Properties

Label 8034.2.a.w.1.9
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(3.23048\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.42840 q^{5} -1.00000 q^{6} -2.28162 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.42840 q^{5} -1.00000 q^{6} -2.28162 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.42840 q^{10} -0.378203 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.28162 q^{14} +1.42840 q^{15} +1.00000 q^{16} -1.07828 q^{17} -1.00000 q^{18} -6.85408 q^{19} +1.42840 q^{20} -2.28162 q^{21} +0.378203 q^{22} -1.64807 q^{23} -1.00000 q^{24} -2.95968 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.28162 q^{28} -5.09337 q^{29} -1.42840 q^{30} +10.3699 q^{31} -1.00000 q^{32} -0.378203 q^{33} +1.07828 q^{34} -3.25906 q^{35} +1.00000 q^{36} +9.76199 q^{37} +6.85408 q^{38} +1.00000 q^{39} -1.42840 q^{40} +0.952622 q^{41} +2.28162 q^{42} +2.65892 q^{43} -0.378203 q^{44} +1.42840 q^{45} +1.64807 q^{46} +7.38329 q^{47} +1.00000 q^{48} -1.79420 q^{49} +2.95968 q^{50} -1.07828 q^{51} +1.00000 q^{52} +7.79122 q^{53} -1.00000 q^{54} -0.540224 q^{55} +2.28162 q^{56} -6.85408 q^{57} +5.09337 q^{58} -0.731526 q^{59} +1.42840 q^{60} -4.05568 q^{61} -10.3699 q^{62} -2.28162 q^{63} +1.00000 q^{64} +1.42840 q^{65} +0.378203 q^{66} -9.03946 q^{67} -1.07828 q^{68} -1.64807 q^{69} +3.25906 q^{70} +3.02171 q^{71} -1.00000 q^{72} -11.2549 q^{73} -9.76199 q^{74} -2.95968 q^{75} -6.85408 q^{76} +0.862916 q^{77} -1.00000 q^{78} +5.25463 q^{79} +1.42840 q^{80} +1.00000 q^{81} -0.952622 q^{82} -11.2812 q^{83} -2.28162 q^{84} -1.54021 q^{85} -2.65892 q^{86} -5.09337 q^{87} +0.378203 q^{88} -5.26838 q^{89} -1.42840 q^{90} -2.28162 q^{91} -1.64807 q^{92} +10.3699 q^{93} -7.38329 q^{94} -9.79035 q^{95} -1.00000 q^{96} -10.0419 q^{97} +1.79420 q^{98} -0.378203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.42840 0.638798 0.319399 0.947620i \(-0.396519\pi\)
0.319399 + 0.947620i \(0.396519\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.28162 −0.862372 −0.431186 0.902263i \(-0.641905\pi\)
−0.431186 + 0.902263i \(0.641905\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.42840 −0.451698
\(11\) −0.378203 −0.114033 −0.0570163 0.998373i \(-0.518159\pi\)
−0.0570163 + 0.998373i \(0.518159\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 2.28162 0.609789
\(15\) 1.42840 0.368810
\(16\) 1.00000 0.250000
\(17\) −1.07828 −0.261521 −0.130761 0.991414i \(-0.541742\pi\)
−0.130761 + 0.991414i \(0.541742\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.85408 −1.57243 −0.786217 0.617950i \(-0.787963\pi\)
−0.786217 + 0.617950i \(0.787963\pi\)
\(20\) 1.42840 0.319399
\(21\) −2.28162 −0.497891
\(22\) 0.378203 0.0806332
\(23\) −1.64807 −0.343647 −0.171824 0.985128i \(-0.554966\pi\)
−0.171824 + 0.985128i \(0.554966\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.95968 −0.591937
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −2.28162 −0.431186
\(29\) −5.09337 −0.945816 −0.472908 0.881112i \(-0.656796\pi\)
−0.472908 + 0.881112i \(0.656796\pi\)
\(30\) −1.42840 −0.260788
\(31\) 10.3699 1.86248 0.931240 0.364406i \(-0.118728\pi\)
0.931240 + 0.364406i \(0.118728\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.378203 −0.0658367
\(34\) 1.07828 0.184923
\(35\) −3.25906 −0.550881
\(36\) 1.00000 0.166667
\(37\) 9.76199 1.60486 0.802430 0.596746i \(-0.203540\pi\)
0.802430 + 0.596746i \(0.203540\pi\)
\(38\) 6.85408 1.11188
\(39\) 1.00000 0.160128
\(40\) −1.42840 −0.225849
\(41\) 0.952622 0.148775 0.0743873 0.997229i \(-0.476300\pi\)
0.0743873 + 0.997229i \(0.476300\pi\)
\(42\) 2.28162 0.352062
\(43\) 2.65892 0.405482 0.202741 0.979232i \(-0.435015\pi\)
0.202741 + 0.979232i \(0.435015\pi\)
\(44\) −0.378203 −0.0570163
\(45\) 1.42840 0.212933
\(46\) 1.64807 0.242995
\(47\) 7.38329 1.07696 0.538482 0.842637i \(-0.318998\pi\)
0.538482 + 0.842637i \(0.318998\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.79420 −0.256315
\(50\) 2.95968 0.418563
\(51\) −1.07828 −0.150989
\(52\) 1.00000 0.138675
\(53\) 7.79122 1.07021 0.535103 0.844787i \(-0.320273\pi\)
0.535103 + 0.844787i \(0.320273\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.540224 −0.0728438
\(56\) 2.28162 0.304894
\(57\) −6.85408 −0.907846
\(58\) 5.09337 0.668793
\(59\) −0.731526 −0.0952366 −0.0476183 0.998866i \(-0.515163\pi\)
−0.0476183 + 0.998866i \(0.515163\pi\)
\(60\) 1.42840 0.184405
\(61\) −4.05568 −0.519277 −0.259638 0.965706i \(-0.583603\pi\)
−0.259638 + 0.965706i \(0.583603\pi\)
\(62\) −10.3699 −1.31697
\(63\) −2.28162 −0.287457
\(64\) 1.00000 0.125000
\(65\) 1.42840 0.177171
\(66\) 0.378203 0.0465536
\(67\) −9.03946 −1.10435 −0.552173 0.833730i \(-0.686201\pi\)
−0.552173 + 0.833730i \(0.686201\pi\)
\(68\) −1.07828 −0.130761
\(69\) −1.64807 −0.198405
\(70\) 3.25906 0.389532
\(71\) 3.02171 0.358611 0.179306 0.983793i \(-0.442615\pi\)
0.179306 + 0.983793i \(0.442615\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.2549 −1.31728 −0.658641 0.752457i \(-0.728868\pi\)
−0.658641 + 0.752457i \(0.728868\pi\)
\(74\) −9.76199 −1.13481
\(75\) −2.95968 −0.341755
\(76\) −6.85408 −0.786217
\(77\) 0.862916 0.0983384
\(78\) −1.00000 −0.113228
\(79\) 5.25463 0.591193 0.295596 0.955313i \(-0.404482\pi\)
0.295596 + 0.955313i \(0.404482\pi\)
\(80\) 1.42840 0.159700
\(81\) 1.00000 0.111111
\(82\) −0.952622 −0.105200
\(83\) −11.2812 −1.23827 −0.619136 0.785284i \(-0.712517\pi\)
−0.619136 + 0.785284i \(0.712517\pi\)
\(84\) −2.28162 −0.248945
\(85\) −1.54021 −0.167059
\(86\) −2.65892 −0.286719
\(87\) −5.09337 −0.546067
\(88\) 0.378203 0.0403166
\(89\) −5.26838 −0.558447 −0.279223 0.960226i \(-0.590077\pi\)
−0.279223 + 0.960226i \(0.590077\pi\)
\(90\) −1.42840 −0.150566
\(91\) −2.28162 −0.239179
\(92\) −1.64807 −0.171824
\(93\) 10.3699 1.07530
\(94\) −7.38329 −0.761528
\(95\) −9.79035 −1.00447
\(96\) −1.00000 −0.102062
\(97\) −10.0419 −1.01960 −0.509802 0.860292i \(-0.670281\pi\)
−0.509802 + 0.860292i \(0.670281\pi\)
\(98\) 1.79420 0.181242
\(99\) −0.378203 −0.0380108
\(100\) −2.95968 −0.295968
\(101\) 6.99307 0.695836 0.347918 0.937525i \(-0.386889\pi\)
0.347918 + 0.937525i \(0.386889\pi\)
\(102\) 1.07828 0.106766
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −3.25906 −0.318052
\(106\) −7.79122 −0.756751
\(107\) −10.4990 −1.01497 −0.507487 0.861660i \(-0.669425\pi\)
−0.507487 + 0.861660i \(0.669425\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.860752 0.0824450 0.0412225 0.999150i \(-0.486875\pi\)
0.0412225 + 0.999150i \(0.486875\pi\)
\(110\) 0.540224 0.0515083
\(111\) 9.76199 0.926567
\(112\) −2.28162 −0.215593
\(113\) −2.84635 −0.267762 −0.133881 0.990997i \(-0.542744\pi\)
−0.133881 + 0.990997i \(0.542744\pi\)
\(114\) 6.85408 0.641944
\(115\) −2.35410 −0.219521
\(116\) −5.09337 −0.472908
\(117\) 1.00000 0.0924500
\(118\) 0.731526 0.0673424
\(119\) 2.46023 0.225528
\(120\) −1.42840 −0.130394
\(121\) −10.8570 −0.986997
\(122\) 4.05568 0.367184
\(123\) 0.952622 0.0858951
\(124\) 10.3699 0.931240
\(125\) −11.3696 −1.01693
\(126\) 2.28162 0.203263
\(127\) −1.00129 −0.0888504 −0.0444252 0.999013i \(-0.514146\pi\)
−0.0444252 + 0.999013i \(0.514146\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.65892 0.234105
\(130\) −1.42840 −0.125279
\(131\) −7.05133 −0.616078 −0.308039 0.951374i \(-0.599673\pi\)
−0.308039 + 0.951374i \(0.599673\pi\)
\(132\) −0.378203 −0.0329183
\(133\) 15.6384 1.35602
\(134\) 9.03946 0.780890
\(135\) 1.42840 0.122937
\(136\) 1.07828 0.0924617
\(137\) −6.33167 −0.540951 −0.270476 0.962727i \(-0.587181\pi\)
−0.270476 + 0.962727i \(0.587181\pi\)
\(138\) 1.64807 0.140293
\(139\) −15.7204 −1.33339 −0.666693 0.745332i \(-0.732291\pi\)
−0.666693 + 0.745332i \(0.732291\pi\)
\(140\) −3.25906 −0.275441
\(141\) 7.38329 0.621785
\(142\) −3.02171 −0.253577
\(143\) −0.378203 −0.0316269
\(144\) 1.00000 0.0833333
\(145\) −7.27536 −0.604185
\(146\) 11.2549 0.931459
\(147\) −1.79420 −0.147984
\(148\) 9.76199 0.802430
\(149\) −10.5290 −0.862570 −0.431285 0.902216i \(-0.641940\pi\)
−0.431285 + 0.902216i \(0.641940\pi\)
\(150\) 2.95968 0.241657
\(151\) −5.65260 −0.460002 −0.230001 0.973190i \(-0.573873\pi\)
−0.230001 + 0.973190i \(0.573873\pi\)
\(152\) 6.85408 0.555940
\(153\) −1.07828 −0.0871737
\(154\) −0.862916 −0.0695358
\(155\) 14.8123 1.18975
\(156\) 1.00000 0.0800641
\(157\) 13.3002 1.06147 0.530736 0.847537i \(-0.321916\pi\)
0.530736 + 0.847537i \(0.321916\pi\)
\(158\) −5.25463 −0.418036
\(159\) 7.79122 0.617884
\(160\) −1.42840 −0.112925
\(161\) 3.76028 0.296352
\(162\) −1.00000 −0.0785674
\(163\) −6.40325 −0.501542 −0.250771 0.968046i \(-0.580684\pi\)
−0.250771 + 0.968046i \(0.580684\pi\)
\(164\) 0.952622 0.0743873
\(165\) −0.540224 −0.0420564
\(166\) 11.2812 0.875591
\(167\) 4.33757 0.335651 0.167826 0.985817i \(-0.446325\pi\)
0.167826 + 0.985817i \(0.446325\pi\)
\(168\) 2.28162 0.176031
\(169\) 1.00000 0.0769231
\(170\) 1.54021 0.118129
\(171\) −6.85408 −0.524145
\(172\) 2.65892 0.202741
\(173\) −9.30906 −0.707755 −0.353878 0.935292i \(-0.615137\pi\)
−0.353878 + 0.935292i \(0.615137\pi\)
\(174\) 5.09337 0.386128
\(175\) 6.75288 0.510470
\(176\) −0.378203 −0.0285081
\(177\) −0.731526 −0.0549849
\(178\) 5.26838 0.394882
\(179\) 6.22986 0.465642 0.232821 0.972520i \(-0.425204\pi\)
0.232821 + 0.972520i \(0.425204\pi\)
\(180\) 1.42840 0.106466
\(181\) −5.27430 −0.392035 −0.196018 0.980600i \(-0.562801\pi\)
−0.196018 + 0.980600i \(0.562801\pi\)
\(182\) 2.28162 0.169125
\(183\) −4.05568 −0.299805
\(184\) 1.64807 0.121498
\(185\) 13.9440 1.02518
\(186\) −10.3699 −0.760354
\(187\) 0.407809 0.0298219
\(188\) 7.38329 0.538482
\(189\) −2.28162 −0.165964
\(190\) 9.79035 0.710266
\(191\) −2.75574 −0.199399 −0.0996993 0.995018i \(-0.531788\pi\)
−0.0996993 + 0.995018i \(0.531788\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.08283 −0.437852 −0.218926 0.975742i \(-0.570255\pi\)
−0.218926 + 0.975742i \(0.570255\pi\)
\(194\) 10.0419 0.720969
\(195\) 1.42840 0.102290
\(196\) −1.79420 −0.128157
\(197\) −10.1965 −0.726468 −0.363234 0.931698i \(-0.618328\pi\)
−0.363234 + 0.931698i \(0.618328\pi\)
\(198\) 0.378203 0.0268777
\(199\) 1.25424 0.0889107 0.0444554 0.999011i \(-0.485845\pi\)
0.0444554 + 0.999011i \(0.485845\pi\)
\(200\) 2.95968 0.209281
\(201\) −9.03946 −0.637594
\(202\) −6.99307 −0.492030
\(203\) 11.6211 0.815645
\(204\) −1.07828 −0.0754947
\(205\) 1.36072 0.0950369
\(206\) 1.00000 0.0696733
\(207\) −1.64807 −0.114549
\(208\) 1.00000 0.0693375
\(209\) 2.59224 0.179309
\(210\) 3.25906 0.224896
\(211\) −16.2651 −1.11973 −0.559867 0.828582i \(-0.689148\pi\)
−0.559867 + 0.828582i \(0.689148\pi\)
\(212\) 7.79122 0.535103
\(213\) 3.02171 0.207044
\(214\) 10.4990 0.717695
\(215\) 3.79799 0.259021
\(216\) −1.00000 −0.0680414
\(217\) −23.6601 −1.60615
\(218\) −0.860752 −0.0582974
\(219\) −11.2549 −0.760533
\(220\) −0.540224 −0.0364219
\(221\) −1.07828 −0.0725329
\(222\) −9.76199 −0.655182
\(223\) −21.2527 −1.42319 −0.711593 0.702592i \(-0.752026\pi\)
−0.711593 + 0.702592i \(0.752026\pi\)
\(224\) 2.28162 0.152447
\(225\) −2.95968 −0.197312
\(226\) 2.84635 0.189336
\(227\) −25.4900 −1.69183 −0.845914 0.533319i \(-0.820945\pi\)
−0.845914 + 0.533319i \(0.820945\pi\)
\(228\) −6.85408 −0.453923
\(229\) 25.8974 1.71135 0.855674 0.517515i \(-0.173143\pi\)
0.855674 + 0.517515i \(0.173143\pi\)
\(230\) 2.35410 0.155225
\(231\) 0.862916 0.0567757
\(232\) 5.09337 0.334396
\(233\) −0.935399 −0.0612800 −0.0306400 0.999530i \(-0.509755\pi\)
−0.0306400 + 0.999530i \(0.509755\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 10.5463 0.687962
\(236\) −0.731526 −0.0476183
\(237\) 5.25463 0.341325
\(238\) −2.46023 −0.159473
\(239\) −7.83371 −0.506720 −0.253360 0.967372i \(-0.581536\pi\)
−0.253360 + 0.967372i \(0.581536\pi\)
\(240\) 1.42840 0.0922026
\(241\) 23.8091 1.53368 0.766841 0.641837i \(-0.221827\pi\)
0.766841 + 0.641837i \(0.221827\pi\)
\(242\) 10.8570 0.697912
\(243\) 1.00000 0.0641500
\(244\) −4.05568 −0.259638
\(245\) −2.56284 −0.163734
\(246\) −0.952622 −0.0607370
\(247\) −6.85408 −0.436115
\(248\) −10.3699 −0.658486
\(249\) −11.2812 −0.714917
\(250\) 11.3696 0.719076
\(251\) −7.47104 −0.471568 −0.235784 0.971805i \(-0.575766\pi\)
−0.235784 + 0.971805i \(0.575766\pi\)
\(252\) −2.28162 −0.143729
\(253\) 0.623307 0.0391870
\(254\) 1.00129 0.0628267
\(255\) −1.54021 −0.0964517
\(256\) 1.00000 0.0625000
\(257\) −29.3198 −1.82892 −0.914459 0.404679i \(-0.867383\pi\)
−0.914459 + 0.404679i \(0.867383\pi\)
\(258\) −2.65892 −0.165537
\(259\) −22.2732 −1.38399
\(260\) 1.42840 0.0885854
\(261\) −5.09337 −0.315272
\(262\) 7.05133 0.435633
\(263\) −25.4645 −1.57021 −0.785106 0.619362i \(-0.787391\pi\)
−0.785106 + 0.619362i \(0.787391\pi\)
\(264\) 0.378203 0.0232768
\(265\) 11.1290 0.683646
\(266\) −15.6384 −0.958853
\(267\) −5.26838 −0.322420
\(268\) −9.03946 −0.552173
\(269\) −23.3373 −1.42290 −0.711451 0.702736i \(-0.751961\pi\)
−0.711451 + 0.702736i \(0.751961\pi\)
\(270\) −1.42840 −0.0869294
\(271\) −6.22070 −0.377881 −0.188940 0.981989i \(-0.560505\pi\)
−0.188940 + 0.981989i \(0.560505\pi\)
\(272\) −1.07828 −0.0653803
\(273\) −2.28162 −0.138090
\(274\) 6.33167 0.382510
\(275\) 1.11936 0.0675001
\(276\) −1.64807 −0.0992024
\(277\) 10.9355 0.657053 0.328526 0.944495i \(-0.393448\pi\)
0.328526 + 0.944495i \(0.393448\pi\)
\(278\) 15.7204 0.942846
\(279\) 10.3699 0.620827
\(280\) 3.25906 0.194766
\(281\) 16.4408 0.980778 0.490389 0.871504i \(-0.336855\pi\)
0.490389 + 0.871504i \(0.336855\pi\)
\(282\) −7.38329 −0.439668
\(283\) 17.3544 1.03161 0.515805 0.856706i \(-0.327493\pi\)
0.515805 + 0.856706i \(0.327493\pi\)
\(284\) 3.02171 0.179306
\(285\) −9.79035 −0.579930
\(286\) 0.378203 0.0223636
\(287\) −2.17352 −0.128299
\(288\) −1.00000 −0.0589256
\(289\) −15.8373 −0.931607
\(290\) 7.27536 0.427224
\(291\) −10.0419 −0.588669
\(292\) −11.2549 −0.658641
\(293\) 9.58413 0.559911 0.279956 0.960013i \(-0.409680\pi\)
0.279956 + 0.960013i \(0.409680\pi\)
\(294\) 1.79420 0.104640
\(295\) −1.04491 −0.0608369
\(296\) −9.76199 −0.567404
\(297\) −0.378203 −0.0219456
\(298\) 10.5290 0.609929
\(299\) −1.64807 −0.0953106
\(300\) −2.95968 −0.170877
\(301\) −6.06665 −0.349676
\(302\) 5.65260 0.325271
\(303\) 6.99307 0.401741
\(304\) −6.85408 −0.393109
\(305\) −5.79312 −0.331713
\(306\) 1.07828 0.0616411
\(307\) −12.5014 −0.713495 −0.356747 0.934201i \(-0.616114\pi\)
−0.356747 + 0.934201i \(0.616114\pi\)
\(308\) 0.862916 0.0491692
\(309\) −1.00000 −0.0568880
\(310\) −14.8123 −0.841279
\(311\) −26.1921 −1.48522 −0.742609 0.669725i \(-0.766412\pi\)
−0.742609 + 0.669725i \(0.766412\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 7.61225 0.430270 0.215135 0.976584i \(-0.430981\pi\)
0.215135 + 0.976584i \(0.430981\pi\)
\(314\) −13.3002 −0.750574
\(315\) −3.25906 −0.183627
\(316\) 5.25463 0.295596
\(317\) 7.73483 0.434431 0.217216 0.976124i \(-0.430303\pi\)
0.217216 + 0.976124i \(0.430303\pi\)
\(318\) −7.79122 −0.436910
\(319\) 1.92633 0.107854
\(320\) 1.42840 0.0798498
\(321\) −10.4990 −0.585995
\(322\) −3.76028 −0.209552
\(323\) 7.39062 0.411225
\(324\) 1.00000 0.0555556
\(325\) −2.95968 −0.164174
\(326\) 6.40325 0.354643
\(327\) 0.860752 0.0475997
\(328\) −0.952622 −0.0525998
\(329\) −16.8459 −0.928743
\(330\) 0.540224 0.0297383
\(331\) 25.0994 1.37959 0.689794 0.724006i \(-0.257701\pi\)
0.689794 + 0.724006i \(0.257701\pi\)
\(332\) −11.2812 −0.619136
\(333\) 9.76199 0.534954
\(334\) −4.33757 −0.237341
\(335\) −12.9119 −0.705454
\(336\) −2.28162 −0.124473
\(337\) −13.9072 −0.757576 −0.378788 0.925483i \(-0.623659\pi\)
−0.378788 + 0.925483i \(0.623659\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.84635 −0.154592
\(340\) −1.54021 −0.0835296
\(341\) −3.92191 −0.212383
\(342\) 6.85408 0.370626
\(343\) 20.0650 1.08341
\(344\) −2.65892 −0.143359
\(345\) −2.35410 −0.126741
\(346\) 9.30906 0.500458
\(347\) −29.0558 −1.55980 −0.779898 0.625907i \(-0.784729\pi\)
−0.779898 + 0.625907i \(0.784729\pi\)
\(348\) −5.09337 −0.273033
\(349\) 23.8454 1.27642 0.638209 0.769864i \(-0.279676\pi\)
0.638209 + 0.769864i \(0.279676\pi\)
\(350\) −6.75288 −0.360957
\(351\) 1.00000 0.0533761
\(352\) 0.378203 0.0201583
\(353\) 25.2391 1.34334 0.671671 0.740850i \(-0.265577\pi\)
0.671671 + 0.740850i \(0.265577\pi\)
\(354\) 0.731526 0.0388802
\(355\) 4.31620 0.229080
\(356\) −5.26838 −0.279223
\(357\) 2.46023 0.130209
\(358\) −6.22986 −0.329259
\(359\) −10.1016 −0.533144 −0.266572 0.963815i \(-0.585891\pi\)
−0.266572 + 0.963815i \(0.585891\pi\)
\(360\) −1.42840 −0.0752831
\(361\) 27.9785 1.47255
\(362\) 5.27430 0.277211
\(363\) −10.8570 −0.569843
\(364\) −2.28162 −0.119589
\(365\) −16.0764 −0.841477
\(366\) 4.05568 0.211994
\(367\) 7.36016 0.384197 0.192099 0.981376i \(-0.438471\pi\)
0.192099 + 0.981376i \(0.438471\pi\)
\(368\) −1.64807 −0.0859118
\(369\) 0.952622 0.0495915
\(370\) −13.9440 −0.724913
\(371\) −17.7766 −0.922916
\(372\) 10.3699 0.537652
\(373\) −11.0954 −0.574496 −0.287248 0.957856i \(-0.592740\pi\)
−0.287248 + 0.957856i \(0.592740\pi\)
\(374\) −0.407809 −0.0210873
\(375\) −11.3696 −0.587123
\(376\) −7.38329 −0.380764
\(377\) −5.09337 −0.262322
\(378\) 2.28162 0.117354
\(379\) 2.69689 0.138530 0.0692649 0.997598i \(-0.477935\pi\)
0.0692649 + 0.997598i \(0.477935\pi\)
\(380\) −9.79035 −0.502234
\(381\) −1.00129 −0.0512978
\(382\) 2.75574 0.140996
\(383\) −3.30815 −0.169038 −0.0845192 0.996422i \(-0.526935\pi\)
−0.0845192 + 0.996422i \(0.526935\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.23259 0.0628184
\(386\) 6.08283 0.309608
\(387\) 2.65892 0.135161
\(388\) −10.0419 −0.509802
\(389\) 23.1855 1.17555 0.587776 0.809024i \(-0.300004\pi\)
0.587776 + 0.809024i \(0.300004\pi\)
\(390\) −1.42840 −0.0723296
\(391\) 1.77708 0.0898711
\(392\) 1.79420 0.0906210
\(393\) −7.05133 −0.355693
\(394\) 10.1965 0.513691
\(395\) 7.50570 0.377653
\(396\) −0.378203 −0.0190054
\(397\) 32.2041 1.61628 0.808138 0.588994i \(-0.200476\pi\)
0.808138 + 0.588994i \(0.200476\pi\)
\(398\) −1.25424 −0.0628694
\(399\) 15.6384 0.782900
\(400\) −2.95968 −0.147984
\(401\) −0.522173 −0.0260761 −0.0130380 0.999915i \(-0.504150\pi\)
−0.0130380 + 0.999915i \(0.504150\pi\)
\(402\) 9.03946 0.450847
\(403\) 10.3699 0.516559
\(404\) 6.99307 0.347918
\(405\) 1.42840 0.0709776
\(406\) −11.6211 −0.576748
\(407\) −3.69201 −0.183006
\(408\) 1.07828 0.0533828
\(409\) 28.7704 1.42260 0.711302 0.702887i \(-0.248106\pi\)
0.711302 + 0.702887i \(0.248106\pi\)
\(410\) −1.36072 −0.0672013
\(411\) −6.33167 −0.312318
\(412\) −1.00000 −0.0492665
\(413\) 1.66907 0.0821293
\(414\) 1.64807 0.0809984
\(415\) −16.1140 −0.791006
\(416\) −1.00000 −0.0490290
\(417\) −15.7204 −0.769831
\(418\) −2.59224 −0.126790
\(419\) −37.5198 −1.83296 −0.916482 0.400075i \(-0.868984\pi\)
−0.916482 + 0.400075i \(0.868984\pi\)
\(420\) −3.25906 −0.159026
\(421\) 2.98562 0.145510 0.0727552 0.997350i \(-0.476821\pi\)
0.0727552 + 0.997350i \(0.476821\pi\)
\(422\) 16.2651 0.791772
\(423\) 7.38329 0.358988
\(424\) −7.79122 −0.378375
\(425\) 3.19137 0.154804
\(426\) −3.02171 −0.146403
\(427\) 9.25353 0.447810
\(428\) −10.4990 −0.507487
\(429\) −0.378203 −0.0182598
\(430\) −3.79799 −0.183156
\(431\) −11.3097 −0.544770 −0.272385 0.962188i \(-0.587812\pi\)
−0.272385 + 0.962188i \(0.587812\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.35879 −0.113356 −0.0566781 0.998393i \(-0.518051\pi\)
−0.0566781 + 0.998393i \(0.518051\pi\)
\(434\) 23.6601 1.13572
\(435\) −7.27536 −0.348827
\(436\) 0.860752 0.0412225
\(437\) 11.2960 0.540363
\(438\) 11.2549 0.537778
\(439\) 26.2481 1.25275 0.626376 0.779521i \(-0.284538\pi\)
0.626376 + 0.779521i \(0.284538\pi\)
\(440\) 0.540224 0.0257542
\(441\) −1.79420 −0.0854383
\(442\) 1.07828 0.0512885
\(443\) 6.78372 0.322304 0.161152 0.986930i \(-0.448479\pi\)
0.161152 + 0.986930i \(0.448479\pi\)
\(444\) 9.76199 0.463283
\(445\) −7.52533 −0.356735
\(446\) 21.2527 1.00634
\(447\) −10.5290 −0.498005
\(448\) −2.28162 −0.107796
\(449\) 37.5659 1.77284 0.886422 0.462879i \(-0.153183\pi\)
0.886422 + 0.462879i \(0.153183\pi\)
\(450\) 2.95968 0.139521
\(451\) −0.360285 −0.0169651
\(452\) −2.84635 −0.133881
\(453\) −5.65260 −0.265582
\(454\) 25.4900 1.19630
\(455\) −3.25906 −0.152787
\(456\) 6.85408 0.320972
\(457\) −7.90872 −0.369954 −0.184977 0.982743i \(-0.559221\pi\)
−0.184977 + 0.982743i \(0.559221\pi\)
\(458\) −25.8974 −1.21011
\(459\) −1.07828 −0.0503298
\(460\) −2.35410 −0.109761
\(461\) 0.520571 0.0242454 0.0121227 0.999927i \(-0.496141\pi\)
0.0121227 + 0.999927i \(0.496141\pi\)
\(462\) −0.862916 −0.0401465
\(463\) −9.28449 −0.431487 −0.215744 0.976450i \(-0.569218\pi\)
−0.215744 + 0.976450i \(0.569218\pi\)
\(464\) −5.09337 −0.236454
\(465\) 14.8123 0.686902
\(466\) 0.935399 0.0433315
\(467\) −0.410904 −0.0190144 −0.00950719 0.999955i \(-0.503026\pi\)
−0.00950719 + 0.999955i \(0.503026\pi\)
\(468\) 1.00000 0.0462250
\(469\) 20.6246 0.952357
\(470\) −10.5463 −0.486463
\(471\) 13.3002 0.612841
\(472\) 0.731526 0.0336712
\(473\) −1.00561 −0.0462381
\(474\) −5.25463 −0.241353
\(475\) 20.2859 0.930782
\(476\) 2.46023 0.112764
\(477\) 7.79122 0.356736
\(478\) 7.83371 0.358305
\(479\) −35.7133 −1.63178 −0.815890 0.578207i \(-0.803753\pi\)
−0.815890 + 0.578207i \(0.803753\pi\)
\(480\) −1.42840 −0.0651971
\(481\) 9.76199 0.445108
\(482\) −23.8091 −1.08448
\(483\) 3.76028 0.171099
\(484\) −10.8570 −0.493498
\(485\) −14.3439 −0.651322
\(486\) −1.00000 −0.0453609
\(487\) 9.35930 0.424110 0.212055 0.977258i \(-0.431984\pi\)
0.212055 + 0.977258i \(0.431984\pi\)
\(488\) 4.05568 0.183592
\(489\) −6.40325 −0.289565
\(490\) 2.56284 0.115777
\(491\) −24.5502 −1.10793 −0.553967 0.832539i \(-0.686887\pi\)
−0.553967 + 0.832539i \(0.686887\pi\)
\(492\) 0.952622 0.0429475
\(493\) 5.49208 0.247351
\(494\) 6.85408 0.308380
\(495\) −0.540224 −0.0242813
\(496\) 10.3699 0.465620
\(497\) −6.89441 −0.309256
\(498\) 11.2812 0.505523
\(499\) 0.932509 0.0417448 0.0208724 0.999782i \(-0.493356\pi\)
0.0208724 + 0.999782i \(0.493356\pi\)
\(500\) −11.3696 −0.508463
\(501\) 4.33757 0.193788
\(502\) 7.47104 0.333449
\(503\) −34.4919 −1.53792 −0.768958 0.639299i \(-0.779225\pi\)
−0.768958 + 0.639299i \(0.779225\pi\)
\(504\) 2.28162 0.101631
\(505\) 9.98887 0.444499
\(506\) −0.623307 −0.0277094
\(507\) 1.00000 0.0444116
\(508\) −1.00129 −0.0444252
\(509\) −24.9502 −1.10590 −0.552948 0.833215i \(-0.686497\pi\)
−0.552948 + 0.833215i \(0.686497\pi\)
\(510\) 1.54021 0.0682017
\(511\) 25.6793 1.13599
\(512\) −1.00000 −0.0441942
\(513\) −6.85408 −0.302615
\(514\) 29.3198 1.29324
\(515\) −1.42840 −0.0629426
\(516\) 2.65892 0.117053
\(517\) −2.79238 −0.122809
\(518\) 22.2732 0.978626
\(519\) −9.30906 −0.408623
\(520\) −1.42840 −0.0626393
\(521\) 0.944534 0.0413808 0.0206904 0.999786i \(-0.493414\pi\)
0.0206904 + 0.999786i \(0.493414\pi\)
\(522\) 5.09337 0.222931
\(523\) 37.5492 1.64191 0.820955 0.570993i \(-0.193442\pi\)
0.820955 + 0.570993i \(0.193442\pi\)
\(524\) −7.05133 −0.308039
\(525\) 6.75288 0.294720
\(526\) 25.4645 1.11031
\(527\) −11.1816 −0.487078
\(528\) −0.378203 −0.0164592
\(529\) −20.2839 −0.881907
\(530\) −11.1290 −0.483411
\(531\) −0.731526 −0.0317455
\(532\) 15.6384 0.678012
\(533\) 0.952622 0.0412626
\(534\) 5.26838 0.227985
\(535\) −14.9967 −0.648363
\(536\) 9.03946 0.390445
\(537\) 6.22986 0.268839
\(538\) 23.3373 1.00614
\(539\) 0.678574 0.0292282
\(540\) 1.42840 0.0614684
\(541\) −8.92867 −0.383874 −0.191937 0.981407i \(-0.561477\pi\)
−0.191937 + 0.981407i \(0.561477\pi\)
\(542\) 6.22070 0.267202
\(543\) −5.27430 −0.226342
\(544\) 1.07828 0.0462309
\(545\) 1.22949 0.0526657
\(546\) 2.28162 0.0976444
\(547\) 1.16295 0.0497243 0.0248622 0.999691i \(-0.492085\pi\)
0.0248622 + 0.999691i \(0.492085\pi\)
\(548\) −6.33167 −0.270476
\(549\) −4.05568 −0.173092
\(550\) −1.11936 −0.0477297
\(551\) 34.9104 1.48723
\(552\) 1.64807 0.0701467
\(553\) −11.9891 −0.509828
\(554\) −10.9355 −0.464606
\(555\) 13.9440 0.591889
\(556\) −15.7204 −0.666693
\(557\) 39.8724 1.68945 0.844723 0.535203i \(-0.179765\pi\)
0.844723 + 0.535203i \(0.179765\pi\)
\(558\) −10.3699 −0.438991
\(559\) 2.65892 0.112460
\(560\) −3.25906 −0.137720
\(561\) 0.407809 0.0172177
\(562\) −16.4408 −0.693515
\(563\) 20.4700 0.862707 0.431353 0.902183i \(-0.358036\pi\)
0.431353 + 0.902183i \(0.358036\pi\)
\(564\) 7.38329 0.310893
\(565\) −4.06571 −0.171046
\(566\) −17.3544 −0.729458
\(567\) −2.28162 −0.0958191
\(568\) −3.02171 −0.126788
\(569\) 44.0758 1.84775 0.923876 0.382692i \(-0.125003\pi\)
0.923876 + 0.382692i \(0.125003\pi\)
\(570\) 9.79035 0.410072
\(571\) −21.6029 −0.904053 −0.452026 0.892005i \(-0.649299\pi\)
−0.452026 + 0.892005i \(0.649299\pi\)
\(572\) −0.378203 −0.0158135
\(573\) −2.75574 −0.115123
\(574\) 2.17352 0.0907211
\(575\) 4.87778 0.203418
\(576\) 1.00000 0.0416667
\(577\) −15.4529 −0.643314 −0.321657 0.946856i \(-0.604240\pi\)
−0.321657 + 0.946856i \(0.604240\pi\)
\(578\) 15.8373 0.658745
\(579\) −6.08283 −0.252794
\(580\) −7.27536 −0.302093
\(581\) 25.7394 1.06785
\(582\) 10.0419 0.416252
\(583\) −2.94666 −0.122038
\(584\) 11.2549 0.465730
\(585\) 1.42840 0.0590569
\(586\) −9.58413 −0.395917
\(587\) 36.1032 1.49014 0.745068 0.666988i \(-0.232417\pi\)
0.745068 + 0.666988i \(0.232417\pi\)
\(588\) −1.79420 −0.0739918
\(589\) −71.0758 −2.92863
\(590\) 1.04491 0.0430182
\(591\) −10.1965 −0.419427
\(592\) 9.76199 0.401215
\(593\) −1.69894 −0.0697669 −0.0348835 0.999391i \(-0.511106\pi\)
−0.0348835 + 0.999391i \(0.511106\pi\)
\(594\) 0.378203 0.0155179
\(595\) 3.51418 0.144067
\(596\) −10.5290 −0.431285
\(597\) 1.25424 0.0513326
\(598\) 1.64807 0.0673948
\(599\) −20.6681 −0.844477 −0.422238 0.906485i \(-0.638755\pi\)
−0.422238 + 0.906485i \(0.638755\pi\)
\(600\) 2.95968 0.120829
\(601\) 2.38180 0.0971556 0.0485778 0.998819i \(-0.484531\pi\)
0.0485778 + 0.998819i \(0.484531\pi\)
\(602\) 6.06665 0.247258
\(603\) −9.03946 −0.368115
\(604\) −5.65260 −0.230001
\(605\) −15.5080 −0.630492
\(606\) −6.99307 −0.284074
\(607\) 1.91188 0.0776006 0.0388003 0.999247i \(-0.487646\pi\)
0.0388003 + 0.999247i \(0.487646\pi\)
\(608\) 6.85408 0.277970
\(609\) 11.6211 0.470913
\(610\) 5.79312 0.234557
\(611\) 7.38329 0.298696
\(612\) −1.07828 −0.0435869
\(613\) 3.62282 0.146324 0.0731621 0.997320i \(-0.476691\pi\)
0.0731621 + 0.997320i \(0.476691\pi\)
\(614\) 12.5014 0.504517
\(615\) 1.36072 0.0548696
\(616\) −0.862916 −0.0347679
\(617\) 15.5713 0.626876 0.313438 0.949609i \(-0.398519\pi\)
0.313438 + 0.949609i \(0.398519\pi\)
\(618\) 1.00000 0.0402259
\(619\) 3.00732 0.120874 0.0604371 0.998172i \(-0.480751\pi\)
0.0604371 + 0.998172i \(0.480751\pi\)
\(620\) 14.8123 0.594874
\(621\) −1.64807 −0.0661350
\(622\) 26.1921 1.05021
\(623\) 12.0204 0.481589
\(624\) 1.00000 0.0400320
\(625\) −1.44184 −0.0576737
\(626\) −7.61225 −0.304247
\(627\) 2.59224 0.103524
\(628\) 13.3002 0.530736
\(629\) −10.5262 −0.419705
\(630\) 3.25906 0.129844
\(631\) −7.36563 −0.293221 −0.146610 0.989194i \(-0.546836\pi\)
−0.146610 + 0.989194i \(0.546836\pi\)
\(632\) −5.25463 −0.209018
\(633\) −16.2651 −0.646479
\(634\) −7.73483 −0.307189
\(635\) −1.43024 −0.0567575
\(636\) 7.79122 0.308942
\(637\) −1.79420 −0.0710890
\(638\) −1.92633 −0.0762641
\(639\) 3.02171 0.119537
\(640\) −1.42840 −0.0564623
\(641\) −38.7570 −1.53081 −0.765405 0.643549i \(-0.777461\pi\)
−0.765405 + 0.643549i \(0.777461\pi\)
\(642\) 10.4990 0.414361
\(643\) 8.06468 0.318040 0.159020 0.987275i \(-0.449167\pi\)
0.159020 + 0.987275i \(0.449167\pi\)
\(644\) 3.76028 0.148176
\(645\) 3.79799 0.149546
\(646\) −7.39062 −0.290780
\(647\) −19.2342 −0.756176 −0.378088 0.925770i \(-0.623418\pi\)
−0.378088 + 0.925770i \(0.623418\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0.276665 0.0108601
\(650\) 2.95968 0.116088
\(651\) −23.6601 −0.927311
\(652\) −6.40325 −0.250771
\(653\) −44.8678 −1.75581 −0.877906 0.478834i \(-0.841060\pi\)
−0.877906 + 0.478834i \(0.841060\pi\)
\(654\) −0.860752 −0.0336580
\(655\) −10.0721 −0.393549
\(656\) 0.952622 0.0371936
\(657\) −11.2549 −0.439094
\(658\) 16.8459 0.656720
\(659\) −7.62830 −0.297156 −0.148578 0.988901i \(-0.547470\pi\)
−0.148578 + 0.988901i \(0.547470\pi\)
\(660\) −0.540224 −0.0210282
\(661\) 35.8892 1.39593 0.697963 0.716133i \(-0.254090\pi\)
0.697963 + 0.716133i \(0.254090\pi\)
\(662\) −25.0994 −0.975516
\(663\) −1.07828 −0.0418769
\(664\) 11.2812 0.437795
\(665\) 22.3379 0.866225
\(666\) −9.76199 −0.378269
\(667\) 8.39426 0.325027
\(668\) 4.33757 0.167826
\(669\) −21.2527 −0.821677
\(670\) 12.9119 0.498831
\(671\) 1.53387 0.0592144
\(672\) 2.28162 0.0880154
\(673\) 43.3613 1.67145 0.835727 0.549145i \(-0.185047\pi\)
0.835727 + 0.549145i \(0.185047\pi\)
\(674\) 13.9072 0.535687
\(675\) −2.95968 −0.113918
\(676\) 1.00000 0.0384615
\(677\) 31.0734 1.19425 0.597125 0.802148i \(-0.296310\pi\)
0.597125 + 0.802148i \(0.296310\pi\)
\(678\) 2.84635 0.109313
\(679\) 22.9119 0.879278
\(680\) 1.54021 0.0590644
\(681\) −25.4900 −0.976778
\(682\) 3.92191 0.150178
\(683\) 15.9518 0.610379 0.305190 0.952292i \(-0.401280\pi\)
0.305190 + 0.952292i \(0.401280\pi\)
\(684\) −6.85408 −0.262072
\(685\) −9.04413 −0.345559
\(686\) −20.0650 −0.766087
\(687\) 25.8974 0.988047
\(688\) 2.65892 0.101370
\(689\) 7.79122 0.296822
\(690\) 2.35410 0.0896192
\(691\) −21.3599 −0.812568 −0.406284 0.913747i \(-0.633176\pi\)
−0.406284 + 0.913747i \(0.633176\pi\)
\(692\) −9.30906 −0.353878
\(693\) 0.862916 0.0327795
\(694\) 29.0558 1.10294
\(695\) −22.4549 −0.851765
\(696\) 5.09337 0.193064
\(697\) −1.02719 −0.0389077
\(698\) −23.8454 −0.902563
\(699\) −0.935399 −0.0353801
\(700\) 6.75288 0.255235
\(701\) 23.2980 0.879954 0.439977 0.898009i \(-0.354987\pi\)
0.439977 + 0.898009i \(0.354987\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −66.9095 −2.52354
\(704\) −0.378203 −0.0142541
\(705\) 10.5463 0.397195
\(706\) −25.2391 −0.949886
\(707\) −15.9555 −0.600069
\(708\) −0.731526 −0.0274924
\(709\) −18.2292 −0.684612 −0.342306 0.939589i \(-0.611208\pi\)
−0.342306 + 0.939589i \(0.611208\pi\)
\(710\) −4.31620 −0.161984
\(711\) 5.25463 0.197064
\(712\) 5.26838 0.197441
\(713\) −17.0903 −0.640036
\(714\) −2.46023 −0.0920716
\(715\) −0.540224 −0.0202032
\(716\) 6.22986 0.232821
\(717\) −7.83371 −0.292555
\(718\) 10.1016 0.376989
\(719\) −17.6809 −0.659388 −0.329694 0.944088i \(-0.606946\pi\)
−0.329694 + 0.944088i \(0.606946\pi\)
\(720\) 1.42840 0.0532332
\(721\) 2.28162 0.0849720
\(722\) −27.9785 −1.04125
\(723\) 23.8091 0.885472
\(724\) −5.27430 −0.196018
\(725\) 15.0748 0.559863
\(726\) 10.8570 0.402940
\(727\) 21.3312 0.791129 0.395564 0.918438i \(-0.370549\pi\)
0.395564 + 0.918438i \(0.370549\pi\)
\(728\) 2.28162 0.0845625
\(729\) 1.00000 0.0370370
\(730\) 16.0764 0.595014
\(731\) −2.86706 −0.106042
\(732\) −4.05568 −0.149902
\(733\) −24.7459 −0.914010 −0.457005 0.889464i \(-0.651078\pi\)
−0.457005 + 0.889464i \(0.651078\pi\)
\(734\) −7.36016 −0.271669
\(735\) −2.56284 −0.0945316
\(736\) 1.64807 0.0607488
\(737\) 3.41875 0.125931
\(738\) −0.952622 −0.0350665
\(739\) −41.3679 −1.52174 −0.760871 0.648903i \(-0.775228\pi\)
−0.760871 + 0.648903i \(0.775228\pi\)
\(740\) 13.9440 0.512591
\(741\) −6.85408 −0.251791
\(742\) 17.7766 0.652600
\(743\) −11.8013 −0.432949 −0.216475 0.976288i \(-0.569456\pi\)
−0.216475 + 0.976288i \(0.569456\pi\)
\(744\) −10.3699 −0.380177
\(745\) −15.0396 −0.551008
\(746\) 11.0954 0.406230
\(747\) −11.2812 −0.412757
\(748\) 0.407809 0.0149110
\(749\) 23.9547 0.875284
\(750\) 11.3696 0.415158
\(751\) −40.1233 −1.46412 −0.732060 0.681240i \(-0.761441\pi\)
−0.732060 + 0.681240i \(0.761441\pi\)
\(752\) 7.38329 0.269241
\(753\) −7.47104 −0.272260
\(754\) 5.09337 0.185490
\(755\) −8.07415 −0.293848
\(756\) −2.28162 −0.0829818
\(757\) −6.41345 −0.233101 −0.116550 0.993185i \(-0.537184\pi\)
−0.116550 + 0.993185i \(0.537184\pi\)
\(758\) −2.69689 −0.0979553
\(759\) 0.623307 0.0226246
\(760\) 9.79035 0.355133
\(761\) −1.13877 −0.0412802 −0.0206401 0.999787i \(-0.506570\pi\)
−0.0206401 + 0.999787i \(0.506570\pi\)
\(762\) 1.00129 0.0362730
\(763\) −1.96391 −0.0710983
\(764\) −2.75574 −0.0996993
\(765\) −1.54021 −0.0556864
\(766\) 3.30815 0.119528
\(767\) −0.731526 −0.0264139
\(768\) 1.00000 0.0360844
\(769\) 20.4055 0.735842 0.367921 0.929857i \(-0.380070\pi\)
0.367921 + 0.929857i \(0.380070\pi\)
\(770\) −1.23259 −0.0444193
\(771\) −29.3198 −1.05593
\(772\) −6.08283 −0.218926
\(773\) −18.8481 −0.677921 −0.338960 0.940801i \(-0.610075\pi\)
−0.338960 + 0.940801i \(0.610075\pi\)
\(774\) −2.65892 −0.0955730
\(775\) −30.6915 −1.10247
\(776\) 10.0419 0.360485
\(777\) −22.2732 −0.799045
\(778\) −23.1855 −0.831240
\(779\) −6.52935 −0.233938
\(780\) 1.42840 0.0511448
\(781\) −1.14282 −0.0408934
\(782\) −1.77708 −0.0635484
\(783\) −5.09337 −0.182022
\(784\) −1.79420 −0.0640787
\(785\) 18.9980 0.678066
\(786\) 7.05133 0.251513
\(787\) 17.2663 0.615476 0.307738 0.951471i \(-0.400428\pi\)
0.307738 + 0.951471i \(0.400428\pi\)
\(788\) −10.1965 −0.363234
\(789\) −25.4645 −0.906562
\(790\) −7.50570 −0.267041
\(791\) 6.49428 0.230910
\(792\) 0.378203 0.0134389
\(793\) −4.05568 −0.144021
\(794\) −32.2041 −1.14288
\(795\) 11.1290 0.394703
\(796\) 1.25424 0.0444554
\(797\) −20.5117 −0.726561 −0.363280 0.931680i \(-0.618343\pi\)
−0.363280 + 0.931680i \(0.618343\pi\)
\(798\) −15.6384 −0.553594
\(799\) −7.96125 −0.281649
\(800\) 2.95968 0.104641
\(801\) −5.26838 −0.186149
\(802\) 0.522173 0.0184386
\(803\) 4.25662 0.150213
\(804\) −9.03946 −0.318797
\(805\) 5.37117 0.189309
\(806\) −10.3699 −0.365262
\(807\) −23.3373 −0.821512
\(808\) −6.99307 −0.246015
\(809\) 1.44475 0.0507948 0.0253974 0.999677i \(-0.491915\pi\)
0.0253974 + 0.999677i \(0.491915\pi\)
\(810\) −1.42840 −0.0501887
\(811\) 21.9751 0.771651 0.385825 0.922572i \(-0.373917\pi\)
0.385825 + 0.922572i \(0.373917\pi\)
\(812\) 11.6211 0.407822
\(813\) −6.22070 −0.218170
\(814\) 3.69201 0.129405
\(815\) −9.14638 −0.320384
\(816\) −1.07828 −0.0377473
\(817\) −18.2245 −0.637594
\(818\) −28.7704 −1.00593
\(819\) −2.28162 −0.0797263
\(820\) 1.36072 0.0475185
\(821\) 36.6375 1.27866 0.639329 0.768934i \(-0.279212\pi\)
0.639329 + 0.768934i \(0.279212\pi\)
\(822\) 6.33167 0.220842
\(823\) 45.4973 1.58594 0.792969 0.609262i \(-0.208534\pi\)
0.792969 + 0.609262i \(0.208534\pi\)
\(824\) 1.00000 0.0348367
\(825\) 1.11936 0.0389712
\(826\) −1.66907 −0.0580742
\(827\) 26.1671 0.909921 0.454960 0.890512i \(-0.349653\pi\)
0.454960 + 0.890512i \(0.349653\pi\)
\(828\) −1.64807 −0.0572746
\(829\) −44.0402 −1.52958 −0.764790 0.644279i \(-0.777158\pi\)
−0.764790 + 0.644279i \(0.777158\pi\)
\(830\) 16.1140 0.559326
\(831\) 10.9355 0.379350
\(832\) 1.00000 0.0346688
\(833\) 1.93465 0.0670318
\(834\) 15.7204 0.544353
\(835\) 6.19576 0.214413
\(836\) 2.59224 0.0896543
\(837\) 10.3699 0.358434
\(838\) 37.5198 1.29610
\(839\) −3.26703 −0.112790 −0.0563952 0.998409i \(-0.517961\pi\)
−0.0563952 + 0.998409i \(0.517961\pi\)
\(840\) 3.25906 0.112448
\(841\) −3.05754 −0.105433
\(842\) −2.98562 −0.102891
\(843\) 16.4408 0.566252
\(844\) −16.2651 −0.559867
\(845\) 1.42840 0.0491383
\(846\) −7.38329 −0.253843
\(847\) 24.7715 0.851158
\(848\) 7.79122 0.267552
\(849\) 17.3544 0.595600
\(850\) −3.19137 −0.109463
\(851\) −16.0885 −0.551506
\(852\) 3.02171 0.103522
\(853\) 53.1514 1.81987 0.909935 0.414751i \(-0.136131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(854\) −9.25353 −0.316649
\(855\) −9.79035 −0.334823
\(856\) 10.4990 0.358847
\(857\) 25.3084 0.864519 0.432259 0.901749i \(-0.357717\pi\)
0.432259 + 0.901749i \(0.357717\pi\)
\(858\) 0.378203 0.0129116
\(859\) −29.3493 −1.00138 −0.500692 0.865625i \(-0.666921\pi\)
−0.500692 + 0.865625i \(0.666921\pi\)
\(860\) 3.79799 0.129511
\(861\) −2.17352 −0.0740735
\(862\) 11.3097 0.385210
\(863\) −41.2065 −1.40268 −0.701342 0.712825i \(-0.747415\pi\)
−0.701342 + 0.712825i \(0.747415\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.2970 −0.452113
\(866\) 2.35879 0.0801549
\(867\) −15.8373 −0.537863
\(868\) −23.6601 −0.803075
\(869\) −1.98732 −0.0674152
\(870\) 7.27536 0.246658
\(871\) −9.03946 −0.306290
\(872\) −0.860752 −0.0291487
\(873\) −10.0419 −0.339868
\(874\) −11.2960 −0.382094
\(875\) 25.9411 0.876969
\(876\) −11.2549 −0.380267
\(877\) 23.5574 0.795477 0.397738 0.917499i \(-0.369795\pi\)
0.397738 + 0.917499i \(0.369795\pi\)
\(878\) −26.2481 −0.885829
\(879\) 9.58413 0.323265
\(880\) −0.540224 −0.0182109
\(881\) −13.5194 −0.455481 −0.227741 0.973722i \(-0.573134\pi\)
−0.227741 + 0.973722i \(0.573134\pi\)
\(882\) 1.79420 0.0604140
\(883\) 51.2268 1.72392 0.861959 0.506978i \(-0.169237\pi\)
0.861959 + 0.506978i \(0.169237\pi\)
\(884\) −1.07828 −0.0362665
\(885\) −1.04491 −0.0351242
\(886\) −6.78372 −0.227903
\(887\) 7.59434 0.254993 0.127496 0.991839i \(-0.459306\pi\)
0.127496 + 0.991839i \(0.459306\pi\)
\(888\) −9.76199 −0.327591
\(889\) 2.28457 0.0766221
\(890\) 7.52533 0.252250
\(891\) −0.378203 −0.0126703
\(892\) −21.2527 −0.711593
\(893\) −50.6057 −1.69345
\(894\) 10.5290 0.352143
\(895\) 8.89871 0.297451
\(896\) 2.28162 0.0762236
\(897\) −1.64807 −0.0550276
\(898\) −37.5659 −1.25359
\(899\) −52.8175 −1.76156
\(900\) −2.95968 −0.0986562
\(901\) −8.40112 −0.279882
\(902\) 0.360285 0.0119962
\(903\) −6.06665 −0.201886
\(904\) 2.84635 0.0946681
\(905\) −7.53379 −0.250432
\(906\) 5.65260 0.187795
\(907\) −50.9773 −1.69267 −0.846336 0.532649i \(-0.821196\pi\)
−0.846336 + 0.532649i \(0.821196\pi\)
\(908\) −25.4900 −0.845914
\(909\) 6.99307 0.231945
\(910\) 3.25906 0.108037
\(911\) −19.8064 −0.656216 −0.328108 0.944640i \(-0.606411\pi\)
−0.328108 + 0.944640i \(0.606411\pi\)
\(912\) −6.85408 −0.226961
\(913\) 4.26658 0.141203
\(914\) 7.90872 0.261597
\(915\) −5.79312 −0.191515
\(916\) 25.8974 0.855674
\(917\) 16.0885 0.531288
\(918\) 1.07828 0.0355885
\(919\) −54.6613 −1.80311 −0.901555 0.432666i \(-0.857573\pi\)
−0.901555 + 0.432666i \(0.857573\pi\)
\(920\) 2.35410 0.0776125
\(921\) −12.5014 −0.411936
\(922\) −0.520571 −0.0171441
\(923\) 3.02171 0.0994609
\(924\) 0.862916 0.0283879
\(925\) −28.8924 −0.949976
\(926\) 9.28449 0.305107
\(927\) −1.00000 −0.0328443
\(928\) 5.09337 0.167198
\(929\) −33.8676 −1.11116 −0.555580 0.831463i \(-0.687504\pi\)
−0.555580 + 0.831463i \(0.687504\pi\)
\(930\) −14.8123 −0.485713
\(931\) 12.2976 0.403039
\(932\) −0.935399 −0.0306400
\(933\) −26.1921 −0.857491
\(934\) 0.410904 0.0134452
\(935\) 0.582512 0.0190502
\(936\) −1.00000 −0.0326860
\(937\) −26.9085 −0.879062 −0.439531 0.898227i \(-0.644855\pi\)
−0.439531 + 0.898227i \(0.644855\pi\)
\(938\) −20.6246 −0.673418
\(939\) 7.61225 0.248416
\(940\) 10.5463 0.343981
\(941\) 7.77064 0.253316 0.126658 0.991946i \(-0.459575\pi\)
0.126658 + 0.991946i \(0.459575\pi\)
\(942\) −13.3002 −0.433344
\(943\) −1.56999 −0.0511260
\(944\) −0.731526 −0.0238091
\(945\) −3.25906 −0.106017
\(946\) 1.00561 0.0326953
\(947\) −0.0677911 −0.00220291 −0.00110146 0.999999i \(-0.500351\pi\)
−0.00110146 + 0.999999i \(0.500351\pi\)
\(948\) 5.25463 0.170663
\(949\) −11.2549 −0.365348
\(950\) −20.2859 −0.658162
\(951\) 7.73483 0.250819
\(952\) −2.46023 −0.0797364
\(953\) 1.00211 0.0324615 0.0162308 0.999868i \(-0.494833\pi\)
0.0162308 + 0.999868i \(0.494833\pi\)
\(954\) −7.79122 −0.252250
\(955\) −3.93629 −0.127375
\(956\) −7.83371 −0.253360
\(957\) 1.92633 0.0622694
\(958\) 35.7133 1.15384
\(959\) 14.4465 0.466501
\(960\) 1.42840 0.0461013
\(961\) 76.5338 2.46883
\(962\) −9.76199 −0.314739
\(963\) −10.4990 −0.338324
\(964\) 23.8091 0.766841
\(965\) −8.68869 −0.279699
\(966\) −3.76028 −0.120985
\(967\) −17.9495 −0.577218 −0.288609 0.957447i \(-0.593193\pi\)
−0.288609 + 0.957447i \(0.593193\pi\)
\(968\) 10.8570 0.348956
\(969\) 7.39062 0.237421
\(970\) 14.3439 0.460554
\(971\) −12.1550 −0.390072 −0.195036 0.980796i \(-0.562482\pi\)
−0.195036 + 0.980796i \(0.562482\pi\)
\(972\) 1.00000 0.0320750
\(973\) 35.8680 1.14987
\(974\) −9.35930 −0.299891
\(975\) −2.95968 −0.0947858
\(976\) −4.05568 −0.129819
\(977\) −9.80531 −0.313700 −0.156850 0.987622i \(-0.550134\pi\)
−0.156850 + 0.987622i \(0.550134\pi\)
\(978\) 6.40325 0.204753
\(979\) 1.99252 0.0636811
\(980\) −2.56284 −0.0818668
\(981\) 0.860752 0.0274817
\(982\) 24.5502 0.783427
\(983\) 27.1478 0.865880 0.432940 0.901423i \(-0.357476\pi\)
0.432940 + 0.901423i \(0.357476\pi\)
\(984\) −0.952622 −0.0303685
\(985\) −14.5646 −0.464067
\(986\) −5.49208 −0.174903
\(987\) −16.8459 −0.536210
\(988\) −6.85408 −0.218057
\(989\) −4.38210 −0.139343
\(990\) 0.540224 0.0171694
\(991\) 53.6231 1.70339 0.851697 0.524035i \(-0.175574\pi\)
0.851697 + 0.524035i \(0.175574\pi\)
\(992\) −10.3699 −0.329243
\(993\) 25.0994 0.796505
\(994\) 6.89441 0.218677
\(995\) 1.79155 0.0567960
\(996\) −11.2812 −0.357458
\(997\) −32.0001 −1.01345 −0.506727 0.862106i \(-0.669145\pi\)
−0.506727 + 0.862106i \(0.669145\pi\)
\(998\) −0.932509 −0.0295181
\(999\) 9.76199 0.308856
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 8034.2.a.w.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.9 12 1.1 even 1 trivial