Properties

Label 4598.2.a.bx.1.8
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.20254\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.01451 q^{3} +1.00000 q^{4} -2.00989 q^{5} -3.01451 q^{6} -2.42872 q^{7} -1.00000 q^{8} +6.08725 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.01451 q^{3} +1.00000 q^{4} -2.00989 q^{5} -3.01451 q^{6} -2.42872 q^{7} -1.00000 q^{8} +6.08725 q^{9} +2.00989 q^{10} +3.01451 q^{12} -3.03557 q^{13} +2.42872 q^{14} -6.05881 q^{15} +1.00000 q^{16} +4.31675 q^{17} -6.08725 q^{18} +1.00000 q^{19} -2.00989 q^{20} -7.32140 q^{21} -2.27832 q^{23} -3.01451 q^{24} -0.960362 q^{25} +3.03557 q^{26} +9.30653 q^{27} -2.42872 q^{28} -8.81144 q^{29} +6.05881 q^{30} +9.29488 q^{31} -1.00000 q^{32} -4.31675 q^{34} +4.88145 q^{35} +6.08725 q^{36} +1.06805 q^{37} -1.00000 q^{38} -9.15074 q^{39} +2.00989 q^{40} +5.15603 q^{41} +7.32140 q^{42} -11.9841 q^{43} -12.2347 q^{45} +2.27832 q^{46} -9.00012 q^{47} +3.01451 q^{48} -1.10131 q^{49} +0.960362 q^{50} +13.0129 q^{51} -3.03557 q^{52} +5.09532 q^{53} -9.30653 q^{54} +2.42872 q^{56} +3.01451 q^{57} +8.81144 q^{58} -4.79071 q^{59} -6.05881 q^{60} -9.26197 q^{61} -9.29488 q^{62} -14.7842 q^{63} +1.00000 q^{64} +6.10114 q^{65} +9.19479 q^{67} +4.31675 q^{68} -6.86802 q^{69} -4.88145 q^{70} -10.7608 q^{71} -6.08725 q^{72} -8.93977 q^{73} -1.06805 q^{74} -2.89502 q^{75} +1.00000 q^{76} +9.15074 q^{78} -6.52103 q^{79} -2.00989 q^{80} +9.79285 q^{81} -5.15603 q^{82} +4.09108 q^{83} -7.32140 q^{84} -8.67617 q^{85} +11.9841 q^{86} -26.5621 q^{87} +6.04832 q^{89} +12.2347 q^{90} +7.37255 q^{91} -2.27832 q^{92} +28.0195 q^{93} +9.00012 q^{94} -2.00989 q^{95} -3.01451 q^{96} -9.42693 q^{97} +1.10131 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{7} - 8 q^{8} + 20 q^{9} - 2 q^{10} - 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} + 8 q^{19} + 2 q^{20} - 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} - 8 q^{28} - 14 q^{29} - 10 q^{30} - 2 q^{31} - 8 q^{32} + 4 q^{34} - 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} + 4 q^{39} - 2 q^{40} - 8 q^{41} + 14 q^{42} - 28 q^{43} - 28 q^{45} - 12 q^{46} + 6 q^{47} + 32 q^{49} + 12 q^{51} - 18 q^{52} - 24 q^{53} + 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} + 24 q^{61} + 2 q^{62} - 30 q^{63} + 8 q^{64} + 16 q^{65} - 22 q^{67} - 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} - 20 q^{72} - 16 q^{73} + 22 q^{74} + 6 q^{75} + 8 q^{76} - 4 q^{78} - 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} - 12 q^{83} - 14 q^{84} - 48 q^{85} + 28 q^{86} - 42 q^{87} - 28 q^{89} + 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} - 6 q^{94} + 2 q^{95} - 22 q^{97} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.01451 1.74043 0.870213 0.492676i \(-0.163981\pi\)
0.870213 + 0.492676i \(0.163981\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00989 −0.898848 −0.449424 0.893319i \(-0.648371\pi\)
−0.449424 + 0.893319i \(0.648371\pi\)
\(6\) −3.01451 −1.23067
\(7\) −2.42872 −0.917971 −0.458985 0.888444i \(-0.651787\pi\)
−0.458985 + 0.888444i \(0.651787\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.08725 2.02908
\(10\) 2.00989 0.635581
\(11\) 0 0
\(12\) 3.01451 0.870213
\(13\) −3.03557 −0.841915 −0.420957 0.907080i \(-0.638306\pi\)
−0.420957 + 0.907080i \(0.638306\pi\)
\(14\) 2.42872 0.649103
\(15\) −6.05881 −1.56438
\(16\) 1.00000 0.250000
\(17\) 4.31675 1.04697 0.523483 0.852036i \(-0.324632\pi\)
0.523483 + 0.852036i \(0.324632\pi\)
\(18\) −6.08725 −1.43478
\(19\) 1.00000 0.229416
\(20\) −2.00989 −0.449424
\(21\) −7.32140 −1.59766
\(22\) 0 0
\(23\) −2.27832 −0.475064 −0.237532 0.971380i \(-0.576338\pi\)
−0.237532 + 0.971380i \(0.576338\pi\)
\(24\) −3.01451 −0.615334
\(25\) −0.960362 −0.192072
\(26\) 3.03557 0.595324
\(27\) 9.30653 1.79104
\(28\) −2.42872 −0.458985
\(29\) −8.81144 −1.63624 −0.818121 0.575046i \(-0.804984\pi\)
−0.818121 + 0.575046i \(0.804984\pi\)
\(30\) 6.05881 1.10618
\(31\) 9.29488 1.66941 0.834705 0.550698i \(-0.185638\pi\)
0.834705 + 0.550698i \(0.185638\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.31675 −0.740316
\(35\) 4.88145 0.825116
\(36\) 6.08725 1.01454
\(37\) 1.06805 0.175587 0.0877933 0.996139i \(-0.472018\pi\)
0.0877933 + 0.996139i \(0.472018\pi\)
\(38\) −1.00000 −0.162221
\(39\) −9.15074 −1.46529
\(40\) 2.00989 0.317791
\(41\) 5.15603 0.805236 0.402618 0.915368i \(-0.368100\pi\)
0.402618 + 0.915368i \(0.368100\pi\)
\(42\) 7.32140 1.12972
\(43\) −11.9841 −1.82756 −0.913779 0.406212i \(-0.866849\pi\)
−0.913779 + 0.406212i \(0.866849\pi\)
\(44\) 0 0
\(45\) −12.2347 −1.82384
\(46\) 2.27832 0.335921
\(47\) −9.00012 −1.31280 −0.656401 0.754412i \(-0.727922\pi\)
−0.656401 + 0.754412i \(0.727922\pi\)
\(48\) 3.01451 0.435107
\(49\) −1.10131 −0.157330
\(50\) 0.960362 0.135816
\(51\) 13.0129 1.82217
\(52\) −3.03557 −0.420957
\(53\) 5.09532 0.699896 0.349948 0.936769i \(-0.386199\pi\)
0.349948 + 0.936769i \(0.386199\pi\)
\(54\) −9.30653 −1.26646
\(55\) 0 0
\(56\) 2.42872 0.324552
\(57\) 3.01451 0.399281
\(58\) 8.81144 1.15700
\(59\) −4.79071 −0.623698 −0.311849 0.950132i \(-0.600948\pi\)
−0.311849 + 0.950132i \(0.600948\pi\)
\(60\) −6.05881 −0.782189
\(61\) −9.26197 −1.18587 −0.592937 0.805249i \(-0.702032\pi\)
−0.592937 + 0.805249i \(0.702032\pi\)
\(62\) −9.29488 −1.18045
\(63\) −14.7842 −1.86264
\(64\) 1.00000 0.125000
\(65\) 6.10114 0.756753
\(66\) 0 0
\(67\) 9.19479 1.12332 0.561661 0.827367i \(-0.310162\pi\)
0.561661 + 0.827367i \(0.310162\pi\)
\(68\) 4.31675 0.523483
\(69\) −6.86802 −0.826813
\(70\) −4.88145 −0.583445
\(71\) −10.7608 −1.27707 −0.638535 0.769593i \(-0.720459\pi\)
−0.638535 + 0.769593i \(0.720459\pi\)
\(72\) −6.08725 −0.717389
\(73\) −8.93977 −1.04632 −0.523161 0.852234i \(-0.675247\pi\)
−0.523161 + 0.852234i \(0.675247\pi\)
\(74\) −1.06805 −0.124158
\(75\) −2.89502 −0.334288
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 9.15074 1.03612
\(79\) −6.52103 −0.733673 −0.366837 0.930285i \(-0.619559\pi\)
−0.366837 + 0.930285i \(0.619559\pi\)
\(80\) −2.00989 −0.224712
\(81\) 9.79285 1.08809
\(82\) −5.15603 −0.569388
\(83\) 4.09108 0.449054 0.224527 0.974468i \(-0.427916\pi\)
0.224527 + 0.974468i \(0.427916\pi\)
\(84\) −7.32140 −0.798830
\(85\) −8.67617 −0.941063
\(86\) 11.9841 1.29228
\(87\) −26.5621 −2.84776
\(88\) 0 0
\(89\) 6.04832 0.641121 0.320560 0.947228i \(-0.396129\pi\)
0.320560 + 0.947228i \(0.396129\pi\)
\(90\) 12.2347 1.28965
\(91\) 7.37255 0.772853
\(92\) −2.27832 −0.237532
\(93\) 28.0195 2.90548
\(94\) 9.00012 0.928291
\(95\) −2.00989 −0.206210
\(96\) −3.01451 −0.307667
\(97\) −9.42693 −0.957159 −0.478580 0.878044i \(-0.658848\pi\)
−0.478580 + 0.878044i \(0.658848\pi\)
\(98\) 1.10131 0.111249
\(99\) 0 0
\(100\) −0.960362 −0.0960362
\(101\) 8.78589 0.874229 0.437114 0.899406i \(-0.356001\pi\)
0.437114 + 0.899406i \(0.356001\pi\)
\(102\) −13.0129 −1.28847
\(103\) −10.5453 −1.03906 −0.519528 0.854454i \(-0.673892\pi\)
−0.519528 + 0.854454i \(0.673892\pi\)
\(104\) 3.03557 0.297662
\(105\) 14.7152 1.43605
\(106\) −5.09532 −0.494901
\(107\) 7.69542 0.743944 0.371972 0.928244i \(-0.378682\pi\)
0.371972 + 0.928244i \(0.378682\pi\)
\(108\) 9.30653 0.895521
\(109\) 14.2327 1.36324 0.681622 0.731705i \(-0.261275\pi\)
0.681622 + 0.731705i \(0.261275\pi\)
\(110\) 0 0
\(111\) 3.21965 0.305596
\(112\) −2.42872 −0.229493
\(113\) −9.50553 −0.894205 −0.447103 0.894483i \(-0.647544\pi\)
−0.447103 + 0.894483i \(0.647544\pi\)
\(114\) −3.01451 −0.282334
\(115\) 4.57917 0.427010
\(116\) −8.81144 −0.818121
\(117\) −18.4783 −1.70831
\(118\) 4.79071 0.441021
\(119\) −10.4842 −0.961084
\(120\) 6.05881 0.553091
\(121\) 0 0
\(122\) 9.26197 0.838540
\(123\) 15.5429 1.40145
\(124\) 9.29488 0.834705
\(125\) 11.9796 1.07149
\(126\) 14.7842 1.31708
\(127\) 10.0590 0.892593 0.446296 0.894885i \(-0.352743\pi\)
0.446296 + 0.894885i \(0.352743\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −36.1261 −3.18073
\(130\) −6.10114 −0.535105
\(131\) −8.33735 −0.728437 −0.364219 0.931313i \(-0.618664\pi\)
−0.364219 + 0.931313i \(0.618664\pi\)
\(132\) 0 0
\(133\) −2.42872 −0.210597
\(134\) −9.19479 −0.794309
\(135\) −18.7051 −1.60987
\(136\) −4.31675 −0.370158
\(137\) −18.8549 −1.61088 −0.805440 0.592677i \(-0.798071\pi\)
−0.805440 + 0.592677i \(0.798071\pi\)
\(138\) 6.86802 0.584645
\(139\) −3.62541 −0.307503 −0.153752 0.988110i \(-0.549136\pi\)
−0.153752 + 0.988110i \(0.549136\pi\)
\(140\) 4.88145 0.412558
\(141\) −27.1309 −2.28483
\(142\) 10.7608 0.903025
\(143\) 0 0
\(144\) 6.08725 0.507271
\(145\) 17.7100 1.47073
\(146\) 8.93977 0.739861
\(147\) −3.31990 −0.273821
\(148\) 1.06805 0.0877933
\(149\) −5.42145 −0.444143 −0.222071 0.975030i \(-0.571282\pi\)
−0.222071 + 0.975030i \(0.571282\pi\)
\(150\) 2.89502 0.236377
\(151\) −7.67465 −0.624554 −0.312277 0.949991i \(-0.601092\pi\)
−0.312277 + 0.949991i \(0.601092\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 26.2771 2.12438
\(154\) 0 0
\(155\) −18.6816 −1.50055
\(156\) −9.15074 −0.732645
\(157\) −24.0027 −1.91562 −0.957810 0.287401i \(-0.907209\pi\)
−0.957810 + 0.287401i \(0.907209\pi\)
\(158\) 6.52103 0.518785
\(159\) 15.3599 1.21812
\(160\) 2.00989 0.158895
\(161\) 5.53342 0.436095
\(162\) −9.79285 −0.769399
\(163\) 0.148660 0.0116440 0.00582198 0.999983i \(-0.498147\pi\)
0.00582198 + 0.999983i \(0.498147\pi\)
\(164\) 5.15603 0.402618
\(165\) 0 0
\(166\) −4.09108 −0.317529
\(167\) −0.501818 −0.0388319 −0.0194159 0.999811i \(-0.506181\pi\)
−0.0194159 + 0.999811i \(0.506181\pi\)
\(168\) 7.32140 0.564858
\(169\) −3.78533 −0.291179
\(170\) 8.67617 0.665432
\(171\) 6.08725 0.465504
\(172\) −11.9841 −0.913779
\(173\) −2.46671 −0.187540 −0.0937701 0.995594i \(-0.529892\pi\)
−0.0937701 + 0.995594i \(0.529892\pi\)
\(174\) 26.5621 2.01367
\(175\) 2.33245 0.176317
\(176\) 0 0
\(177\) −14.4416 −1.08550
\(178\) −6.04832 −0.453341
\(179\) 19.2904 1.44183 0.720915 0.693023i \(-0.243722\pi\)
0.720915 + 0.693023i \(0.243722\pi\)
\(180\) −12.2347 −0.911918
\(181\) −8.54815 −0.635379 −0.317690 0.948195i \(-0.602907\pi\)
−0.317690 + 0.948195i \(0.602907\pi\)
\(182\) −7.37255 −0.546490
\(183\) −27.9203 −2.06393
\(184\) 2.27832 0.167960
\(185\) −2.14666 −0.157826
\(186\) −28.0195 −2.05449
\(187\) 0 0
\(188\) −9.00012 −0.656401
\(189\) −22.6030 −1.64412
\(190\) 2.00989 0.145812
\(191\) −0.999674 −0.0723339 −0.0361669 0.999346i \(-0.511515\pi\)
−0.0361669 + 0.999346i \(0.511515\pi\)
\(192\) 3.01451 0.217553
\(193\) −5.02121 −0.361434 −0.180717 0.983535i \(-0.557842\pi\)
−0.180717 + 0.983535i \(0.557842\pi\)
\(194\) 9.42693 0.676814
\(195\) 18.3919 1.31707
\(196\) −1.10131 −0.0786648
\(197\) 11.0791 0.789355 0.394677 0.918820i \(-0.370856\pi\)
0.394677 + 0.918820i \(0.370856\pi\)
\(198\) 0 0
\(199\) −10.4170 −0.738442 −0.369221 0.929342i \(-0.620375\pi\)
−0.369221 + 0.929342i \(0.620375\pi\)
\(200\) 0.960362 0.0679079
\(201\) 27.7178 1.95506
\(202\) −8.78589 −0.618173
\(203\) 21.4005 1.50202
\(204\) 13.0129 0.911083
\(205\) −10.3630 −0.723785
\(206\) 10.5453 0.734723
\(207\) −13.8687 −0.963943
\(208\) −3.03557 −0.210479
\(209\) 0 0
\(210\) −14.7152 −1.01544
\(211\) −21.2337 −1.46179 −0.730894 0.682491i \(-0.760896\pi\)
−0.730894 + 0.682491i \(0.760896\pi\)
\(212\) 5.09532 0.349948
\(213\) −32.4385 −2.22265
\(214\) −7.69542 −0.526048
\(215\) 24.0867 1.64270
\(216\) −9.30653 −0.633229
\(217\) −22.5747 −1.53247
\(218\) −14.2327 −0.963959
\(219\) −26.9490 −1.82104
\(220\) 0 0
\(221\) −13.1038 −0.881456
\(222\) −3.21965 −0.216089
\(223\) −5.98764 −0.400962 −0.200481 0.979698i \(-0.564251\pi\)
−0.200481 + 0.979698i \(0.564251\pi\)
\(224\) 2.42872 0.162276
\(225\) −5.84596 −0.389731
\(226\) 9.50553 0.632298
\(227\) −13.6499 −0.905973 −0.452987 0.891517i \(-0.649641\pi\)
−0.452987 + 0.891517i \(0.649641\pi\)
\(228\) 3.01451 0.199641
\(229\) −11.3245 −0.748347 −0.374173 0.927359i \(-0.622074\pi\)
−0.374173 + 0.927359i \(0.622074\pi\)
\(230\) −4.57917 −0.301942
\(231\) 0 0
\(232\) 8.81144 0.578499
\(233\) −7.52089 −0.492710 −0.246355 0.969180i \(-0.579233\pi\)
−0.246355 + 0.969180i \(0.579233\pi\)
\(234\) 18.4783 1.20796
\(235\) 18.0892 1.18001
\(236\) −4.79071 −0.311849
\(237\) −19.6577 −1.27690
\(238\) 10.4842 0.679589
\(239\) 0.510794 0.0330405 0.0165203 0.999864i \(-0.494741\pi\)
0.0165203 + 0.999864i \(0.494741\pi\)
\(240\) −6.05881 −0.391095
\(241\) 22.6208 1.45713 0.728566 0.684975i \(-0.240187\pi\)
0.728566 + 0.684975i \(0.240187\pi\)
\(242\) 0 0
\(243\) 1.60101 0.102705
\(244\) −9.26197 −0.592937
\(245\) 2.21350 0.141415
\(246\) −15.5429 −0.990978
\(247\) −3.03557 −0.193149
\(248\) −9.29488 −0.590225
\(249\) 12.3326 0.781546
\(250\) −11.9796 −0.757659
\(251\) −12.6321 −0.797333 −0.398667 0.917096i \(-0.630527\pi\)
−0.398667 + 0.917096i \(0.630527\pi\)
\(252\) −14.7842 −0.931319
\(253\) 0 0
\(254\) −10.0590 −0.631158
\(255\) −26.1544 −1.63785
\(256\) 1.00000 0.0625000
\(257\) 25.0497 1.56256 0.781278 0.624183i \(-0.214568\pi\)
0.781278 + 0.624183i \(0.214568\pi\)
\(258\) 36.1261 2.24911
\(259\) −2.59400 −0.161183
\(260\) 6.10114 0.378377
\(261\) −53.6374 −3.32007
\(262\) 8.33735 0.515083
\(263\) 1.50368 0.0927212 0.0463606 0.998925i \(-0.485238\pi\)
0.0463606 + 0.998925i \(0.485238\pi\)
\(264\) 0 0
\(265\) −10.2410 −0.629100
\(266\) 2.42872 0.148915
\(267\) 18.2327 1.11582
\(268\) 9.19479 0.561661
\(269\) −10.5493 −0.643201 −0.321601 0.946875i \(-0.604221\pi\)
−0.321601 + 0.946875i \(0.604221\pi\)
\(270\) 18.7051 1.13835
\(271\) −1.95259 −0.118611 −0.0593056 0.998240i \(-0.518889\pi\)
−0.0593056 + 0.998240i \(0.518889\pi\)
\(272\) 4.31675 0.261741
\(273\) 22.2246 1.34509
\(274\) 18.8549 1.13906
\(275\) 0 0
\(276\) −6.86802 −0.413407
\(277\) −32.5196 −1.95391 −0.976957 0.213434i \(-0.931535\pi\)
−0.976957 + 0.213434i \(0.931535\pi\)
\(278\) 3.62541 0.217438
\(279\) 56.5802 3.38737
\(280\) −4.88145 −0.291723
\(281\) −13.9545 −0.832454 −0.416227 0.909261i \(-0.636648\pi\)
−0.416227 + 0.909261i \(0.636648\pi\)
\(282\) 27.1309 1.61562
\(283\) 22.5828 1.34241 0.671205 0.741272i \(-0.265777\pi\)
0.671205 + 0.741272i \(0.265777\pi\)
\(284\) −10.7608 −0.638535
\(285\) −6.05881 −0.358893
\(286\) 0 0
\(287\) −12.5226 −0.739183
\(288\) −6.08725 −0.358695
\(289\) 1.63432 0.0961362
\(290\) −17.7100 −1.03997
\(291\) −28.4175 −1.66586
\(292\) −8.93977 −0.523161
\(293\) −26.1229 −1.52612 −0.763058 0.646330i \(-0.776303\pi\)
−0.763058 + 0.646330i \(0.776303\pi\)
\(294\) 3.31990 0.193620
\(295\) 9.62879 0.560610
\(296\) −1.06805 −0.0620792
\(297\) 0 0
\(298\) 5.42145 0.314056
\(299\) 6.91601 0.399963
\(300\) −2.89502 −0.167144
\(301\) 29.1060 1.67764
\(302\) 7.67465 0.441626
\(303\) 26.4851 1.52153
\(304\) 1.00000 0.0573539
\(305\) 18.6155 1.06592
\(306\) −26.2771 −1.50216
\(307\) 17.6119 1.00516 0.502581 0.864530i \(-0.332384\pi\)
0.502581 + 0.864530i \(0.332384\pi\)
\(308\) 0 0
\(309\) −31.7888 −1.80840
\(310\) 18.6816 1.06105
\(311\) −8.78558 −0.498185 −0.249092 0.968480i \(-0.580132\pi\)
−0.249092 + 0.968480i \(0.580132\pi\)
\(312\) 9.15074 0.518058
\(313\) 32.2305 1.82177 0.910886 0.412657i \(-0.135399\pi\)
0.910886 + 0.412657i \(0.135399\pi\)
\(314\) 24.0027 1.35455
\(315\) 29.7146 1.67423
\(316\) −6.52103 −0.366837
\(317\) 0.754060 0.0423522 0.0211761 0.999776i \(-0.493259\pi\)
0.0211761 + 0.999776i \(0.493259\pi\)
\(318\) −15.3599 −0.861339
\(319\) 0 0
\(320\) −2.00989 −0.112356
\(321\) 23.1979 1.29478
\(322\) −5.53342 −0.308365
\(323\) 4.31675 0.240190
\(324\) 9.79285 0.544047
\(325\) 2.91524 0.161709
\(326\) −0.148660 −0.00823352
\(327\) 42.9045 2.37263
\(328\) −5.15603 −0.284694
\(329\) 21.8588 1.20511
\(330\) 0 0
\(331\) 6.33131 0.348000 0.174000 0.984746i \(-0.444331\pi\)
0.174000 + 0.984746i \(0.444331\pi\)
\(332\) 4.09108 0.224527
\(333\) 6.50150 0.356280
\(334\) 0.501818 0.0274583
\(335\) −18.4805 −1.00970
\(336\) −7.32140 −0.399415
\(337\) −9.10922 −0.496211 −0.248105 0.968733i \(-0.579808\pi\)
−0.248105 + 0.968733i \(0.579808\pi\)
\(338\) 3.78533 0.205895
\(339\) −28.6545 −1.55630
\(340\) −8.67617 −0.470531
\(341\) 0 0
\(342\) −6.08725 −0.329161
\(343\) 19.6758 1.06239
\(344\) 11.9841 0.646139
\(345\) 13.8039 0.743179
\(346\) 2.46671 0.132611
\(347\) 0.810769 0.0435243 0.0217622 0.999763i \(-0.493072\pi\)
0.0217622 + 0.999763i \(0.493072\pi\)
\(348\) −26.5621 −1.42388
\(349\) 25.9076 1.38680 0.693401 0.720552i \(-0.256111\pi\)
0.693401 + 0.720552i \(0.256111\pi\)
\(350\) −2.33245 −0.124675
\(351\) −28.2506 −1.50791
\(352\) 0 0
\(353\) 30.9371 1.64662 0.823309 0.567594i \(-0.192125\pi\)
0.823309 + 0.567594i \(0.192125\pi\)
\(354\) 14.4416 0.767564
\(355\) 21.6279 1.14789
\(356\) 6.04832 0.320560
\(357\) −31.6046 −1.67269
\(358\) −19.2904 −1.01953
\(359\) 33.8838 1.78832 0.894158 0.447751i \(-0.147775\pi\)
0.894158 + 0.447751i \(0.147775\pi\)
\(360\) 12.2347 0.644824
\(361\) 1.00000 0.0526316
\(362\) 8.54815 0.449281
\(363\) 0 0
\(364\) 7.37255 0.386427
\(365\) 17.9679 0.940484
\(366\) 27.9203 1.45942
\(367\) −25.7084 −1.34197 −0.670983 0.741473i \(-0.734128\pi\)
−0.670983 + 0.741473i \(0.734128\pi\)
\(368\) −2.27832 −0.118766
\(369\) 31.3860 1.63389
\(370\) 2.14666 0.111600
\(371\) −12.3751 −0.642484
\(372\) 28.0195 1.45274
\(373\) −12.1401 −0.628589 −0.314294 0.949326i \(-0.601768\pi\)
−0.314294 + 0.949326i \(0.601768\pi\)
\(374\) 0 0
\(375\) 36.1127 1.86485
\(376\) 9.00012 0.464146
\(377\) 26.7477 1.37758
\(378\) 22.6030 1.16257
\(379\) 31.9986 1.64366 0.821829 0.569734i \(-0.192954\pi\)
0.821829 + 0.569734i \(0.192954\pi\)
\(380\) −2.00989 −0.103105
\(381\) 30.3229 1.55349
\(382\) 0.999674 0.0511478
\(383\) −19.5901 −1.00101 −0.500504 0.865734i \(-0.666852\pi\)
−0.500504 + 0.865734i \(0.666852\pi\)
\(384\) −3.01451 −0.153833
\(385\) 0 0
\(386\) 5.02121 0.255573
\(387\) −72.9502 −3.70827
\(388\) −9.42693 −0.478580
\(389\) −10.9428 −0.554824 −0.277412 0.960751i \(-0.589477\pi\)
−0.277412 + 0.960751i \(0.589477\pi\)
\(390\) −18.3919 −0.931311
\(391\) −9.83496 −0.497375
\(392\) 1.10131 0.0556244
\(393\) −25.1330 −1.26779
\(394\) −11.0791 −0.558158
\(395\) 13.1065 0.659460
\(396\) 0 0
\(397\) 17.4801 0.877303 0.438651 0.898657i \(-0.355456\pi\)
0.438651 + 0.898657i \(0.355456\pi\)
\(398\) 10.4170 0.522157
\(399\) −7.32140 −0.366528
\(400\) −0.960362 −0.0480181
\(401\) 16.7907 0.838485 0.419243 0.907874i \(-0.362296\pi\)
0.419243 + 0.907874i \(0.362296\pi\)
\(402\) −27.7178 −1.38244
\(403\) −28.2152 −1.40550
\(404\) 8.78589 0.437114
\(405\) −19.6825 −0.978031
\(406\) −21.4005 −1.06209
\(407\) 0 0
\(408\) −13.0129 −0.644233
\(409\) 7.01163 0.346703 0.173351 0.984860i \(-0.444540\pi\)
0.173351 + 0.984860i \(0.444540\pi\)
\(410\) 10.3630 0.511793
\(411\) −56.8381 −2.80362
\(412\) −10.5453 −0.519528
\(413\) 11.6353 0.572536
\(414\) 13.8687 0.681611
\(415\) −8.22260 −0.403632
\(416\) 3.03557 0.148831
\(417\) −10.9288 −0.535186
\(418\) 0 0
\(419\) −22.9541 −1.12138 −0.560690 0.828026i \(-0.689464\pi\)
−0.560690 + 0.828026i \(0.689464\pi\)
\(420\) 14.7152 0.718027
\(421\) 5.13342 0.250188 0.125094 0.992145i \(-0.460077\pi\)
0.125094 + 0.992145i \(0.460077\pi\)
\(422\) 21.2337 1.03364
\(423\) −54.7859 −2.66378
\(424\) −5.09532 −0.247450
\(425\) −4.14564 −0.201093
\(426\) 32.4385 1.57165
\(427\) 22.4948 1.08860
\(428\) 7.69542 0.371972
\(429\) 0 0
\(430\) −24.0867 −1.16156
\(431\) 39.2597 1.89107 0.945536 0.325518i \(-0.105539\pi\)
0.945536 + 0.325518i \(0.105539\pi\)
\(432\) 9.30653 0.447761
\(433\) −23.8406 −1.14571 −0.572853 0.819658i \(-0.694163\pi\)
−0.572853 + 0.819658i \(0.694163\pi\)
\(434\) 22.5747 1.08362
\(435\) 53.3868 2.55970
\(436\) 14.2327 0.681622
\(437\) −2.27832 −0.108987
\(438\) 26.9490 1.28767
\(439\) 20.4659 0.976783 0.488391 0.872625i \(-0.337584\pi\)
0.488391 + 0.872625i \(0.337584\pi\)
\(440\) 0 0
\(441\) −6.70393 −0.319235
\(442\) 13.1038 0.623283
\(443\) 20.4954 0.973768 0.486884 0.873467i \(-0.338133\pi\)
0.486884 + 0.873467i \(0.338133\pi\)
\(444\) 3.21965 0.152798
\(445\) −12.1564 −0.576270
\(446\) 5.98764 0.283523
\(447\) −16.3430 −0.772998
\(448\) −2.42872 −0.114746
\(449\) 29.9575 1.41378 0.706891 0.707323i \(-0.250097\pi\)
0.706891 + 0.707323i \(0.250097\pi\)
\(450\) 5.84596 0.275581
\(451\) 0 0
\(452\) −9.50553 −0.447103
\(453\) −23.1353 −1.08699
\(454\) 13.6499 0.640620
\(455\) −14.8180 −0.694678
\(456\) −3.01451 −0.141167
\(457\) 29.4491 1.37757 0.688786 0.724964i \(-0.258144\pi\)
0.688786 + 0.724964i \(0.258144\pi\)
\(458\) 11.3245 0.529161
\(459\) 40.1739 1.87516
\(460\) 4.57917 0.213505
\(461\) 12.7886 0.595624 0.297812 0.954624i \(-0.403743\pi\)
0.297812 + 0.954624i \(0.403743\pi\)
\(462\) 0 0
\(463\) −22.8327 −1.06112 −0.530562 0.847646i \(-0.678019\pi\)
−0.530562 + 0.847646i \(0.678019\pi\)
\(464\) −8.81144 −0.409061
\(465\) −56.3159 −2.61159
\(466\) 7.52089 0.348398
\(467\) 23.2585 1.07627 0.538137 0.842857i \(-0.319128\pi\)
0.538137 + 0.842857i \(0.319128\pi\)
\(468\) −18.4783 −0.854157
\(469\) −22.3316 −1.03118
\(470\) −18.0892 −0.834392
\(471\) −72.3562 −3.33400
\(472\) 4.79071 0.220511
\(473\) 0 0
\(474\) 19.6577 0.902907
\(475\) −0.960362 −0.0440644
\(476\) −10.4842 −0.480542
\(477\) 31.0165 1.42015
\(478\) −0.510794 −0.0233632
\(479\) 12.5716 0.574410 0.287205 0.957869i \(-0.407274\pi\)
0.287205 + 0.957869i \(0.407274\pi\)
\(480\) 6.05881 0.276546
\(481\) −3.24214 −0.147829
\(482\) −22.6208 −1.03035
\(483\) 16.6805 0.758990
\(484\) 0 0
\(485\) 18.9470 0.860341
\(486\) −1.60101 −0.0726232
\(487\) −20.6412 −0.935342 −0.467671 0.883903i \(-0.654907\pi\)
−0.467671 + 0.883903i \(0.654907\pi\)
\(488\) 9.26197 0.419270
\(489\) 0.448137 0.0202654
\(490\) −2.21350 −0.0999958
\(491\) 30.2650 1.36584 0.682919 0.730494i \(-0.260710\pi\)
0.682919 + 0.730494i \(0.260710\pi\)
\(492\) 15.5429 0.700727
\(493\) −38.0368 −1.71309
\(494\) 3.03557 0.136577
\(495\) 0 0
\(496\) 9.29488 0.417352
\(497\) 26.1350 1.17231
\(498\) −12.3326 −0.552636
\(499\) −36.2503 −1.62279 −0.811394 0.584499i \(-0.801291\pi\)
−0.811394 + 0.584499i \(0.801291\pi\)
\(500\) 11.9796 0.535746
\(501\) −1.51273 −0.0675840
\(502\) 12.6321 0.563800
\(503\) −14.4032 −0.642208 −0.321104 0.947044i \(-0.604054\pi\)
−0.321104 + 0.947044i \(0.604054\pi\)
\(504\) 14.7842 0.658542
\(505\) −17.6586 −0.785799
\(506\) 0 0
\(507\) −11.4109 −0.506776
\(508\) 10.0590 0.446296
\(509\) 35.9803 1.59480 0.797400 0.603451i \(-0.206208\pi\)
0.797400 + 0.603451i \(0.206208\pi\)
\(510\) 26.1544 1.15813
\(511\) 21.7122 0.960492
\(512\) −1.00000 −0.0441942
\(513\) 9.30653 0.410893
\(514\) −25.0497 −1.10489
\(515\) 21.1948 0.933953
\(516\) −36.1261 −1.59036
\(517\) 0 0
\(518\) 2.59400 0.113974
\(519\) −7.43590 −0.326400
\(520\) −6.10114 −0.267553
\(521\) −24.0468 −1.05351 −0.526756 0.850017i \(-0.676592\pi\)
−0.526756 + 0.850017i \(0.676592\pi\)
\(522\) 53.6374 2.34765
\(523\) −33.7621 −1.47631 −0.738157 0.674629i \(-0.764304\pi\)
−0.738157 + 0.674629i \(0.764304\pi\)
\(524\) −8.33735 −0.364219
\(525\) 7.03120 0.306867
\(526\) −1.50368 −0.0655638
\(527\) 40.1237 1.74781
\(528\) 0 0
\(529\) −17.8092 −0.774315
\(530\) 10.2410 0.444841
\(531\) −29.1623 −1.26553
\(532\) −2.42872 −0.105298
\(533\) −15.6515 −0.677940
\(534\) −18.2327 −0.789006
\(535\) −15.4669 −0.668693
\(536\) −9.19479 −0.397155
\(537\) 58.1510 2.50940
\(538\) 10.5493 0.454812
\(539\) 0 0
\(540\) −18.7051 −0.804937
\(541\) 8.63940 0.371437 0.185718 0.982603i \(-0.440539\pi\)
0.185718 + 0.982603i \(0.440539\pi\)
\(542\) 1.95259 0.0838708
\(543\) −25.7685 −1.10583
\(544\) −4.31675 −0.185079
\(545\) −28.6061 −1.22535
\(546\) −22.2246 −0.951125
\(547\) 19.6355 0.839552 0.419776 0.907628i \(-0.362109\pi\)
0.419776 + 0.907628i \(0.362109\pi\)
\(548\) −18.8549 −0.805440
\(549\) −56.3799 −2.40624
\(550\) 0 0
\(551\) −8.81144 −0.375380
\(552\) 6.86802 0.292323
\(553\) 15.8378 0.673490
\(554\) 32.5196 1.38163
\(555\) −6.47112 −0.274684
\(556\) −3.62541 −0.153752
\(557\) 42.2771 1.79134 0.895669 0.444722i \(-0.146697\pi\)
0.895669 + 0.444722i \(0.146697\pi\)
\(558\) −56.5802 −2.39523
\(559\) 36.3785 1.53865
\(560\) 4.88145 0.206279
\(561\) 0 0
\(562\) 13.9545 0.588634
\(563\) −15.9375 −0.671688 −0.335844 0.941918i \(-0.609021\pi\)
−0.335844 + 0.941918i \(0.609021\pi\)
\(564\) −27.1309 −1.14242
\(565\) 19.1050 0.803754
\(566\) −22.5828 −0.949227
\(567\) −23.7841 −0.998839
\(568\) 10.7608 0.451513
\(569\) 21.4299 0.898387 0.449193 0.893435i \(-0.351711\pi\)
0.449193 + 0.893435i \(0.351711\pi\)
\(570\) 6.05881 0.253776
\(571\) −21.2358 −0.888691 −0.444346 0.895855i \(-0.646564\pi\)
−0.444346 + 0.895855i \(0.646564\pi\)
\(572\) 0 0
\(573\) −3.01352 −0.125892
\(574\) 12.5226 0.522682
\(575\) 2.18802 0.0912466
\(576\) 6.08725 0.253635
\(577\) 5.86811 0.244292 0.122146 0.992512i \(-0.461022\pi\)
0.122146 + 0.992512i \(0.461022\pi\)
\(578\) −1.63432 −0.0679786
\(579\) −15.1365 −0.629050
\(580\) 17.7100 0.735367
\(581\) −9.93610 −0.412219
\(582\) 28.4175 1.17794
\(583\) 0 0
\(584\) 8.93977 0.369930
\(585\) 37.1392 1.53552
\(586\) 26.1229 1.07913
\(587\) −41.6241 −1.71801 −0.859006 0.511966i \(-0.828917\pi\)
−0.859006 + 0.511966i \(0.828917\pi\)
\(588\) −3.31990 −0.136910
\(589\) 9.29488 0.382989
\(590\) −9.62879 −0.396411
\(591\) 33.3981 1.37381
\(592\) 1.06805 0.0438967
\(593\) 13.4779 0.553469 0.276735 0.960946i \(-0.410748\pi\)
0.276735 + 0.960946i \(0.410748\pi\)
\(594\) 0 0
\(595\) 21.0720 0.863868
\(596\) −5.42145 −0.222071
\(597\) −31.4021 −1.28520
\(598\) −6.91601 −0.282817
\(599\) 8.59259 0.351084 0.175542 0.984472i \(-0.443832\pi\)
0.175542 + 0.984472i \(0.443832\pi\)
\(600\) 2.89502 0.118189
\(601\) −8.48908 −0.346277 −0.173138 0.984898i \(-0.555391\pi\)
−0.173138 + 0.984898i \(0.555391\pi\)
\(602\) −29.1060 −1.18627
\(603\) 55.9710 2.27932
\(604\) −7.67465 −0.312277
\(605\) 0 0
\(606\) −26.4851 −1.07588
\(607\) 4.75233 0.192891 0.0964457 0.995338i \(-0.469253\pi\)
0.0964457 + 0.995338i \(0.469253\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 64.5120 2.61416
\(610\) −18.6155 −0.753719
\(611\) 27.3205 1.10527
\(612\) 26.2771 1.06219
\(613\) −29.3790 −1.18661 −0.593304 0.804979i \(-0.702177\pi\)
−0.593304 + 0.804979i \(0.702177\pi\)
\(614\) −17.6119 −0.710758
\(615\) −31.2394 −1.25969
\(616\) 0 0
\(617\) 14.0808 0.566870 0.283435 0.958991i \(-0.408526\pi\)
0.283435 + 0.958991i \(0.408526\pi\)
\(618\) 31.7888 1.27873
\(619\) −22.3677 −0.899034 −0.449517 0.893272i \(-0.648404\pi\)
−0.449517 + 0.893272i \(0.648404\pi\)
\(620\) −18.6816 −0.750273
\(621\) −21.2033 −0.850859
\(622\) 8.78558 0.352270
\(623\) −14.6897 −0.588530
\(624\) −9.15074 −0.366323
\(625\) −19.2759 −0.771036
\(626\) −32.2305 −1.28819
\(627\) 0 0
\(628\) −24.0027 −0.957810
\(629\) 4.61051 0.183833
\(630\) −29.7146 −1.18386
\(631\) −36.4436 −1.45080 −0.725399 0.688328i \(-0.758345\pi\)
−0.725399 + 0.688328i \(0.758345\pi\)
\(632\) 6.52103 0.259393
\(633\) −64.0091 −2.54413
\(634\) −0.754060 −0.0299476
\(635\) −20.2175 −0.802305
\(636\) 15.3599 0.609058
\(637\) 3.34309 0.132458
\(638\) 0 0
\(639\) −65.5036 −2.59128
\(640\) 2.00989 0.0794477
\(641\) −15.3418 −0.605964 −0.302982 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(642\) −23.1979 −0.915547
\(643\) −29.2007 −1.15156 −0.575781 0.817604i \(-0.695302\pi\)
−0.575781 + 0.817604i \(0.695302\pi\)
\(644\) 5.53342 0.218047
\(645\) 72.6094 2.85899
\(646\) −4.31675 −0.169840
\(647\) 19.9255 0.783350 0.391675 0.920104i \(-0.371896\pi\)
0.391675 + 0.920104i \(0.371896\pi\)
\(648\) −9.79285 −0.384699
\(649\) 0 0
\(650\) −2.91524 −0.114345
\(651\) −68.0515 −2.66715
\(652\) 0.148660 0.00582198
\(653\) −3.46955 −0.135774 −0.0678870 0.997693i \(-0.521626\pi\)
−0.0678870 + 0.997693i \(0.521626\pi\)
\(654\) −42.9045 −1.67770
\(655\) 16.7571 0.654754
\(656\) 5.15603 0.201309
\(657\) −54.4186 −2.12307
\(658\) −21.8588 −0.852144
\(659\) 3.67907 0.143316 0.0716582 0.997429i \(-0.477171\pi\)
0.0716582 + 0.997429i \(0.477171\pi\)
\(660\) 0 0
\(661\) 45.8391 1.78294 0.891468 0.453084i \(-0.149676\pi\)
0.891468 + 0.453084i \(0.149676\pi\)
\(662\) −6.33131 −0.246073
\(663\) −39.5014 −1.53411
\(664\) −4.09108 −0.158765
\(665\) 4.88145 0.189295
\(666\) −6.50150 −0.251928
\(667\) 20.0753 0.777319
\(668\) −0.501818 −0.0194159
\(669\) −18.0498 −0.697845
\(670\) 18.4805 0.713963
\(671\) 0 0
\(672\) 7.32140 0.282429
\(673\) −13.9610 −0.538156 −0.269078 0.963118i \(-0.586719\pi\)
−0.269078 + 0.963118i \(0.586719\pi\)
\(674\) 9.10922 0.350874
\(675\) −8.93764 −0.344010
\(676\) −3.78533 −0.145590
\(677\) 9.42523 0.362241 0.181121 0.983461i \(-0.442028\pi\)
0.181121 + 0.983461i \(0.442028\pi\)
\(678\) 28.6545 1.10047
\(679\) 22.8954 0.878644
\(680\) 8.67617 0.332716
\(681\) −41.1476 −1.57678
\(682\) 0 0
\(683\) 45.4023 1.73727 0.868635 0.495453i \(-0.164998\pi\)
0.868635 + 0.495453i \(0.164998\pi\)
\(684\) 6.08725 0.232752
\(685\) 37.8961 1.44794
\(686\) −19.6758 −0.751227
\(687\) −34.1379 −1.30244
\(688\) −11.9841 −0.456889
\(689\) −15.4672 −0.589253
\(690\) −13.8039 −0.525507
\(691\) −42.3347 −1.61049 −0.805244 0.592944i \(-0.797966\pi\)
−0.805244 + 0.592944i \(0.797966\pi\)
\(692\) −2.46671 −0.0937701
\(693\) 0 0
\(694\) −0.810769 −0.0307764
\(695\) 7.28665 0.276399
\(696\) 26.5621 1.00683
\(697\) 22.2573 0.843054
\(698\) −25.9076 −0.980617
\(699\) −22.6718 −0.857525
\(700\) 2.33245 0.0881585
\(701\) 30.8992 1.16705 0.583524 0.812096i \(-0.301673\pi\)
0.583524 + 0.812096i \(0.301673\pi\)
\(702\) 28.2506 1.06625
\(703\) 1.06805 0.0402823
\(704\) 0 0
\(705\) 54.5300 2.05372
\(706\) −30.9371 −1.16433
\(707\) −21.3385 −0.802516
\(708\) −14.4416 −0.542750
\(709\) 13.4148 0.503804 0.251902 0.967753i \(-0.418944\pi\)
0.251902 + 0.967753i \(0.418944\pi\)
\(710\) −21.6279 −0.811682
\(711\) −39.6951 −1.48868
\(712\) −6.04832 −0.226670
\(713\) −21.1768 −0.793076
\(714\) 31.6046 1.18277
\(715\) 0 0
\(716\) 19.2904 0.720915
\(717\) 1.53979 0.0575046
\(718\) −33.8838 −1.26453
\(719\) 40.6614 1.51641 0.758207 0.652014i \(-0.226076\pi\)
0.758207 + 0.652014i \(0.226076\pi\)
\(720\) −12.2347 −0.455959
\(721\) 25.6115 0.953823
\(722\) −1.00000 −0.0372161
\(723\) 68.1905 2.53603
\(724\) −8.54815 −0.317690
\(725\) 8.46217 0.314277
\(726\) 0 0
\(727\) −21.0063 −0.779080 −0.389540 0.921010i \(-0.627366\pi\)
−0.389540 + 0.921010i \(0.627366\pi\)
\(728\) −7.37255 −0.273245
\(729\) −24.5523 −0.909344
\(730\) −17.9679 −0.665022
\(731\) −51.7323 −1.91339
\(732\) −27.9203 −1.03196
\(733\) −35.6221 −1.31573 −0.657867 0.753134i \(-0.728541\pi\)
−0.657867 + 0.753134i \(0.728541\pi\)
\(734\) 25.7084 0.948913
\(735\) 6.67261 0.246123
\(736\) 2.27832 0.0839802
\(737\) 0 0
\(738\) −31.3860 −1.15534
\(739\) −38.8113 −1.42770 −0.713848 0.700300i \(-0.753049\pi\)
−0.713848 + 0.700300i \(0.753049\pi\)
\(740\) −2.14666 −0.0789128
\(741\) −9.15074 −0.336161
\(742\) 12.3751 0.454305
\(743\) 20.4994 0.752051 0.376025 0.926609i \(-0.377291\pi\)
0.376025 + 0.926609i \(0.377291\pi\)
\(744\) −28.0195 −1.02724
\(745\) 10.8965 0.399217
\(746\) 12.1401 0.444479
\(747\) 24.9034 0.911168
\(748\) 0 0
\(749\) −18.6900 −0.682919
\(750\) −36.1127 −1.31865
\(751\) 37.0819 1.35314 0.676568 0.736380i \(-0.263466\pi\)
0.676568 + 0.736380i \(0.263466\pi\)
\(752\) −9.00012 −0.328200
\(753\) −38.0796 −1.38770
\(754\) −26.7477 −0.974094
\(755\) 15.4252 0.561379
\(756\) −22.6030 −0.822062
\(757\) 1.99625 0.0725549 0.0362774 0.999342i \(-0.488450\pi\)
0.0362774 + 0.999342i \(0.488450\pi\)
\(758\) −31.9986 −1.16224
\(759\) 0 0
\(760\) 2.00989 0.0729062
\(761\) −15.0734 −0.546411 −0.273205 0.961956i \(-0.588084\pi\)
−0.273205 + 0.961956i \(0.588084\pi\)
\(762\) −30.3229 −1.09848
\(763\) −34.5672 −1.25142
\(764\) −0.999674 −0.0361669
\(765\) −52.8140 −1.90949
\(766\) 19.5901 0.707820
\(767\) 14.5425 0.525101
\(768\) 3.01451 0.108777
\(769\) 30.4602 1.09842 0.549212 0.835683i \(-0.314928\pi\)
0.549212 + 0.835683i \(0.314928\pi\)
\(770\) 0 0
\(771\) 75.5125 2.71951
\(772\) −5.02121 −0.180717
\(773\) −23.3781 −0.840851 −0.420425 0.907327i \(-0.638119\pi\)
−0.420425 + 0.907327i \(0.638119\pi\)
\(774\) 72.9502 2.62214
\(775\) −8.92645 −0.320648
\(776\) 9.42693 0.338407
\(777\) −7.81963 −0.280528
\(778\) 10.9428 0.392320
\(779\) 5.15603 0.184734
\(780\) 18.3919 0.658537
\(781\) 0 0
\(782\) 9.83496 0.351697
\(783\) −82.0039 −2.93058
\(784\) −1.10131 −0.0393324
\(785\) 48.2426 1.72185
\(786\) 25.1330 0.896464
\(787\) 6.11169 0.217858 0.108929 0.994050i \(-0.465258\pi\)
0.108929 + 0.994050i \(0.465258\pi\)
\(788\) 11.0791 0.394677
\(789\) 4.53287 0.161374
\(790\) −13.1065 −0.466309
\(791\) 23.0863 0.820854
\(792\) 0 0
\(793\) 28.1153 0.998405
\(794\) −17.4801 −0.620347
\(795\) −30.8716 −1.09490
\(796\) −10.4170 −0.369221
\(797\) −1.27382 −0.0451210 −0.0225605 0.999745i \(-0.507182\pi\)
−0.0225605 + 0.999745i \(0.507182\pi\)
\(798\) 7.32140 0.259175
\(799\) −38.8512 −1.37446
\(800\) 0.960362 0.0339539
\(801\) 36.8176 1.30089
\(802\) −16.7907 −0.592899
\(803\) 0 0
\(804\) 27.7178 0.977530
\(805\) −11.1215 −0.391983
\(806\) 28.2152 0.993839
\(807\) −31.8009 −1.11944
\(808\) −8.78589 −0.309086
\(809\) 39.4787 1.38800 0.693999 0.719976i \(-0.255847\pi\)
0.693999 + 0.719976i \(0.255847\pi\)
\(810\) 19.6825 0.691572
\(811\) −4.05599 −0.142425 −0.0712125 0.997461i \(-0.522687\pi\)
−0.0712125 + 0.997461i \(0.522687\pi\)
\(812\) 21.4005 0.751011
\(813\) −5.88609 −0.206434
\(814\) 0 0
\(815\) −0.298790 −0.0104661
\(816\) 13.0129 0.455541
\(817\) −11.9841 −0.419270
\(818\) −7.01163 −0.245156
\(819\) 44.8785 1.56818
\(820\) −10.3630 −0.361892
\(821\) −5.11544 −0.178530 −0.0892650 0.996008i \(-0.528452\pi\)
−0.0892650 + 0.996008i \(0.528452\pi\)
\(822\) 56.8381 1.98246
\(823\) −36.6100 −1.27614 −0.638072 0.769976i \(-0.720268\pi\)
−0.638072 + 0.769976i \(0.720268\pi\)
\(824\) 10.5453 0.367362
\(825\) 0 0
\(826\) −11.6353 −0.404844
\(827\) −25.8761 −0.899800 −0.449900 0.893079i \(-0.648540\pi\)
−0.449900 + 0.893079i \(0.648540\pi\)
\(828\) −13.8687 −0.481972
\(829\) −32.3503 −1.12357 −0.561785 0.827283i \(-0.689885\pi\)
−0.561785 + 0.827283i \(0.689885\pi\)
\(830\) 8.22260 0.285411
\(831\) −98.0306 −3.40064
\(832\) −3.03557 −0.105239
\(833\) −4.75407 −0.164719
\(834\) 10.9288 0.378434
\(835\) 1.00860 0.0349039
\(836\) 0 0
\(837\) 86.5031 2.98998
\(838\) 22.9541 0.792935
\(839\) 29.3647 1.01378 0.506891 0.862010i \(-0.330795\pi\)
0.506891 + 0.862010i \(0.330795\pi\)
\(840\) −14.7152 −0.507722
\(841\) 48.6414 1.67729
\(842\) −5.13342 −0.176909
\(843\) −42.0659 −1.44883
\(844\) −21.2337 −0.730894
\(845\) 7.60808 0.261726
\(846\) 54.7859 1.88358
\(847\) 0 0
\(848\) 5.09532 0.174974
\(849\) 68.0761 2.33636
\(850\) 4.14564 0.142194
\(851\) −2.43337 −0.0834148
\(852\) −32.4385 −1.11132
\(853\) −32.8026 −1.12314 −0.561570 0.827429i \(-0.689803\pi\)
−0.561570 + 0.827429i \(0.689803\pi\)
\(854\) −22.4948 −0.769755
\(855\) −12.2347 −0.418417
\(856\) −7.69542 −0.263024
\(857\) 10.5746 0.361222 0.180611 0.983555i \(-0.442193\pi\)
0.180611 + 0.983555i \(0.442193\pi\)
\(858\) 0 0
\(859\) 31.7422 1.08303 0.541514 0.840692i \(-0.317851\pi\)
0.541514 + 0.840692i \(0.317851\pi\)
\(860\) 24.0867 0.821348
\(861\) −37.7493 −1.28649
\(862\) −39.2597 −1.33719
\(863\) −29.2335 −0.995120 −0.497560 0.867430i \(-0.665771\pi\)
−0.497560 + 0.867430i \(0.665771\pi\)
\(864\) −9.30653 −0.316615
\(865\) 4.95780 0.168570
\(866\) 23.8406 0.810137
\(867\) 4.92666 0.167318
\(868\) −22.5747 −0.766235
\(869\) 0 0
\(870\) −53.3868 −1.80998
\(871\) −27.9114 −0.945742
\(872\) −14.2327 −0.481979
\(873\) −57.3840 −1.94216
\(874\) 2.27832 0.0770655
\(875\) −29.0952 −0.983598
\(876\) −26.9490 −0.910522
\(877\) −0.282048 −0.00952409 −0.00476205 0.999989i \(-0.501516\pi\)
−0.00476205 + 0.999989i \(0.501516\pi\)
\(878\) −20.4659 −0.690690
\(879\) −78.7476 −2.65609
\(880\) 0 0
\(881\) −16.1017 −0.542480 −0.271240 0.962512i \(-0.587434\pi\)
−0.271240 + 0.962512i \(0.587434\pi\)
\(882\) 6.70393 0.225733
\(883\) −22.6230 −0.761326 −0.380663 0.924714i \(-0.624304\pi\)
−0.380663 + 0.924714i \(0.624304\pi\)
\(884\) −13.1038 −0.440728
\(885\) 29.0260 0.975699
\(886\) −20.4954 −0.688558
\(887\) −9.68158 −0.325076 −0.162538 0.986702i \(-0.551968\pi\)
−0.162538 + 0.986702i \(0.551968\pi\)
\(888\) −3.21965 −0.108044
\(889\) −24.4305 −0.819374
\(890\) 12.1564 0.407484
\(891\) 0 0
\(892\) −5.98764 −0.200481
\(893\) −9.00012 −0.301177
\(894\) 16.3430 0.546592
\(895\) −38.7714 −1.29599
\(896\) 2.42872 0.0811379
\(897\) 20.8484 0.696106
\(898\) −29.9575 −0.999695
\(899\) −81.9012 −2.73156
\(900\) −5.84596 −0.194865
\(901\) 21.9952 0.732766
\(902\) 0 0
\(903\) 87.7404 2.91982
\(904\) 9.50553 0.316149
\(905\) 17.1808 0.571109
\(906\) 23.1353 0.768618
\(907\) 49.2582 1.63559 0.817796 0.575508i \(-0.195196\pi\)
0.817796 + 0.575508i \(0.195196\pi\)
\(908\) −13.6499 −0.452987
\(909\) 53.4819 1.77388
\(910\) 14.8180 0.491211
\(911\) 46.4531 1.53906 0.769529 0.638612i \(-0.220491\pi\)
0.769529 + 0.638612i \(0.220491\pi\)
\(912\) 3.01451 0.0998203
\(913\) 0 0
\(914\) −29.4491 −0.974091
\(915\) 56.1165 1.85516
\(916\) −11.3245 −0.374173
\(917\) 20.2491 0.668684
\(918\) −40.1739 −1.32594
\(919\) −5.90424 −0.194763 −0.0973814 0.995247i \(-0.531047\pi\)
−0.0973814 + 0.995247i \(0.531047\pi\)
\(920\) −4.57917 −0.150971
\(921\) 53.0911 1.74941
\(922\) −12.7886 −0.421170
\(923\) 32.6651 1.07518
\(924\) 0 0
\(925\) −1.02572 −0.0337254
\(926\) 22.8327 0.750328
\(927\) −64.1916 −2.10833
\(928\) 8.81144 0.289250
\(929\) 27.7607 0.910799 0.455400 0.890287i \(-0.349496\pi\)
0.455400 + 0.890287i \(0.349496\pi\)
\(930\) 56.3159 1.84667
\(931\) −1.10131 −0.0360939
\(932\) −7.52089 −0.246355
\(933\) −26.4842 −0.867054
\(934\) −23.2585 −0.761041
\(935\) 0 0
\(936\) 18.4783 0.603981
\(937\) −14.2974 −0.467077 −0.233539 0.972348i \(-0.575031\pi\)
−0.233539 + 0.972348i \(0.575031\pi\)
\(938\) 22.3316 0.729153
\(939\) 97.1589 3.17066
\(940\) 18.0892 0.590005
\(941\) −41.9545 −1.36768 −0.683839 0.729633i \(-0.739691\pi\)
−0.683839 + 0.729633i \(0.739691\pi\)
\(942\) 72.3562 2.35749
\(943\) −11.7471 −0.382538
\(944\) −4.79071 −0.155924
\(945\) 45.4294 1.47782
\(946\) 0 0
\(947\) 20.2850 0.659174 0.329587 0.944125i \(-0.393090\pi\)
0.329587 + 0.944125i \(0.393090\pi\)
\(948\) −19.6577 −0.638452
\(949\) 27.1373 0.880913
\(950\) 0.960362 0.0311583
\(951\) 2.27312 0.0737109
\(952\) 10.4842 0.339794
\(953\) −33.3967 −1.08182 −0.540912 0.841079i \(-0.681921\pi\)
−0.540912 + 0.841079i \(0.681921\pi\)
\(954\) −31.0165 −1.00420
\(955\) 2.00923 0.0650171
\(956\) 0.510794 0.0165203
\(957\) 0 0
\(958\) −12.5716 −0.406169
\(959\) 45.7933 1.47874
\(960\) −6.05881 −0.195547
\(961\) 55.3948 1.78693
\(962\) 3.24214 0.104531
\(963\) 46.8439 1.50952
\(964\) 22.6208 0.728566
\(965\) 10.0920 0.324874
\(966\) −16.6805 −0.536687
\(967\) −27.7370 −0.891963 −0.445981 0.895042i \(-0.647145\pi\)
−0.445981 + 0.895042i \(0.647145\pi\)
\(968\) 0 0
\(969\) 13.0129 0.418033
\(970\) −18.9470 −0.608353
\(971\) −6.22298 −0.199705 −0.0998524 0.995002i \(-0.531837\pi\)
−0.0998524 + 0.995002i \(0.531837\pi\)
\(972\) 1.60101 0.0513524
\(973\) 8.80511 0.282279
\(974\) 20.6412 0.661387
\(975\) 8.78802 0.281442
\(976\) −9.26197 −0.296468
\(977\) 26.2137 0.838650 0.419325 0.907836i \(-0.362267\pi\)
0.419325 + 0.907836i \(0.362267\pi\)
\(978\) −0.448137 −0.0143298
\(979\) 0 0
\(980\) 2.21350 0.0707077
\(981\) 86.6379 2.76613
\(982\) −30.2650 −0.965794
\(983\) 16.2834 0.519360 0.259680 0.965695i \(-0.416383\pi\)
0.259680 + 0.965695i \(0.416383\pi\)
\(984\) −15.5429 −0.495489
\(985\) −22.2678 −0.709510
\(986\) 38.0368 1.21134
\(987\) 65.8934 2.09741
\(988\) −3.03557 −0.0965743
\(989\) 27.3037 0.868206
\(990\) 0 0
\(991\) 22.4663 0.713664 0.356832 0.934169i \(-0.383857\pi\)
0.356832 + 0.934169i \(0.383857\pi\)
\(992\) −9.29488 −0.295113
\(993\) 19.0858 0.605669
\(994\) −26.1350 −0.828951
\(995\) 20.9370 0.663747
\(996\) 12.3326 0.390773
\(997\) 19.3207 0.611892 0.305946 0.952049i \(-0.401027\pi\)
0.305946 + 0.952049i \(0.401027\pi\)
\(998\) 36.2503 1.14748
\(999\) 9.93985 0.314483
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 4598.2.a.bx.1.8 8
11.2 odd 10 418.2.f.f.191.4 16
11.6 odd 10 418.2.f.f.267.4 yes 16
11.10 odd 2 4598.2.a.ca.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.f.191.4 16 11.2 odd 10
418.2.f.f.267.4 yes 16 11.6 odd 10
4598.2.a.bx.1.8 8 1.1 even 1 trivial
4598.2.a.ca.1.8 8 11.10 odd 2