −3.16353e12 −0.223998
\(837\) 3.02576e12 0.213093
\(838\) −1.67782e13 −1.17530
\(839\) −2.38915e13 −1.66462 −0.832310 0.554311i \(-0.812982\pi\)
−0.832310 + 0.554311i \(0.812982\pi\)
\(840\) 8.02348e12 0.556040
\(841\) 2.53993e13 1.75081
\(842\) −6.55980e12 −0.449766
\(843\) −2.68682e12 −0.183238
\(844\) −1.00686e13 −0.683015
\(845\) −2.06823e13 −1.39554
\(846\) −3.44398e12 −0.231150
\(847\) 3.70461e12 0.247324
\(848\) −3.55242e12 −0.235908
\(849\) 2.01995e13 1.33431
\(850\) 6.81588e12 0.447855
\(851\) −1.66974e12 −0.109136
\(852\) −7.78264e12 −0.505998
\(853\) 1.68078e13 1.08703 0.543514 0.839400i \(-0.317094\pi\)
0.543514 + 0.839400i \(0.317094\pi\)
\(854\) −1.01442e12 −0.0652615
\(855\) −9.08988e12 −0.581716
\(856\) 2.27012e13 1.44516
\(857\) −6.68278e12 −0.423198 −0.211599 0.977357i \(-0.567867\pi\)
−0.211599 + 0.977357i \(0.567867\pi\)
\(858\) 7.31625e12 0.460888
\(859\) 2.04541e13 1.28177 0.640887 0.767635i \(-0.278567\pi\)
0.640887 + 0.767635i \(0.278567\pi\)
\(860\) 9.17308e12 0.571837
\(861\) 5.67170e12 0.351722
\(862\) −3.52198e12 −0.217272
\(863\) −1.87505e13 −1.15071 −0.575353 0.817905i \(-0.695135\pi\)
−0.575353 + 0.817905i \(0.695135\pi\)
\(864\) 1.43584e13 0.876583
\(865\) −1.92632e13 −1.16992
\(866\) 5.79867e12 0.350347
\(867\) −1.19025e13 −0.715405
\(868\) −5.49088e11 −0.0328324
\(869\) −1.33801e13 −0.795924
\(870\) −2.85796e13 −1.69130
\(871\) −3.01954e13 −1.77771
\(872\) −2.88762e12 −0.169128
\(873\) 5.61630e12 0.327255
\(874\) −1.15688e12 −0.0670634
\(875\) 1.19535e13 0.689380
\(876\) 7.19400e12 0.412764
\(877\) 2.02759e13 1.15740 0.578699 0.815542i \(-0.303561\pi\)
0.578699 + 0.815542i \(0.303561\pi\)
\(878\) −1.01162e13 −0.574500
\(879\) 1.53568e13 0.867665
\(880\) −6.59563e12 −0.370753
\(881\) −2.17613e13 −1.21701 −0.608505 0.793550i \(-0.708230\pi\)
−0.608505 + 0.793550i \(0.708230\pi\)
\(882\) 7.42364e11 0.0413056
\(883\) −3.44540e13 −1.90729 −0.953645 0.300933i \(-0.902702\pi\)
−0.953645 + 0.300933i \(0.902702\pi\)
\(884\) 3.16270e12 0.174190
\(885\) −4.87183e12 −0.266961
\(886\) 5.15505e12 0.281048
\(887\) 5.98069e12 0.324410 0.162205 0.986757i \(-0.448139\pi\)
0.162205 + 0.986757i \(0.448139\pi\)
\(888\) 1.63342e13 0.881534
\(889\) −1.01070e13 −0.542707
\(890\) 1.39114e13 0.743217
\(891\) 5.11751e12 0.272025
\(892\) −5.45638e12 −0.288578
\(893\) 1.30759e13 0.688084
\(894\) 1.42430e12 0.0745732
\(895\) −1.12153e13 −0.584260
\(896\) −6.38384e11 −0.0330900
\(897\) −2.12503e12 −0.109597
\(898\) −4.06774e12 −0.208742
\(899\) 6.37418e12 0.325466
\(900\) 6.90165e12 0.350640
\(901\) −3.78778e12 −0.191480
\(902\) −1.03735e13 −0.521789
\(903\) −4.37579e12 −0.219009
\(904\) −1.89637e13 −0.944422
\(905\) 1.63641e12 0.0810910
\(906\) 1.82679e13 0.900766
\(907\) −2.51683e12 −0.123487 −0.0617434 0.998092i \(-0.519666\pi\)
−0.0617434 + 0.998092i \(0.519666\pi\)
\(908\) 3.88897e10 0.00189867
\(909\) −1.50504e12 −0.0731158
\(910\) −1.36648e13 −0.660569
\(911\) 3.96021e13 1.90496 0.952478 0.304607i \(-0.0985252\pi\)
0.952478 + 0.304607i \(0.0985252\pi\)
\(912\) 5.08648e12 0.243467
\(913\) 2.21365e12 0.105437
\(914\) −3.99044e12 −0.189131
\(915\) 6.69846e12 0.315922
\(916\) 2.53338e12 0.118897
\(917\) 9.89525e12 0.462131
\(918\) −5.11667e12 −0.237791
\(919\) −4.00850e13 −1.85380 −0.926899 0.375310i \(-0.877536\pi\)
−0.926899 + 0.375310i \(0.877536\pi\)
\(920\) 4.26237e12 0.196158
\(921\) −1.34671e13 −0.616746
\(922\) 1.97844e13 0.901640
\(923\) 4.31975e13 1.95907
\(924\) −1.70603e12 −0.0769953
\(925\) 4.76174e13 2.13859
\(926\) 3.02263e13 1.35094
\(927\) 9.71579e12 0.432135
\(928\) 3.02479e13 1.33884
\(929\) −8.38470e12 −0.369332 −0.184666 0.982801i \(-0.559120\pi\)
−0.184666 + 0.982801i \(0.559120\pi\)
\(930\) −4.56497e12 −0.200108
\(931\) −2.81857e12 −0.122958
\(932\) −1.48336e12 −0.0643986
\(933\) −1.58265e13 −0.683783
\(934\) 6.43944e12 0.276877
\(935\) −7.03261e12 −0.300929
\(936\) −1.31406e13 −0.559596
\(937\) 2.33983e12 0.0991645 0.0495822 0.998770i \(-0.484211\pi\)
0.0495822 + 0.998770i \(0.484211\pi\)
\(938\) −8.86501e12 −0.373909
\(939\) −3.01725e13 −1.26653
\(940\) −1.47825e13 −0.617550
\(941\) −6.45893e12 −0.268539 −0.134270 0.990945i \(-0.542869\pi\)
−0.134270 + 0.990945i \(0.542869\pi\)
\(942\) −1.49166e13 −0.617221
\(943\) 3.01302e12 0.124079
\(944\) −1.72327e12 −0.0706285
\(945\) −1.75589e13 −0.716231
\(946\) 8.00327e12 0.324906
\(947\) −2.43323e13 −0.983123 −0.491561 0.870843i \(-0.663574\pi\)
−0.491561 + 0.870843i \(0.663574\pi\)
\(948\) −1.16653e13 −0.469093
\(949\) −3.99302e13 −1.59810
\(950\) 3.29916e13 1.31416
\(951\) −1.35228e13 −0.536109
\(952\) 3.02611e12 0.119404
\(953\) 2.50351e12 0.0983177 0.0491589 0.998791i \(-0.484346\pi\)
0.0491589 + 0.998791i \(0.484346\pi\)
\(954\) 4.82896e12 0.188749
\(955\) −2.45551e13 −0.955269
\(956\) 2.05932e12 0.0797377
\(957\) 1.98048e13 0.763250
\(958\) 2.89108e13 1.10896
\(959\) 3.07189e12 0.117279
\(960\) −3.46527e13 −1.31679
\(961\) −2.54215e13 −0.961492
\(962\) −2.78188e13 −1.04725
\(963\) −1.38695e13 −0.519687
\(964\) 1.21573e13 0.453408
\(965\) 4.82576e13 1.79140
\(966\) −6.23883e11 −0.0230519
\(967\) 2.19887e13 0.808686 0.404343 0.914607i \(-0.367500\pi\)
0.404343 + 0.914607i \(0.367500\pi\)
\(968\) −1.92521e13 −0.704757
\(969\) 5.42348e12 0.197615
\(970\) −3.03512e13 −1.10079
\(971\) −9.07743e12 −0.327700 −0.163850 0.986485i \(-0.552391\pi\)
−0.163850 + 0.986485i \(0.552391\pi\)
\(972\) −8.91568e12 −0.320373
\(973\) 9.66019e12 0.345524
\(974\) −1.24787e13 −0.444276
\(975\) 6.06013e13 2.14764
\(976\) 2.36939e12 0.0835818
\(977\) 2.73152e13 0.959134 0.479567 0.877505i \(-0.340794\pi\)
0.479567 + 0.877505i \(0.340794\pi\)
\(978\) 1.68320e13 0.588316
\(979\) −9.64018e12 −0.335400
\(980\) 3.18643e12 0.110354
\(981\) 1.76421e12 0.0608192
\(982\) −2.56962e13 −0.881795
\(983\) 1.07541e13 0.367351 0.183676 0.982987i \(-0.441200\pi\)
0.183676 + 0.982987i \(0.441200\pi\)
\(984\) −2.94747e13 −1.00224
\(985\) 2.62188e13 0.887464
\(986\) −1.07790e13 −0.363188
\(987\) 7.05162e12 0.236517
\(988\) 1.53087e13 0.511131
\(989\) −2.32458e12 −0.0772612
\(990\) 8.96573e12 0.296638
\(991\) −2.06047e13 −0.678633 −0.339317 0.940672i \(-0.610196\pi\)
−0.339317 + 0.940672i \(0.610196\pi\)
\(992\) 4.83144e12 0.158407
\(993\) 3.70352e13 1.20877
\(994\) 1.26822e13 0.412057
\(995\) 3.85474e12 0.124678
\(996\) 1.92995e12 0.0621411
\(997\) 2.30380e13 0.738444 0.369222 0.929341i \(-0.379624\pi\)
0.369222 + 0.929341i \(0.379624\pi\)
\(998\) 2.83320e13 0.904046
\(999\) −3.57463e13 −1.13550
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 7.10.a.a.1.1 2
3.2 odd 2 63.10.a.d.1.2 2
4.3 odd 2 112.10.a.e.1.1 2
5.2 odd 4 175.10.b.b.99.1 4
5.3 odd 4 175.10.b.b.99.4 4
5.4 even 2 175.10.a.b.1.2 2
7.2 even 3 49.10.c.c.18.2 4
7.3 odd 6 49.10.c.b.30.2 4
7.4 even 3 49.10.c.c.30.2 4
7.5 odd 6 49.10.c.b.18.2 4
7.6 odd 2 49.10.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.1 2 1.1 even 1 trivial
49.10.a.b.1.1 2 7.6 odd 2
49.10.c.b.18.2 4 7.5 odd 6
49.10.c.b.30.2 4 7.3 odd 6
49.10.c.c.18.2 4 7.2 even 3
49.10.c.c.30.2 4 7.4 even 3
63.10.a.d.1.2 2 3.2 odd 2
112.10.a.e.1.1 2 4.3 odd 2
175.10.a.b.1.2 2 5.4 even 2
175.10.b.b.99.1 4 5.2 odd 4
175.10.b.b.99.4 4 5.3 odd 4