Properties

Label 3234.2.a.bg.1.2
Level 3234
Weight 2
Character 3234.1
Self dual yes
Analytic conductor 25.824
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.454904 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.454904 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.454904 q^{10} +1.00000 q^{11} -1.00000 q^{12} -0.909808 q^{13} +0.454904 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -3.88325 q^{19} -0.454904 q^{20} +1.00000 q^{22} +8.33816 q^{23} -1.00000 q^{24} -4.79306 q^{25} -0.909808 q^{26} -1.00000 q^{27} -2.79306 q^{29} +0.454904 q^{30} +9.58612 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +7.88325 q^{37} -3.88325 q^{38} +0.909808 q^{39} -0.454904 q^{40} +1.79306 q^{41} +7.88325 q^{43} +1.00000 q^{44} -0.454904 q^{45} +8.33816 q^{46} -0.338158 q^{47} -1.00000 q^{48} -4.79306 q^{50} +1.00000 q^{51} -0.909808 q^{52} +0.909808 q^{53} -1.00000 q^{54} -0.454904 q^{55} +3.88325 q^{57} -2.79306 q^{58} -6.97345 q^{59} +0.454904 q^{60} +14.0410 q^{61} +9.58612 q^{62} +1.00000 q^{64} +0.413875 q^{65} -1.00000 q^{66} -3.79306 q^{67} -1.00000 q^{68} -8.33816 q^{69} -4.79306 q^{71} +1.00000 q^{72} -10.6763 q^{73} +7.88325 q^{74} +4.79306 q^{75} -3.88325 q^{76} +0.909808 q^{78} +5.36471 q^{79} -0.454904 q^{80} +1.00000 q^{81} +1.79306 q^{82} -1.97345 q^{83} +0.454904 q^{85} +7.88325 q^{86} +2.79306 q^{87} +1.00000 q^{88} +11.0902 q^{89} -0.454904 q^{90} +8.33816 q^{92} -9.58612 q^{93} -0.338158 q^{94} +1.76651 q^{95} -1.00000 q^{96} -14.5861 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + 3q^{16} - 3q^{17} + 3q^{18} + 3q^{19} + 3q^{22} + 9q^{23} - 3q^{24} + 3q^{25} - 3q^{27} + 9q^{29} - 6q^{31} + 3q^{32} - 3q^{33} - 3q^{34} + 3q^{36} + 9q^{37} + 3q^{38} - 12q^{41} + 9q^{43} + 3q^{44} + 9q^{46} + 15q^{47} - 3q^{48} + 3q^{50} + 3q^{51} - 3q^{54} - 3q^{57} + 9q^{58} - 9q^{59} + 6q^{61} - 6q^{62} + 3q^{64} + 36q^{65} - 3q^{66} + 6q^{67} - 3q^{68} - 9q^{69} + 3q^{71} + 3q^{72} + 9q^{74} - 3q^{75} + 3q^{76} + 12q^{79} + 3q^{81} - 12q^{82} + 6q^{83} + 9q^{86} - 9q^{87} + 3q^{88} + 36q^{89} + 9q^{92} + 6q^{93} + 15q^{94} - 24q^{95} - 3q^{96} - 9q^{97} + 3q^{99} + O(q^{100}) \)

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
\(5\) −0.454904 −0.203439 −0.101720 0.994813i \(-0.532434\pi\)
−0.101720 + 0.994813i \(0.532434\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.454904 −0.143853
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −0.909808 −0.252335 −0.126168 0.992009i \(-0.540268\pi\)
−0.126168 + 0.992009i \(0.540268\pi\)
\(14\) 0 0
\(15\) 0.454904 0.117456
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.88325 −0.890880 −0.445440 0.895312i \(-0.646953\pi\)
−0.445440 + 0.895312i \(0.646953\pi\)
\(20\) −0.454904 −0.101720
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.33816 1.73863 0.869313 0.494262i \(-0.164562\pi\)
0.869313 + 0.494262i \(0.164562\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.79306 −0.958612
\(26\) −0.909808 −0.178428
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.79306 −0.518659 −0.259329 0.965789i \(-0.583502\pi\)
−0.259329 + 0.965789i \(0.583502\pi\)
\(30\) 0.454904 0.0830537
\(31\) 9.58612 1.72172 0.860859 0.508843i \(-0.169927\pi\)
0.860859 + 0.508843i \(0.169927\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.88325 1.29600 0.647999 0.761641i \(-0.275606\pi\)
0.647999 + 0.761641i \(0.275606\pi\)
\(38\) −3.88325 −0.629947
\(39\) 0.909808 0.145686
\(40\) −0.454904 −0.0719267
\(41\) 1.79306 0.280029 0.140015 0.990149i \(-0.455285\pi\)
0.140015 + 0.990149i \(0.455285\pi\)
\(42\) 0 0
\(43\) 7.88325 1.20218 0.601092 0.799179i \(-0.294732\pi\)
0.601092 + 0.799179i \(0.294732\pi\)
\(44\) 1.00000 0.150756
\(45\) −0.454904 −0.0678131
\(46\) 8.33816 1.22939
\(47\) −0.338158 −0.0493254 −0.0246627 0.999696i \(-0.507851\pi\)
−0.0246627 + 0.999696i \(0.507851\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.79306 −0.677841
\(51\) 1.00000 0.140028
\(52\) −0.909808 −0.126168
\(53\) 0.909808 0.124972 0.0624859 0.998046i \(-0.480097\pi\)
0.0624859 + 0.998046i \(0.480097\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.454904 −0.0613393
\(56\) 0 0
\(57\) 3.88325 0.514350
\(58\) −2.79306 −0.366747
\(59\) −6.97345 −0.907865 −0.453933 0.891036i \(-0.649979\pi\)
−0.453933 + 0.891036i \(0.649979\pi\)
\(60\) 0.454904 0.0587279
\(61\) 14.0410 1.79777 0.898885 0.438185i \(-0.144379\pi\)
0.898885 + 0.438185i \(0.144379\pi\)
\(62\) 9.58612 1.21744
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.413875 0.0513349
\(66\) −1.00000 −0.123091
\(67\) −3.79306 −0.463396 −0.231698 0.972788i \(-0.574428\pi\)
−0.231698 + 0.972788i \(0.574428\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.33816 −1.00380
\(70\) 0 0
\(71\) −4.79306 −0.568832 −0.284416 0.958701i \(-0.591800\pi\)
−0.284416 + 0.958701i \(0.591800\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.6763 −1.24957 −0.624784 0.780798i \(-0.714813\pi\)
−0.624784 + 0.780798i \(0.714813\pi\)
\(74\) 7.88325 0.916410
\(75\) 4.79306 0.553455
\(76\) −3.88325 −0.445440
\(77\) 0 0
\(78\) 0.909808 0.103015
\(79\) 5.36471 0.603577 0.301789 0.953375i \(-0.402416\pi\)
0.301789 + 0.953375i \(0.402416\pi\)
\(80\) −0.454904 −0.0508598
\(81\) 1.00000 0.111111
\(82\) 1.79306 0.198011
\(83\) −1.97345 −0.216614 −0.108307 0.994118i \(-0.534543\pi\)
−0.108307 + 0.994118i \(0.534543\pi\)
\(84\) 0 0
\(85\) 0.454904 0.0493413
\(86\) 7.88325 0.850073
\(87\) 2.79306 0.299448
\(88\) 1.00000 0.106600
\(89\) 11.0902 1.17556 0.587779 0.809022i \(-0.300002\pi\)
0.587779 + 0.809022i \(0.300002\pi\)
\(90\) −0.454904 −0.0479511
\(91\) 0 0
\(92\) 8.33816 0.869313
\(93\) −9.58612 −0.994035
\(94\) −0.338158 −0.0348784
\(95\) 1.76651 0.181240
\(96\) −1.00000 −0.102062
\(97\) −14.5861 −1.48100 −0.740498 0.672058i \(-0.765410\pi\)
−0.740498 + 0.672058i \(0.765410\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −4.79306 −0.479306
\(101\) 8.06364 0.802362 0.401181 0.915999i \(-0.368600\pi\)
0.401181 + 0.915999i \(0.368600\pi\)
\(102\) 1.00000 0.0990148
\(103\) −5.76651 −0.568191 −0.284095 0.958796i \(-0.591693\pi\)
−0.284095 + 0.958796i \(0.591693\pi\)
\(104\) −0.909808 −0.0892140
\(105\) 0 0
\(106\) 0.909808 0.0883684
\(107\) 15.6127 1.50933 0.754667 0.656108i \(-0.227798\pi\)
0.754667 + 0.656108i \(0.227798\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0410 1.34489 0.672443 0.740149i \(-0.265245\pi\)
0.672443 + 0.740149i \(0.265245\pi\)
\(110\) −0.454904 −0.0433734
\(111\) −7.88325 −0.748245
\(112\) 0 0
\(113\) 7.58612 0.713643 0.356821 0.934173i \(-0.383861\pi\)
0.356821 + 0.934173i \(0.383861\pi\)
\(114\) 3.88325 0.363700
\(115\) −3.79306 −0.353705
\(116\) −2.79306 −0.259329
\(117\) −0.909808 −0.0841118
\(118\) −6.97345 −0.641958
\(119\) 0 0
\(120\) 0.454904 0.0415269
\(121\) 1.00000 0.0909091
\(122\) 14.0410 1.27122
\(123\) −1.79306 −0.161675
\(124\) 9.58612 0.860859
\(125\) 4.45490 0.398459
\(126\) 0 0
\(127\) 7.42835 0.659159 0.329580 0.944128i \(-0.393093\pi\)
0.329580 + 0.944128i \(0.393093\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.88325 −0.694082
\(130\) 0.413875 0.0362993
\(131\) 19.4057 1.69549 0.847744 0.530406i \(-0.177961\pi\)
0.847744 + 0.530406i \(0.177961\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −3.79306 −0.327671
\(135\) 0.454904 0.0391519
\(136\) −1.00000 −0.0857493
\(137\) 0.909808 0.0777302 0.0388651 0.999244i \(-0.487626\pi\)
0.0388651 + 0.999244i \(0.487626\pi\)
\(138\) −8.33816 −0.709791
\(139\) 8.37919 0.710713 0.355357 0.934731i \(-0.384359\pi\)
0.355357 + 0.934731i \(0.384359\pi\)
\(140\) 0 0
\(141\) 0.338158 0.0284781
\(142\) −4.79306 −0.402225
\(143\) −0.909808 −0.0760820
\(144\) 1.00000 0.0833333
\(145\) 1.27058 0.105516
\(146\) −10.6763 −0.883578
\(147\) 0 0
\(148\) 7.88325 0.647999
\(149\) 4.11675 0.337257 0.168628 0.985680i \(-0.446066\pi\)
0.168628 + 0.985680i \(0.446066\pi\)
\(150\) 4.79306 0.391352
\(151\) −6.15777 −0.501113 −0.250556 0.968102i \(-0.580614\pi\)
−0.250556 + 0.968102i \(0.580614\pi\)
\(152\) −3.88325 −0.314973
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −4.36077 −0.350265
\(156\) 0.909808 0.0728430
\(157\) 17.6498 1.40860 0.704302 0.709900i \(-0.251260\pi\)
0.704302 + 0.709900i \(0.251260\pi\)
\(158\) 5.36471 0.426794
\(159\) −0.909808 −0.0721525
\(160\) −0.454904 −0.0359633
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −11.6127 −0.909575 −0.454788 0.890600i \(-0.650285\pi\)
−0.454788 + 0.890600i \(0.650285\pi\)
\(164\) 1.79306 0.140015
\(165\) 0.454904 0.0354142
\(166\) −1.97345 −0.153169
\(167\) 4.85670 0.375823 0.187911 0.982186i \(-0.439828\pi\)
0.187911 + 0.982186i \(0.439828\pi\)
\(168\) 0 0
\(169\) −12.1722 −0.936327
\(170\) 0.454904 0.0348896
\(171\) −3.88325 −0.296960
\(172\) 7.88325 0.601092
\(173\) 25.5861 1.94528 0.972639 0.232324i \(-0.0746329\pi\)
0.972639 + 0.232324i \(0.0746329\pi\)
\(174\) 2.79306 0.211742
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 6.97345 0.524156
\(178\) 11.0902 0.831245
\(179\) −15.7029 −1.17369 −0.586844 0.809700i \(-0.699630\pi\)
−0.586844 + 0.809700i \(0.699630\pi\)
\(180\) −0.454904 −0.0339065
\(181\) 2.49593 0.185521 0.0927606 0.995688i \(-0.470431\pi\)
0.0927606 + 0.995688i \(0.470431\pi\)
\(182\) 0 0
\(183\) −14.0410 −1.03794
\(184\) 8.33816 0.614697
\(185\) −3.58612 −0.263657
\(186\) −9.58612 −0.702889
\(187\) −1.00000 −0.0731272
\(188\) −0.338158 −0.0246627
\(189\) 0 0
\(190\) 1.76651 0.128156
\(191\) 9.81962 0.710523 0.355261 0.934767i \(-0.384392\pi\)
0.355261 + 0.934767i \(0.384392\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.85670 0.493556 0.246778 0.969072i \(-0.420628\pi\)
0.246778 + 0.969072i \(0.420628\pi\)
\(194\) −14.5861 −1.04722
\(195\) −0.413875 −0.0296382
\(196\) 0 0
\(197\) 23.4694 1.67212 0.836062 0.548635i \(-0.184852\pi\)
0.836062 + 0.548635i \(0.184852\pi\)
\(198\) 1.00000 0.0710669
\(199\) 2.23349 0.158328 0.0791640 0.996862i \(-0.474775\pi\)
0.0791640 + 0.996862i \(0.474775\pi\)
\(200\) −4.79306 −0.338921
\(201\) 3.79306 0.267542
\(202\) 8.06364 0.567356
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −0.815671 −0.0569690
\(206\) −5.76651 −0.401772
\(207\) 8.33816 0.579542
\(208\) −0.909808 −0.0630838
\(209\) −3.88325 −0.268610
\(210\) 0 0
\(211\) −18.2624 −1.25724 −0.628619 0.777713i \(-0.716380\pi\)
−0.628619 + 0.777713i \(0.716380\pi\)
\(212\) 0.909808 0.0624859
\(213\) 4.79306 0.328415
\(214\) 15.6127 1.06726
\(215\) −3.58612 −0.244572
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0410 0.950978
\(219\) 10.6763 0.721439
\(220\) −0.454904 −0.0306696
\(221\) 0.909808 0.0612003
\(222\) −7.88325 −0.529089
\(223\) −12.4959 −0.836790 −0.418395 0.908265i \(-0.637407\pi\)
−0.418395 + 0.908265i \(0.637407\pi\)
\(224\) 0 0
\(225\) −4.79306 −0.319537
\(226\) 7.58612 0.504621
\(227\) −9.79306 −0.649988 −0.324994 0.945716i \(-0.605362\pi\)
−0.324994 + 0.945716i \(0.605362\pi\)
\(228\) 3.88325 0.257175
\(229\) −8.67632 −0.573347 −0.286674 0.958028i \(-0.592550\pi\)
−0.286674 + 0.958028i \(0.592550\pi\)
\(230\) −3.79306 −0.250107
\(231\) 0 0
\(232\) −2.79306 −0.183374
\(233\) 16.7665 1.09841 0.549205 0.835688i \(-0.314931\pi\)
0.549205 + 0.835688i \(0.314931\pi\)
\(234\) −0.909808 −0.0594760
\(235\) 0.153830 0.0100347
\(236\) −6.97345 −0.453933
\(237\) −5.36471 −0.348476
\(238\) 0 0
\(239\) −26.2624 −1.69878 −0.849388 0.527769i \(-0.823029\pi\)
−0.849388 + 0.527769i \(0.823029\pi\)
\(240\) 0.454904 0.0293639
\(241\) −7.94689 −0.511904 −0.255952 0.966689i \(-0.582389\pi\)
−0.255952 + 0.966689i \(0.582389\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 14.0410 0.898885
\(245\) 0 0
\(246\) −1.79306 −0.114321
\(247\) 3.53302 0.224800
\(248\) 9.58612 0.608720
\(249\) 1.97345 0.125062
\(250\) 4.45490 0.281753
\(251\) 3.93636 0.248461 0.124230 0.992253i \(-0.460354\pi\)
0.124230 + 0.992253i \(0.460354\pi\)
\(252\) 0 0
\(253\) 8.33816 0.524216
\(254\) 7.42835 0.466096
\(255\) −0.454904 −0.0284872
\(256\) 1.00000 0.0625000
\(257\) 15.0902 0.941300 0.470650 0.882320i \(-0.344019\pi\)
0.470650 + 0.882320i \(0.344019\pi\)
\(258\) −7.88325 −0.490790
\(259\) 0 0
\(260\) 0.413875 0.0256675
\(261\) −2.79306 −0.172886
\(262\) 19.4057 1.19889
\(263\) −28.6232 −1.76498 −0.882491 0.470329i \(-0.844135\pi\)
−0.882491 + 0.470329i \(0.844135\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −0.413875 −0.0254242
\(266\) 0 0
\(267\) −11.0902 −0.678709
\(268\) −3.79306 −0.231698
\(269\) 22.9508 1.39934 0.699669 0.714468i \(-0.253331\pi\)
0.699669 + 0.714468i \(0.253331\pi\)
\(270\) 0.454904 0.0276846
\(271\) −9.81962 −0.596499 −0.298250 0.954488i \(-0.596403\pi\)
−0.298250 + 0.954488i \(0.596403\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 0.909808 0.0549635
\(275\) −4.79306 −0.289033
\(276\) −8.33816 −0.501898
\(277\) 10.7294 0.644669 0.322334 0.946626i \(-0.395532\pi\)
0.322334 + 0.946626i \(0.395532\pi\)
\(278\) 8.37919 0.502550
\(279\) 9.58612 0.573906
\(280\) 0 0
\(281\) 2.58612 0.154275 0.0771376 0.997020i \(-0.475422\pi\)
0.0771376 + 0.997020i \(0.475422\pi\)
\(282\) 0.338158 0.0201370
\(283\) 20.0821 1.19375 0.596877 0.802333i \(-0.296408\pi\)
0.596877 + 0.802333i \(0.296408\pi\)
\(284\) −4.79306 −0.284416
\(285\) −1.76651 −0.104639
\(286\) −0.909808 −0.0537981
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 1.27058 0.0746108
\(291\) 14.5861 0.855054
\(292\) −10.6763 −0.624784
\(293\) −10.3792 −0.606359 −0.303179 0.952934i \(-0.598048\pi\)
−0.303179 + 0.952934i \(0.598048\pi\)
\(294\) 0 0
\(295\) 3.17225 0.184695
\(296\) 7.88325 0.458205
\(297\) −1.00000 −0.0580259
\(298\) 4.11675 0.238477
\(299\) −7.58612 −0.438717
\(300\) 4.79306 0.276728
\(301\) 0 0
\(302\) −6.15777 −0.354340
\(303\) −8.06364 −0.463244
\(304\) −3.88325 −0.222720
\(305\) −6.38732 −0.365737
\(306\) −1.00000 −0.0571662
\(307\) −9.76651 −0.557404 −0.278702 0.960378i \(-0.589904\pi\)
−0.278702 + 0.960378i \(0.589904\pi\)
\(308\) 0 0
\(309\) 5.76651 0.328045
\(310\) −4.36077 −0.247675
\(311\) 6.51854 0.369633 0.184816 0.982773i \(-0.440831\pi\)
0.184816 + 0.982773i \(0.440831\pi\)
\(312\) 0.909808 0.0515077
\(313\) −30.7931 −1.74053 −0.870263 0.492587i \(-0.836051\pi\)
−0.870263 + 0.492587i \(0.836051\pi\)
\(314\) 17.6498 0.996034
\(315\) 0 0
\(316\) 5.36471 0.301789
\(317\) −27.6272 −1.55170 −0.775848 0.630920i \(-0.782678\pi\)
−0.775848 + 0.630920i \(0.782678\pi\)
\(318\) −0.909808 −0.0510195
\(319\) −2.79306 −0.156381
\(320\) −0.454904 −0.0254299
\(321\) −15.6127 −0.871415
\(322\) 0 0
\(323\) 3.88325 0.216070
\(324\) 1.00000 0.0555556
\(325\) 4.36077 0.241892
\(326\) −11.6127 −0.643167
\(327\) −14.0410 −0.776471
\(328\) 1.79306 0.0990053
\(329\) 0 0
\(330\) 0.454904 0.0250416
\(331\) −28.9653 −1.59208 −0.796039 0.605246i \(-0.793075\pi\)
−0.796039 + 0.605246i \(0.793075\pi\)
\(332\) −1.97345 −0.108307
\(333\) 7.88325 0.432000
\(334\) 4.85670 0.265747
\(335\) 1.72548 0.0942730
\(336\) 0 0
\(337\) −9.03708 −0.492281 −0.246141 0.969234i \(-0.579163\pi\)
−0.246141 + 0.969234i \(0.579163\pi\)
\(338\) −12.1722 −0.662083
\(339\) −7.58612 −0.412022
\(340\) 0.454904 0.0246706
\(341\) 9.58612 0.519118
\(342\) −3.88325 −0.209982
\(343\) 0 0
\(344\) 7.88325 0.425037
\(345\) 3.79306 0.204212
\(346\) 25.5861 1.37552
\(347\) −21.6127 −1.16023 −0.580115 0.814535i \(-0.696992\pi\)
−0.580115 + 0.814535i \(0.696992\pi\)
\(348\) 2.79306 0.149724
\(349\) −14.2745 −0.764098 −0.382049 0.924142i \(-0.624781\pi\)
−0.382049 + 0.924142i \(0.624781\pi\)
\(350\) 0 0
\(351\) 0.909808 0.0485620
\(352\) 1.00000 0.0533002
\(353\) 28.4959 1.51669 0.758343 0.651856i \(-0.226009\pi\)
0.758343 + 0.651856i \(0.226009\pi\)
\(354\) 6.97345 0.370634
\(355\) 2.18038 0.115723
\(356\) 11.0902 0.587779
\(357\) 0 0
\(358\) −15.7029 −0.829922
\(359\) −13.6392 −0.719851 −0.359926 0.932981i \(-0.617198\pi\)
−0.359926 + 0.932981i \(0.617198\pi\)
\(360\) −0.454904 −0.0239756
\(361\) −3.92034 −0.206334
\(362\) 2.49593 0.131183
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 4.85670 0.254211
\(366\) −14.0410 −0.733936
\(367\) −22.7294 −1.18647 −0.593233 0.805031i \(-0.702149\pi\)
−0.593233 + 0.805031i \(0.702149\pi\)
\(368\) 8.33816 0.434657
\(369\) 1.79306 0.0933431
\(370\) −3.58612 −0.186434
\(371\) 0 0
\(372\) −9.58612 −0.497017
\(373\) 10.4018 0.538585 0.269292 0.963058i \(-0.413210\pi\)
0.269292 + 0.963058i \(0.413210\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −4.45490 −0.230050
\(376\) −0.338158 −0.0174392
\(377\) 2.54115 0.130876
\(378\) 0 0
\(379\) 7.61268 0.391037 0.195519 0.980700i \(-0.437361\pi\)
0.195519 + 0.980700i \(0.437361\pi\)
\(380\) 1.76651 0.0906200
\(381\) −7.42835 −0.380566
\(382\) 9.81962 0.502415
\(383\) 22.1457 1.13159 0.565796 0.824545i \(-0.308569\pi\)
0.565796 + 0.824545i \(0.308569\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.85670 0.348997
\(387\) 7.88325 0.400728
\(388\) −14.5861 −0.740498
\(389\) −3.54510 −0.179743 −0.0898717 0.995953i \(-0.528646\pi\)
−0.0898717 + 0.995953i \(0.528646\pi\)
\(390\) −0.413875 −0.0209574
\(391\) −8.33816 −0.421679
\(392\) 0 0
\(393\) −19.4057 −0.978890
\(394\) 23.4694 1.18237
\(395\) −2.44043 −0.122791
\(396\) 1.00000 0.0502519
\(397\) 2.55957 0.128461 0.0642306 0.997935i \(-0.479541\pi\)
0.0642306 + 0.997935i \(0.479541\pi\)
\(398\) 2.23349 0.111955
\(399\) 0 0
\(400\) −4.79306 −0.239653
\(401\) 10.4959 0.524142 0.262071 0.965049i \(-0.415595\pi\)
0.262071 + 0.965049i \(0.415595\pi\)
\(402\) 3.79306 0.189181
\(403\) −8.72153 −0.434451
\(404\) 8.06364 0.401181
\(405\) −0.454904 −0.0226044
\(406\) 0 0
\(407\) 7.88325 0.390758
\(408\) 1.00000 0.0495074
\(409\) −15.8486 −0.783661 −0.391831 0.920037i \(-0.628158\pi\)
−0.391831 + 0.920037i \(0.628158\pi\)
\(410\) −0.815671 −0.0402831
\(411\) −0.909808 −0.0448775
\(412\) −5.76651 −0.284095
\(413\) 0 0
\(414\) 8.33816 0.409798
\(415\) 0.897729 0.0440678
\(416\) −0.909808 −0.0446070
\(417\) −8.37919 −0.410331
\(418\) −3.88325 −0.189936
\(419\) 9.52249 0.465204 0.232602 0.972572i \(-0.425276\pi\)
0.232602 + 0.972572i \(0.425276\pi\)
\(420\) 0 0
\(421\) −0.530621 −0.0258609 −0.0129305 0.999916i \(-0.504116\pi\)
−0.0129305 + 0.999916i \(0.504116\pi\)
\(422\) −18.2624 −0.889002
\(423\) −0.338158 −0.0164418
\(424\) 0.909808 0.0441842
\(425\) 4.79306 0.232498
\(426\) 4.79306 0.232225
\(427\) 0 0
\(428\) 15.6127 0.754667
\(429\) 0.909808 0.0439260
\(430\) −3.58612 −0.172938
\(431\) 20.8036 1.00207 0.501037 0.865426i \(-0.332952\pi\)
0.501037 + 0.865426i \(0.332952\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.40574 0.211726 0.105863 0.994381i \(-0.466239\pi\)
0.105863 + 0.994381i \(0.466239\pi\)
\(434\) 0 0
\(435\) −1.27058 −0.0609194
\(436\) 14.0410 0.672443
\(437\) −32.3792 −1.54891
\(438\) 10.6763 0.510134
\(439\) 15.1949 0.725211 0.362606 0.931943i \(-0.381887\pi\)
0.362606 + 0.931943i \(0.381887\pi\)
\(440\) −0.454904 −0.0216867
\(441\) 0 0
\(442\) 0.909808 0.0432752
\(443\) −11.3873 −0.541028 −0.270514 0.962716i \(-0.587194\pi\)
−0.270514 + 0.962716i \(0.587194\pi\)
\(444\) −7.88325 −0.374123
\(445\) −5.04497 −0.239155
\(446\) −12.4959 −0.591700
\(447\) −4.11675 −0.194715
\(448\) 0 0
\(449\) −24.0821 −1.13650 −0.568251 0.822855i \(-0.692380\pi\)
−0.568251 + 0.822855i \(0.692380\pi\)
\(450\) −4.79306 −0.225947
\(451\) 1.79306 0.0844320
\(452\) 7.58612 0.356821
\(453\) 6.15777 0.289317
\(454\) −9.79306 −0.459611
\(455\) 0 0
\(456\) 3.88325 0.181850
\(457\) −29.5330 −1.38150 −0.690748 0.723095i \(-0.742719\pi\)
−0.690748 + 0.723095i \(0.742719\pi\)
\(458\) −8.67632 −0.405418
\(459\) 1.00000 0.0466760
\(460\) −3.79306 −0.176852
\(461\) −4.14569 −0.193084 −0.0965421 0.995329i \(-0.530778\pi\)
−0.0965421 + 0.995329i \(0.530778\pi\)
\(462\) 0 0
\(463\) −9.32368 −0.433308 −0.216654 0.976248i \(-0.569514\pi\)
−0.216654 + 0.976248i \(0.569514\pi\)
\(464\) −2.79306 −0.129665
\(465\) 4.36077 0.202226
\(466\) 16.7665 0.776693
\(467\) −36.5596 −1.69178 −0.845888 0.533361i \(-0.820929\pi\)
−0.845888 + 0.533361i \(0.820929\pi\)
\(468\) −0.909808 −0.0420559
\(469\) 0 0
\(470\) 0.153830 0.00709563
\(471\) −17.6498 −0.813258
\(472\) −6.97345 −0.320979
\(473\) 7.88325 0.362472
\(474\) −5.36471 −0.246409
\(475\) 18.6127 0.854008
\(476\) 0 0
\(477\) 0.909808 0.0416573
\(478\) −26.2624 −1.20122
\(479\) −0.729425 −0.0333283 −0.0166641 0.999861i \(-0.505305\pi\)
−0.0166641 + 0.999861i \(0.505305\pi\)
\(480\) 0.454904 0.0207634
\(481\) −7.17225 −0.327026
\(482\) −7.94689 −0.361971
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 6.63529 0.301293
\(486\) −1.00000 −0.0453609
\(487\) −36.8567 −1.67014 −0.835068 0.550146i \(-0.814572\pi\)
−0.835068 + 0.550146i \(0.814572\pi\)
\(488\) 14.0410 0.635608
\(489\) 11.6127 0.525143
\(490\) 0 0
\(491\) 25.5065 1.15109 0.575545 0.817770i \(-0.304790\pi\)
0.575545 + 0.817770i \(0.304790\pi\)
\(492\) −1.79306 −0.0808375
\(493\) 2.79306 0.125793
\(494\) 3.53302 0.158958
\(495\) −0.454904 −0.0204464
\(496\) 9.58612 0.430430
\(497\) 0 0
\(498\) 1.97345 0.0884322
\(499\) −4.12728 −0.184762 −0.0923811 0.995724i \(-0.529448\pi\)
−0.0923811 + 0.995724i \(0.529448\pi\)
\(500\) 4.45490 0.199229
\(501\) −4.85670 −0.216981
\(502\) 3.93636 0.175688
\(503\) 6.85670 0.305725 0.152863 0.988247i \(-0.451151\pi\)
0.152863 + 0.988247i \(0.451151\pi\)
\(504\) 0 0
\(505\) −3.66818 −0.163232
\(506\) 8.33816 0.370676
\(507\) 12.1722 0.540589
\(508\) 7.42835 0.329580
\(509\) −16.4428 −0.728815 −0.364408 0.931240i \(-0.618729\pi\)
−0.364408 + 0.931240i \(0.618729\pi\)
\(510\) −0.454904 −0.0201435
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.88325 0.171450
\(514\) 15.0902 0.665600
\(515\) 2.62321 0.115592
\(516\) −7.88325 −0.347041
\(517\) −0.338158 −0.0148722
\(518\) 0 0
\(519\) −25.5861 −1.12311
\(520\) 0.413875 0.0181496
\(521\) 5.63923 0.247059 0.123530 0.992341i \(-0.460579\pi\)
0.123530 + 0.992341i \(0.460579\pi\)
\(522\) −2.79306 −0.122249
\(523\) 4.08206 0.178496 0.0892480 0.996009i \(-0.471554\pi\)
0.0892480 + 0.996009i \(0.471554\pi\)
\(524\) 19.4057 0.847744
\(525\) 0 0
\(526\) −28.6232 −1.24803
\(527\) −9.58612 −0.417578
\(528\) −1.00000 −0.0435194
\(529\) 46.5249 2.02282
\(530\) −0.413875 −0.0179776
\(531\) −6.97345 −0.302622
\(532\) 0 0
\(533\) −1.63134 −0.0706613
\(534\) −11.0902 −0.479920
\(535\) −7.10227 −0.307058
\(536\) −3.79306 −0.163835
\(537\) 15.7029 0.677629
\(538\) 22.9508 0.989481
\(539\) 0 0
\(540\) 0.454904 0.0195760
\(541\) −4.89773 −0.210570 −0.105285 0.994442i \(-0.533575\pi\)
−0.105285 + 0.994442i \(0.533575\pi\)
\(542\) −9.81962 −0.421789
\(543\) −2.49593 −0.107111
\(544\) −1.00000 −0.0428746
\(545\) −6.38732 −0.273603
\(546\) 0 0
\(547\) 21.4694 0.917964 0.458982 0.888445i \(-0.348214\pi\)
0.458982 + 0.888445i \(0.348214\pi\)
\(548\) 0.909808 0.0388651
\(549\) 14.0410 0.599256
\(550\) −4.79306 −0.204377
\(551\) 10.8462 0.462062
\(552\) −8.33816 −0.354896
\(553\) 0 0
\(554\) 10.7294 0.455850
\(555\) 3.58612 0.152223
\(556\) 8.37919 0.355357
\(557\) −39.4694 −1.67237 −0.836186 0.548447i \(-0.815219\pi\)
−0.836186 + 0.548447i \(0.815219\pi\)
\(558\) 9.58612 0.405813
\(559\) −7.17225 −0.303354
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) 2.58612 0.109089
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0.338158 0.0142390
\(565\) −3.45096 −0.145183
\(566\) 20.0821 0.844112
\(567\) 0 0
\(568\) −4.79306 −0.201112
\(569\) 34.1988 1.43369 0.716844 0.697233i \(-0.245586\pi\)
0.716844 + 0.697233i \(0.245586\pi\)
\(570\) −1.76651 −0.0739909
\(571\) 14.7931 0.619070 0.309535 0.950888i \(-0.399827\pi\)
0.309535 + 0.950888i \(0.399827\pi\)
\(572\) −0.909808 −0.0380410
\(573\) −9.81962 −0.410220
\(574\) 0 0
\(575\) −39.9653 −1.66667
\(576\) 1.00000 0.0416667
\(577\) 7.19880 0.299690 0.149845 0.988709i \(-0.452123\pi\)
0.149845 + 0.988709i \(0.452123\pi\)
\(578\) −16.0000 −0.665512
\(579\) −6.85670 −0.284955
\(580\) 1.27058 0.0527578
\(581\) 0 0
\(582\) 14.5861 0.604614
\(583\) 0.909808 0.0376804
\(584\) −10.6763 −0.441789
\(585\) 0.413875 0.0171116
\(586\) −10.3792 −0.428760
\(587\) 44.4959 1.83654 0.918272 0.395951i \(-0.129585\pi\)
0.918272 + 0.395951i \(0.129585\pi\)
\(588\) 0 0
\(589\) −37.2254 −1.53384
\(590\) 3.17225 0.130599
\(591\) −23.4694 −0.965401
\(592\) 7.88325 0.324000
\(593\) −17.9653 −0.737747 −0.368873 0.929480i \(-0.620256\pi\)
−0.368873 + 0.929480i \(0.620256\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 4.11675 0.168628
\(597\) −2.23349 −0.0914107
\(598\) −7.58612 −0.310220
\(599\) −25.8075 −1.05447 −0.527234 0.849720i \(-0.676771\pi\)
−0.527234 + 0.849720i \(0.676771\pi\)
\(600\) 4.79306 0.195676
\(601\) −10.4139 −0.424791 −0.212395 0.977184i \(-0.568126\pi\)
−0.212395 + 0.977184i \(0.568126\pi\)
\(602\) 0 0
\(603\) −3.79306 −0.154465
\(604\) −6.15777 −0.250556
\(605\) −0.454904 −0.0184945
\(606\) −8.06364 −0.327563
\(607\) 8.89773 0.361148 0.180574 0.983561i \(-0.442205\pi\)
0.180574 + 0.983561i \(0.442205\pi\)
\(608\) −3.88325 −0.157487
\(609\) 0 0
\(610\) −6.38732 −0.258615
\(611\) 0.307659 0.0124466
\(612\) −1.00000 −0.0404226
\(613\) −23.6272 −0.954292 −0.477146 0.878824i \(-0.658329\pi\)
−0.477146 + 0.878824i \(0.658329\pi\)
\(614\) −9.76651 −0.394144
\(615\) 0.815671 0.0328910
\(616\) 0 0
\(617\) −35.8486 −1.44321 −0.721604 0.692306i \(-0.756595\pi\)
−0.721604 + 0.692306i \(0.756595\pi\)
\(618\) 5.76651 0.231963
\(619\) −27.9203 −1.12221 −0.561107 0.827744i \(-0.689624\pi\)
−0.561107 + 0.827744i \(0.689624\pi\)
\(620\) −4.36077 −0.175133
\(621\) −8.33816 −0.334599
\(622\) 6.51854 0.261370
\(623\) 0 0
\(624\) 0.909808 0.0364215
\(625\) 21.9388 0.877550
\(626\) −30.7931 −1.23074
\(627\) 3.88325 0.155082
\(628\) 17.6498 0.704302
\(629\) −7.88325 −0.314326
\(630\) 0 0
\(631\) 43.6682 1.73840 0.869201 0.494458i \(-0.164633\pi\)
0.869201 + 0.494458i \(0.164633\pi\)
\(632\) 5.36471 0.213397
\(633\) 18.2624 0.725867
\(634\) −27.6272 −1.09721
\(635\) −3.37919 −0.134099
\(636\) −0.909808 −0.0360762
\(637\) 0 0
\(638\) −2.79306 −0.110578
\(639\) −4.79306 −0.189611
\(640\) −0.454904 −0.0179817
\(641\) −19.7665 −0.780730 −0.390365 0.920660i \(-0.627651\pi\)
−0.390365 + 0.920660i \(0.627651\pi\)
\(642\) −15.6127 −0.616183
\(643\) −24.7584 −0.976375 −0.488187 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(644\) 0 0
\(645\) 3.58612 0.141204
\(646\) 3.88325 0.152785
\(647\) −31.0781 −1.22181 −0.610903 0.791705i \(-0.709194\pi\)
−0.610903 + 0.791705i \(0.709194\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.97345 −0.273732
\(650\) 4.36077 0.171043
\(651\) 0 0
\(652\) −11.6127 −0.454788
\(653\) 22.7173 0.888998 0.444499 0.895779i \(-0.353382\pi\)
0.444499 + 0.895779i \(0.353382\pi\)
\(654\) −14.0410 −0.549048
\(655\) −8.82775 −0.344929
\(656\) 1.79306 0.0700073
\(657\) −10.6763 −0.416523
\(658\) 0 0
\(659\) 43.6127 1.69891 0.849454 0.527662i \(-0.176931\pi\)
0.849454 + 0.527662i \(0.176931\pi\)
\(660\) 0.454904 0.0177071
\(661\) 7.93636 0.308689 0.154344 0.988017i \(-0.450673\pi\)
0.154344 + 0.988017i \(0.450673\pi\)
\(662\) −28.9653 −1.12577
\(663\) −0.909808 −0.0353340
\(664\) −1.97345 −0.0765846
\(665\) 0 0
\(666\) 7.88325 0.305470
\(667\) −23.2890 −0.901753
\(668\) 4.85670 0.187911
\(669\) 12.4959 0.483121
\(670\) 1.72548 0.0666611
\(671\) 14.0410 0.542048
\(672\) 0 0
\(673\) 36.9388 1.42388 0.711942 0.702238i \(-0.247816\pi\)
0.711942 + 0.702238i \(0.247816\pi\)
\(674\) −9.03708 −0.348095
\(675\) 4.79306 0.184485
\(676\) −12.1722 −0.468163
\(677\) 39.6498 1.52386 0.761932 0.647657i \(-0.224251\pi\)
0.761932 + 0.647657i \(0.224251\pi\)
\(678\) −7.58612 −0.291343
\(679\) 0 0
\(680\) 0.454904 0.0174448
\(681\) 9.79306 0.375271
\(682\) 9.58612 0.367072
\(683\) −45.5514 −1.74298 −0.871489 0.490416i \(-0.836845\pi\)
−0.871489 + 0.490416i \(0.836845\pi\)
\(684\) −3.88325 −0.148480
\(685\) −0.413875 −0.0158134
\(686\) 0 0
\(687\) 8.67632 0.331022
\(688\) 7.88325 0.300546
\(689\) −0.827751 −0.0315348
\(690\) 3.79306 0.144399
\(691\) 19.1988 0.730357 0.365178 0.930938i \(-0.381008\pi\)
0.365178 + 0.930938i \(0.381008\pi\)
\(692\) 25.5861 0.972639
\(693\) 0 0
\(694\) −21.6127 −0.820406
\(695\) −3.81173 −0.144587
\(696\) 2.79306 0.105871
\(697\) −1.79306 −0.0679171
\(698\) −14.2745 −0.540299
\(699\) −16.7665 −0.634168
\(700\) 0 0
\(701\) 23.2890 0.879613 0.439807 0.898093i \(-0.355047\pi\)
0.439807 + 0.898093i \(0.355047\pi\)
\(702\) 0.909808 0.0343385
\(703\) −30.6127 −1.15458
\(704\) 1.00000 0.0376889
\(705\) −0.153830 −0.00579356
\(706\) 28.4959 1.07246
\(707\) 0 0
\(708\) 6.97345 0.262078
\(709\) −20.7931 −0.780900 −0.390450 0.920624i \(-0.627681\pi\)
−0.390450 + 0.920624i \(0.627681\pi\)
\(710\) 2.18038 0.0818283
\(711\) 5.36471 0.201192
\(712\) 11.0902 0.415623
\(713\) 79.9306 2.99343
\(714\) 0 0
\(715\) 0.413875 0.0154781
\(716\) −15.7029 −0.586844
\(717\) 26.2624 0.980789
\(718\) −13.6392 −0.509012
\(719\) 5.48146 0.204424 0.102212 0.994763i \(-0.467408\pi\)
0.102212 + 0.994763i \(0.467408\pi\)
\(720\) −0.454904 −0.0169533
\(721\) 0 0
\(722\) −3.92034 −0.145900
\(723\) 7.94689 0.295548
\(724\) 2.49593 0.0927606
\(725\) 13.3873 0.497193
\(726\) −1.00000 −0.0371135
\(727\) −25.5330 −0.946967 −0.473484 0.880803i \(-0.657004\pi\)
−0.473484 + 0.880803i \(0.657004\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.85670 0.179755
\(731\) −7.88325 −0.291573
\(732\) −14.0410 −0.518971
\(733\) −16.2214 −0.599152 −0.299576 0.954073i \(-0.596845\pi\)
−0.299576 + 0.954073i \(0.596845\pi\)
\(734\) −22.7294 −0.838958
\(735\) 0 0
\(736\) 8.33816 0.307349
\(737\) −3.79306 −0.139719
\(738\) 1.79306 0.0660035
\(739\) 18.7584 0.690038 0.345019 0.938596i \(-0.387872\pi\)
0.345019 + 0.938596i \(0.387872\pi\)
\(740\) −3.58612 −0.131829
\(741\) −3.53302 −0.129789
\(742\) 0 0
\(743\) −44.6763 −1.63902 −0.819508 0.573068i \(-0.805753\pi\)
−0.819508 + 0.573068i \(0.805753\pi\)
\(744\) −9.58612 −0.351444
\(745\) −1.87272 −0.0686113
\(746\) 10.4018 0.380837
\(747\) −1.97345 −0.0722046
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −4.45490 −0.162670
\(751\) −2.96292 −0.108118 −0.0540592 0.998538i \(-0.517216\pi\)
−0.0540592 + 0.998538i \(0.517216\pi\)
\(752\) −0.338158 −0.0123314
\(753\) −3.93636 −0.143449
\(754\) 2.54115 0.0925433
\(755\) 2.80120 0.101946
\(756\) 0 0
\(757\) 26.1988 0.952212 0.476106 0.879388i \(-0.342048\pi\)
0.476106 + 0.879388i \(0.342048\pi\)
\(758\) 7.61268 0.276505
\(759\) −8.33816 −0.302656
\(760\) 1.76651 0.0640780
\(761\) −25.7931 −0.934998 −0.467499 0.883994i \(-0.654845\pi\)
−0.467499 + 0.883994i \(0.654845\pi\)
\(762\) −7.42835 −0.269101
\(763\) 0 0
\(764\) 9.81962 0.355261
\(765\) 0.454904 0.0164471
\(766\) 22.1457 0.800156
\(767\) 6.34450 0.229087
\(768\) −1.00000 −0.0360844
\(769\) 37.2012 1.34151 0.670755 0.741679i \(-0.265970\pi\)
0.670755 + 0.741679i \(0.265970\pi\)
\(770\) 0 0
\(771\) −15.0902 −0.543460
\(772\) 6.85670 0.246778
\(773\) 40.7705 1.46641 0.733206 0.680007i \(-0.238023\pi\)
0.733206 + 0.680007i \(0.238023\pi\)
\(774\) 7.88325 0.283358
\(775\) −45.9469 −1.65046
\(776\) −14.5861 −0.523611
\(777\) 0 0
\(778\) −3.54510 −0.127098
\(779\) −6.96292 −0.249472
\(780\) −0.413875 −0.0148191
\(781\) −4.79306 −0.171509
\(782\) −8.33816 −0.298172
\(783\) 2.79306 0.0998159
\(784\) 0 0
\(785\) −8.02895 −0.286565
\(786\) −19.4057 −0.692180
\(787\) −54.5885 −1.94587 −0.972935 0.231078i \(-0.925775\pi\)
−0.972935 + 0.231078i \(0.925775\pi\)
\(788\) 23.4694 0.836062
\(789\) 28.6232 1.01901
\(790\) −2.44043 −0.0868266
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −12.7746 −0.453641
\(794\) 2.55957 0.0908358
\(795\) 0.413875 0.0146787
\(796\) 2.23349 0.0791640
\(797\) 16.7705 0.594040 0.297020 0.954871i \(-0.404007\pi\)
0.297020 + 0.954871i \(0.404007\pi\)
\(798\) 0 0
\(799\) 0.338158 0.0119632
\(800\) −4.79306 −0.169460
\(801\) 11.0902 0.391853
\(802\) 10.4959 0.370624
\(803\) −10.6763 −0.376759
\(804\) 3.79306 0.133771
\(805\) 0 0
\(806\) −8.72153 −0.307203
\(807\) −22.9508 −0.807908
\(808\) 8.06364 0.283678
\(809\) −10.2069 −0.358857 −0.179428 0.983771i \(-0.557425\pi\)
−0.179428 + 0.983771i \(0.557425\pi\)
\(810\) −0.454904 −0.0159837
\(811\) 28.4139 0.997746 0.498873 0.866675i \(-0.333747\pi\)
0.498873 + 0.866675i \(0.333747\pi\)
\(812\) 0 0
\(813\) 9.81962 0.344389
\(814\) 7.88325 0.276308
\(815\) 5.28266 0.185043
\(816\) 1.00000 0.0350070
\(817\) −30.6127 −1.07100
\(818\) −15.8486 −0.554132
\(819\) 0 0
\(820\) −0.815671 −0.0284845
\(821\) −3.40574 −0.118861 −0.0594306 0.998232i \(-0.518928\pi\)
−0.0594306 + 0.998232i \(0.518928\pi\)
\(822\) −0.909808 −0.0317332
\(823\) −38.0289 −1.32561 −0.662803 0.748794i \(-0.730633\pi\)
−0.662803 + 0.748794i \(0.730633\pi\)
\(824\) −5.76651 −0.200886
\(825\) 4.79306 0.166873
\(826\) 0 0
\(827\) −12.9653 −0.450848 −0.225424 0.974261i \(-0.572377\pi\)
−0.225424 + 0.974261i \(0.572377\pi\)
\(828\) 8.33816 0.289771
\(829\) −51.8833 −1.80198 −0.900990 0.433840i \(-0.857158\pi\)
−0.900990 + 0.433840i \(0.857158\pi\)
\(830\) 0.897729 0.0311606
\(831\) −10.7294 −0.372200
\(832\) −0.909808 −0.0315419
\(833\) 0 0
\(834\) −8.37919 −0.290148
\(835\) −2.20933 −0.0764571
\(836\) −3.88325 −0.134305
\(837\) −9.58612 −0.331345
\(838\) 9.52249 0.328949
\(839\) 50.2504 1.73484 0.867418 0.497581i \(-0.165778\pi\)
0.867418 + 0.497581i \(0.165778\pi\)
\(840\) 0 0
\(841\) −21.1988 −0.730993
\(842\) −0.530621 −0.0182864
\(843\) −2.58612 −0.0890709
\(844\) −18.2624 −0.628619
\(845\) 5.53721 0.190486
\(846\) −0.338158 −0.0116261
\(847\) 0 0
\(848\) 0.909808 0.0312429
\(849\) −20.0821 −0.689214
\(850\) 4.79306 0.164401
\(851\) 65.7318 2.25326
\(852\) 4.79306 0.164208
\(853\) −13.3647 −0.457599 −0.228800 0.973474i \(-0.573480\pi\)
−0.228800 + 0.973474i \(0.573480\pi\)
\(854\) 0 0
\(855\) 1.76651 0.0604133
\(856\) 15.6127 0.533630
\(857\) −26.3526 −0.900189 −0.450094 0.892981i \(-0.648610\pi\)
−0.450094 + 0.892981i \(0.648610\pi\)
\(858\) 0.909808 0.0310603
\(859\) −1.97345 −0.0673331 −0.0336666 0.999433i \(-0.510718\pi\)
−0.0336666 + 0.999433i \(0.510718\pi\)
\(860\) −3.58612 −0.122286
\(861\) 0 0
\(862\) 20.8036 0.708573
\(863\) −49.8075 −1.69547 −0.847734 0.530421i \(-0.822034\pi\)
−0.847734 + 0.530421i \(0.822034\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −11.6392 −0.395746
\(866\) 4.40574 0.149713
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 5.36471 0.181985
\(870\) −1.27058 −0.0430765
\(871\) 3.45096 0.116931
\(872\) 14.0410 0.475489
\(873\) −14.5861 −0.493666
\(874\) −32.3792 −1.09524
\(875\) 0 0
\(876\) 10.6763 0.360719
\(877\) −24.5370 −0.828554 −0.414277 0.910151i \(-0.635966\pi\)
−0.414277 + 0.910151i \(0.635966\pi\)
\(878\) 15.1949 0.512802
\(879\) 10.3792 0.350081
\(880\) −0.454904 −0.0153348
\(881\) −11.5861 −0.390346 −0.195173 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(882\) 0 0
\(883\) −44.1376 −1.48535 −0.742674 0.669654i \(-0.766443\pi\)
−0.742674 + 0.669654i \(0.766443\pi\)
\(884\) 0.909808 0.0306002
\(885\) −3.17225 −0.106634
\(886\) −11.3873 −0.382565
\(887\) 20.0289 0.672506 0.336253 0.941772i \(-0.390840\pi\)
0.336253 + 0.941772i \(0.390840\pi\)
\(888\) −7.88325 −0.264545
\(889\) 0 0
\(890\) −5.04497 −0.169108
\(891\) 1.00000 0.0335013
\(892\) −12.4959 −0.418395
\(893\) 1.31315 0.0439430
\(894\) −4.11675 −0.137685
\(895\) 7.14330 0.238774
\(896\) 0 0
\(897\) 7.58612 0.253293
\(898\) −24.0821 −0.803629
\(899\) −26.7746 −0.892984
\(900\) −4.79306 −0.159769
\(901\) −0.909808 −0.0303101
\(902\) 1.79306 0.0597024
\(903\) 0 0
\(904\) 7.58612 0.252311
\(905\) −1.13541 −0.0377423
\(906\) 6.15777 0.204578
\(907\) −17.9203 −0.595035 −0.297518 0.954716i \(-0.596159\pi\)
−0.297518 + 0.954716i \(0.596159\pi\)
\(908\) −9.79306 −0.324994
\(909\) 8.06364 0.267454
\(910\) 0 0
\(911\) 13.9695 0.462830 0.231415 0.972855i \(-0.425664\pi\)
0.231415 + 0.972855i \(0.425664\pi\)
\(912\) 3.88325 0.128587
\(913\) −1.97345 −0.0653115
\(914\) −29.5330 −0.976865
\(915\) 6.38732 0.211158
\(916\) −8.67632 −0.286674
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −42.9614 −1.41716 −0.708582 0.705628i \(-0.750665\pi\)
−0.708582 + 0.705628i \(0.750665\pi\)
\(920\) −3.79306 −0.125054
\(921\) 9.76651 0.321817
\(922\) −4.14569 −0.136531
\(923\) 4.36077 0.143536
\(924\) 0 0
\(925\) −37.7849 −1.24236
\(926\) −9.32368 −0.306395
\(927\) −5.76651 −0.189397
\(928\) −2.79306 −0.0916868
\(929\) 25.6682 0.842146 0.421073 0.907027i \(-0.361654\pi\)
0.421073 + 0.907027i \(0.361654\pi\)
\(930\) 4.36077 0.142995
\(931\) 0 0
\(932\) 16.7665 0.549205
\(933\) −6.51854 −0.213407
\(934\) −36.5596 −1.19627
\(935\) 0.454904 0.0148770
\(936\) −0.909808 −0.0297380
\(937\) −26.7584 −0.874158 −0.437079 0.899423i \(-0.643987\pi\)
−0.437079 + 0.899423i \(0.643987\pi\)
\(938\) 0 0
\(939\) 30.7931 1.00489
\(940\) 0.153830 0.00501737
\(941\) 15.6208 0.509224 0.254612 0.967043i \(-0.418052\pi\)
0.254612 + 0.967043i \(0.418052\pi\)
\(942\) −17.6498 −0.575060
\(943\) 14.9508 0.486866
\(944\) −6.97345 −0.226966
\(945\) 0 0
\(946\) 7.88325 0.256307
\(947\) 19.7849 0.642924 0.321462 0.946923i \(-0.395826\pi\)
0.321462 + 0.946923i \(0.395826\pi\)
\(948\) −5.36471 −0.174238
\(949\) 9.71340 0.315310
\(950\) 18.6127 0.603875
\(951\) 27.6272 0.895872
\(952\) 0 0
\(953\) 43.5065 1.40931 0.704656 0.709549i \(-0.251101\pi\)
0.704656 + 0.709549i \(0.251101\pi\)
\(954\) 0.909808 0.0294561
\(955\) −4.46698 −0.144548
\(956\) −26.2624 −0.849388
\(957\) 2.79306 0.0902869
\(958\) −0.729425 −0.0235666
\(959\) 0 0
\(960\) 0.454904 0.0146820
\(961\) 60.8938 1.96432
\(962\) −7.17225 −0.231243
\(963\) 15.6127 0.503112
\(964\) −7.94689 −0.255952
\(965\) −3.11914 −0.100409
\(966\) 0 0
\(967\) 34.5185 1.11004 0.555021 0.831837i \(-0.312710\pi\)
0.555021 + 0.831837i \(0.312710\pi\)
\(968\) 1.00000 0.0321412
\(969\) −3.88325 −0.124748
\(970\) 6.63529 0.213046
\(971\) 44.9388 1.44215 0.721077 0.692855i \(-0.243648\pi\)
0.721077 + 0.692855i \(0.243648\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −36.8567 −1.18096
\(975\) −4.36077 −0.139656
\(976\) 14.0410 0.449442
\(977\) 32.5249 1.04056 0.520282 0.853995i \(-0.325827\pi\)
0.520282 + 0.853995i \(0.325827\pi\)
\(978\) 11.6127 0.371333
\(979\) 11.0902 0.354444
\(980\) 0 0
\(981\) 14.0410 0.448296
\(982\) 25.5065 0.813944
\(983\) 47.1949 1.50528 0.752641 0.658431i \(-0.228780\pi\)
0.752641 + 0.658431i \(0.228780\pi\)
\(984\) −1.79306 −0.0571607
\(985\) −10.6763 −0.340176
\(986\) 2.79306 0.0889492
\(987\) 0 0
\(988\) 3.53302 0.112400
\(989\) 65.7318 2.09015
\(990\) −0.454904 −0.0144578
\(991\) 12.5490 0.398633 0.199317 0.979935i \(-0.436128\pi\)
0.199317 + 0.979935i \(0.436128\pi\)
\(992\) 9.58612 0.304360
\(993\) 28.9653 0.919186
\(994\) 0 0
\(995\) −1.01602 −0.0322101
\(996\) 1.97345 0.0625310
\(997\) −40.9098 −1.29563 −0.647813 0.761799i \(-0.724316\pi\)
−0.647813 + 0.761799i \(0.724316\pi\)
\(998\) −4.12728 −0.130647
\(999\) −7.88325 −0.249415
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 3234.2.a.bg.1.2 3
3.2 odd 2 9702.2.a.du.1.2 3
7.3 odd 6 462.2.i.f.331.2 yes 6
7.5 odd 6 462.2.i.f.67.2 6
7.6 odd 2 3234.2.a.bi.1.2 3
21.5 even 6 1386.2.k.w.991.2 6
21.17 even 6 1386.2.k.w.793.2 6
21.20 even 2 9702.2.a.dt.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.f.67.2 6 7.5 odd 6
462.2.i.f.331.2 yes 6 7.3 odd 6
1386.2.k.w.793.2 6 21.17 even 6
1386.2.k.w.991.2 6 21.5 even 6
3234.2.a.bg.1.2 3 1.1 even 1 trivial
3234.2.a.bi.1.2 3 7.6 odd 2
9702.2.a.dt.1.2 3 21.20 even 2
9702.2.a.du.1.2 3 3.2 odd 2