Properties

Label 4598.2.a.ca.1.8
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
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: \(0\)
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 \(-0.348213\pi\)
0.458985 + 0.888444i \(0.348213\pi\)
\(8\) 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 \(-0.361694\pi\)
0.420957 + 0.907080i \(0.361694\pi\)
\(14\) 2.42872 0.649103
\(15\) −6.05881 −1.56438
\(16\) 1.00000 0.250000
\(17\) −4.31675 −1.04697 −0.523483 0.852036i \(-0.675368\pi\)
−0.523483 + 0.852036i \(0.675368\pi\)
\(18\) 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 \(-0.195016\pi\)
0.818121 + 0.575046i \(0.195016\pi\)
\(30\) −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 \(-0.631900\pi\)
−0.402618 + 0.915368i \(0.631900\pi\)
\(42\) 7.32140 1.12972
\(43\) 11.9841 1.82756 0.913779 0.406212i \(-0.133151\pi\)
0.913779 + 0.406212i \(0.133151\pi\)
\(44\) 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 \(-0.297968\pi\)
0.592937 + 0.805249i \(0.297968\pi\)
\(62\) 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 \(-0.324753\pi\)
0.523161 + 0.852234i \(0.324753\pi\)
\(74\) 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 \(-0.380441\pi\)
0.366837 + 0.930285i \(0.380441\pi\)
\(80\) −2.00989 −0.224712
\(81\) 9.79285 1.08809
\(82\) −5.15603 −0.569388
\(83\) −4.09108 −0.449054 −0.224527 0.974468i \(-0.572084\pi\)
−0.224527 + 0.974468i \(0.572084\pi\)
\(84\) 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 \(-0.643999\pi\)
−0.437114 + 0.899406i \(0.643999\pi\)
\(102\) −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 \(-0.621318\pi\)
−0.371972 + 0.928244i \(0.621318\pi\)
\(108\) 9.30653 0.895521
\(109\) −14.2327 −1.36324 −0.681622 0.731705i \(-0.738725\pi\)
−0.681622 + 0.731705i \(0.738725\pi\)
\(110\) 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 \(-0.647257\pi\)
−0.446296 + 0.894885i \(0.647257\pi\)
\(128\) 1.00000 0.0883883
\(129\) 36.1261 3.18073
\(130\) −6.10114 −0.535105
\(131\) 8.33735 0.728437 0.364219 0.931313i \(-0.381336\pi\)
0.364219 + 0.931313i \(0.381336\pi\)
\(132\) 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 \(-0.450864\pi\)
0.153752 + 0.988110i \(0.450864\pi\)
\(140\) −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 \(-0.428718\pi\)
0.222071 + 0.975030i \(0.428718\pi\)
\(150\) −2.89502 −0.236377
\(151\) 7.67465 0.624554 0.312277 0.949991i \(-0.398908\pi\)
0.312277 + 0.949991i \(0.398908\pi\)
\(152\) −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 \(-0.493819\pi\)
0.0194159 + 0.999811i \(0.493819\pi\)
\(168\) 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 \(-0.470108\pi\)
0.0937701 + 0.995594i \(0.470108\pi\)
\(174\) 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 \(-0.442158\pi\)
0.180717 + 0.983535i \(0.442158\pi\)
\(194\) −9.42693 −0.676814
\(195\) −18.3919 −1.31707
\(196\) −1.10131 −0.0786648
\(197\) −11.0791 −0.789355 −0.394677 0.918820i \(-0.629144\pi\)
−0.394677 + 0.918820i \(0.629144\pi\)
\(198\) 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 \(-0.239104\pi\)
0.730894 + 0.682491i \(0.239104\pi\)
\(212\) 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 \(-0.350359\pi\)
0.452987 + 0.891517i \(0.350359\pi\)
\(228\) −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 \(-0.420767\pi\)
0.246355 + 0.969180i \(0.420767\pi\)
\(234\) 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 \(-0.505259\pi\)
−0.0165203 + 0.999864i \(0.505259\pi\)
\(240\) −6.05881 −0.391095
\(241\) −22.6208 −1.45713 −0.728566 0.684975i \(-0.759813\pi\)
−0.728566 + 0.684975i \(0.759813\pi\)
\(242\) 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 \(-0.514762\pi\)
−0.0463606 + 0.998925i \(0.514762\pi\)
\(264\) 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 \(-0.481111\pi\)
0.0593056 + 0.998240i \(0.481111\pi\)
\(272\) −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 \(-0.0684648\pi\)
0.976957 + 0.213434i \(0.0684648\pi\)
\(278\) 3.62541 0.217438
\(279\) 56.5802 3.38737
\(280\) −4.88145 −0.291723
\(281\) 13.9545 0.832454 0.416227 0.909261i \(-0.363352\pi\)
0.416227 + 0.909261i \(0.363352\pi\)
\(282\) −27.1309 −1.61562
\(283\) −22.5828 −1.34241 −0.671205 0.741272i \(-0.734223\pi\)
−0.671205 + 0.741272i \(0.734223\pi\)
\(284\) −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 \(-0.223697\pi\)
0.763058 + 0.646330i \(0.223697\pi\)
\(294\) −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 \(-0.667616\pi\)
−0.502581 + 0.864530i \(0.667616\pi\)
\(308\) 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 \(-0.420192\pi\)
0.248105 + 0.968733i \(0.420192\pi\)
\(338\) −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 \(-0.506928\pi\)
−0.0217622 + 0.999763i \(0.506928\pi\)
\(348\) 26.5621 1.42388
\(349\) −25.9076 −1.38680 −0.693401 0.720552i \(-0.743889\pi\)
−0.693401 + 0.720552i \(0.743889\pi\)
\(350\) −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 \(-0.852225\pi\)
−0.894158 + 0.447751i \(0.852225\pi\)
\(360\) −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 \(-0.398232\pi\)
0.314294 + 0.949326i \(0.398232\pi\)
\(374\) 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 \(-0.555460\pi\)
−0.173351 + 0.984860i \(0.555460\pi\)
\(410\) 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 \(-0.894461\pi\)
−0.945536 + 0.325518i \(0.894461\pi\)
\(432\) 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 \(-0.662416\pi\)
−0.488391 + 0.872625i \(0.662416\pi\)
\(440\) 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 \(-0.741856\pi\)
−0.688786 + 0.724964i \(0.741856\pi\)
\(458\) −11.3245 −0.529161
\(459\) −40.1739 −1.87516
\(460\) 4.57917 0.213505
\(461\) −12.7886 −0.595624 −0.297812 0.954624i \(-0.596257\pi\)
−0.297812 + 0.954624i \(0.596257\pi\)
\(462\) 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 \(-0.592726\pi\)
−0.287205 + 0.957869i \(0.592726\pi\)
\(480\) −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 \(-0.739290\pi\)
−0.682919 + 0.730494i \(0.739290\pi\)
\(492\) −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 \(-0.395946\pi\)
0.321104 + 0.947044i \(0.395946\pi\)
\(504\) 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 \(-0.235696\pi\)
0.738157 + 0.674629i \(0.235696\pi\)
\(524\) 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 \(-0.559461\pi\)
−0.185718 + 0.982603i \(0.559461\pi\)
\(542\) 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 \(-0.637891\pi\)
−0.419776 + 0.907628i \(0.637891\pi\)
\(548\) −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 \(-0.853303\pi\)
−0.895669 + 0.444722i \(0.853303\pi\)
\(558\) 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 \(-0.390979\pi\)
0.335844 + 0.941918i \(0.390979\pi\)
\(564\) −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 \(-0.648289\pi\)
−0.449193 + 0.893435i \(0.648289\pi\)
\(570\) 6.05881 0.253776
\(571\) 21.2358 0.888691 0.444346 0.895855i \(-0.353436\pi\)
0.444346 + 0.895855i \(0.353436\pi\)
\(572\) 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 \(-0.589252\pi\)
−0.276735 + 0.960946i \(0.589252\pi\)
\(594\) 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 \(-0.444609\pi\)
0.173138 + 0.984898i \(0.444609\pi\)
\(602\) 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 \(-0.530747\pi\)
−0.0964457 + 0.995338i \(0.530747\pi\)
\(608\) −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 \(-0.297823\pi\)
0.593304 + 0.804979i \(0.297823\pi\)
\(614\) −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 \(-0.522829\pi\)
−0.0716582 + 0.997429i \(0.522829\pi\)
\(660\) 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 \(-0.413281\pi\)
0.269078 + 0.963118i \(0.413281\pi\)
\(674\) 9.10922 0.350874
\(675\) −8.93764 −0.344010
\(676\) −3.78533 −0.145590
\(677\) −9.42523 −0.362241 −0.181121 0.983461i \(-0.557972\pi\)
−0.181121 + 0.983461i \(0.557972\pi\)
\(678\) −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 \(-0.698327\pi\)
−0.583524 + 0.812096i \(0.698327\pi\)
\(702\) 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 \(-0.271459\pi\)
0.657867 + 0.753134i \(0.271459\pi\)
\(734\) −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 \(-0.246951\pi\)
0.713848 + 0.700300i \(0.246951\pi\)
\(740\) −2.14666 −0.0789128
\(741\) −9.15074 −0.336161
\(742\) 12.3751 0.454305
\(743\) −20.4994 −0.752051 −0.376025 0.926609i \(-0.622709\pi\)
−0.376025 + 0.926609i \(0.622709\pi\)
\(744\) 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 \(-0.411916\pi\)
0.273205 + 0.961956i \(0.411916\pi\)
\(762\) −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 \(-0.685072\pi\)
−0.549212 + 0.835683i \(0.685072\pi\)
\(770\) 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 \(-0.534742\pi\)
−0.108929 + 0.994050i \(0.534742\pi\)
\(788\) −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 \(-0.744153\pi\)
−0.693999 + 0.719976i \(0.744153\pi\)
\(810\) −19.6825 −0.691572
\(811\) 4.05599 0.142425 0.0712125 0.997461i \(-0.477313\pi\)
0.0712125 + 0.997461i \(0.477313\pi\)
\(812\) 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 \(-0.471548\pi\)
0.0892650 + 0.996008i \(0.471548\pi\)
\(822\) −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 \(-0.351460\pi\)
0.449900 + 0.893079i \(0.351460\pi\)
\(828\) −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 \(-0.310197\pi\)
0.561570 + 0.827429i \(0.310197\pi\)
\(854\) 22.4948 0.769755
\(855\) 12.2347 0.418417
\(856\) −7.69542 −0.263024
\(857\) −10.5746 −0.361222 −0.180611 0.983555i \(-0.557807\pi\)
−0.180611 + 0.983555i \(0.557807\pi\)
\(858\) 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 \(-0.498484\pi\)
0.00476205 + 0.999989i \(0.498484\pi\)
\(878\) −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 \(-0.448032\pi\)
0.162538 + 0.986702i \(0.448032\pi\)
\(888\) 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 \(-0.468953\pi\)
0.0973814 + 0.995247i \(0.468953\pi\)
\(920\) 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 \(-0.424969\pi\)
0.233539 + 0.972348i \(0.424969\pi\)
\(938\) 22.3316 0.729153
\(939\) 97.1589 3.17066
\(940\) 18.0892 0.590005
\(941\) 41.9545 1.36768 0.683839 0.729633i \(-0.260309\pi\)
0.683839 + 0.729633i \(0.260309\pi\)
\(942\) −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 \(-0.318079\pi\)
0.540912 + 0.841079i \(0.318079\pi\)
\(954\) 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 \(-0.352855\pi\)
0.445981 + 0.895042i \(0.352855\pi\)
\(968\) 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 \(-0.598973\pi\)
−0.305946 + 0.952049i \(0.598973\pi\)
\(998\) −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.ca.1.8 8
11.5 even 5 418.2.f.f.267.4 yes 16
11.9 even 5 418.2.f.f.191.4 16
11.10 odd 2 4598.2.a.bx.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.f.191.4 16 11.9 even 5
418.2.f.f.267.4 yes 16 11.5 even 5
4598.2.a.bx.1.8 8 11.10 odd 2
4598.2.a.ca.1.8 8 1.1 even 1 trivial