Properties

Label 69.8.a.a.1.4
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5545667584\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(12.4672\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} -509.647 q^{10} +500.265 q^{11} -740.672 q^{12} -12895.9 q^{13} +14046.8 q^{14} +1103.73 q^{15} -19142.8 q^{16} -23798.2 q^{17} +9088.62 q^{18} -32879.5 q^{19} -1121.40 q^{20} -30420.8 q^{21} +6236.93 q^{22} +12167.0 q^{23} +33852.7 q^{24} -76453.9 q^{25} -160777. q^{26} -19683.0 q^{27} +30907.8 q^{28} -152213. q^{29} +13760.5 q^{30} +252935. q^{31} -78171.4 q^{32} -13507.2 q^{33} -296698. q^{34} -46058.0 q^{35} +19998.1 q^{36} +89429.5 q^{37} -409916. q^{38} +348190. q^{39} +51254.0 q^{40} +549274. q^{41} -379263. q^{42} -330324. q^{43} +13723.4 q^{44} -29800.7 q^{45} +151689. q^{46} +834394. q^{47} +516856. q^{48} +445899. q^{49} -953170. q^{50} +642551. q^{51} -353765. q^{52} -1.73470e6 q^{53} -245393. q^{54} -20450.3 q^{55} -1.41265e6 q^{56} +887745. q^{57} -1.89768e6 q^{58} -2.58024e6 q^{59} +30277.8 q^{60} -2.40055e6 q^{61} +3.15340e6 q^{62} +821361. q^{63} +1.47570e6 q^{64} +527171. q^{65} -168397. q^{66} +3.57070e6 q^{67} -652839. q^{68} -328509. q^{69} -574216. q^{70} +709607. q^{71} -914022. q^{72} +797398. q^{73} +1.11494e6 q^{74} +2.06426e6 q^{75} -901959. q^{76} +563647. q^{77} +4.34098e6 q^{78} -1.98775e6 q^{79} +782535. q^{80} +531441. q^{81} +6.84793e6 q^{82} +8.20364e6 q^{83} -834512. q^{84} +972842. q^{85} -4.11823e6 q^{86} +4.10976e6 q^{87} -627234. q^{88} +8.23613e6 q^{89} -371532. q^{90} -1.45298e7 q^{91} +333769. q^{92} -6.82924e6 q^{93} +1.04026e7 q^{94} +1.34407e6 q^{95} +2.11063e6 q^{96} +1.54790e6 q^{97} +5.55914e6 q^{98} +364694. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 135q^{3} + 270q^{4} - 266q^{5} - 496q^{7} + 1422q^{8} + 3645q^{9} + O(q^{10}) \) \( 5q - 135q^{3} + 270q^{4} - 266q^{5} - 496q^{7} + 1422q^{8} + 3645q^{9} + 1452q^{10} - 1148q^{11} - 7290q^{12} - 642q^{13} + 5756q^{14} + 7182q^{15} - 22606q^{16} - 5798q^{17} - 6036q^{19} - 27376q^{20} + 13392q^{21} - 97896q^{22} + 60835q^{23} - 38394q^{24} - 262477q^{25} - 355992q^{26} - 98415q^{27} - 507124q^{28} - 169162q^{29} - 39204q^{30} - 199640q^{31} - 284794q^{32} + 30996q^{33} - 1027740q^{34} - 137680q^{35} + 196830q^{36} - 202002q^{37} - 554924q^{38} + 17334q^{39} - 340904q^{40} + 541282q^{41} - 155412q^{42} - 909596q^{43} - 1236032q^{44} - 193914q^{45} + 80208q^{47} + 610362q^{48} + 850589q^{49} - 941416q^{50} + 156546q^{51} + 146940q^{52} - 278138q^{53} - 933560q^{55} - 539932q^{56} + 162972q^{57} - 3522712q^{58} - 3177380q^{59} + 739152q^{60} + 147782q^{61} + 4606456q^{62} - 361584q^{63} - 4142622q^{64} + 3877332q^{65} + 2643192q^{66} - 464916q^{67} + 7513072q^{68} - 1642545q^{69} + 2093200q^{70} + 1576792q^{71} + 1036638q^{72} - 38190q^{73} + 12164864q^{74} + 7086879q^{75} + 6889436q^{76} + 10332384q^{77} + 9611784q^{78} - 3913336q^{79} + 6334776q^{80} + 2657205q^{81} + 6799360q^{82} + 15774716q^{83} + 13692348q^{84} - 8520740q^{85} + 24874084q^{86} + 4567374q^{87} + 53216q^{88} + 1116482q^{89} + 1058508q^{90} - 27369552q^{91} + 3285090q^{92} + 5390280q^{93} - 7153744q^{94} - 6067832q^{95} + 7689438q^{96} - 15738566q^{97} + 11730488q^{98} - 836892q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.4672 1.10196 0.550980 0.834519i \(-0.314254\pi\)
0.550980 + 0.834519i \(0.314254\pi\)
\(3\) −27.0000 −0.577350
\(4\) 27.4323 0.214315
\(5\) −40.8788 −0.146253 −0.0731263 0.997323i \(-0.523298\pi\)
−0.0731263 + 0.997323i \(0.523298\pi\)
\(6\) −336.616 −0.636217
\(7\) 1126.70 1.24155 0.620774 0.783990i \(-0.286819\pi\)
0.620774 + 0.783990i \(0.286819\pi\)
\(8\) −1253.80 −0.865793
\(9\) 729.000 0.333333
\(10\) −509.647 −0.161164
\(11\) 500.265 0.113325 0.0566626 0.998393i \(-0.481954\pi\)
0.0566626 + 0.998393i \(0.481954\pi\)
\(12\) −740.672 −0.123735
\(13\) −12895.9 −1.62799 −0.813994 0.580873i \(-0.802711\pi\)
−0.813994 + 0.580873i \(0.802711\pi\)
\(14\) 14046.8 1.36814
\(15\) 1103.73 0.0844390
\(16\) −19142.8 −1.16838
\(17\) −23798.2 −1.17482 −0.587412 0.809288i \(-0.699853\pi\)
−0.587412 + 0.809288i \(0.699853\pi\)
\(18\) 9088.62 0.367320
\(19\) −32879.5 −1.09973 −0.549866 0.835253i \(-0.685321\pi\)
−0.549866 + 0.835253i \(0.685321\pi\)
\(20\) −1121.40 −0.0313441
\(21\) −30420.8 −0.716808
\(22\) 6236.93 0.124880
\(23\) 12167.0 0.208514
\(24\) 33852.7 0.499866
\(25\) −76453.9 −0.978610
\(26\) −160777. −1.79398
\(27\) −19683.0 −0.192450
\(28\) 30907.8 0.266082
\(29\) −152213. −1.15894 −0.579468 0.814995i \(-0.696740\pi\)
−0.579468 + 0.814995i \(0.696740\pi\)
\(30\) 13760.5 0.0930483
\(31\) 252935. 1.52490 0.762452 0.647045i \(-0.223995\pi\)
0.762452 + 0.647045i \(0.223995\pi\)
\(32\) −78171.4 −0.421719
\(33\) −13507.2 −0.0654283
\(34\) −296698. −1.29461
\(35\) −46058.0 −0.181580
\(36\) 19998.1 0.0714383
\(37\) 89429.5 0.290252 0.145126 0.989413i \(-0.453641\pi\)
0.145126 + 0.989413i \(0.453641\pi\)
\(38\) −409916. −1.21186
\(39\) 348190. 0.939920
\(40\) 51254.0 0.126624
\(41\) 549274. 1.24464 0.622322 0.782761i \(-0.286189\pi\)
0.622322 + 0.782761i \(0.286189\pi\)
\(42\) −379263. −0.789893
\(43\) −330324. −0.633578 −0.316789 0.948496i \(-0.602605\pi\)
−0.316789 + 0.948496i \(0.602605\pi\)
\(44\) 13723.4 0.0242873
\(45\) −29800.7 −0.0487509
\(46\) 151689. 0.229774
\(47\) 834394. 1.17227 0.586136 0.810213i \(-0.300648\pi\)
0.586136 + 0.810213i \(0.300648\pi\)
\(48\) 516856. 0.674567
\(49\) 445899. 0.541440
\(50\) −953170. −1.07839
\(51\) 642551. 0.678284
\(52\) −353765. −0.348902
\(53\) −1.73470e6 −1.60051 −0.800257 0.599658i \(-0.795303\pi\)
−0.800257 + 0.599658i \(0.795303\pi\)
\(54\) −245393. −0.212072
\(55\) −20450.3 −0.0165741
\(56\) −1.41265e6 −1.07492
\(57\) 887745. 0.634931
\(58\) −1.89768e6 −1.27710
\(59\) −2.58024e6 −1.63560 −0.817801 0.575502i \(-0.804807\pi\)
−0.817801 + 0.575502i \(0.804807\pi\)
\(60\) 30277.8 0.0180965
\(61\) −2.40055e6 −1.35412 −0.677060 0.735928i \(-0.736746\pi\)
−0.677060 + 0.735928i \(0.736746\pi\)
\(62\) 3.15340e6 1.68038
\(63\) 821361. 0.413849
\(64\) 1.47570e6 0.703667
\(65\) 527171. 0.238097
\(66\) −168397. −0.0720993
\(67\) 3.57070e6 1.45041 0.725206 0.688532i \(-0.241744\pi\)
0.725206 + 0.688532i \(0.241744\pi\)
\(68\) −652839. −0.251782
\(69\) −328509. −0.120386
\(70\) −574216. −0.200093
\(71\) 709607. 0.235296 0.117648 0.993055i \(-0.462465\pi\)
0.117648 + 0.993055i \(0.462465\pi\)
\(72\) −914022. −0.288598
\(73\) 797398. 0.239908 0.119954 0.992779i \(-0.461725\pi\)
0.119954 + 0.992779i \(0.461725\pi\)
\(74\) 1.11494e6 0.319846
\(75\) 2.06426e6 0.565001
\(76\) −901959. −0.235689
\(77\) 563647. 0.140699
\(78\) 4.34098e6 1.03575
\(79\) −1.98775e6 −0.453594 −0.226797 0.973942i \(-0.572825\pi\)
−0.226797 + 0.973942i \(0.572825\pi\)
\(80\) 782535. 0.170879
\(81\) 531441. 0.111111
\(82\) 6.84793e6 1.37155
\(83\) 8.20364e6 1.57483 0.787414 0.616425i \(-0.211419\pi\)
0.787414 + 0.616425i \(0.211419\pi\)
\(84\) −834512. −0.153623
\(85\) 972842. 0.171821
\(86\) −4.11823e6 −0.698177
\(87\) 4.10976e6 0.669112
\(88\) −627234. −0.0981161
\(89\) 8.23613e6 1.23839 0.619196 0.785236i \(-0.287459\pi\)
0.619196 + 0.785236i \(0.287459\pi\)
\(90\) −371532. −0.0537215
\(91\) −1.45298e7 −2.02123
\(92\) 333769. 0.0446877
\(93\) −6.82924e6 −0.880404
\(94\) 1.04026e7 1.29180
\(95\) 1.34407e6 0.160839
\(96\) 2.11063e6 0.243479
\(97\) 1.54790e6 0.172203 0.0861015 0.996286i \(-0.472559\pi\)
0.0861015 + 0.996286i \(0.472559\pi\)
\(98\) 5.55914e6 0.596645
\(99\) 364694. 0.0377750
\(100\) −2.09731e6 −0.209731
\(101\) 1.26141e7 1.21824 0.609119 0.793079i \(-0.291523\pi\)
0.609119 + 0.793079i \(0.291523\pi\)
\(102\) 8.01084e6 0.747442
\(103\) −1.53226e7 −1.38166 −0.690830 0.723017i \(-0.742755\pi\)
−0.690830 + 0.723017i \(0.742755\pi\)
\(104\) 1.61690e7 1.40950
\(105\) 1.24357e6 0.104835
\(106\) −2.16270e7 −1.76370
\(107\) 3.58516e6 0.282921 0.141461 0.989944i \(-0.454820\pi\)
0.141461 + 0.989944i \(0.454820\pi\)
\(108\) −539950. −0.0412449
\(109\) −1.14000e7 −0.843163 −0.421582 0.906790i \(-0.638525\pi\)
−0.421582 + 0.906790i \(0.638525\pi\)
\(110\) −254959. −0.0182640
\(111\) −2.41460e6 −0.167577
\(112\) −2.15681e7 −1.45060
\(113\) −1.16966e7 −0.762579 −0.381290 0.924456i \(-0.624520\pi\)
−0.381290 + 0.924456i \(0.624520\pi\)
\(114\) 1.10677e7 0.699668
\(115\) −497373. −0.0304958
\(116\) −4.17556e6 −0.248377
\(117\) −9.40114e6 −0.542663
\(118\) −3.21685e7 −1.80237
\(119\) −2.68133e7 −1.45860
\(120\) −1.38386e6 −0.0731067
\(121\) −1.92369e7 −0.987157
\(122\) −2.99283e7 −1.49219
\(123\) −1.48304e7 −0.718596
\(124\) 6.93858e6 0.326810
\(125\) 6.31901e6 0.289377
\(126\) 1.02401e7 0.456045
\(127\) 4.09153e7 1.77244 0.886222 0.463262i \(-0.153321\pi\)
0.886222 + 0.463262i \(0.153321\pi\)
\(128\) 2.84038e7 1.19713
\(129\) 8.91874e6 0.365796
\(130\) 6.57237e6 0.262374
\(131\) 9.54150e6 0.370823 0.185412 0.982661i \(-0.440638\pi\)
0.185412 + 0.982661i \(0.440638\pi\)
\(132\) −370533. −0.0140223
\(133\) −3.70451e7 −1.36537
\(134\) 4.45168e7 1.59830
\(135\) 804618. 0.0281463
\(136\) 2.98382e7 1.01715
\(137\) −3.72077e7 −1.23626 −0.618132 0.786075i \(-0.712110\pi\)
−0.618132 + 0.786075i \(0.712110\pi\)
\(138\) −4.09560e6 −0.132660
\(139\) −4.43290e7 −1.40003 −0.700013 0.714130i \(-0.746823\pi\)
−0.700013 + 0.714130i \(0.746823\pi\)
\(140\) −1.26348e6 −0.0389152
\(141\) −2.25286e7 −0.676812
\(142\) 8.84685e6 0.259286
\(143\) −6.45139e6 −0.184492
\(144\) −1.39551e7 −0.389461
\(145\) 6.22230e6 0.169497
\(146\) 9.94137e6 0.264369
\(147\) −1.20393e7 −0.312601
\(148\) 2.45326e6 0.0622053
\(149\) 2.13993e7 0.529965 0.264983 0.964253i \(-0.414634\pi\)
0.264983 + 0.964253i \(0.414634\pi\)
\(150\) 2.57356e7 0.622608
\(151\) −5.39107e7 −1.27425 −0.637126 0.770759i \(-0.719877\pi\)
−0.637126 + 0.770759i \(0.719877\pi\)
\(152\) 4.12243e7 0.952141
\(153\) −1.73489e7 −0.391608
\(154\) 7.02712e6 0.155044
\(155\) −1.03397e7 −0.223021
\(156\) 9.55167e6 0.201439
\(157\) −4.92467e6 −0.101561 −0.0507807 0.998710i \(-0.516171\pi\)
−0.0507807 + 0.998710i \(0.516171\pi\)
\(158\) −2.47818e7 −0.499842
\(159\) 4.68369e7 0.924057
\(160\) 3.19555e6 0.0616774
\(161\) 1.37085e7 0.258881
\(162\) 6.62561e6 0.122440
\(163\) −5.24341e7 −0.948326 −0.474163 0.880437i \(-0.657249\pi\)
−0.474163 + 0.880437i \(0.657249\pi\)
\(164\) 1.50678e7 0.266746
\(165\) 552157. 0.00956905
\(166\) 1.02277e8 1.73540
\(167\) 5.66790e7 0.941704 0.470852 0.882212i \(-0.343947\pi\)
0.470852 + 0.882212i \(0.343947\pi\)
\(168\) 3.81416e7 0.620607
\(169\) 1.03557e8 1.65035
\(170\) 1.21287e7 0.189340
\(171\) −2.39691e7 −0.366577
\(172\) −9.06154e6 −0.135785
\(173\) −4.71933e7 −0.692977 −0.346489 0.938054i \(-0.612626\pi\)
−0.346489 + 0.938054i \(0.612626\pi\)
\(174\) 5.12374e7 0.737335
\(175\) −8.61403e7 −1.21499
\(176\) −9.57648e6 −0.132407
\(177\) 6.96664e7 0.944315
\(178\) 1.02682e8 1.36466
\(179\) 9.63068e6 0.125508 0.0627540 0.998029i \(-0.480012\pi\)
0.0627540 + 0.998029i \(0.480012\pi\)
\(180\) −817501. −0.0104480
\(181\) 1.36975e8 1.71699 0.858494 0.512823i \(-0.171400\pi\)
0.858494 + 0.512823i \(0.171400\pi\)
\(182\) −1.81147e8 −2.22731
\(183\) 6.48150e7 0.781802
\(184\) −1.52550e7 −0.180530
\(185\) −3.65577e6 −0.0424501
\(186\) −8.51418e7 −0.970169
\(187\) −1.19054e7 −0.133137
\(188\) 2.28893e7 0.251235
\(189\) −2.21767e7 −0.238936
\(190\) 1.67569e7 0.177238
\(191\) 5.87953e7 0.610556 0.305278 0.952263i \(-0.401251\pi\)
0.305278 + 0.952263i \(0.401251\pi\)
\(192\) −3.98438e7 −0.406262
\(193\) 1.44972e6 0.0145155 0.00725776 0.999974i \(-0.497690\pi\)
0.00725776 + 0.999974i \(0.497690\pi\)
\(194\) 1.92980e7 0.189761
\(195\) −1.42336e7 −0.137466
\(196\) 1.22320e7 0.116039
\(197\) −6.62738e7 −0.617604 −0.308802 0.951126i \(-0.599928\pi\)
−0.308802 + 0.951126i \(0.599928\pi\)
\(198\) 4.54672e6 0.0416266
\(199\) −1.78732e8 −1.60774 −0.803871 0.594803i \(-0.797230\pi\)
−0.803871 + 0.594803i \(0.797230\pi\)
\(200\) 9.58581e7 0.847274
\(201\) −9.64089e7 −0.837396
\(202\) 1.57263e8 1.34245
\(203\) −1.71498e8 −1.43888
\(204\) 1.76266e7 0.145366
\(205\) −2.24537e7 −0.182032
\(206\) −1.91030e8 −1.52253
\(207\) 8.86974e6 0.0695048
\(208\) 2.46864e8 1.90212
\(209\) −1.64485e7 −0.124627
\(210\) 1.55038e7 0.115524
\(211\) −1.07572e7 −0.0788335 −0.0394167 0.999223i \(-0.512550\pi\)
−0.0394167 + 0.999223i \(0.512550\pi\)
\(212\) −4.75869e7 −0.343014
\(213\) −1.91594e7 −0.135848
\(214\) 4.46971e7 0.311768
\(215\) 1.35032e7 0.0926624
\(216\) 2.46786e7 0.166622
\(217\) 2.84980e8 1.89324
\(218\) −1.42126e8 −0.929132
\(219\) −2.15298e7 −0.138511
\(220\) −560998. −0.00355207
\(221\) 3.06900e8 1.91260
\(222\) −3.01034e7 −0.184663
\(223\) −2.14285e8 −1.29397 −0.646986 0.762502i \(-0.723971\pi\)
−0.646986 + 0.762502i \(0.723971\pi\)
\(224\) −8.80753e7 −0.523584
\(225\) −5.57349e7 −0.326203
\(226\) −1.45824e8 −0.840331
\(227\) −7.33851e7 −0.416406 −0.208203 0.978086i \(-0.566762\pi\)
−0.208203 + 0.978086i \(0.566762\pi\)
\(228\) 2.43529e7 0.136075
\(229\) −2.83323e7 −0.155904 −0.0779519 0.996957i \(-0.524838\pi\)
−0.0779519 + 0.996957i \(0.524838\pi\)
\(230\) −6.20087e6 −0.0336051
\(231\) −1.52185e7 −0.0812323
\(232\) 1.90845e8 1.00340
\(233\) 1.15251e8 0.596896 0.298448 0.954426i \(-0.403531\pi\)
0.298448 + 0.954426i \(0.403531\pi\)
\(234\) −1.17206e8 −0.597993
\(235\) −3.41090e7 −0.171448
\(236\) −7.07818e7 −0.350534
\(237\) 5.36693e7 0.261883
\(238\) −3.34288e8 −1.60732
\(239\) 5.89008e7 0.279080 0.139540 0.990216i \(-0.455438\pi\)
0.139540 + 0.990216i \(0.455438\pi\)
\(240\) −2.11285e7 −0.0986571
\(241\) −5.12886e7 −0.236027 −0.118013 0.993012i \(-0.537653\pi\)
−0.118013 + 0.993012i \(0.537653\pi\)
\(242\) −2.39831e8 −1.08781
\(243\) −1.43489e7 −0.0641500
\(244\) −6.58527e7 −0.290208
\(245\) −1.82278e7 −0.0791870
\(246\) −1.84894e8 −0.791863
\(247\) 4.24012e8 1.79035
\(248\) −3.17130e8 −1.32025
\(249\) −2.21498e8 −0.909227
\(250\) 7.87806e7 0.318882
\(251\) −3.11308e7 −0.124260 −0.0621300 0.998068i \(-0.519789\pi\)
−0.0621300 + 0.998068i \(0.519789\pi\)
\(252\) 2.25318e7 0.0886940
\(253\) 6.08673e6 0.0236299
\(254\) 5.10101e8 1.95316
\(255\) −2.62667e7 −0.0992008
\(256\) 1.65228e8 0.615523
\(257\) 2.38296e8 0.875690 0.437845 0.899051i \(-0.355742\pi\)
0.437845 + 0.899051i \(0.355742\pi\)
\(258\) 1.11192e8 0.403093
\(259\) 1.00760e8 0.360361
\(260\) 1.44615e7 0.0510278
\(261\) −1.10964e8 −0.386312
\(262\) 1.18956e8 0.408632
\(263\) 2.84255e8 0.963525 0.481762 0.876302i \(-0.339997\pi\)
0.481762 + 0.876302i \(0.339997\pi\)
\(264\) 1.69353e7 0.0566474
\(265\) 7.09126e7 0.234079
\(266\) −4.61851e8 −1.50458
\(267\) −2.22376e8 −0.714986
\(268\) 9.79525e7 0.310845
\(269\) −1.94289e8 −0.608576 −0.304288 0.952580i \(-0.598419\pi\)
−0.304288 + 0.952580i \(0.598419\pi\)
\(270\) 1.00314e7 0.0310161
\(271\) −3.69086e8 −1.12651 −0.563254 0.826284i \(-0.690451\pi\)
−0.563254 + 0.826284i \(0.690451\pi\)
\(272\) 4.55564e8 1.37264
\(273\) 3.92305e8 1.16695
\(274\) −4.63878e8 −1.36231
\(275\) −3.82473e7 −0.110901
\(276\) −9.01176e6 −0.0258005
\(277\) −5.28946e8 −1.49531 −0.747657 0.664086i \(-0.768821\pi\)
−0.747657 + 0.664086i \(0.768821\pi\)
\(278\) −5.52661e8 −1.54277
\(279\) 1.84389e8 0.508301
\(280\) 5.77476e7 0.157210
\(281\) 4.54274e7 0.122137 0.0610683 0.998134i \(-0.480549\pi\)
0.0610683 + 0.998134i \(0.480549\pi\)
\(282\) −2.80870e8 −0.745819
\(283\) 2.28233e8 0.598585 0.299293 0.954161i \(-0.403249\pi\)
0.299293 + 0.954161i \(0.403249\pi\)
\(284\) 1.94662e7 0.0504274
\(285\) −3.62900e7 −0.0928603
\(286\) −8.04311e7 −0.203303
\(287\) 6.18864e8 1.54528
\(288\) −5.69869e7 −0.140573
\(289\) 1.56015e8 0.380210
\(290\) 7.75750e7 0.186779
\(291\) −4.17932e7 −0.0994214
\(292\) 2.18745e7 0.0514159
\(293\) 4.49349e8 1.04363 0.521815 0.853058i \(-0.325255\pi\)
0.521815 + 0.853058i \(0.325255\pi\)
\(294\) −1.50097e8 −0.344473
\(295\) 1.05477e8 0.239211
\(296\) −1.12127e8 −0.251298
\(297\) −9.84673e6 −0.0218094
\(298\) 2.66790e8 0.584000
\(299\) −1.56905e8 −0.339459
\(300\) 5.66273e7 0.121088
\(301\) −3.72174e8 −0.786617
\(302\) −6.72118e8 −1.40418
\(303\) −3.40581e8 −0.703350
\(304\) 6.29405e8 1.28491
\(305\) 9.81319e7 0.198044
\(306\) −2.16293e8 −0.431536
\(307\) −3.56204e8 −0.702609 −0.351305 0.936261i \(-0.614262\pi\)
−0.351305 + 0.936261i \(0.614262\pi\)
\(308\) 1.54621e7 0.0301538
\(309\) 4.13709e8 0.797702
\(310\) −1.28907e8 −0.245760
\(311\) 7.14279e8 1.34650 0.673250 0.739415i \(-0.264898\pi\)
0.673250 + 0.739415i \(0.264898\pi\)
\(312\) −4.36562e8 −0.813776
\(313\) 9.57174e7 0.176435 0.0882177 0.996101i \(-0.471883\pi\)
0.0882177 + 0.996101i \(0.471883\pi\)
\(314\) −6.13971e7 −0.111916
\(315\) −3.35763e7 −0.0605265
\(316\) −5.45286e7 −0.0972120
\(317\) −6.88984e8 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(318\) 5.83928e8 1.01827
\(319\) −7.61471e7 −0.131337
\(320\) −6.03248e7 −0.102913
\(321\) −9.67993e7 −0.163345
\(322\) 1.70907e8 0.285276
\(323\) 7.82471e8 1.29199
\(324\) 1.45787e7 0.0238128
\(325\) 9.85945e8 1.59317
\(326\) −6.53709e8 −1.04502
\(327\) 3.07800e8 0.486801
\(328\) −6.88681e8 −1.07760
\(329\) 9.40108e8 1.45543
\(330\) 6.88388e6 0.0105447
\(331\) 8.37330e8 1.26911 0.634554 0.772878i \(-0.281184\pi\)
0.634554 + 0.772878i \(0.281184\pi\)
\(332\) 2.25045e8 0.337509
\(333\) 6.51941e7 0.0967506
\(334\) 7.06631e8 1.03772
\(335\) −1.45966e8 −0.212127
\(336\) 5.82339e8 0.837507
\(337\) −1.24811e9 −1.77643 −0.888213 0.459431i \(-0.848053\pi\)
−0.888213 + 0.459431i \(0.848053\pi\)
\(338\) 1.29107e9 1.81862
\(339\) 3.15808e8 0.440275
\(340\) 2.66873e7 0.0368238
\(341\) 1.26534e8 0.172810
\(342\) −2.98829e8 −0.403954
\(343\) −4.25489e8 −0.569324
\(344\) 4.14161e8 0.548547
\(345\) 1.34291e7 0.0176067
\(346\) −5.88371e8 −0.763633
\(347\) −2.11977e8 −0.272356 −0.136178 0.990684i \(-0.543482\pi\)
−0.136178 + 0.990684i \(0.543482\pi\)
\(348\) 1.12740e8 0.143401
\(349\) −1.03849e8 −0.130772 −0.0653858 0.997860i \(-0.520828\pi\)
−0.0653858 + 0.997860i \(0.520828\pi\)
\(350\) −1.07393e9 −1.33887
\(351\) 2.53831e8 0.313307
\(352\) −3.91064e7 −0.0477913
\(353\) 2.33920e8 0.283045 0.141523 0.989935i \(-0.454800\pi\)
0.141523 + 0.989935i \(0.454800\pi\)
\(354\) 8.68548e8 1.04060
\(355\) −2.90079e7 −0.0344126
\(356\) 2.25936e8 0.265406
\(357\) 7.23959e8 0.842122
\(358\) 1.20068e8 0.138305
\(359\) −1.66286e9 −1.89682 −0.948408 0.317054i \(-0.897306\pi\)
−0.948408 + 0.317054i \(0.897306\pi\)
\(360\) 3.73642e7 0.0422082
\(361\) 1.87187e8 0.209412
\(362\) 1.70771e9 1.89205
\(363\) 5.19396e8 0.569936
\(364\) −3.98586e8 −0.433179
\(365\) −3.25967e7 −0.0350872
\(366\) 8.08065e8 0.861514
\(367\) −5.37695e8 −0.567812 −0.283906 0.958852i \(-0.591630\pi\)
−0.283906 + 0.958852i \(0.591630\pi\)
\(368\) −2.32910e8 −0.243625
\(369\) 4.00420e8 0.414881
\(370\) −4.55774e7 −0.0467782
\(371\) −1.95448e9 −1.98711
\(372\) −1.87342e8 −0.188684
\(373\) −2.19239e8 −0.218744 −0.109372 0.994001i \(-0.534884\pi\)
−0.109372 + 0.994001i \(0.534884\pi\)
\(374\) −1.48428e8 −0.146712
\(375\) −1.70613e8 −0.167072
\(376\) −1.04617e9 −1.01495
\(377\) 1.96293e9 1.88674
\(378\) −2.76483e8 −0.263298
\(379\) 1.09529e9 1.03346 0.516730 0.856148i \(-0.327149\pi\)
0.516730 + 0.856148i \(0.327149\pi\)
\(380\) 3.68710e7 0.0344701
\(381\) −1.10471e9 −1.02332
\(382\) 7.33015e8 0.672808
\(383\) −1.49495e9 −1.35966 −0.679831 0.733368i \(-0.737947\pi\)
−0.679831 + 0.733368i \(0.737947\pi\)
\(384\) −7.66903e8 −0.691164
\(385\) −2.30412e7 −0.0205775
\(386\) 1.80740e7 0.0159955
\(387\) −2.40806e8 −0.211193
\(388\) 4.24623e7 0.0369057
\(389\) −1.06612e9 −0.918296 −0.459148 0.888360i \(-0.651845\pi\)
−0.459148 + 0.888360i \(0.651845\pi\)
\(390\) −1.77454e8 −0.151482
\(391\) −2.89552e8 −0.244968
\(392\) −5.59070e8 −0.468775
\(393\) −2.57620e8 −0.214095
\(394\) −8.26252e8 −0.680575
\(395\) 8.12570e7 0.0663393
\(396\) 1.00044e7 0.00809575
\(397\) 1.02859e9 0.825044 0.412522 0.910948i \(-0.364648\pi\)
0.412522 + 0.910948i \(0.364648\pi\)
\(398\) −2.22830e9 −1.77167
\(399\) 1.00022e9 0.788297
\(400\) 1.46354e9 1.14339
\(401\) 2.90455e8 0.224943 0.112472 0.993655i \(-0.464123\pi\)
0.112472 + 0.993655i \(0.464123\pi\)
\(402\) −1.20195e9 −0.922776
\(403\) −3.26183e9 −2.48253
\(404\) 3.46034e8 0.261087
\(405\) −2.17247e7 −0.0162503
\(406\) −2.13811e9 −1.58558
\(407\) 4.47385e7 0.0328928
\(408\) −8.05632e8 −0.587254
\(409\) −6.42289e8 −0.464193 −0.232097 0.972693i \(-0.574559\pi\)
−0.232097 + 0.972693i \(0.574559\pi\)
\(410\) −2.79935e8 −0.200592
\(411\) 1.00461e9 0.713757
\(412\) −4.20333e8 −0.296110
\(413\) −2.90714e9 −2.03068
\(414\) 1.10581e8 0.0765915
\(415\) −3.35355e8 −0.230323
\(416\) 1.00809e9 0.686553
\(417\) 1.19688e9 0.808306
\(418\) −2.05067e8 −0.137334
\(419\) 1.95033e9 1.29526 0.647632 0.761953i \(-0.275759\pi\)
0.647632 + 0.761953i \(0.275759\pi\)
\(420\) 3.41139e7 0.0224677
\(421\) −2.66647e9 −1.74160 −0.870801 0.491635i \(-0.836399\pi\)
−0.870801 + 0.491635i \(0.836399\pi\)
\(422\) −1.34113e8 −0.0868713
\(423\) 6.08273e8 0.390757
\(424\) 2.17497e9 1.38571
\(425\) 1.81946e9 1.14969
\(426\) −2.38865e8 −0.149699
\(427\) −2.70469e9 −1.68120
\(428\) 9.83492e7 0.0606342
\(429\) 1.74188e8 0.106516
\(430\) 1.68348e8 0.102110
\(431\) −1.45000e9 −0.872366 −0.436183 0.899858i \(-0.643670\pi\)
−0.436183 + 0.899858i \(0.643670\pi\)
\(432\) 3.76788e8 0.224856
\(433\) 8.59337e8 0.508693 0.254347 0.967113i \(-0.418140\pi\)
0.254347 + 0.967113i \(0.418140\pi\)
\(434\) 3.55292e9 2.08627
\(435\) −1.68002e8 −0.0978594
\(436\) −3.12728e8 −0.180702
\(437\) −4.00044e8 −0.229310
\(438\) −2.68417e8 −0.152634
\(439\) −2.27752e9 −1.28480 −0.642402 0.766368i \(-0.722062\pi\)
−0.642402 + 0.766368i \(0.722062\pi\)
\(440\) 2.56406e7 0.0143497
\(441\) 3.25061e8 0.180480
\(442\) 3.82620e9 2.10761
\(443\) 1.83034e8 0.100027 0.0500136 0.998749i \(-0.484074\pi\)
0.0500136 + 0.998749i \(0.484074\pi\)
\(444\) −6.62379e7 −0.0359142
\(445\) −3.36683e8 −0.181118
\(446\) −2.67154e9 −1.42590
\(447\) −5.77781e8 −0.305976
\(448\) 1.66266e9 0.873636
\(449\) 1.10637e9 0.576819 0.288409 0.957507i \(-0.406874\pi\)
0.288409 + 0.957507i \(0.406874\pi\)
\(450\) −6.94861e8 −0.359463
\(451\) 2.74783e8 0.141049
\(452\) −3.20865e8 −0.163432
\(453\) 1.45559e9 0.735690
\(454\) −9.14910e8 −0.458863
\(455\) 5.93961e8 0.295609
\(456\) −1.11306e9 −0.549719
\(457\) −1.67986e8 −0.0823314 −0.0411657 0.999152i \(-0.513107\pi\)
−0.0411657 + 0.999152i \(0.513107\pi\)
\(458\) −3.53225e8 −0.171800
\(459\) 4.68420e8 0.226095
\(460\) −1.36441e7 −0.00653570
\(461\) 2.56850e9 1.22103 0.610516 0.792004i \(-0.290962\pi\)
0.610516 + 0.792004i \(0.290962\pi\)
\(462\) −1.89732e8 −0.0895147
\(463\) −1.10660e9 −0.518150 −0.259075 0.965857i \(-0.583418\pi\)
−0.259075 + 0.965857i \(0.583418\pi\)
\(464\) 2.91379e9 1.35408
\(465\) 2.79171e8 0.128761
\(466\) 1.43686e9 0.657756
\(467\) 2.79649e9 1.27059 0.635294 0.772271i \(-0.280879\pi\)
0.635294 + 0.772271i \(0.280879\pi\)
\(468\) −2.57895e8 −0.116301
\(469\) 4.02309e9 1.80076
\(470\) −4.25246e8 −0.188929
\(471\) 1.32966e8 0.0586365
\(472\) 3.23511e9 1.41609
\(473\) −1.65249e8 −0.0718003
\(474\) 6.69109e8 0.288584
\(475\) 2.51376e9 1.07621
\(476\) −7.35551e8 −0.312599
\(477\) −1.26460e9 −0.533504
\(478\) 7.34331e8 0.307535
\(479\) 2.19334e9 0.911866 0.455933 0.890014i \(-0.349306\pi\)
0.455933 + 0.890014i \(0.349306\pi\)
\(480\) −8.62800e7 −0.0356095
\(481\) −1.15328e9 −0.472526
\(482\) −6.39428e8 −0.260092
\(483\) −3.70130e8 −0.149465
\(484\) −5.27713e8 −0.211563
\(485\) −6.32762e7 −0.0251851
\(486\) −1.78891e8 −0.0706907
\(487\) −1.48441e9 −0.582373 −0.291187 0.956666i \(-0.594050\pi\)
−0.291187 + 0.956666i \(0.594050\pi\)
\(488\) 3.00982e9 1.17239
\(489\) 1.41572e9 0.547516
\(490\) −2.27251e8 −0.0872609
\(491\) −3.58617e9 −1.36724 −0.683621 0.729837i \(-0.739596\pi\)
−0.683621 + 0.729837i \(0.739596\pi\)
\(492\) −4.06832e8 −0.154006
\(493\) 3.62240e9 1.36155
\(494\) 5.28626e9 1.97290
\(495\) −1.49082e7 −0.00552470
\(496\) −4.84188e9 −1.78167
\(497\) 7.99511e8 0.292131
\(498\) −2.76147e9 −1.00193
\(499\) −2.64545e9 −0.953122 −0.476561 0.879141i \(-0.658117\pi\)
−0.476561 + 0.879141i \(0.658117\pi\)
\(500\) 1.73345e8 0.0620177
\(501\) −1.53033e9 −0.543693
\(502\) −3.88115e8 −0.136930
\(503\) 4.47733e8 0.156867 0.0784334 0.996919i \(-0.475008\pi\)
0.0784334 + 0.996919i \(0.475008\pi\)
\(504\) −1.02982e9 −0.358308
\(505\) −5.15651e8 −0.178170
\(506\) 7.58848e7 0.0260392
\(507\) −2.79603e9 −0.952828
\(508\) 1.12240e9 0.379861
\(509\) −1.50184e9 −0.504792 −0.252396 0.967624i \(-0.581218\pi\)
−0.252396 + 0.967624i \(0.581218\pi\)
\(510\) −3.27474e8 −0.109315
\(511\) 8.98425e8 0.297858
\(512\) −1.57575e9 −0.518850
\(513\) 6.47166e8 0.211644
\(514\) 2.97089e9 0.964975
\(515\) 6.26368e8 0.202071
\(516\) 2.44661e8 0.0783956
\(517\) 4.17418e8 0.132848
\(518\) 1.25620e9 0.397104
\(519\) 1.27422e9 0.400091
\(520\) −6.60968e8 −0.206143
\(521\) 1.31386e9 0.407020 0.203510 0.979073i \(-0.434765\pi\)
0.203510 + 0.979073i \(0.434765\pi\)
\(522\) −1.38341e9 −0.425700
\(523\) 4.83261e9 1.47715 0.738577 0.674169i \(-0.235498\pi\)
0.738577 + 0.674169i \(0.235498\pi\)
\(524\) 2.61745e8 0.0794729
\(525\) 2.32579e9 0.701475
\(526\) 3.54388e9 1.06177
\(527\) −6.01938e9 −1.79149
\(528\) 2.58565e8 0.0764454
\(529\) 1.48036e8 0.0434783
\(530\) 8.84085e8 0.257946
\(531\) −1.88099e9 −0.545200
\(532\) −1.01623e9 −0.292619
\(533\) −7.08340e9 −2.02627
\(534\) −2.77241e9 −0.787886
\(535\) −1.46557e8 −0.0413779
\(536\) −4.47695e9 −1.25576
\(537\) −2.60028e8 −0.0724621
\(538\) −2.42225e9 −0.670626
\(539\) 2.23068e8 0.0613588
\(540\) 2.20725e7 0.00603217
\(541\) 8.09486e8 0.219796 0.109898 0.993943i \(-0.464948\pi\)
0.109898 + 0.993943i \(0.464948\pi\)
\(542\) −4.60148e9 −1.24137
\(543\) −3.69833e9 −0.991304
\(544\) 1.86034e9 0.495445
\(545\) 4.66018e8 0.123315
\(546\) 4.89096e9 1.28594
\(547\) 2.70806e9 0.707460 0.353730 0.935348i \(-0.384913\pi\)
0.353730 + 0.935348i \(0.384913\pi\)
\(548\) −1.02069e9 −0.264950
\(549\) −1.75000e9 −0.451373
\(550\) −4.76838e8 −0.122209
\(551\) 5.00469e9 1.27452
\(552\) 4.11885e8 0.104229
\(553\) −2.23959e9 −0.563159
\(554\) −6.59450e9 −1.64777
\(555\) 9.87059e7 0.0245086
\(556\) −1.21605e9 −0.300047
\(557\) −2.74885e9 −0.673997 −0.336999 0.941505i \(-0.609412\pi\)
−0.336999 + 0.941505i \(0.609412\pi\)
\(558\) 2.29883e9 0.560127
\(559\) 4.25983e9 1.03146
\(560\) 8.81679e8 0.212155
\(561\) 3.21446e8 0.0768667
\(562\) 5.66355e8 0.134590
\(563\) −4.75796e9 −1.12368 −0.561838 0.827247i \(-0.689906\pi\)
−0.561838 + 0.827247i \(0.689906\pi\)
\(564\) −6.18012e8 −0.145051
\(565\) 4.78143e8 0.111529
\(566\) 2.84544e9 0.659617
\(567\) 5.98772e8 0.137950
\(568\) −8.89708e8 −0.203717
\(569\) 3.85644e9 0.877595 0.438797 0.898586i \(-0.355405\pi\)
0.438797 + 0.898586i \(0.355405\pi\)
\(570\) −4.52436e8 −0.102328
\(571\) −1.05862e9 −0.237966 −0.118983 0.992896i \(-0.537963\pi\)
−0.118983 + 0.992896i \(0.537963\pi\)
\(572\) −1.76977e8 −0.0395394
\(573\) −1.58747e9 −0.352505
\(574\) 7.71553e9 1.70284
\(575\) −9.30215e8 −0.204054
\(576\) 1.07578e9 0.234556
\(577\) −5.02825e9 −1.08969 −0.544843 0.838538i \(-0.683411\pi\)
−0.544843 + 0.838538i \(0.683411\pi\)
\(578\) 1.94507e9 0.418976
\(579\) −3.91424e7 −0.00838054
\(580\) 1.70692e8 0.0363258
\(581\) 9.24300e9 1.95522
\(582\) −5.21046e8 −0.109558
\(583\) −8.67811e8 −0.181378
\(584\) −9.99780e8 −0.207711
\(585\) 3.84308e8 0.0793658
\(586\) 5.60214e9 1.15004
\(587\) −8.27222e9 −1.68806 −0.844032 0.536293i \(-0.819824\pi\)
−0.844032 + 0.536293i \(0.819824\pi\)
\(588\) −3.30265e8 −0.0669950
\(589\) −8.31636e9 −1.67699
\(590\) 1.31501e9 0.263601
\(591\) 1.78939e9 0.356574
\(592\) −1.71193e9 −0.339125
\(593\) −2.13907e9 −0.421245 −0.210622 0.977568i \(-0.567549\pi\)
−0.210622 + 0.977568i \(0.567549\pi\)
\(594\) −1.22762e8 −0.0240331
\(595\) 1.09610e9 0.213324
\(596\) 5.87032e8 0.113579
\(597\) 4.82576e9 0.928230
\(598\) −1.95617e9 −0.374070
\(599\) −6.33073e9 −1.20354 −0.601769 0.798670i \(-0.705537\pi\)
−0.601769 + 0.798670i \(0.705537\pi\)
\(600\) −2.58817e9 −0.489174
\(601\) −4.50671e8 −0.0846835 −0.0423418 0.999103i \(-0.513482\pi\)
−0.0423418 + 0.999103i \(0.513482\pi\)
\(602\) −4.63999e9 −0.866820
\(603\) 2.60304e9 0.483471
\(604\) −1.47889e9 −0.273091
\(605\) 7.86382e8 0.144374
\(606\) −4.24611e9 −0.775064
\(607\) −6.88005e9 −1.24862 −0.624311 0.781176i \(-0.714620\pi\)
−0.624311 + 0.781176i \(0.714620\pi\)
\(608\) 2.57023e9 0.463778
\(609\) 4.63045e9 0.830735
\(610\) 1.22343e9 0.218236
\(611\) −1.07603e10 −1.90845
\(612\) −4.75920e8 −0.0839274
\(613\) 8.99737e9 1.57763 0.788813 0.614633i \(-0.210696\pi\)
0.788813 + 0.614633i \(0.210696\pi\)
\(614\) −4.44088e9 −0.774247
\(615\) 6.06249e8 0.105096
\(616\) −7.06702e8 −0.121816
\(617\) −2.14587e9 −0.367794 −0.183897 0.982946i \(-0.558871\pi\)
−0.183897 + 0.982946i \(0.558871\pi\)
\(618\) 5.15782e9 0.879035
\(619\) 3.60649e9 0.611177 0.305589 0.952164i \(-0.401147\pi\)
0.305589 + 0.952164i \(0.401147\pi\)
\(620\) −2.83641e8 −0.0477967
\(621\) −2.39483e8 −0.0401286
\(622\) 8.90509e9 1.48379
\(623\) 9.27961e9 1.53752
\(624\) −6.66534e9 −1.09819
\(625\) 5.71465e9 0.936288
\(626\) 1.19333e9 0.194425
\(627\) 4.44108e8 0.0719536
\(628\) −1.35095e8 −0.0217661
\(629\) −2.12826e9 −0.340994
\(630\) −4.18604e8 −0.0666978
\(631\) −1.96983e9 −0.312123 −0.156062 0.987747i \(-0.549880\pi\)
−0.156062 + 0.987747i \(0.549880\pi\)
\(632\) 2.49225e9 0.392719
\(633\) 2.90444e8 0.0455145
\(634\) −8.58973e9 −1.33865
\(635\) −1.67257e9 −0.259224
\(636\) 1.28485e9 0.198039
\(637\) −5.75029e9 −0.881459
\(638\) −9.49345e8 −0.144728
\(639\) 5.17304e8 0.0784319
\(640\) −1.16111e9 −0.175084
\(641\) 1.07839e9 0.161724 0.0808620 0.996725i \(-0.474233\pi\)
0.0808620 + 0.996725i \(0.474233\pi\)
\(642\) −1.20682e9 −0.179999
\(643\) −9.43215e9 −1.39918 −0.699588 0.714547i \(-0.746633\pi\)
−0.699588 + 0.714547i \(0.746633\pi\)
\(644\) 3.76056e8 0.0554820
\(645\) −3.64588e8 −0.0534986
\(646\) 9.75526e9 1.42372
\(647\) −1.91605e9 −0.278126 −0.139063 0.990284i \(-0.544409\pi\)
−0.139063 + 0.990284i \(0.544409\pi\)
\(648\) −6.66322e8 −0.0961992
\(649\) −1.29080e9 −0.185355
\(650\) 1.22920e10 1.75560
\(651\) −7.69447e9 −1.09306
\(652\) −1.43839e9 −0.203240
\(653\) 1.28782e10 1.80991 0.904956 0.425505i \(-0.139904\pi\)
0.904956 + 0.425505i \(0.139904\pi\)
\(654\) 3.83742e9 0.536434
\(655\) −3.90045e8 −0.0542339
\(656\) −1.05146e10 −1.45422
\(657\) 5.81303e8 0.0799695
\(658\) 1.17206e10 1.60383
\(659\) −1.04515e10 −1.42259 −0.711294 0.702895i \(-0.751890\pi\)
−0.711294 + 0.702895i \(0.751890\pi\)
\(660\) 1.51469e7 0.00205079
\(661\) 9.36029e9 1.26062 0.630310 0.776344i \(-0.282928\pi\)
0.630310 + 0.776344i \(0.282928\pi\)
\(662\) 1.04392e10 1.39851
\(663\) −8.28630e9 −1.10424
\(664\) −1.02857e10 −1.36348
\(665\) 1.51436e9 0.199689
\(666\) 8.12791e8 0.106615
\(667\) −1.85198e9 −0.241655
\(668\) 1.55484e9 0.201821
\(669\) 5.78569e9 0.747075
\(670\) −1.81979e9 −0.233755
\(671\) −1.20091e9 −0.153456
\(672\) 2.37803e9 0.302291
\(673\) −2.34925e9 −0.297083 −0.148541 0.988906i \(-0.547458\pi\)
−0.148541 + 0.988906i \(0.547458\pi\)
\(674\) −1.55605e10 −1.95755
\(675\) 1.50484e9 0.188334
\(676\) 2.84080e9 0.353694
\(677\) 1.18065e10 1.46238 0.731188 0.682176i \(-0.238966\pi\)
0.731188 + 0.682176i \(0.238966\pi\)
\(678\) 3.93726e9 0.485166
\(679\) 1.74401e9 0.213798
\(680\) −1.21975e9 −0.148761
\(681\) 1.98140e9 0.240412
\(682\) 1.57754e9 0.190429
\(683\) 5.88145e9 0.706337 0.353169 0.935560i \(-0.385104\pi\)
0.353169 + 0.935560i \(0.385104\pi\)
\(684\) −6.57528e8 −0.0785630
\(685\) 1.52101e9 0.180807
\(686\) −5.30468e9 −0.627372
\(687\) 7.64971e8 0.0900112
\(688\) 6.32332e9 0.740262
\(689\) 2.23706e10 2.60562
\(690\) 1.67423e8 0.0194019
\(691\) −4.16582e9 −0.480316 −0.240158 0.970734i \(-0.577199\pi\)
−0.240158 + 0.970734i \(0.577199\pi\)
\(692\) −1.29462e9 −0.148515
\(693\) 4.10898e8 0.0468995
\(694\) −2.64277e9 −0.300125
\(695\) 1.81212e9 0.204757
\(696\) −5.15283e9 −0.579313
\(697\) −1.30717e10 −1.46224
\(698\) −1.29471e9 −0.144105
\(699\) −3.11178e9 −0.344618
\(700\) −2.36303e9 −0.260391
\(701\) −1.38729e10 −1.52109 −0.760546 0.649284i \(-0.775069\pi\)
−0.760546 + 0.649284i \(0.775069\pi\)
\(702\) 3.16457e9 0.345251
\(703\) −2.94039e9 −0.319199
\(704\) 7.38240e8 0.0797431
\(705\) 9.20944e8 0.0989855
\(706\) 2.91634e9 0.311905
\(707\) 1.42123e10 1.51250
\(708\) 1.91111e9 0.202381
\(709\) 3.27840e9 0.345462 0.172731 0.984969i \(-0.444741\pi\)
0.172731 + 0.984969i \(0.444741\pi\)
\(710\) −3.61649e8 −0.0379213
\(711\) −1.44907e9 −0.151198
\(712\) −1.03265e10 −1.07219
\(713\) 3.07746e9 0.317964
\(714\) 9.02578e9 0.927985
\(715\) 2.63725e8 0.0269824
\(716\) 2.64192e8 0.0268982
\(717\) −1.59032e9 −0.161127
\(718\) −2.07313e10 −2.09021
\(719\) −6.37084e9 −0.639213 −0.319607 0.947550i \(-0.603551\pi\)
−0.319607 + 0.947550i \(0.603551\pi\)
\(720\) 5.70468e8 0.0569597
\(721\) −1.72639e10 −1.71540
\(722\) 2.33371e9 0.230763
\(723\) 1.38479e9 0.136270
\(724\) 3.75755e9 0.367976
\(725\) 1.16373e10 1.13415
\(726\) 6.47544e9 0.628046
\(727\) 1.86871e10 1.80373 0.901867 0.432013i \(-0.142197\pi\)
0.901867 + 0.432013i \(0.142197\pi\)
\(728\) 1.82175e10 1.74996
\(729\) 3.87420e8 0.0370370
\(730\) −4.06391e8 −0.0386647
\(731\) 7.86110e9 0.744342
\(732\) 1.77802e9 0.167552
\(733\) 1.28678e10 1.20682 0.603409 0.797432i \(-0.293809\pi\)
0.603409 + 0.797432i \(0.293809\pi\)
\(734\) −6.70358e9 −0.625706
\(735\) 4.92152e8 0.0457187
\(736\) −9.51111e8 −0.0879344
\(737\) 1.78630e9 0.164368
\(738\) 4.99214e9 0.457182
\(739\) −5.37369e9 −0.489798 −0.244899 0.969549i \(-0.578755\pi\)
−0.244899 + 0.969549i \(0.578755\pi\)
\(740\) −1.00286e8 −0.00909768
\(741\) −1.14483e10 −1.03366
\(742\) −2.43670e10 −2.18972
\(743\) −5.20876e9 −0.465879 −0.232940 0.972491i \(-0.574834\pi\)
−0.232940 + 0.972491i \(0.574834\pi\)
\(744\) 8.56251e9 0.762248
\(745\) −8.74778e8 −0.0775088
\(746\) −2.73331e9 −0.241047
\(747\) 5.98045e9 0.524943
\(748\) −3.26593e8 −0.0285332
\(749\) 4.03938e9 0.351260
\(750\) −2.12708e9 −0.184106
\(751\) 7.73437e9 0.666324 0.333162 0.942870i \(-0.391884\pi\)
0.333162 + 0.942870i \(0.391884\pi\)
\(752\) −1.59726e10 −1.36966
\(753\) 8.40530e8 0.0717416
\(754\) 2.44724e10 2.07911
\(755\) 2.20381e9 0.186363
\(756\) −6.08359e8 −0.0512075
\(757\) −1.32525e9 −0.111036 −0.0555178 0.998458i \(-0.517681\pi\)
−0.0555178 + 0.998458i \(0.517681\pi\)
\(758\) 1.36553e10 1.13883
\(759\) −1.64342e8 −0.0136427
\(760\) −1.68520e9 −0.139253
\(761\) 1.43521e10 1.18050 0.590252 0.807219i \(-0.299028\pi\)
0.590252 + 0.807219i \(0.299028\pi\)
\(762\) −1.37727e10 −1.12766
\(763\) −1.28443e10 −1.04683
\(764\) 1.61289e9 0.130851
\(765\) 7.09202e8 0.0572736
\(766\) −1.86379e10 −1.49829
\(767\) 3.32746e10 2.66274
\(768\) −4.46116e9 −0.355373
\(769\) 1.52167e10 1.20664 0.603320 0.797499i \(-0.293844\pi\)
0.603320 + 0.797499i \(0.293844\pi\)
\(770\) −2.87261e8 −0.0226756
\(771\) −6.43398e9 −0.505580
\(772\) 3.97691e7 0.00311089
\(773\) 1.74767e10 1.36092 0.680459 0.732786i \(-0.261781\pi\)
0.680459 + 0.732786i \(0.261781\pi\)
\(774\) −3.00219e9 −0.232726
\(775\) −1.93378e10 −1.49229
\(776\) −1.94076e9 −0.149092
\(777\) −2.72051e9 −0.208055
\(778\) −1.32916e10 −1.01193
\(779\) −1.80598e10 −1.36878
\(780\) −3.90461e8 −0.0294609
\(781\) 3.54992e8 0.0266649
\(782\) −3.60992e9 −0.269944
\(783\) 2.99602e9 0.223037
\(784\) −8.53576e9 −0.632610
\(785\) 2.01315e8 0.0148536
\(786\) −3.21182e9 −0.235924
\(787\) −7.75898e8 −0.0567405 −0.0283702 0.999597i \(-0.509032\pi\)
−0.0283702 + 0.999597i \(0.509032\pi\)
\(788\) −1.81804e9 −0.132362
\(789\) −7.67488e9 −0.556291
\(790\) 1.01305e9 0.0731032
\(791\) −1.31785e10 −0.946778
\(792\) −4.57254e8 −0.0327054
\(793\) 3.09574e10 2.20449
\(794\) 1.28237e10 0.909165
\(795\) −1.91464e9 −0.135146
\(796\) −4.90303e9 −0.344563
\(797\) −3.17179e9 −0.221922 −0.110961 0.993825i \(-0.535393\pi\)
−0.110961 + 0.993825i \(0.535393\pi\)
\(798\) 1.24700e10 0.868671
\(799\) −1.98571e10 −1.37721
\(800\) 5.97651e9 0.412698
\(801\) 6.00414e9 0.412797
\(802\) 3.62117e9 0.247879
\(803\) 3.98911e8 0.0271876
\(804\) −2.64472e9 −0.179466
\(805\) −5.60388e8 −0.0378619
\(806\) −4.06661e10 −2.73564
\(807\) 5.24580e9 0.351362
\(808\) −1.58156e10 −1.05474
\(809\) −2.96650e10 −1.96981 −0.984905 0.173096i \(-0.944623\pi\)
−0.984905 + 0.173096i \(0.944623\pi\)
\(810\) −2.70847e8 −0.0179072
\(811\) 2.28837e10 1.50645 0.753223 0.657765i \(-0.228498\pi\)
0.753223 + 0.657765i \(0.228498\pi\)
\(812\) −4.70459e9 −0.308372
\(813\) 9.96531e9 0.650390
\(814\) 5.57766e8 0.0362465
\(815\) 2.14345e9 0.138695
\(816\) −1.23002e10 −0.792497
\(817\) 1.08609e10 0.696766
\(818\) −8.00757e9 −0.511522
\(819\) −1.05922e10 −0.673742
\(820\) −6.15956e8 −0.0390122
\(821\) −9.31507e9 −0.587469 −0.293734 0.955887i \(-0.594898\pi\)
−0.293734 + 0.955887i \(0.594898\pi\)
\(822\) 1.25247e10 0.786531
\(823\) 2.28072e10 1.42618 0.713088 0.701074i \(-0.247296\pi\)
0.713088 + 0.701074i \(0.247296\pi\)
\(824\) 1.92115e10 1.19623
\(825\) 1.03268e9 0.0640288
\(826\) −3.62440e10 −2.23772
\(827\) 1.04877e10 0.644781 0.322391 0.946607i \(-0.395514\pi\)
0.322391 + 0.946607i \(0.395514\pi\)
\(828\) 2.43317e8 0.0148959
\(829\) −1.95304e10 −1.19061 −0.595307 0.803498i \(-0.702970\pi\)
−0.595307 + 0.803498i \(0.702970\pi\)
\(830\) −4.18096e9 −0.253806
\(831\) 1.42815e10 0.863319
\(832\) −1.90305e10 −1.14556
\(833\) −1.06116e10 −0.636097
\(834\) 1.49218e10 0.890720
\(835\) −2.31697e9 −0.137727
\(836\) −4.51219e8 −0.0267095
\(837\) −4.97851e9 −0.293468
\(838\) 2.43152e10 1.42733
\(839\) −1.46981e10 −0.859200 −0.429600 0.903019i \(-0.641345\pi\)
−0.429600 + 0.903019i \(0.641345\pi\)
\(840\) −1.55919e9 −0.0907654
\(841\) 5.91903e9 0.343134
\(842\) −3.32435e10 −1.91918
\(843\) −1.22654e9 −0.0705156
\(844\) −2.95095e8 −0.0168952
\(845\) −4.23328e9 −0.241367
\(846\) 7.58349e9 0.430599
\(847\) −2.16741e10 −1.22560
\(848\) 3.32071e10 1.87001
\(849\) −6.16229e9 −0.345593
\(850\) 2.26837e10 1.26692
\(851\) 1.08809e9 0.0605217
\(852\) −5.25587e8 −0.0291143
\(853\) 1.80045e10 0.993253 0.496626 0.867964i \(-0.334572\pi\)
0.496626 + 0.867964i \(0.334572\pi\)
\(854\) −3.37201e10 −1.85262
\(855\) 9.79830e8 0.0536129
\(856\) −4.49508e9 −0.244951
\(857\) −1.06997e10 −0.580681 −0.290340 0.956923i \(-0.593768\pi\)
−0.290340 + 0.956923i \(0.593768\pi\)
\(858\) 2.17164e9 0.117377
\(859\) 2.30270e10 1.23954 0.619771 0.784783i \(-0.287225\pi\)
0.619771 + 0.784783i \(0.287225\pi\)
\(860\) 3.70425e8 0.0198589
\(861\) −1.67093e10 −0.892171
\(862\) −1.80776e10 −0.961312
\(863\) 1.75568e10 0.929837 0.464919 0.885353i \(-0.346084\pi\)
0.464919 + 0.885353i \(0.346084\pi\)
\(864\) 1.53865e9 0.0811598
\(865\) 1.92921e9 0.101350
\(866\) 1.07136e10 0.560560
\(867\) −4.21240e9 −0.219514
\(868\) 7.81767e9 0.405750
\(869\) −9.94404e8 −0.0514036
\(870\) −2.09453e9 −0.107837
\(871\) −4.60475e10 −2.36125
\(872\) 1.42933e10 0.730005
\(873\) 1.12842e9 0.0574010
\(874\) −4.98745e9 −0.252690
\(875\) 7.11959e9 0.359275
\(876\) −5.90611e8 −0.0296850
\(877\) −3.55406e10 −1.77920 −0.889602 0.456737i \(-0.849018\pi\)
−0.889602 + 0.456737i \(0.849018\pi\)
\(878\) −2.83944e10 −1.41580
\(879\) −1.21324e10 −0.602541
\(880\) 3.91475e8 0.0193649
\(881\) −1.43619e10 −0.707616 −0.353808 0.935318i \(-0.615113\pi\)
−0.353808 + 0.935318i \(0.615113\pi\)
\(882\) 4.05261e9 0.198882
\(883\) 7.01212e9 0.342757 0.171379 0.985205i \(-0.445178\pi\)
0.171379 + 0.985205i \(0.445178\pi\)
\(884\) 8.41897e9 0.409898
\(885\) −2.84788e9 −0.138108
\(886\) 2.28193e9 0.110226
\(887\) 1.65555e10 0.796546 0.398273 0.917267i \(-0.369610\pi\)
0.398273 + 0.917267i \(0.369610\pi\)
\(888\) 3.02743e9 0.145087
\(889\) 4.60990e10 2.20057
\(890\) −4.19752e9 −0.199585
\(891\) 2.65862e8 0.0125917
\(892\) −5.87833e9 −0.277317
\(893\) −2.74344e10 −1.28919
\(894\) −7.20334e9 −0.337173
\(895\) −3.93691e8 −0.0183559
\(896\) 3.20024e10 1.48630
\(897\) 4.23643e9 0.195987
\(898\) 1.37934e10 0.635631
\(899\) −3.85000e10 −1.76727
\(900\) −1.52894e9 −0.0699102
\(901\) 4.12827e10 1.88032
\(902\) 3.42578e9 0.155431
\(903\) 1.00487e10 0.454154
\(904\) 1.46652e10 0.660236
\(905\) −5.59939e9 −0.251114
\(906\) 1.81472e10 0.810701
\(907\) 3.79998e10 1.69105 0.845525 0.533936i \(-0.179288\pi\)
0.845525 + 0.533936i \(0.179288\pi\)
\(908\) −2.01312e9 −0.0892421
\(909\) 9.19570e9 0.406079
\(910\) 7.40506e9 0.325750
\(911\) 9.57936e9 0.419781 0.209890 0.977725i \(-0.432689\pi\)
0.209890 + 0.977725i \(0.432689\pi\)
\(912\) −1.69939e10 −0.741843
\(913\) 4.10400e9 0.178468
\(914\) −2.09432e9 −0.0907258
\(915\) −2.64956e9 −0.114340
\(916\) −7.77219e8 −0.0334125
\(917\) 1.07504e10 0.460395
\(918\) 5.83990e9 0.249147
\(919\) 2.53060e10 1.07552 0.537760 0.843098i \(-0.319271\pi\)
0.537760 + 0.843098i \(0.319271\pi\)
\(920\) 6.23607e8 0.0264030
\(921\) 9.61750e9 0.405652
\(922\) 3.20222e10 1.34553
\(923\) −9.15106e9 −0.383059
\(924\) −4.17477e8 −0.0174093
\(925\) −6.83724e9 −0.284043
\(926\) −1.37962e10 −0.570980
\(927\) −1.11701e10 −0.460553
\(928\) 1.18987e10 0.488745
\(929\) −8.03531e9 −0.328812 −0.164406 0.986393i \(-0.552571\pi\)
−0.164406 + 0.986393i \(0.552571\pi\)
\(930\) 3.48050e9 0.141890
\(931\) −1.46609e10 −0.595440
\(932\) 3.16160e9 0.127924
\(933\) −1.92855e10 −0.777402
\(934\) 3.48646e10 1.40014
\(935\) 4.86679e8 0.0194716
\(936\) 1.17872e10 0.469834
\(937\) −1.25462e10 −0.498223 −0.249112 0.968475i \(-0.580139\pi\)
−0.249112 + 0.968475i \(0.580139\pi\)
\(938\) 5.01569e10 1.98436
\(939\) −2.58437e9 −0.101865
\(940\) −9.35690e8 −0.0367438
\(941\) −1.99670e10 −0.781176 −0.390588 0.920566i \(-0.627728\pi\)
−0.390588 + 0.920566i \(0.627728\pi\)
\(942\) 1.65772e9 0.0646150
\(943\) 6.68301e9 0.259526
\(944\) 4.93930e10 1.91101
\(945\) 9.06559e8 0.0349450
\(946\) −2.06021e9 −0.0791210
\(947\) 3.32659e10 1.27284 0.636422 0.771341i \(-0.280414\pi\)
0.636422 + 0.771341i \(0.280414\pi\)
\(948\) 1.47227e9 0.0561254
\(949\) −1.02832e10 −0.390568
\(950\) 3.13397e10 1.18594
\(951\) 1.86026e10 0.701360
\(952\) 3.36186e10 1.26285
\(953\) 4.41016e8 0.0165055 0.00825277 0.999966i \(-0.497373\pi\)
0.00825277 + 0.999966i \(0.497373\pi\)
\(954\) −1.57661e10 −0.587900
\(955\) −2.40348e9 −0.0892954
\(956\) 1.61578e9 0.0598109
\(957\) 2.05597e9 0.0758272
\(958\) 2.73449e10 1.00484
\(959\) −4.19218e10 −1.53488
\(960\) 1.62877e9 0.0594169
\(961\) 3.64633e10 1.32533
\(962\) −1.43782e10 −0.520705
\(963\) 2.61358e9 0.0943070
\(964\) −1.40696e9 −0.0505840
\(965\) −5.92628e7 −0.00212293
\(966\) −4.61450e9 −0.164704
\(967\) −4.20440e9 −0.149524 −0.0747620 0.997201i \(-0.523820\pi\)
−0.0747620 + 0.997201i \(0.523820\pi\)
\(968\) 2.41193e10 0.854674
\(969\) −2.11267e10 −0.745931
\(970\) −7.88880e8 −0.0277530
\(971\) −3.02108e10 −1.05900 −0.529499 0.848311i \(-0.677620\pi\)
−0.529499 + 0.848311i \(0.677620\pi\)
\(972\) −3.93624e8 −0.0137483
\(973\) −4.99453e10 −1.73820
\(974\) −1.85065e10 −0.641752
\(975\) −2.66205e10 −0.919815
\(976\) 4.59533e10 1.58213
\(977\) −3.60793e10 −1.23773 −0.618866 0.785496i \(-0.712408\pi\)
−0.618866 + 0.785496i \(0.712408\pi\)
\(978\) 1.76502e10 0.603340
\(979\) 4.12025e9 0.140341
\(980\) −5.00032e8 −0.0169710
\(981\) −8.31059e9 −0.281054
\(982\) −4.47097e10 −1.50665
\(983\) −3.38986e9 −0.113827 −0.0569133 0.998379i \(-0.518126\pi\)
−0.0569133 + 0.998379i \(0.518126\pi\)
\(984\) 1.85944e10 0.622155
\(985\) 2.70920e9 0.0903262
\(986\) 4.51614e10 1.50037
\(987\) −2.53829e10 −0.840294
\(988\) 1.16316e10 0.383699
\(989\) −4.01905e9 −0.132110
\(990\) −1.85865e8 −0.00608799
\(991\) 8.93953e9 0.291781 0.145890 0.989301i \(-0.453395\pi\)
0.145890 + 0.989301i \(0.453395\pi\)
\(992\) −1.97723e10 −0.643080
\(993\) −2.26079e10 −0.732720
\(994\) 9.96771e9 0.321916
\(995\) 7.30635e9 0.235136
\(996\) −6.07621e9 −0.194861
\(997\) −4.01968e10 −1.28457 −0.642286 0.766465i \(-0.722014\pi\)
−0.642286 + 0.766465i \(0.722014\pi\)
\(998\) −3.29815e10 −1.05030
\(999\) −1.76024e9 −0.0558590
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 69.8.a.a.1.4 5
3.2 odd 2 207.8.a.a.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.a.1.4 5 1.1 even 1 trivial
207.8.a.a.1.2 5 3.2 odd 2