Properties

Label 289.10.a.a.1.3
Level $289$
Weight $10$
Character 289.1
Self dual yes
Analytic conductor $148.845$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.77274\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.22726 q^{2} -177.437 q^{3} -510.494 q^{4} +1620.18 q^{5} +217.762 q^{6} +1834.42 q^{7} +1254.87 q^{8} +11801.0 q^{9} +O(q^{10})\) \(q-1.22726 q^{2} -177.437 q^{3} -510.494 q^{4} +1620.18 q^{5} +217.762 q^{6} +1834.42 q^{7} +1254.87 q^{8} +11801.0 q^{9} -1988.39 q^{10} +31779.3 q^{11} +90580.6 q^{12} -132363. q^{13} -2251.32 q^{14} -287481. q^{15} +259833. q^{16} -14482.9 q^{18} -1603.79 q^{19} -827094. q^{20} -325495. q^{21} -39001.6 q^{22} -23254.0 q^{23} -222661. q^{24} +671873. q^{25} +162444. q^{26} +1.39856e6 q^{27} -936463. q^{28} -3.73267e6 q^{29} +352815. q^{30} -8.91479e6 q^{31} -961376. q^{32} -5.63884e6 q^{33} +2.97211e6 q^{35} -6.02433e6 q^{36} +1.20475e7 q^{37} +1968.28 q^{38} +2.34861e7 q^{39} +2.03312e6 q^{40} +1.26779e7 q^{41} +399468. q^{42} +2.86352e7 q^{43} -1.62231e7 q^{44} +1.91198e7 q^{45} +28538.8 q^{46} -7.17761e6 q^{47} -4.61040e7 q^{48} -3.69885e7 q^{49} -824565. q^{50} +6.75705e7 q^{52} -5.96065e7 q^{53} -1.71640e6 q^{54} +5.14884e7 q^{55} +2.30196e6 q^{56} +284573. q^{57} +4.58096e6 q^{58} +1.85990e8 q^{59} +1.46757e8 q^{60} +2.00037e8 q^{61} +1.09408e7 q^{62} +2.16480e7 q^{63} -1.31855e8 q^{64} -2.14453e8 q^{65} +6.92034e6 q^{66} -1.27030e8 q^{67} +4.12613e6 q^{69} -3.64756e6 q^{70} +3.27860e8 q^{71} +1.48087e7 q^{72} +1.48678e8 q^{73} -1.47854e7 q^{74} -1.19215e8 q^{75} +818726. q^{76} +5.82968e7 q^{77} -2.88237e7 q^{78} +2.58778e8 q^{79} +4.20977e8 q^{80} -4.80436e8 q^{81} -1.55591e7 q^{82} +3.45060e8 q^{83} +1.66163e8 q^{84} -3.51430e7 q^{86} +6.62314e8 q^{87} +3.98789e7 q^{88} +4.03936e8 q^{89} -2.34650e7 q^{90} -2.42810e8 q^{91} +1.18710e7 q^{92} +1.58182e9 q^{93} +8.80881e6 q^{94} -2.59844e6 q^{95} +1.70584e8 q^{96} +9.89973e8 q^{97} +4.53946e7 q^{98} +3.75028e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 33q^{2} + 236q^{3} + 853q^{4} - 1480q^{5} - 7578q^{6} + 13202q^{7} - 42423q^{8} + 10981q^{9} + O(q^{10}) \) \( 5q - 33q^{2} + 236q^{3} + 853q^{4} - 1480q^{5} - 7578q^{6} + 13202q^{7} - 42423q^{8} + 10981q^{9} + 89328q^{10} + 68036q^{11} + 406010q^{12} - 158862q^{13} + 84700q^{14} - 687324q^{15} + 350225q^{16} - 1911585q^{18} - 370992q^{19} - 1632640q^{20} + 1783880q^{21} - 122290q^{22} - 1645870q^{23} - 9678702q^{24} + 3270239q^{25} + 734846q^{26} + 2998268q^{27} - 183372q^{28} - 3668616q^{29} + 17048544q^{30} + 7262362q^{31} - 5605919q^{32} - 11334900q^{33} - 26503988q^{35} + 49782133q^{36} + 31420708q^{37} + 18513700q^{38} + 42449884q^{39} + 53930464q^{40} + 7996938q^{41} - 44519496q^{42} - 56908268q^{43} - 43323054q^{44} - 12799536q^{45} + 32063472q^{46} - 16903336q^{47} + 102794498q^{48} - 11784059q^{49} + 85921093q^{50} + 173619082q^{52} - 83362982q^{53} - 386329164q^{54} + 6363364q^{55} - 317409372q^{56} - 136615904q^{57} - 64577488q^{58} - 37946604q^{59} - 223158912q^{60} + 77685452q^{61} - 324855300q^{62} + 191945278q^{63} + 131623105q^{64} + 40321288q^{65} + 298037676q^{66} - 304503600q^{67} - 333409272q^{69} - 122787392q^{70} + 476602922q^{71} - 1301701911q^{72} + 289980486q^{73} - 262289012q^{74} + 153685772q^{75} - 1031276084q^{76} - 143385648q^{77} - 691646196q^{78} + 828240610q^{79} - 912750944q^{80} + 891328609q^{81} + 1109615654q^{82} + 194681148q^{83} + 1541719592q^{84} + 1164707144q^{86} + 158149884q^{87} + 1017979978q^{88} + 376848106q^{89} + 2240087472q^{90} - 194543664q^{91} - 2506713088q^{92} + 3494835920q^{93} - 2244811104q^{94} - 1498679864q^{95} - 2935047582q^{96} - 692035246q^{97} + 871744055q^{98} - 2027106408q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22726 −0.0542379 −0.0271189 0.999632i \(-0.508633\pi\)
−0.0271189 + 0.999632i \(0.508633\pi\)
\(3\) −177.437 −1.26473 −0.632367 0.774669i \(-0.717917\pi\)
−0.632367 + 0.774669i \(0.717917\pi\)
\(4\) −510.494 −0.997058
\(5\) 1620.18 1.15931 0.579655 0.814862i \(-0.303187\pi\)
0.579655 + 0.814862i \(0.303187\pi\)
\(6\) 217.762 0.0685965
\(7\) 1834.42 0.288774 0.144387 0.989521i \(-0.453879\pi\)
0.144387 + 0.989521i \(0.453879\pi\)
\(8\) 1254.87 0.108316
\(9\) 11801.0 0.599553
\(10\) −1988.39 −0.0628785
\(11\) 31779.3 0.654452 0.327226 0.944946i \(-0.393886\pi\)
0.327226 + 0.944946i \(0.393886\pi\)
\(12\) 90580.6 1.26101
\(13\) −132363. −1.28535 −0.642675 0.766139i \(-0.722176\pi\)
−0.642675 + 0.766139i \(0.722176\pi\)
\(14\) −2251.32 −0.0156625
\(15\) −287481. −1.46622
\(16\) 259833. 0.991183
\(17\) 0 0
\(18\) −14482.9 −0.0325185
\(19\) −1603.79 −0.00282330 −0.00141165 0.999999i \(-0.500449\pi\)
−0.00141165 + 0.999999i \(0.500449\pi\)
\(20\) −827094. −1.15590
\(21\) −325495. −0.365223
\(22\) −39001.6 −0.0354961
\(23\) −23254.0 −0.0173270 −0.00866348 0.999962i \(-0.502758\pi\)
−0.00866348 + 0.999962i \(0.502758\pi\)
\(24\) −222661. −0.136991
\(25\) 671873. 0.343999
\(26\) 162444. 0.0697147
\(27\) 1.39856e6 0.506460
\(28\) −936463. −0.287925
\(29\) −3.73267e6 −0.980005 −0.490002 0.871721i \(-0.663004\pi\)
−0.490002 + 0.871721i \(0.663004\pi\)
\(30\) 352815. 0.0795246
\(31\) −8.91479e6 −1.73374 −0.866869 0.498536i \(-0.833871\pi\)
−0.866869 + 0.498536i \(0.833871\pi\)
\(32\) −961376. −0.162076
\(33\) −5.63884e6 −0.827707
\(34\) 0 0
\(35\) 2.97211e6 0.334779
\(36\) −6.02433e6 −0.597789
\(37\) 1.20475e7 1.05679 0.528395 0.848999i \(-0.322794\pi\)
0.528395 + 0.848999i \(0.322794\pi\)
\(38\) 1968.28 0.000153130 0
\(39\) 2.34861e7 1.62563
\(40\) 2.03312e6 0.125572
\(41\) 1.26779e7 0.700680 0.350340 0.936623i \(-0.386066\pi\)
0.350340 + 0.936623i \(0.386066\pi\)
\(42\) 399468. 0.0198089
\(43\) 2.86352e7 1.27730 0.638650 0.769498i \(-0.279493\pi\)
0.638650 + 0.769498i \(0.279493\pi\)
\(44\) −1.62231e7 −0.652526
\(45\) 1.91198e7 0.695067
\(46\) 28538.8 0.000939777 0
\(47\) −7.17761e6 −0.214555 −0.107278 0.994229i \(-0.534213\pi\)
−0.107278 + 0.994229i \(0.534213\pi\)
\(48\) −4.61040e7 −1.25358
\(49\) −3.69885e7 −0.916609
\(50\) −824565. −0.0186578
\(51\) 0 0
\(52\) 6.75705e7 1.28157
\(53\) −5.96065e7 −1.03765 −0.518826 0.854880i \(-0.673631\pi\)
−0.518826 + 0.854880i \(0.673631\pi\)
\(54\) −1.71640e6 −0.0274693
\(55\) 5.14884e7 0.758712
\(56\) 2.30196e6 0.0312790
\(57\) 284573. 0.00357072
\(58\) 4.58096e6 0.0531534
\(59\) 1.85990e8 1.99828 0.999139 0.0414925i \(-0.0132113\pi\)
0.999139 + 0.0414925i \(0.0132113\pi\)
\(60\) 1.46757e8 1.46191
\(61\) 2.00037e8 1.84981 0.924905 0.380198i \(-0.124144\pi\)
0.924905 + 0.380198i \(0.124144\pi\)
\(62\) 1.09408e7 0.0940343
\(63\) 2.16480e7 0.173135
\(64\) −1.31855e8 −0.982393
\(65\) −2.14453e8 −1.49012
\(66\) 6.92034e6 0.0448931
\(67\) −1.27030e8 −0.770137 −0.385069 0.922888i \(-0.625822\pi\)
−0.385069 + 0.922888i \(0.625822\pi\)
\(68\) 0 0
\(69\) 4.12613e6 0.0219140
\(70\) −3.64756e6 −0.0181577
\(71\) 3.27860e8 1.53118 0.765590 0.643329i \(-0.222447\pi\)
0.765590 + 0.643329i \(0.222447\pi\)
\(72\) 1.48087e7 0.0649413
\(73\) 1.48678e8 0.612766 0.306383 0.951908i \(-0.400881\pi\)
0.306383 + 0.951908i \(0.400881\pi\)
\(74\) −1.47854e7 −0.0573181
\(75\) −1.19215e8 −0.435067
\(76\) 818726. 0.00281500
\(77\) 5.82968e7 0.188989
\(78\) −2.88237e7 −0.0881705
\(79\) 2.58778e8 0.747491 0.373746 0.927531i \(-0.378073\pi\)
0.373746 + 0.927531i \(0.378073\pi\)
\(80\) 4.20977e8 1.14909
\(81\) −4.80436e8 −1.24009
\(82\) −1.55591e7 −0.0380034
\(83\) 3.45060e8 0.798073 0.399037 0.916935i \(-0.369345\pi\)
0.399037 + 0.916935i \(0.369345\pi\)
\(84\) 1.66163e8 0.364149
\(85\) 0 0
\(86\) −3.51430e7 −0.0692780
\(87\) 6.62314e8 1.23945
\(88\) 3.98789e7 0.0708877
\(89\) 4.03936e8 0.682428 0.341214 0.939986i \(-0.389162\pi\)
0.341214 + 0.939986i \(0.389162\pi\)
\(90\) −2.34650e7 −0.0376990
\(91\) −2.42810e8 −0.371176
\(92\) 1.18710e7 0.0172760
\(93\) 1.58182e9 2.19272
\(94\) 8.80881e6 0.0116370
\(95\) −2.59844e6 −0.00327308
\(96\) 1.70584e8 0.204983
\(97\) 9.89973e8 1.13540 0.567702 0.823234i \(-0.307833\pi\)
0.567702 + 0.823234i \(0.307833\pi\)
\(98\) 4.53946e7 0.0497149
\(99\) 3.75028e8 0.392378
\(100\) −3.42987e8 −0.342987
\(101\) −1.11917e9 −1.07017 −0.535084 0.844799i \(-0.679720\pi\)
−0.535084 + 0.844799i \(0.679720\pi\)
\(102\) 0 0
\(103\) 6.58796e6 0.00576744 0.00288372 0.999996i \(-0.499082\pi\)
0.00288372 + 0.999996i \(0.499082\pi\)
\(104\) −1.66098e8 −0.139224
\(105\) −5.27363e8 −0.423407
\(106\) 7.31528e7 0.0562801
\(107\) −1.55590e9 −1.14751 −0.573754 0.819028i \(-0.694513\pi\)
−0.573754 + 0.819028i \(0.694513\pi\)
\(108\) −7.13957e8 −0.504970
\(109\) −6.94766e8 −0.471432 −0.235716 0.971822i \(-0.575744\pi\)
−0.235716 + 0.971822i \(0.575744\pi\)
\(110\) −6.31898e7 −0.0411509
\(111\) −2.13767e9 −1.33656
\(112\) 4.76644e8 0.286228
\(113\) −2.09735e9 −1.21009 −0.605047 0.796190i \(-0.706846\pi\)
−0.605047 + 0.796190i \(0.706846\pi\)
\(114\) −349246. −0.000193669 0
\(115\) −3.76758e7 −0.0200873
\(116\) 1.90550e9 0.977122
\(117\) −1.56202e9 −0.770635
\(118\) −2.28259e8 −0.108382
\(119\) 0 0
\(120\) −3.60751e8 −0.158815
\(121\) −1.34802e9 −0.571693
\(122\) −2.45499e8 −0.100330
\(123\) −2.24953e9 −0.886173
\(124\) 4.55095e9 1.72864
\(125\) −2.07586e9 −0.760508
\(126\) −2.65678e7 −0.00939050
\(127\) −4.92229e9 −1.67900 −0.839499 0.543361i \(-0.817151\pi\)
−0.839499 + 0.543361i \(0.817151\pi\)
\(128\) 6.54045e8 0.215359
\(129\) −5.08096e9 −1.61544
\(130\) 2.63190e8 0.0808209
\(131\) −4.54604e9 −1.34869 −0.674345 0.738416i \(-0.735574\pi\)
−0.674345 + 0.738416i \(0.735574\pi\)
\(132\) 2.87859e9 0.825272
\(133\) −2.94204e6 −0.000815297 0
\(134\) 1.55899e8 0.0417706
\(135\) 2.26593e9 0.587143
\(136\) 0 0
\(137\) 5.25313e9 1.27402 0.637010 0.770856i \(-0.280171\pi\)
0.637010 + 0.770856i \(0.280171\pi\)
\(138\) −5.06384e6 −0.00118857
\(139\) −1.55602e9 −0.353549 −0.176774 0.984251i \(-0.556566\pi\)
−0.176774 + 0.984251i \(0.556566\pi\)
\(140\) −1.51724e9 −0.333794
\(141\) 1.27358e9 0.271356
\(142\) −4.02371e8 −0.0830479
\(143\) −4.20641e9 −0.841200
\(144\) 3.06629e9 0.594267
\(145\) −6.04761e9 −1.13613
\(146\) −1.82467e8 −0.0332351
\(147\) 6.56314e9 1.15927
\(148\) −6.15017e9 −1.05368
\(149\) 2.13936e9 0.355587 0.177794 0.984068i \(-0.443104\pi\)
0.177794 + 0.984068i \(0.443104\pi\)
\(150\) 1.46309e8 0.0235971
\(151\) −1.65162e9 −0.258531 −0.129266 0.991610i \(-0.541262\pi\)
−0.129266 + 0.991610i \(0.541262\pi\)
\(152\) −2.01255e6 −0.000305809 0
\(153\) 0 0
\(154\) −7.15455e7 −0.0102504
\(155\) −1.44436e10 −2.00994
\(156\) −1.19895e10 −1.62084
\(157\) −4.44203e9 −0.583490 −0.291745 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(158\) −3.17589e8 −0.0405423
\(159\) 1.05764e10 1.31235
\(160\) −1.55761e9 −0.187896
\(161\) −4.26577e7 −0.00500358
\(162\) 5.89621e8 0.0672598
\(163\) −2.66196e9 −0.295363 −0.147682 0.989035i \(-0.547181\pi\)
−0.147682 + 0.989035i \(0.547181\pi\)
\(164\) −6.47198e9 −0.698618
\(165\) −9.13596e9 −0.959569
\(166\) −4.23479e8 −0.0432858
\(167\) 1.58326e10 1.57518 0.787588 0.616203i \(-0.211330\pi\)
0.787588 + 0.616203i \(0.211330\pi\)
\(168\) −4.08454e8 −0.0395596
\(169\) 6.91547e9 0.652126
\(170\) 0 0
\(171\) −1.89264e7 −0.00169272
\(172\) −1.46181e10 −1.27354
\(173\) 1.27988e10 1.08633 0.543165 0.839626i \(-0.317226\pi\)
0.543165 + 0.839626i \(0.317226\pi\)
\(174\) −8.12834e8 −0.0672249
\(175\) 1.23250e9 0.0993382
\(176\) 8.25731e9 0.648681
\(177\) −3.30016e10 −2.52729
\(178\) −4.95735e8 −0.0370135
\(179\) −8.33273e9 −0.606665 −0.303332 0.952885i \(-0.598099\pi\)
−0.303332 + 0.952885i \(0.598099\pi\)
\(180\) −9.76053e9 −0.693022
\(181\) 8.70236e9 0.602676 0.301338 0.953517i \(-0.402567\pi\)
0.301338 + 0.953517i \(0.402567\pi\)
\(182\) 2.97992e8 0.0201318
\(183\) −3.54941e10 −2.33952
\(184\) −2.91807e7 −0.00187679
\(185\) 1.95192e10 1.22515
\(186\) −1.94130e9 −0.118928
\(187\) 0 0
\(188\) 3.66413e9 0.213924
\(189\) 2.56556e9 0.146253
\(190\) 3.18897e6 0.000177525 0
\(191\) 2.29558e10 1.24808 0.624040 0.781392i \(-0.285490\pi\)
0.624040 + 0.781392i \(0.285490\pi\)
\(192\) 2.33959e10 1.24247
\(193\) −2.04685e10 −1.06189 −0.530943 0.847407i \(-0.678162\pi\)
−0.530943 + 0.847407i \(0.678162\pi\)
\(194\) −1.21496e9 −0.0615819
\(195\) 3.80519e10 1.88460
\(196\) 1.88824e10 0.913913
\(197\) −5.81009e9 −0.274843 −0.137422 0.990513i \(-0.543882\pi\)
−0.137422 + 0.990513i \(0.543882\pi\)
\(198\) −4.60257e8 −0.0212818
\(199\) −4.49526e9 −0.203196 −0.101598 0.994826i \(-0.532396\pi\)
−0.101598 + 0.994826i \(0.532396\pi\)
\(200\) 8.43113e8 0.0372607
\(201\) 2.25398e10 0.974019
\(202\) 1.37352e9 0.0580436
\(203\) −6.84730e9 −0.283000
\(204\) 0 0
\(205\) 2.05405e10 0.812305
\(206\) −8.08516e6 −0.000312814 0
\(207\) −2.74420e8 −0.0103884
\(208\) −3.43922e10 −1.27402
\(209\) −5.09674e7 −0.00184771
\(210\) 6.47213e8 0.0229647
\(211\) −3.47865e10 −1.20820 −0.604101 0.796908i \(-0.706467\pi\)
−0.604101 + 0.796908i \(0.706467\pi\)
\(212\) 3.04287e10 1.03460
\(213\) −5.81746e10 −1.93654
\(214\) 1.90950e9 0.0622384
\(215\) 4.63944e10 1.48079
\(216\) 1.75501e9 0.0548578
\(217\) −1.63535e10 −0.500659
\(218\) 8.52661e8 0.0255695
\(219\) −2.63811e10 −0.774986
\(220\) −2.62845e10 −0.756480
\(221\) 0 0
\(222\) 2.62349e9 0.0724921
\(223\) −2.46348e10 −0.667078 −0.333539 0.942736i \(-0.608243\pi\)
−0.333539 + 0.942736i \(0.608243\pi\)
\(224\) −1.76357e9 −0.0468034
\(225\) 7.92877e9 0.206246
\(226\) 2.57401e9 0.0656329
\(227\) −4.45657e10 −1.11400 −0.556999 0.830513i \(-0.688047\pi\)
−0.556999 + 0.830513i \(0.688047\pi\)
\(228\) −1.45273e8 −0.00356022
\(229\) 6.61817e10 1.59030 0.795148 0.606415i \(-0.207393\pi\)
0.795148 + 0.606415i \(0.207393\pi\)
\(230\) 4.62381e7 0.00108949
\(231\) −1.03440e10 −0.239021
\(232\) −4.68401e9 −0.106150
\(233\) 4.35721e10 0.968516 0.484258 0.874925i \(-0.339090\pi\)
0.484258 + 0.874925i \(0.339090\pi\)
\(234\) 1.91700e9 0.0417976
\(235\) −1.16291e10 −0.248736
\(236\) −9.49468e10 −1.99240
\(237\) −4.59169e10 −0.945378
\(238\) 0 0
\(239\) 3.67128e10 0.727825 0.363912 0.931433i \(-0.381441\pi\)
0.363912 + 0.931433i \(0.381441\pi\)
\(240\) −7.46970e10 −1.45329
\(241\) −3.62693e10 −0.692568 −0.346284 0.938130i \(-0.612557\pi\)
−0.346284 + 0.938130i \(0.612557\pi\)
\(242\) 1.65438e9 0.0310074
\(243\) 5.77194e10 1.06192
\(244\) −1.02118e11 −1.84437
\(245\) −5.99282e10 −1.06263
\(246\) 2.76076e9 0.0480642
\(247\) 2.12283e8 0.00362893
\(248\) −1.11869e10 −0.187792
\(249\) −6.12264e10 −1.00935
\(250\) 2.54763e9 0.0412484
\(251\) −9.34890e10 −1.48672 −0.743359 0.668892i \(-0.766769\pi\)
−0.743359 + 0.668892i \(0.766769\pi\)
\(252\) −1.10512e10 −0.172626
\(253\) −7.38996e8 −0.0113397
\(254\) 6.04094e9 0.0910653
\(255\) 0 0
\(256\) 6.67068e10 0.970712
\(257\) 1.91619e10 0.273993 0.136996 0.990572i \(-0.456255\pi\)
0.136996 + 0.990572i \(0.456255\pi\)
\(258\) 6.23567e9 0.0876183
\(259\) 2.21002e10 0.305174
\(260\) 1.09477e11 1.48574
\(261\) −4.40492e10 −0.587564
\(262\) 5.57918e9 0.0731501
\(263\) 3.75028e10 0.483352 0.241676 0.970357i \(-0.422303\pi\)
0.241676 + 0.970357i \(0.422303\pi\)
\(264\) −7.07600e9 −0.0896541
\(265\) −9.65735e10 −1.20296
\(266\) 3.61065e6 4.42200e−5 0
\(267\) −7.16732e10 −0.863090
\(268\) 6.48478e10 0.767872
\(269\) 9.36804e10 1.09085 0.545423 0.838161i \(-0.316369\pi\)
0.545423 + 0.838161i \(0.316369\pi\)
\(270\) −2.78089e9 −0.0318454
\(271\) −9.81985e10 −1.10597 −0.552985 0.833191i \(-0.686511\pi\)
−0.552985 + 0.833191i \(0.686511\pi\)
\(272\) 0 0
\(273\) 4.30836e10 0.469439
\(274\) −6.44698e9 −0.0691001
\(275\) 2.13517e10 0.225131
\(276\) −2.10636e9 −0.0218495
\(277\) −9.27606e9 −0.0946683 −0.0473341 0.998879i \(-0.515073\pi\)
−0.0473341 + 0.998879i \(0.515073\pi\)
\(278\) 1.90965e9 0.0191757
\(279\) −1.05203e11 −1.03947
\(280\) 3.72960e9 0.0362620
\(281\) −1.04165e11 −0.996649 −0.498325 0.866991i \(-0.666051\pi\)
−0.498325 + 0.866991i \(0.666051\pi\)
\(282\) −1.56301e9 −0.0147177
\(283\) 9.57066e10 0.886958 0.443479 0.896285i \(-0.353744\pi\)
0.443479 + 0.896285i \(0.353744\pi\)
\(284\) −1.67371e11 −1.52668
\(285\) 4.61060e8 0.00413958
\(286\) 5.16237e9 0.0456249
\(287\) 2.32566e10 0.202338
\(288\) −1.13452e10 −0.0971730
\(289\) 0 0
\(290\) 7.42201e9 0.0616212
\(291\) −1.75658e11 −1.43598
\(292\) −7.58993e10 −0.610963
\(293\) −1.62664e11 −1.28940 −0.644698 0.764437i \(-0.723017\pi\)
−0.644698 + 0.764437i \(0.723017\pi\)
\(294\) −8.05470e9 −0.0628762
\(295\) 3.01338e11 2.31662
\(296\) 1.51180e10 0.114468
\(297\) 4.44453e10 0.331453
\(298\) −2.62556e9 −0.0192863
\(299\) 3.07797e9 0.0222712
\(300\) 6.08587e10 0.433788
\(301\) 5.25292e10 0.368851
\(302\) 2.02697e9 0.0140222
\(303\) 1.98583e11 1.35348
\(304\) −4.16718e8 −0.00279841
\(305\) 3.24098e11 2.14450
\(306\) 0 0
\(307\) −2.26286e11 −1.45390 −0.726952 0.686688i \(-0.759064\pi\)
−0.726952 + 0.686688i \(0.759064\pi\)
\(308\) −2.97601e10 −0.188433
\(309\) −1.16895e9 −0.00729428
\(310\) 1.77261e10 0.109015
\(311\) −1.17225e11 −0.710556 −0.355278 0.934761i \(-0.615614\pi\)
−0.355278 + 0.934761i \(0.615614\pi\)
\(312\) 2.94720e10 0.176082
\(313\) −2.85060e11 −1.67876 −0.839378 0.543549i \(-0.817080\pi\)
−0.839378 + 0.543549i \(0.817080\pi\)
\(314\) 5.45154e9 0.0316472
\(315\) 3.50738e10 0.200718
\(316\) −1.32105e11 −0.745292
\(317\) 9.98016e9 0.0555099 0.0277550 0.999615i \(-0.491164\pi\)
0.0277550 + 0.999615i \(0.491164\pi\)
\(318\) −1.29800e10 −0.0711793
\(319\) −1.18622e11 −0.641365
\(320\) −2.13629e11 −1.13890
\(321\) 2.76075e11 1.45129
\(322\) 5.23522e7 0.000271384 0
\(323\) 0 0
\(324\) 2.45260e11 1.23644
\(325\) −8.89312e10 −0.442159
\(326\) 3.26692e9 0.0160199
\(327\) 1.23277e11 0.596236
\(328\) 1.59091e10 0.0758949
\(329\) −1.31668e10 −0.0619581
\(330\) 1.12122e10 0.0520450
\(331\) −2.14841e11 −0.983766 −0.491883 0.870661i \(-0.663691\pi\)
−0.491883 + 0.870661i \(0.663691\pi\)
\(332\) −1.76151e11 −0.795725
\(333\) 1.42172e11 0.633601
\(334\) −1.94308e10 −0.0854342
\(335\) −2.05811e11 −0.892828
\(336\) −8.45744e10 −0.362003
\(337\) 3.33228e11 1.40737 0.703683 0.710514i \(-0.251537\pi\)
0.703683 + 0.710514i \(0.251537\pi\)
\(338\) −8.48710e9 −0.0353699
\(339\) 3.72149e11 1.53045
\(340\) 0 0
\(341\) −2.83306e11 −1.13465
\(342\) 2.32276e7 9.18094e−5 0
\(343\) −1.41878e11 −0.553468
\(344\) 3.59335e10 0.138352
\(345\) 6.68509e9 0.0254051
\(346\) −1.57075e10 −0.0589203
\(347\) 1.63241e10 0.0604432 0.0302216 0.999543i \(-0.490379\pi\)
0.0302216 + 0.999543i \(0.490379\pi\)
\(348\) −3.38107e11 −1.23580
\(349\) 1.61528e11 0.582819 0.291410 0.956598i \(-0.405876\pi\)
0.291410 + 0.956598i \(0.405876\pi\)
\(350\) −1.51260e9 −0.00538789
\(351\) −1.85118e11 −0.650978
\(352\) −3.05519e10 −0.106071
\(353\) 2.89170e11 0.991212 0.495606 0.868547i \(-0.334946\pi\)
0.495606 + 0.868547i \(0.334946\pi\)
\(354\) 4.05016e10 0.137075
\(355\) 5.31194e11 1.77511
\(356\) −2.06207e11 −0.680421
\(357\) 0 0
\(358\) 1.02265e10 0.0329042
\(359\) 2.34120e11 0.743898 0.371949 0.928253i \(-0.378690\pi\)
0.371949 + 0.928253i \(0.378690\pi\)
\(360\) 2.39928e10 0.0752870
\(361\) −3.22685e11 −0.999992
\(362\) −1.06801e10 −0.0326878
\(363\) 2.39190e11 0.723040
\(364\) 1.23953e11 0.370084
\(365\) 2.40886e11 0.710385
\(366\) 4.35606e10 0.126890
\(367\) 1.08932e11 0.313442 0.156721 0.987643i \(-0.449908\pi\)
0.156721 + 0.987643i \(0.449908\pi\)
\(368\) −6.04215e9 −0.0171742
\(369\) 1.49612e11 0.420094
\(370\) −2.39551e10 −0.0664494
\(371\) −1.09344e11 −0.299648
\(372\) −8.07507e11 −2.18627
\(373\) −6.82746e11 −1.82629 −0.913145 0.407635i \(-0.866353\pi\)
−0.913145 + 0.407635i \(0.866353\pi\)
\(374\) 0 0
\(375\) 3.68336e11 0.961841
\(376\) −9.00696e9 −0.0232398
\(377\) 4.94067e11 1.25965
\(378\) −3.14861e9 −0.00793243
\(379\) −1.87645e11 −0.467156 −0.233578 0.972338i \(-0.575043\pi\)
−0.233578 + 0.972338i \(0.575043\pi\)
\(380\) 1.32649e9 0.00326345
\(381\) 8.73397e11 2.12349
\(382\) −2.81728e10 −0.0676932
\(383\) 1.01456e11 0.240925 0.120462 0.992718i \(-0.461562\pi\)
0.120462 + 0.992718i \(0.461562\pi\)
\(384\) −1.16052e11 −0.272372
\(385\) 9.44515e10 0.219097
\(386\) 2.51202e10 0.0575945
\(387\) 3.37924e11 0.765808
\(388\) −5.05375e11 −1.13206
\(389\) −4.62050e11 −1.02310 −0.511548 0.859255i \(-0.670928\pi\)
−0.511548 + 0.859255i \(0.670928\pi\)
\(390\) −4.66997e10 −0.102217
\(391\) 0 0
\(392\) −4.64157e10 −0.0992836
\(393\) 8.06637e11 1.70573
\(394\) 7.13051e9 0.0149069
\(395\) 4.19269e11 0.866574
\(396\) −1.91449e11 −0.391224
\(397\) −8.72806e11 −1.76344 −0.881720 0.471774i \(-0.843614\pi\)
−0.881720 + 0.471774i \(0.843614\pi\)
\(398\) 5.51687e9 0.0110209
\(399\) 5.22027e8 0.00103113
\(400\) 1.74575e11 0.340966
\(401\) 1.68101e11 0.324654 0.162327 0.986737i \(-0.448100\pi\)
0.162327 + 0.986737i \(0.448100\pi\)
\(402\) −2.76622e10 −0.0528287
\(403\) 1.17999e12 2.22846
\(404\) 5.71332e11 1.06702
\(405\) −7.78395e11 −1.43765
\(406\) 8.40343e9 0.0153493
\(407\) 3.82861e11 0.691618
\(408\) 0 0
\(409\) −6.99127e11 −1.23538 −0.617691 0.786421i \(-0.711932\pi\)
−0.617691 + 0.786421i \(0.711932\pi\)
\(410\) −2.52086e10 −0.0440577
\(411\) −9.32102e11 −1.61130
\(412\) −3.36311e9 −0.00575047
\(413\) 3.41185e11 0.577052
\(414\) 3.36786e8 0.000563446 0
\(415\) 5.59060e11 0.925214
\(416\) 1.27251e11 0.208324
\(417\) 2.76097e11 0.447145
\(418\) 6.25505e7 0.000100216 0
\(419\) −9.68451e11 −1.53502 −0.767511 0.641036i \(-0.778505\pi\)
−0.767511 + 0.641036i \(0.778505\pi\)
\(420\) 2.69215e11 0.422161
\(421\) −5.47606e11 −0.849569 −0.424784 0.905295i \(-0.639650\pi\)
−0.424784 + 0.905295i \(0.639650\pi\)
\(422\) 4.26921e10 0.0655303
\(423\) −8.47029e10 −0.128637
\(424\) −7.47983e10 −0.112395
\(425\) 0 0
\(426\) 7.13956e10 0.105034
\(427\) 3.66954e11 0.534178
\(428\) 7.94279e11 1.14413
\(429\) 7.46373e11 1.06389
\(430\) −5.69381e10 −0.0803147
\(431\) −4.80670e11 −0.670964 −0.335482 0.942047i \(-0.608899\pi\)
−0.335482 + 0.942047i \(0.608899\pi\)
\(432\) 3.63392e11 0.501994
\(433\) −7.09547e11 −0.970031 −0.485016 0.874506i \(-0.661186\pi\)
−0.485016 + 0.874506i \(0.661186\pi\)
\(434\) 2.00701e10 0.0271547
\(435\) 1.07307e12 1.43690
\(436\) 3.54674e11 0.470045
\(437\) 3.72946e7 4.89192e−5 0
\(438\) 3.23765e10 0.0420336
\(439\) −1.17101e12 −1.50477 −0.752387 0.658721i \(-0.771098\pi\)
−0.752387 + 0.658721i \(0.771098\pi\)
\(440\) 6.46111e10 0.0821808
\(441\) −4.36501e11 −0.549555
\(442\) 0 0
\(443\) −4.75055e11 −0.586040 −0.293020 0.956106i \(-0.594660\pi\)
−0.293020 + 0.956106i \(0.594660\pi\)
\(444\) 1.09127e12 1.33263
\(445\) 6.54450e11 0.791146
\(446\) 3.02333e10 0.0361809
\(447\) −3.79603e11 −0.449723
\(448\) −2.41877e11 −0.283690
\(449\) 9.33820e11 1.08431 0.542156 0.840278i \(-0.317608\pi\)
0.542156 + 0.840278i \(0.317608\pi\)
\(450\) −9.73069e9 −0.0111863
\(451\) 4.02895e11 0.458561
\(452\) 1.07069e12 1.20653
\(453\) 2.93058e11 0.326973
\(454\) 5.46938e10 0.0604209
\(455\) −3.93397e11 −0.430308
\(456\) 3.57101e8 0.000386767 0
\(457\) 1.28584e12 1.37900 0.689498 0.724288i \(-0.257831\pi\)
0.689498 + 0.724288i \(0.257831\pi\)
\(458\) −8.12223e10 −0.0862543
\(459\) 0 0
\(460\) 1.92332e10 0.0200282
\(461\) 5.18444e11 0.534623 0.267311 0.963610i \(-0.413865\pi\)
0.267311 + 0.963610i \(0.413865\pi\)
\(462\) 1.26948e10 0.0129640
\(463\) −7.76808e11 −0.785596 −0.392798 0.919625i \(-0.628493\pi\)
−0.392798 + 0.919625i \(0.628493\pi\)
\(464\) −9.69869e11 −0.971364
\(465\) 2.56283e12 2.54204
\(466\) −5.34744e10 −0.0525302
\(467\) 8.28768e10 0.0806319 0.0403159 0.999187i \(-0.487164\pi\)
0.0403159 + 0.999187i \(0.487164\pi\)
\(468\) 7.97399e11 0.768368
\(469\) −2.33026e11 −0.222396
\(470\) 1.42719e10 0.0134909
\(471\) 7.88182e11 0.737959
\(472\) 2.33393e11 0.216446
\(473\) 9.10008e11 0.835930
\(474\) 5.63522e10 0.0512753
\(475\) −1.07755e9 −0.000971213 0
\(476\) 0 0
\(477\) −7.03416e11 −0.622127
\(478\) −4.50562e10 −0.0394757
\(479\) 2.08230e12 1.80731 0.903657 0.428257i \(-0.140872\pi\)
0.903657 + 0.428257i \(0.140872\pi\)
\(480\) 2.76378e11 0.237639
\(481\) −1.59464e12 −1.35835
\(482\) 4.45120e10 0.0375634
\(483\) 7.56907e9 0.00632820
\(484\) 6.88157e11 0.570011
\(485\) 1.60394e12 1.31629
\(486\) −7.08369e10 −0.0575965
\(487\) 2.69427e11 0.217051 0.108525 0.994094i \(-0.465387\pi\)
0.108525 + 0.994094i \(0.465387\pi\)
\(488\) 2.51021e11 0.200364
\(489\) 4.72330e11 0.373556
\(490\) 7.35477e10 0.0576350
\(491\) −5.87772e10 −0.0456396 −0.0228198 0.999740i \(-0.507264\pi\)
−0.0228198 + 0.999740i \(0.507264\pi\)
\(492\) 1.14837e12 0.883567
\(493\) 0 0
\(494\) −2.60527e8 −0.000196826 0
\(495\) 6.07614e11 0.454888
\(496\) −2.31635e12 −1.71845
\(497\) 6.01435e11 0.442166
\(498\) 7.51409e10 0.0547450
\(499\) 3.13142e11 0.226094 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(500\) 1.05972e12 0.758271
\(501\) −2.80930e12 −1.99218
\(502\) 1.14736e11 0.0806365
\(503\) −7.31761e11 −0.509699 −0.254849 0.966981i \(-0.582026\pi\)
−0.254849 + 0.966981i \(0.582026\pi\)
\(504\) 2.71654e10 0.0187534
\(505\) −1.81327e12 −1.24066
\(506\) 9.06943e8 0.000615039 0
\(507\) −1.22706e12 −0.824766
\(508\) 2.51280e12 1.67406
\(509\) 2.84601e12 1.87935 0.939673 0.342073i \(-0.111129\pi\)
0.939673 + 0.342073i \(0.111129\pi\)
\(510\) 0 0
\(511\) 2.72739e11 0.176951
\(512\) −4.16738e11 −0.268008
\(513\) −2.24300e9 −0.00142989
\(514\) −2.35167e10 −0.0148608
\(515\) 1.06737e10 0.00668625
\(516\) 2.59380e12 1.61069
\(517\) −2.28100e11 −0.140416
\(518\) −2.71228e10 −0.0165520
\(519\) −2.27099e12 −1.37392
\(520\) −2.69110e11 −0.161404
\(521\) 1.42270e10 0.00845949 0.00422974 0.999991i \(-0.498654\pi\)
0.00422974 + 0.999991i \(0.498654\pi\)
\(522\) 5.40599e10 0.0318682
\(523\) 1.20853e12 0.706316 0.353158 0.935564i \(-0.385108\pi\)
0.353158 + 0.935564i \(0.385108\pi\)
\(524\) 2.32072e12 1.34472
\(525\) −2.18692e11 −0.125636
\(526\) −4.60259e10 −0.0262160
\(527\) 0 0
\(528\) −1.46515e12 −0.820410
\(529\) −1.80061e12 −0.999700
\(530\) 1.18521e11 0.0652460
\(531\) 2.19487e12 1.19807
\(532\) 1.50189e9 0.000812899 0
\(533\) −1.67808e12 −0.900619
\(534\) 8.79619e10 0.0468122
\(535\) −2.52085e12 −1.33032
\(536\) −1.59405e11 −0.0834184
\(537\) 1.47854e12 0.767270
\(538\) −1.14971e11 −0.0591652
\(539\) −1.17547e12 −0.599876
\(540\) −1.15674e12 −0.585416
\(541\) −2.35337e12 −1.18115 −0.590573 0.806985i \(-0.701098\pi\)
−0.590573 + 0.806985i \(0.701098\pi\)
\(542\) 1.20515e11 0.0599854
\(543\) −1.54412e12 −0.762224
\(544\) 0 0
\(545\) −1.12565e12 −0.546536
\(546\) −5.28749e10 −0.0254614
\(547\) −1.30236e12 −0.621996 −0.310998 0.950411i \(-0.600663\pi\)
−0.310998 + 0.950411i \(0.600663\pi\)
\(548\) −2.68169e12 −1.27027
\(549\) 2.36064e12 1.10906
\(550\) −2.62041e10 −0.0122106
\(551\) 5.98642e9 0.00276685
\(552\) 5.17775e9 0.00237364
\(553\) 4.74710e11 0.215856
\(554\) 1.13842e10 0.00513461
\(555\) −3.46343e12 −1.54949
\(556\) 7.94340e11 0.352509
\(557\) 3.33519e12 1.46816 0.734078 0.679065i \(-0.237615\pi\)
0.734078 + 0.679065i \(0.237615\pi\)
\(558\) 1.29112e11 0.0563785
\(559\) −3.79025e12 −1.64178
\(560\) 7.72251e11 0.331827
\(561\) 0 0
\(562\) 1.27838e11 0.0540561
\(563\) −9.51343e11 −0.399070 −0.199535 0.979891i \(-0.563943\pi\)
−0.199535 + 0.979891i \(0.563943\pi\)
\(564\) −6.50152e11 −0.270557
\(565\) −3.39810e12 −1.40287
\(566\) −1.17457e11 −0.0481067
\(567\) −8.81324e11 −0.358106
\(568\) 4.11422e11 0.165852
\(569\) −8.14899e10 −0.0325911 −0.0162955 0.999867i \(-0.505187\pi\)
−0.0162955 + 0.999867i \(0.505187\pi\)
\(570\) −5.65842e8 −0.000224522 0
\(571\) −1.06368e12 −0.418744 −0.209372 0.977836i \(-0.567142\pi\)
−0.209372 + 0.977836i \(0.567142\pi\)
\(572\) 2.14734e12 0.838725
\(573\) −4.07322e12 −1.57849
\(574\) −2.85420e10 −0.0109744
\(575\) −1.56237e10 −0.00596046
\(576\) −1.55601e12 −0.588996
\(577\) −3.44925e12 −1.29549 −0.647743 0.761859i \(-0.724287\pi\)
−0.647743 + 0.761859i \(0.724287\pi\)
\(578\) 0 0
\(579\) 3.63187e12 1.34300
\(580\) 3.08727e12 1.13279
\(581\) 6.32986e11 0.230463
\(582\) 2.15579e11 0.0778848
\(583\) −1.89425e12 −0.679093
\(584\) 1.86572e11 0.0663725
\(585\) −2.53075e12 −0.893405
\(586\) 1.99631e11 0.0699341
\(587\) −3.31180e12 −1.15131 −0.575655 0.817693i \(-0.695253\pi\)
−0.575655 + 0.817693i \(0.695253\pi\)
\(588\) −3.35044e12 −1.15586
\(589\) 1.42975e10 0.00489486
\(590\) −3.69821e11 −0.125649
\(591\) 1.03093e12 0.347604
\(592\) 3.13033e12 1.04747
\(593\) 9.22763e10 0.0306439 0.0153219 0.999883i \(-0.495123\pi\)
0.0153219 + 0.999883i \(0.495123\pi\)
\(594\) −5.45461e10 −0.0179773
\(595\) 0 0
\(596\) −1.09213e12 −0.354541
\(597\) 7.97627e11 0.256989
\(598\) −3.77748e9 −0.00120794
\(599\) −4.28034e11 −0.135849 −0.0679246 0.997690i \(-0.521638\pi\)
−0.0679246 + 0.997690i \(0.521638\pi\)
\(600\) −1.49600e11 −0.0471249
\(601\) −2.92420e12 −0.914266 −0.457133 0.889398i \(-0.651124\pi\)
−0.457133 + 0.889398i \(0.651124\pi\)
\(602\) −6.44671e10 −0.0200057
\(603\) −1.49908e12 −0.461738
\(604\) 8.43140e11 0.257771
\(605\) −2.18405e12 −0.662770
\(606\) −2.43714e11 −0.0734097
\(607\) 1.45237e12 0.434237 0.217118 0.976145i \(-0.430334\pi\)
0.217118 + 0.976145i \(0.430334\pi\)
\(608\) 1.54185e9 0.000457589 0
\(609\) 1.21497e12 0.357920
\(610\) −3.97753e11 −0.116313
\(611\) 9.50050e11 0.275779
\(612\) 0 0
\(613\) 5.13743e12 1.46951 0.734757 0.678331i \(-0.237296\pi\)
0.734757 + 0.678331i \(0.237296\pi\)
\(614\) 2.77713e11 0.0788567
\(615\) −3.64465e12 −1.02735
\(616\) 7.31548e10 0.0204706
\(617\) 5.31905e12 1.47758 0.738790 0.673936i \(-0.235397\pi\)
0.738790 + 0.673936i \(0.235397\pi\)
\(618\) 1.43461e9 0.000395626 0
\(619\) 3.23470e12 0.885577 0.442789 0.896626i \(-0.353989\pi\)
0.442789 + 0.896626i \(0.353989\pi\)
\(620\) 7.37337e12 2.00403
\(621\) −3.25222e10 −0.00877540
\(622\) 1.43866e11 0.0385391
\(623\) 7.40989e11 0.197068
\(624\) 6.10247e12 1.61129
\(625\) −4.67554e12 −1.22566
\(626\) 3.49844e11 0.0910521
\(627\) 9.04353e9 0.00233687
\(628\) 2.26763e12 0.581773
\(629\) 0 0
\(630\) −4.30448e10 −0.0108865
\(631\) −7.91021e12 −1.98635 −0.993175 0.116630i \(-0.962791\pi\)
−0.993175 + 0.116630i \(0.962791\pi\)
\(632\) 3.24733e11 0.0809654
\(633\) 6.17242e12 1.52805
\(634\) −1.22483e10 −0.00301074
\(635\) −7.97501e12 −1.94648
\(636\) −5.39919e12 −1.30849
\(637\) 4.89591e12 1.17816
\(638\) 1.45580e11 0.0347863
\(639\) 3.86908e12 0.918023
\(640\) 1.05967e12 0.249668
\(641\) 2.28438e12 0.534451 0.267225 0.963634i \(-0.413893\pi\)
0.267225 + 0.963634i \(0.413893\pi\)
\(642\) −3.38817e11 −0.0787150
\(643\) −7.52203e11 −0.173534 −0.0867672 0.996229i \(-0.527654\pi\)
−0.0867672 + 0.996229i \(0.527654\pi\)
\(644\) 2.17765e10 0.00498886
\(645\) −8.23209e12 −1.87280
\(646\) 0 0
\(647\) −4.59159e12 −1.03013 −0.515067 0.857150i \(-0.672233\pi\)
−0.515067 + 0.857150i \(0.672233\pi\)
\(648\) −6.02884e11 −0.134322
\(649\) 5.91064e12 1.30778
\(650\) 1.09142e11 0.0239818
\(651\) 2.90172e12 0.633201
\(652\) 1.35891e12 0.294495
\(653\) −4.27664e12 −0.920436 −0.460218 0.887806i \(-0.652229\pi\)
−0.460218 + 0.887806i \(0.652229\pi\)
\(654\) −1.51294e11 −0.0323386
\(655\) −7.36542e12 −1.56355
\(656\) 3.29413e12 0.694502
\(657\) 1.75455e12 0.367385
\(658\) 1.61591e10 0.00336048
\(659\) −8.26860e12 −1.70784 −0.853920 0.520404i \(-0.825782\pi\)
−0.853920 + 0.520404i \(0.825782\pi\)
\(660\) 4.66385e12 0.956746
\(661\) 4.45784e12 0.908276 0.454138 0.890931i \(-0.349947\pi\)
0.454138 + 0.890931i \(0.349947\pi\)
\(662\) 2.63667e11 0.0533574
\(663\) 0 0
\(664\) 4.33005e11 0.0864442
\(665\) −4.76664e9 −0.000945182 0
\(666\) −1.74483e11 −0.0343652
\(667\) 8.67994e10 0.0169805
\(668\) −8.08246e12 −1.57054
\(669\) 4.37112e12 0.843676
\(670\) 2.52585e11 0.0484251
\(671\) 6.35705e12 1.21061
\(672\) 3.12923e11 0.0591938
\(673\) −3.90397e12 −0.733566 −0.366783 0.930307i \(-0.619541\pi\)
−0.366783 + 0.930307i \(0.619541\pi\)
\(674\) −4.08959e11 −0.0763326
\(675\) 9.39656e11 0.174222
\(676\) −3.53030e12 −0.650207
\(677\) −4.84414e12 −0.886273 −0.443137 0.896454i \(-0.646134\pi\)
−0.443137 + 0.896454i \(0.646134\pi\)
\(678\) −4.56725e11 −0.0830082
\(679\) 1.81603e12 0.327876
\(680\) 0 0
\(681\) 7.90761e12 1.40891
\(682\) 3.47691e11 0.0615409
\(683\) 3.67160e12 0.645598 0.322799 0.946468i \(-0.395376\pi\)
0.322799 + 0.946468i \(0.395376\pi\)
\(684\) 9.66179e9 0.00168774
\(685\) 8.51105e12 1.47698
\(686\) 1.74122e11 0.0300189
\(687\) −1.17431e13 −2.01130
\(688\) 7.44037e12 1.26604
\(689\) 7.88969e12 1.33375
\(690\) −8.20436e9 −0.00137792
\(691\) −3.94359e12 −0.658023 −0.329011 0.944326i \(-0.606715\pi\)
−0.329011 + 0.944326i \(0.606715\pi\)
\(692\) −6.53371e12 −1.08314
\(693\) 6.87960e11 0.113309
\(694\) −2.00340e10 −0.00327831
\(695\) −2.52104e12 −0.409873
\(696\) 8.31117e11 0.134252
\(697\) 0 0
\(698\) −1.98238e11 −0.0316109
\(699\) −7.73131e12 −1.22492
\(700\) −6.29184e11 −0.0990459
\(701\) −9.80834e12 −1.53414 −0.767069 0.641564i \(-0.778286\pi\)
−0.767069 + 0.641564i \(0.778286\pi\)
\(702\) 2.27188e11 0.0353077
\(703\) −1.93217e10 −0.00298364
\(704\) −4.19025e12 −0.642928
\(705\) 2.06343e12 0.314585
\(706\) −3.54887e11 −0.0537613
\(707\) −2.05304e12 −0.309037
\(708\) 1.68471e13 2.51986
\(709\) −4.04313e12 −0.600910 −0.300455 0.953796i \(-0.597139\pi\)
−0.300455 + 0.953796i \(0.597139\pi\)
\(710\) −6.51915e11 −0.0962783
\(711\) 3.05384e12 0.448160
\(712\) 5.06886e11 0.0739180
\(713\) 2.07305e11 0.0300404
\(714\) 0 0
\(715\) −6.81515e12 −0.975211
\(716\) 4.25381e12 0.604880
\(717\) −6.51422e12 −0.920505
\(718\) −2.87327e11 −0.0403475
\(719\) 4.48823e12 0.626319 0.313159 0.949701i \(-0.398613\pi\)
0.313159 + 0.949701i \(0.398613\pi\)
\(720\) 4.96795e12 0.688939
\(721\) 1.20851e10 0.00166549
\(722\) 3.96020e11 0.0542374
\(723\) 6.43553e12 0.875915
\(724\) −4.44250e12 −0.600903
\(725\) −2.50788e12 −0.337121
\(726\) −2.93548e11 −0.0392162
\(727\) −1.02503e13 −1.36092 −0.680459 0.732786i \(-0.738220\pi\)
−0.680459 + 0.732786i \(0.738220\pi\)
\(728\) −3.04695e11 −0.0402044
\(729\) −7.85147e11 −0.102962
\(730\) −2.95631e11 −0.0385298
\(731\) 0 0
\(732\) 1.81195e13 2.33264
\(733\) −1.77332e12 −0.226892 −0.113446 0.993544i \(-0.536189\pi\)
−0.113446 + 0.993544i \(0.536189\pi\)
\(734\) −1.33688e11 −0.0170004
\(735\) 1.06335e13 1.34395
\(736\) 2.23558e10 0.00280828
\(737\) −4.03691e12 −0.504018
\(738\) −1.83613e11 −0.0227850
\(739\) 2.07204e12 0.255563 0.127781 0.991802i \(-0.459214\pi\)
0.127781 + 0.991802i \(0.459214\pi\)
\(740\) −9.96441e12 −1.22154
\(741\) −3.76669e10 −0.00458963
\(742\) 1.34193e11 0.0162523
\(743\) −7.60966e12 −0.916042 −0.458021 0.888941i \(-0.651442\pi\)
−0.458021 + 0.888941i \(0.651442\pi\)
\(744\) 1.98497e12 0.237507
\(745\) 3.46616e12 0.412236
\(746\) 8.37909e11 0.0990541
\(747\) 4.07205e12 0.478487
\(748\) 0 0
\(749\) −2.85419e12 −0.331371
\(750\) −4.52045e11 −0.0521682
\(751\) 7.56877e12 0.868252 0.434126 0.900852i \(-0.357057\pi\)
0.434126 + 0.900852i \(0.357057\pi\)
\(752\) −1.86498e12 −0.212664
\(753\) 1.65884e13 1.88030
\(754\) −6.06350e11 −0.0683207
\(755\) −2.67592e12 −0.299718
\(756\) −1.30970e12 −0.145822
\(757\) −6.86633e12 −0.759965 −0.379982 0.924994i \(-0.624070\pi\)
−0.379982 + 0.924994i \(0.624070\pi\)
\(758\) 2.30290e11 0.0253375
\(759\) 1.31125e11 0.0143416
\(760\) −3.26070e9 −0.000354528 0
\(761\) 9.51816e12 1.02878 0.514389 0.857557i \(-0.328019\pi\)
0.514389 + 0.857557i \(0.328019\pi\)
\(762\) −1.07189e12 −0.115173
\(763\) −1.27450e12 −0.136138
\(764\) −1.17188e13 −1.24441
\(765\) 0 0
\(766\) −1.24513e11 −0.0130672
\(767\) −2.46182e13 −2.56849
\(768\) −1.18363e13 −1.22769
\(769\) −6.30061e12 −0.649702 −0.324851 0.945765i \(-0.605314\pi\)
−0.324851 + 0.945765i \(0.605314\pi\)
\(770\) −1.15917e11 −0.0118833
\(771\) −3.40003e12 −0.346528
\(772\) 1.04490e13 1.05876
\(773\) −1.40318e13 −1.41354 −0.706768 0.707446i \(-0.749847\pi\)
−0.706768 + 0.707446i \(0.749847\pi\)
\(774\) −4.14722e11 −0.0415358
\(775\) −5.98961e12 −0.596404
\(776\) 1.24229e12 0.122983
\(777\) −3.92140e12 −0.385964
\(778\) 5.67057e11 0.0554905
\(779\) −2.03327e10 −0.00197823
\(780\) −1.94252e13 −1.87906
\(781\) 1.04192e13 1.00208
\(782\) 0 0
\(783\) −5.22036e12 −0.496333
\(784\) −9.61082e12 −0.908528
\(785\) −7.19691e12 −0.676445
\(786\) −9.89955e11 −0.0925154
\(787\) −1.03038e13 −0.957440 −0.478720 0.877968i \(-0.658899\pi\)
−0.478720 + 0.877968i \(0.658899\pi\)
\(788\) 2.96602e12 0.274035
\(789\) −6.65440e12 −0.611312
\(790\) −5.14553e11 −0.0470011
\(791\) −3.84744e12 −0.349444
\(792\) 4.70610e11 0.0425009
\(793\) −2.64776e13 −2.37765
\(794\) 1.07116e12 0.0956452
\(795\) 1.71357e13 1.52143
\(796\) 2.29480e12 0.202599
\(797\) 1.57641e13 1.38390 0.691952 0.721944i \(-0.256751\pi\)
0.691952 + 0.721944i \(0.256751\pi\)
\(798\) −6.40665e8 −5.59265e−5 0
\(799\) 0 0
\(800\) −6.45923e11 −0.0557540
\(801\) 4.76684e12 0.409152
\(802\) −2.06304e11 −0.0176086
\(803\) 4.72489e12 0.401026
\(804\) −1.15064e13 −0.971154
\(805\) −6.91134e10 −0.00580070
\(806\) −1.44816e12 −0.120867
\(807\) −1.66224e13 −1.37963
\(808\) −1.40442e12 −0.115916
\(809\) −1.37214e13 −1.12623 −0.563117 0.826377i \(-0.690398\pi\)
−0.563117 + 0.826377i \(0.690398\pi\)
\(810\) 9.55295e11 0.0779750
\(811\) −1.55198e13 −1.25977 −0.629887 0.776687i \(-0.716899\pi\)
−0.629887 + 0.776687i \(0.716899\pi\)
\(812\) 3.49550e12 0.282168
\(813\) 1.74241e13 1.39876
\(814\) −4.69871e11 −0.0375119
\(815\) −4.31286e12 −0.342418
\(816\) 0 0
\(817\) −4.59250e10 −0.00360620
\(818\) 8.58013e11 0.0670045
\(819\) −2.86540e12 −0.222540
\(820\) −1.04858e13 −0.809915
\(821\) −3.40134e12 −0.261280 −0.130640 0.991430i \(-0.541703\pi\)
−0.130640 + 0.991430i \(0.541703\pi\)
\(822\) 1.14393e12 0.0873933
\(823\) −4.01848e12 −0.305325 −0.152663 0.988278i \(-0.548785\pi\)
−0.152663 + 0.988278i \(0.548785\pi\)
\(824\) 8.26702e9 0.000624707 0
\(825\) −3.78858e12 −0.284731
\(826\) −4.18724e11 −0.0312981
\(827\) 4.31147e12 0.320517 0.160258 0.987075i \(-0.448767\pi\)
0.160258 + 0.987075i \(0.448767\pi\)
\(828\) 1.40090e11 0.0103579
\(829\) 4.42248e12 0.325215 0.162608 0.986691i \(-0.448010\pi\)
0.162608 + 0.986691i \(0.448010\pi\)
\(830\) −6.86114e11 −0.0501816
\(831\) 1.64592e12 0.119730
\(832\) 1.74527e13 1.26272
\(833\) 0 0
\(834\) −3.38843e11 −0.0242522
\(835\) 2.56518e13 1.82612
\(836\) 2.60186e10 0.00184228
\(837\) −1.24679e13 −0.878068
\(838\) 1.18854e12 0.0832563
\(839\) −1.12787e13 −0.785835 −0.392917 0.919574i \(-0.628534\pi\)
−0.392917 + 0.919574i \(0.628534\pi\)
\(840\) −6.61771e11 −0.0458618
\(841\) −5.74353e11 −0.0395910
\(842\) 6.72056e11 0.0460788
\(843\) 1.84827e13 1.26050
\(844\) 1.77583e13 1.20465
\(845\) 1.12043e13 0.756016
\(846\) 1.03953e11 0.00697701
\(847\) −2.47285e12 −0.165090
\(848\) −1.54877e13 −1.02850
\(849\) −1.69819e13 −1.12177
\(850\) 0 0
\(851\) −2.80152e11 −0.0183110
\(852\) 2.96978e13 1.93084
\(853\) 3.07683e12 0.198991 0.0994954 0.995038i \(-0.468277\pi\)
0.0994954 + 0.995038i \(0.468277\pi\)
\(854\) −4.50349e11 −0.0289727
\(855\) −3.06642e10 −0.00196238
\(856\) −1.95246e12 −0.124294
\(857\) 3.10840e13 1.96845 0.984223 0.176930i \(-0.0566166\pi\)
0.984223 + 0.176930i \(0.0566166\pi\)
\(858\) −9.15996e11 −0.0577033
\(859\) −7.61893e12 −0.477446 −0.238723 0.971088i \(-0.576729\pi\)
−0.238723 + 0.971088i \(0.576729\pi\)
\(860\) −2.36840e13 −1.47643
\(861\) −4.12659e12 −0.255904
\(862\) 5.89908e11 0.0363917
\(863\) −2.61427e13 −1.60436 −0.802179 0.597083i \(-0.796326\pi\)
−0.802179 + 0.597083i \(0.796326\pi\)
\(864\) −1.34454e12 −0.0820849
\(865\) 2.07364e13 1.25939
\(866\) 8.70801e11 0.0526124
\(867\) 0 0
\(868\) 8.34837e12 0.499186
\(869\) 8.22380e12 0.489197
\(870\) −1.31694e12 −0.0779345
\(871\) 1.68140e13 0.989896
\(872\) −8.71840e11 −0.0510638
\(873\) 1.16827e13 0.680735
\(874\) −4.57703e7 −2.65327e−6 0
\(875\) −3.80802e12 −0.219615
\(876\) 1.34674e13 0.772706
\(877\) 3.00641e13 1.71613 0.858065 0.513541i \(-0.171667\pi\)
0.858065 + 0.513541i \(0.171667\pi\)
\(878\) 1.43714e12 0.0816158
\(879\) 2.88626e13 1.63074
\(880\) 1.33784e13 0.752023
\(881\) 2.06409e13 1.15435 0.577175 0.816620i \(-0.304155\pi\)
0.577175 + 0.816620i \(0.304155\pi\)
\(882\) 5.35702e11 0.0298067
\(883\) 1.85711e13 1.02805 0.514026 0.857774i \(-0.328153\pi\)
0.514026 + 0.857774i \(0.328153\pi\)
\(884\) 0 0
\(885\) −5.34687e13 −2.92991
\(886\) 5.83017e11 0.0317855
\(887\) −1.35590e11 −0.00735482 −0.00367741 0.999993i \(-0.501171\pi\)
−0.00367741 + 0.999993i \(0.501171\pi\)
\(888\) −2.68250e12 −0.144771
\(889\) −9.02957e12 −0.484852
\(890\) −8.03182e11 −0.0429101
\(891\) −1.52679e13 −0.811578
\(892\) 1.25759e13 0.665115
\(893\) 1.15114e10 0.000605754 0
\(894\) 4.65872e11 0.0243920
\(895\) −1.35006e13 −0.703313
\(896\) 1.19980e12 0.0621901
\(897\) −5.46146e11 −0.0281672
\(898\) −1.14604e12 −0.0588108
\(899\) 3.32759e13 1.69907
\(900\) −4.04759e12 −0.205639
\(901\) 0 0
\(902\) −4.94458e11 −0.0248714
\(903\) −9.32064e12 −0.466499
\(904\) −2.63190e12 −0.131073
\(905\) 1.40994e13 0.698688
\(906\) −3.59660e11 −0.0177343
\(907\) 1.36972e13 0.672045 0.336022 0.941854i \(-0.390918\pi\)
0.336022 + 0.941854i \(0.390918\pi\)
\(908\) 2.27505e13 1.11072
\(909\) −1.32074e13 −0.641622
\(910\) 4.82802e11 0.0233390
\(911\) 2.24844e13 1.08156 0.540778 0.841165i \(-0.318130\pi\)
0.540778 + 0.841165i \(0.318130\pi\)
\(912\) 7.39413e10 0.00353924
\(913\) 1.09658e13 0.522300
\(914\) −1.57806e12 −0.0747938
\(915\) −5.75070e13 −2.71223
\(916\) −3.37853e13 −1.58562
\(917\) −8.33936e12 −0.389467
\(918\) 0 0
\(919\) 9.66637e12 0.447037 0.223519 0.974700i \(-0.428246\pi\)
0.223519 + 0.974700i \(0.428246\pi\)
\(920\) −4.72781e10 −0.00217578
\(921\) 4.01516e13 1.83880
\(922\) −6.36267e11 −0.0289968
\(923\) −4.33966e13 −1.96810
\(924\) 5.28056e12 0.238318
\(925\) 8.09439e12 0.363535
\(926\) 9.53348e11 0.0426091
\(927\) 7.77444e10 0.00345788
\(928\) 3.58850e12 0.158835
\(929\) −8.19216e12 −0.360851 −0.180425 0.983589i \(-0.557747\pi\)
−0.180425 + 0.983589i \(0.557747\pi\)
\(930\) −3.14527e12 −0.137875
\(931\) 5.93219e10 0.00258786
\(932\) −2.22433e13 −0.965667
\(933\) 2.08001e13 0.898665
\(934\) −1.01712e11 −0.00437330
\(935\) 0 0
\(936\) −1.96012e12 −0.0834723
\(937\) 3.49872e12 0.148279 0.0741396 0.997248i \(-0.476379\pi\)
0.0741396 + 0.997248i \(0.476379\pi\)
\(938\) 2.85984e11 0.0120623
\(939\) 5.05803e13 2.12318
\(940\) 5.93656e12 0.248004
\(941\) 1.51926e12 0.0631655 0.0315827 0.999501i \(-0.489945\pi\)
0.0315827 + 0.999501i \(0.489945\pi\)
\(942\) −9.67307e11 −0.0400254
\(943\) −2.94811e11 −0.0121406
\(944\) 4.83263e13 1.98066
\(945\) 4.15668e12 0.169552
\(946\) −1.11682e12 −0.0453391
\(947\) 2.12785e13 0.859738 0.429869 0.902891i \(-0.358560\pi\)
0.429869 + 0.902891i \(0.358560\pi\)
\(948\) 2.34403e13 0.942597
\(949\) −1.96795e13 −0.787619
\(950\) 1.32243e9 5.26765e−5 0
\(951\) −1.77085e12 −0.0702053
\(952\) 0 0
\(953\) 9.61359e12 0.377544 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(954\) 8.63276e11 0.0337429
\(955\) 3.71927e13 1.44691
\(956\) −1.87417e13 −0.725684
\(957\) 2.10479e13 0.811157
\(958\) −2.55553e12 −0.0980249
\(959\) 9.63648e12 0.367904
\(960\) 3.79057e13 1.44040
\(961\) 5.30339e13 2.00585
\(962\) 1.95705e12 0.0736738
\(963\) −1.83612e13 −0.687992
\(964\) 1.85153e13 0.690531
\(965\) −3.31627e13 −1.23106
\(966\) −9.28924e9 −0.000343228 0
\(967\) −1.08700e13 −0.399772 −0.199886 0.979819i \(-0.564057\pi\)
−0.199886 + 0.979819i \(0.564057\pi\)
\(968\) −1.69159e12 −0.0619236
\(969\) 0 0
\(970\) −1.96846e12 −0.0713925
\(971\) 2.79438e13 1.00879 0.504393 0.863474i \(-0.331716\pi\)
0.504393 + 0.863474i \(0.331716\pi\)
\(972\) −2.94654e13 −1.05880
\(973\) −2.85441e12 −0.102096
\(974\) −3.30658e11 −0.0117724
\(975\) 1.57797e13 0.559214
\(976\) 5.19763e13 1.83350
\(977\) −1.49620e13 −0.525369 −0.262685 0.964882i \(-0.584608\pi\)
−0.262685 + 0.964882i \(0.584608\pi\)
\(978\) −5.79674e11 −0.0202609
\(979\) 1.28368e13 0.446616
\(980\) 3.05930e13 1.05951
\(981\) −8.19893e12 −0.282648
\(982\) 7.21350e10 0.00247540
\(983\) −2.54966e13 −0.870946 −0.435473 0.900202i \(-0.643419\pi\)
−0.435473 + 0.900202i \(0.643419\pi\)
\(984\) −2.82286e12 −0.0959869
\(985\) −9.41343e12 −0.318629
\(986\) 0 0
\(987\) 2.33628e12 0.0783605
\(988\) −1.08369e11 −0.00361826
\(989\) −6.65884e11 −0.0221317
\(990\) −7.45702e11 −0.0246721
\(991\) 9.59893e12 0.316149 0.158074 0.987427i \(-0.449471\pi\)
0.158074 + 0.987427i \(0.449471\pi\)
\(992\) 8.57047e12 0.280997
\(993\) 3.81209e13 1.24420
\(994\) −7.38119e11 −0.0239821
\(995\) −7.28315e12 −0.235568
\(996\) 3.12557e13 1.00638
\(997\) 3.17533e13 1.01780 0.508898 0.860827i \(-0.330053\pi\)
0.508898 + 0.860827i \(0.330053\pi\)
\(998\) −3.84308e11 −0.0122629
\(999\) 1.68492e13 0.535222
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 289.10.a.a.1.3 5
17.16 even 2 17.10.a.a.1.3 5
51.50 odd 2 153.10.a.c.1.3 5
68.67 odd 2 272.10.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.a.a.1.3 5 17.16 even 2
153.10.a.c.1.3 5 51.50 odd 2
272.10.a.f.1.1 5 68.67 odd 2
289.10.a.a.1.3 5 1.1 even 1 trivial