Properties

Label 7623.2.a.ch.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.46673\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} -1.00000 q^{7} -5.14256 q^{8} +O(q^{10})\) \(q-2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} -1.00000 q^{7} -5.14256 q^{8} -8.55150 q^{10} -0.653752 q^{13} +2.46673 q^{14} +4.51578 q^{16} +1.13715 q^{17} -6.07602 q^{19} +14.1608 q^{20} +6.66708 q^{23} +7.01823 q^{25} +1.61263 q^{26} -4.08477 q^{28} +4.57357 q^{29} +2.79631 q^{31} -0.854102 q^{32} -2.80505 q^{34} -3.46673 q^{35} -0.439758 q^{37} +14.9879 q^{38} -17.8279 q^{40} +5.90315 q^{41} +8.70820 q^{43} -16.4459 q^{46} +0.604703 q^{47} +1.00000 q^{49} -17.3121 q^{50} -2.67042 q^{52} -9.82247 q^{53} +5.14256 q^{56} -11.2818 q^{58} -1.69406 q^{59} +6.85818 q^{61} -6.89775 q^{62} -6.92472 q^{64} -2.26638 q^{65} -6.17828 q^{67} +4.64501 q^{68} +8.55150 q^{70} +5.41687 q^{71} +6.70198 q^{73} +1.08477 q^{74} -24.8191 q^{76} +2.65375 q^{79} +15.6550 q^{80} -14.5615 q^{82} -6.69658 q^{83} +3.94221 q^{85} -21.4808 q^{86} +0.698213 q^{89} +0.653752 q^{91} +27.2335 q^{92} -1.49164 q^{94} -21.0639 q^{95} -14.8587 q^{97} -2.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} - 14q^{10} + 2q^{14} - 4q^{16} + 3q^{17} + 3q^{19} + 17q^{20} + 8q^{23} + 12q^{26} - 4q^{28} - 3q^{29} - 3q^{31} + 10q^{32} - 12q^{34} - 6q^{35} - 7q^{37} + 20q^{38} - 13q^{40} - 4q^{41} + 8q^{43} - 3q^{46} + 14q^{47} + 4q^{49} - 33q^{50} - 17q^{52} + 9q^{53} + 9q^{56} + 3q^{58} + 25q^{59} - 19q^{61} - 10q^{62} + 3q^{64} - 12q^{65} - 15q^{67} + q^{68} + 14q^{70} + 7q^{71} - 11q^{73} - 8q^{74} - 26q^{76} + 8q^{79} + 4q^{80} + 3q^{82} + q^{83} + 15q^{85} - 4q^{86} + 17q^{89} + 17q^{92} - 20q^{94} - 17q^{95} - 15q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.46673 −1.74424 −0.872121 0.489290i \(-0.837256\pi\)
−0.872121 + 0.489290i \(0.837256\pi\)
\(3\) 0 0
\(4\) 4.08477 2.04238
\(5\) 3.46673 1.55037 0.775185 0.631735i \(-0.217657\pi\)
0.775185 + 0.631735i \(0.217657\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −5.14256 −1.81817
\(9\) 0 0
\(10\) −8.55150 −2.70422
\(11\) 0 0
\(12\) 0 0
\(13\) −0.653752 −0.181318 −0.0906590 0.995882i \(-0.528897\pi\)
−0.0906590 + 0.995882i \(0.528897\pi\)
\(14\) 2.46673 0.659262
\(15\) 0 0
\(16\) 4.51578 1.12894
\(17\) 1.13715 0.275800 0.137900 0.990446i \(-0.455965\pi\)
0.137900 + 0.990446i \(0.455965\pi\)
\(18\) 0 0
\(19\) −6.07602 −1.39393 −0.696967 0.717103i \(-0.745468\pi\)
−0.696967 + 0.717103i \(0.745468\pi\)
\(20\) 14.1608 3.16645
\(21\) 0 0
\(22\) 0 0
\(23\) 6.66708 1.39018 0.695091 0.718921i \(-0.255364\pi\)
0.695091 + 0.718921i \(0.255364\pi\)
\(24\) 0 0
\(25\) 7.01823 1.40365
\(26\) 1.61263 0.316263
\(27\) 0 0
\(28\) −4.08477 −0.771948
\(29\) 4.57357 0.849291 0.424646 0.905360i \(-0.360399\pi\)
0.424646 + 0.905360i \(0.360399\pi\)
\(30\) 0 0
\(31\) 2.79631 0.502232 0.251116 0.967957i \(-0.419202\pi\)
0.251116 + 0.967957i \(0.419202\pi\)
\(32\) −0.854102 −0.150985
\(33\) 0 0
\(34\) −2.80505 −0.481063
\(35\) −3.46673 −0.585985
\(36\) 0 0
\(37\) −0.439758 −0.0722958 −0.0361479 0.999346i \(-0.511509\pi\)
−0.0361479 + 0.999346i \(0.511509\pi\)
\(38\) 14.9879 2.43136
\(39\) 0 0
\(40\) −17.8279 −2.81883
\(41\) 5.90315 0.921917 0.460959 0.887422i \(-0.347506\pi\)
0.460959 + 0.887422i \(0.347506\pi\)
\(42\) 0 0
\(43\) 8.70820 1.32799 0.663994 0.747738i \(-0.268860\pi\)
0.663994 + 0.747738i \(0.268860\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −16.4459 −2.42482
\(47\) 0.604703 0.0882051 0.0441025 0.999027i \(-0.485957\pi\)
0.0441025 + 0.999027i \(0.485957\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −17.3121 −2.44830
\(51\) 0 0
\(52\) −2.67042 −0.370321
\(53\) −9.82247 −1.34922 −0.674610 0.738175i \(-0.735688\pi\)
−0.674610 + 0.738175i \(0.735688\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.14256 0.687203
\(57\) 0 0
\(58\) −11.2818 −1.48137
\(59\) −1.69406 −0.220547 −0.110274 0.993901i \(-0.535173\pi\)
−0.110274 + 0.993901i \(0.535173\pi\)
\(60\) 0 0
\(61\) 6.85818 0.878100 0.439050 0.898463i \(-0.355315\pi\)
0.439050 + 0.898463i \(0.355315\pi\)
\(62\) −6.89775 −0.876015
\(63\) 0 0
\(64\) −6.92472 −0.865590
\(65\) −2.26638 −0.281110
\(66\) 0 0
\(67\) −6.17828 −0.754797 −0.377398 0.926051i \(-0.623181\pi\)
−0.377398 + 0.926051i \(0.623181\pi\)
\(68\) 4.64501 0.563290
\(69\) 0 0
\(70\) 8.55150 1.02210
\(71\) 5.41687 0.642864 0.321432 0.946933i \(-0.395836\pi\)
0.321432 + 0.946933i \(0.395836\pi\)
\(72\) 0 0
\(73\) 6.70198 0.784408 0.392204 0.919878i \(-0.371713\pi\)
0.392204 + 0.919878i \(0.371713\pi\)
\(74\) 1.08477 0.126101
\(75\) 0 0
\(76\) −24.8191 −2.84695
\(77\) 0 0
\(78\) 0 0
\(79\) 2.65375 0.298570 0.149285 0.988794i \(-0.452303\pi\)
0.149285 + 0.988794i \(0.452303\pi\)
\(80\) 15.6550 1.75028
\(81\) 0 0
\(82\) −14.5615 −1.60805
\(83\) −6.69658 −0.735045 −0.367522 0.930015i \(-0.619794\pi\)
−0.367522 + 0.930015i \(0.619794\pi\)
\(84\) 0 0
\(85\) 3.94221 0.427592
\(86\) −21.4808 −2.31633
\(87\) 0 0
\(88\) 0 0
\(89\) 0.698213 0.0740105 0.0370052 0.999315i \(-0.488218\pi\)
0.0370052 + 0.999315i \(0.488218\pi\)
\(90\) 0 0
\(91\) 0.653752 0.0685318
\(92\) 27.2335 2.83929
\(93\) 0 0
\(94\) −1.49164 −0.153851
\(95\) −21.0639 −2.16111
\(96\) 0 0
\(97\) −14.8587 −1.50867 −0.754336 0.656489i \(-0.772041\pi\)
−0.754336 + 0.656489i \(0.772041\pi\)
\(98\) −2.46673 −0.249178
\(99\) 0 0
\(100\) 28.6678 2.86678
\(101\) −8.61959 −0.857682 −0.428841 0.903380i \(-0.641078\pi\)
−0.428841 + 0.903380i \(0.641078\pi\)
\(102\) 0 0
\(103\) −0.932958 −0.0919271 −0.0459636 0.998943i \(-0.514636\pi\)
−0.0459636 + 0.998943i \(0.514636\pi\)
\(104\) 3.36196 0.329667
\(105\) 0 0
\(106\) 24.2294 2.35337
\(107\) −6.62212 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(108\) 0 0
\(109\) −4.12507 −0.395110 −0.197555 0.980292i \(-0.563300\pi\)
−0.197555 + 0.980292i \(0.563300\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.51578 −0.426701
\(113\) 18.8258 1.77098 0.885491 0.464656i \(-0.153822\pi\)
0.885491 + 0.464656i \(0.153822\pi\)
\(114\) 0 0
\(115\) 23.1130 2.15530
\(116\) 18.6820 1.73458
\(117\) 0 0
\(118\) 4.17878 0.384688
\(119\) −1.13715 −0.104243
\(120\) 0 0
\(121\) 0 0
\(122\) −16.9173 −1.53162
\(123\) 0 0
\(124\) 11.4223 1.02575
\(125\) 6.99666 0.625800
\(126\) 0 0
\(127\) −7.96635 −0.706899 −0.353449 0.935454i \(-0.614991\pi\)
−0.353449 + 0.935454i \(0.614991\pi\)
\(128\) 18.7896 1.66078
\(129\) 0 0
\(130\) 5.59056 0.490324
\(131\) 4.80505 0.419819 0.209910 0.977721i \(-0.432683\pi\)
0.209910 + 0.977721i \(0.432683\pi\)
\(132\) 0 0
\(133\) 6.07602 0.526858
\(134\) 15.2401 1.31655
\(135\) 0 0
\(136\) −5.84788 −0.501452
\(137\) 21.8777 1.86914 0.934571 0.355778i \(-0.115784\pi\)
0.934571 + 0.355778i \(0.115784\pi\)
\(138\) 0 0
\(139\) −19.8137 −1.68058 −0.840289 0.542139i \(-0.817615\pi\)
−0.840289 + 0.542139i \(0.817615\pi\)
\(140\) −14.1608 −1.19680
\(141\) 0 0
\(142\) −13.3620 −1.12131
\(143\) 0 0
\(144\) 0 0
\(145\) 15.8553 1.31672
\(146\) −16.5320 −1.36820
\(147\) 0 0
\(148\) −1.79631 −0.147656
\(149\) −3.16211 −0.259050 −0.129525 0.991576i \(-0.541345\pi\)
−0.129525 + 0.991576i \(0.541345\pi\)
\(150\) 0 0
\(151\) 8.92806 0.726555 0.363278 0.931681i \(-0.381658\pi\)
0.363278 + 0.931681i \(0.381658\pi\)
\(152\) 31.2463 2.53441
\(153\) 0 0
\(154\) 0 0
\(155\) 9.69406 0.778645
\(156\) 0 0
\(157\) 1.13968 0.0909561 0.0454780 0.998965i \(-0.485519\pi\)
0.0454780 + 0.998965i \(0.485519\pi\)
\(158\) −6.54609 −0.520779
\(159\) 0 0
\(160\) −2.96094 −0.234083
\(161\) −6.66708 −0.525440
\(162\) 0 0
\(163\) −5.06728 −0.396900 −0.198450 0.980111i \(-0.563591\pi\)
−0.198450 + 0.980111i \(0.563591\pi\)
\(164\) 24.1130 1.88291
\(165\) 0 0
\(166\) 16.5187 1.28210
\(167\) 19.7069 1.52496 0.762482 0.647009i \(-0.223981\pi\)
0.762482 + 0.647009i \(0.223981\pi\)
\(168\) 0 0
\(169\) −12.5726 −0.967124
\(170\) −9.72437 −0.745825
\(171\) 0 0
\(172\) 35.5710 2.71226
\(173\) 14.3948 1.09442 0.547208 0.836997i \(-0.315691\pi\)
0.547208 + 0.836997i \(0.315691\pi\)
\(174\) 0 0
\(175\) −7.01823 −0.530528
\(176\) 0 0
\(177\) 0 0
\(178\) −1.72230 −0.129092
\(179\) 4.66420 0.348619 0.174309 0.984691i \(-0.444231\pi\)
0.174309 + 0.984691i \(0.444231\pi\)
\(180\) 0 0
\(181\) 9.90805 0.736459 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(182\) −1.61263 −0.119536
\(183\) 0 0
\(184\) −34.2859 −2.52759
\(185\) −1.52452 −0.112085
\(186\) 0 0
\(187\) 0 0
\(188\) 2.47007 0.180148
\(189\) 0 0
\(190\) 51.9591 3.76951
\(191\) −9.94295 −0.719447 −0.359723 0.933059i \(-0.617129\pi\)
−0.359723 + 0.933059i \(0.617129\pi\)
\(192\) 0 0
\(193\) 3.70665 0.266810 0.133405 0.991062i \(-0.457409\pi\)
0.133405 + 0.991062i \(0.457409\pi\)
\(194\) 36.6524 2.63149
\(195\) 0 0
\(196\) 4.08477 0.291769
\(197\) −5.91982 −0.421770 −0.210885 0.977511i \(-0.567635\pi\)
−0.210885 + 0.977511i \(0.567635\pi\)
\(198\) 0 0
\(199\) 11.4842 0.814095 0.407047 0.913407i \(-0.366558\pi\)
0.407047 + 0.913407i \(0.366558\pi\)
\(200\) −36.0917 −2.55207
\(201\) 0 0
\(202\) 21.2622 1.49600
\(203\) −4.57357 −0.321002
\(204\) 0 0
\(205\) 20.4646 1.42931
\(206\) 2.30136 0.160343
\(207\) 0 0
\(208\) −2.95220 −0.204698
\(209\) 0 0
\(210\) 0 0
\(211\) 8.72487 0.600645 0.300323 0.953838i \(-0.402906\pi\)
0.300323 + 0.953838i \(0.402906\pi\)
\(212\) −40.1225 −2.75562
\(213\) 0 0
\(214\) 16.3350 1.11664
\(215\) 30.1890 2.05887
\(216\) 0 0
\(217\) −2.79631 −0.189826
\(218\) 10.1754 0.689168
\(219\) 0 0
\(220\) 0 0
\(221\) −0.743416 −0.0500076
\(222\) 0 0
\(223\) 10.3328 0.691938 0.345969 0.938246i \(-0.387550\pi\)
0.345969 + 0.938246i \(0.387550\pi\)
\(224\) 0.854102 0.0570671
\(225\) 0 0
\(226\) −46.4382 −3.08902
\(227\) 13.2690 0.880691 0.440346 0.897828i \(-0.354856\pi\)
0.440346 + 0.897828i \(0.354856\pi\)
\(228\) 0 0
\(229\) −2.49623 −0.164955 −0.0824777 0.996593i \(-0.526283\pi\)
−0.0824777 + 0.996593i \(0.526283\pi\)
\(230\) −57.0135 −3.75936
\(231\) 0 0
\(232\) −23.5199 −1.54415
\(233\) 9.60389 0.629171 0.314586 0.949229i \(-0.398134\pi\)
0.314586 + 0.949229i \(0.398134\pi\)
\(234\) 0 0
\(235\) 2.09634 0.136750
\(236\) −6.91982 −0.450442
\(237\) 0 0
\(238\) 2.80505 0.181825
\(239\) −5.56327 −0.359858 −0.179929 0.983680i \(-0.557587\pi\)
−0.179929 + 0.983680i \(0.557587\pi\)
\(240\) 0 0
\(241\) 15.0208 0.967572 0.483786 0.875186i \(-0.339261\pi\)
0.483786 + 0.875186i \(0.339261\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 28.0141 1.79342
\(245\) 3.46673 0.221481
\(246\) 0 0
\(247\) 3.97221 0.252746
\(248\) −14.3802 −0.913143
\(249\) 0 0
\(250\) −17.2589 −1.09155
\(251\) −27.6131 −1.74292 −0.871460 0.490466i \(-0.836827\pi\)
−0.871460 + 0.490466i \(0.836827\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 19.6508 1.23300
\(255\) 0 0
\(256\) −32.4995 −2.03122
\(257\) 28.4680 1.77578 0.887892 0.460052i \(-0.152169\pi\)
0.887892 + 0.460052i \(0.152169\pi\)
\(258\) 0 0
\(259\) 0.439758 0.0273253
\(260\) −9.25764 −0.574134
\(261\) 0 0
\(262\) −11.8528 −0.732267
\(263\) −14.1803 −0.874397 −0.437199 0.899365i \(-0.644029\pi\)
−0.437199 + 0.899365i \(0.644029\pi\)
\(264\) 0 0
\(265\) −34.0519 −2.09179
\(266\) −14.9879 −0.918968
\(267\) 0 0
\(268\) −25.2368 −1.54158
\(269\) 24.1937 1.47511 0.737557 0.675285i \(-0.235979\pi\)
0.737557 + 0.675285i \(0.235979\pi\)
\(270\) 0 0
\(271\) 7.44975 0.452540 0.226270 0.974065i \(-0.427347\pi\)
0.226270 + 0.974065i \(0.427347\pi\)
\(272\) 5.13514 0.311363
\(273\) 0 0
\(274\) −53.9665 −3.26024
\(275\) 0 0
\(276\) 0 0
\(277\) −19.1890 −1.15296 −0.576478 0.817113i \(-0.695573\pi\)
−0.576478 + 0.817113i \(0.695573\pi\)
\(278\) 48.8751 2.93134
\(279\) 0 0
\(280\) 17.8279 1.06542
\(281\) 1.90063 0.113382 0.0566910 0.998392i \(-0.481945\pi\)
0.0566910 + 0.998392i \(0.481945\pi\)
\(282\) 0 0
\(283\) 7.25178 0.431073 0.215537 0.976496i \(-0.430850\pi\)
0.215537 + 0.976496i \(0.430850\pi\)
\(284\) 22.1266 1.31297
\(285\) 0 0
\(286\) 0 0
\(287\) −5.90315 −0.348452
\(288\) 0 0
\(289\) −15.7069 −0.923934
\(290\) −39.1109 −2.29667
\(291\) 0 0
\(292\) 27.3760 1.60206
\(293\) 3.27097 0.191092 0.0955460 0.995425i \(-0.469540\pi\)
0.0955460 + 0.995425i \(0.469540\pi\)
\(294\) 0 0
\(295\) −5.87284 −0.341930
\(296\) 2.26148 0.131446
\(297\) 0 0
\(298\) 7.80008 0.451846
\(299\) −4.35862 −0.252065
\(300\) 0 0
\(301\) −8.70820 −0.501933
\(302\) −22.0231 −1.26729
\(303\) 0 0
\(304\) −27.4380 −1.57368
\(305\) 23.7755 1.36138
\(306\) 0 0
\(307\) 31.6121 1.80420 0.902099 0.431530i \(-0.142026\pi\)
0.902099 + 0.431530i \(0.142026\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −23.9126 −1.35815
\(311\) 9.03829 0.512514 0.256257 0.966609i \(-0.417511\pi\)
0.256257 + 0.966609i \(0.417511\pi\)
\(312\) 0 0
\(313\) 14.4990 0.819534 0.409767 0.912190i \(-0.365610\pi\)
0.409767 + 0.912190i \(0.365610\pi\)
\(314\) −2.81128 −0.158649
\(315\) 0 0
\(316\) 10.8400 0.609795
\(317\) 18.4174 1.03442 0.517211 0.855858i \(-0.326970\pi\)
0.517211 + 0.855858i \(0.326970\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −24.0061 −1.34198
\(321\) 0 0
\(322\) 16.4459 0.916494
\(323\) −6.90937 −0.384448
\(324\) 0 0
\(325\) −4.58818 −0.254506
\(326\) 12.4996 0.692290
\(327\) 0 0
\(328\) −30.3573 −1.67620
\(329\) −0.604703 −0.0333384
\(330\) 0 0
\(331\) −6.47653 −0.355982 −0.177991 0.984032i \(-0.556960\pi\)
−0.177991 + 0.984032i \(0.556960\pi\)
\(332\) −27.3540 −1.50124
\(333\) 0 0
\(334\) −48.6116 −2.65991
\(335\) −21.4184 −1.17021
\(336\) 0 0
\(337\) 6.25682 0.340831 0.170415 0.985372i \(-0.445489\pi\)
0.170415 + 0.985372i \(0.445489\pi\)
\(338\) 31.0133 1.68690
\(339\) 0 0
\(340\) 16.1030 0.873308
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −44.7824 −2.41451
\(345\) 0 0
\(346\) −35.5081 −1.90893
\(347\) −22.7156 −1.21944 −0.609719 0.792617i \(-0.708718\pi\)
−0.609719 + 0.792617i \(0.708718\pi\)
\(348\) 0 0
\(349\) 29.6941 1.58949 0.794743 0.606946i \(-0.207606\pi\)
0.794743 + 0.606946i \(0.207606\pi\)
\(350\) 17.3121 0.925370
\(351\) 0 0
\(352\) 0 0
\(353\) 6.82506 0.363262 0.181631 0.983367i \(-0.441862\pi\)
0.181631 + 0.983367i \(0.441862\pi\)
\(354\) 0 0
\(355\) 18.7788 0.996676
\(356\) 2.85204 0.151158
\(357\) 0 0
\(358\) −11.5053 −0.608076
\(359\) 7.25970 0.383152 0.191576 0.981478i \(-0.438640\pi\)
0.191576 + 0.981478i \(0.438640\pi\)
\(360\) 0 0
\(361\) 17.9180 0.943055
\(362\) −24.4405 −1.28456
\(363\) 0 0
\(364\) 2.67042 0.139968
\(365\) 23.2340 1.21612
\(366\) 0 0
\(367\) −36.3366 −1.89676 −0.948378 0.317143i \(-0.897277\pi\)
−0.948378 + 0.317143i \(0.897277\pi\)
\(368\) 30.1071 1.56944
\(369\) 0 0
\(370\) 3.76059 0.195504
\(371\) 9.82247 0.509957
\(372\) 0 0
\(373\) −14.2913 −0.739977 −0.369989 0.929036i \(-0.620638\pi\)
−0.369989 + 0.929036i \(0.620638\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.10972 −0.160372
\(377\) −2.98998 −0.153992
\(378\) 0 0
\(379\) −2.54528 −0.130742 −0.0653710 0.997861i \(-0.520823\pi\)
−0.0653710 + 0.997861i \(0.520823\pi\)
\(380\) −86.0413 −4.41382
\(381\) 0 0
\(382\) 24.5266 1.25489
\(383\) 23.1367 1.18223 0.591115 0.806587i \(-0.298688\pi\)
0.591115 + 0.806587i \(0.298688\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.14330 −0.465382
\(387\) 0 0
\(388\) −60.6943 −3.08128
\(389\) −30.2615 −1.53432 −0.767158 0.641458i \(-0.778330\pi\)
−0.767158 + 0.641458i \(0.778330\pi\)
\(390\) 0 0
\(391\) 7.58150 0.383413
\(392\) −5.14256 −0.259738
\(393\) 0 0
\(394\) 14.6026 0.735669
\(395\) 9.19985 0.462894
\(396\) 0 0
\(397\) −22.6740 −1.13798 −0.568989 0.822345i \(-0.692665\pi\)
−0.568989 + 0.822345i \(0.692665\pi\)
\(398\) −28.3285 −1.41998
\(399\) 0 0
\(400\) 31.6928 1.58464
\(401\) 16.2186 0.809917 0.404959 0.914335i \(-0.367286\pi\)
0.404959 + 0.914335i \(0.367286\pi\)
\(402\) 0 0
\(403\) −1.82809 −0.0910637
\(404\) −35.2090 −1.75171
\(405\) 0 0
\(406\) 11.2818 0.559905
\(407\) 0 0
\(408\) 0 0
\(409\) −35.0614 −1.73368 −0.866838 0.498590i \(-0.833851\pi\)
−0.866838 + 0.498590i \(0.833851\pi\)
\(410\) −50.4808 −2.49307
\(411\) 0 0
\(412\) −3.81092 −0.187750
\(413\) 1.69406 0.0833590
\(414\) 0 0
\(415\) −23.2152 −1.13959
\(416\) 0.558371 0.0273764
\(417\) 0 0
\(418\) 0 0
\(419\) −28.2633 −1.38075 −0.690376 0.723451i \(-0.742555\pi\)
−0.690376 + 0.723451i \(0.742555\pi\)
\(420\) 0 0
\(421\) −13.7947 −0.672311 −0.336156 0.941806i \(-0.609127\pi\)
−0.336156 + 0.941806i \(0.609127\pi\)
\(422\) −21.5219 −1.04767
\(423\) 0 0
\(424\) 50.5126 2.45311
\(425\) 7.98081 0.387126
\(426\) 0 0
\(427\) −6.85818 −0.331891
\(428\) −27.0498 −1.30750
\(429\) 0 0
\(430\) −74.4682 −3.59117
\(431\) −28.1700 −1.35690 −0.678451 0.734645i \(-0.737349\pi\)
−0.678451 + 0.734645i \(0.737349\pi\)
\(432\) 0 0
\(433\) 14.1793 0.681415 0.340708 0.940169i \(-0.389333\pi\)
0.340708 + 0.940169i \(0.389333\pi\)
\(434\) 6.89775 0.331102
\(435\) 0 0
\(436\) −16.8499 −0.806966
\(437\) −40.5093 −1.93782
\(438\) 0 0
\(439\) 28.0185 1.33725 0.668625 0.743599i \(-0.266883\pi\)
0.668625 + 0.743599i \(0.266883\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.83381 0.0872254
\(443\) −17.3370 −0.823706 −0.411853 0.911250i \(-0.635118\pi\)
−0.411853 + 0.911250i \(0.635118\pi\)
\(444\) 0 0
\(445\) 2.42052 0.114744
\(446\) −25.4883 −1.20691
\(447\) 0 0
\(448\) 6.92472 0.327162
\(449\) 29.5215 1.39321 0.696603 0.717457i \(-0.254694\pi\)
0.696603 + 0.717457i \(0.254694\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 76.8990 3.61702
\(453\) 0 0
\(454\) −32.7309 −1.53614
\(455\) 2.26638 0.106250
\(456\) 0 0
\(457\) 9.69725 0.453618 0.226809 0.973939i \(-0.427171\pi\)
0.226809 + 0.973939i \(0.427171\pi\)
\(458\) 6.15752 0.287722
\(459\) 0 0
\(460\) 94.4111 4.40194
\(461\) 19.2216 0.895240 0.447620 0.894224i \(-0.352272\pi\)
0.447620 + 0.894224i \(0.352272\pi\)
\(462\) 0 0
\(463\) 20.5327 0.954235 0.477117 0.878840i \(-0.341682\pi\)
0.477117 + 0.878840i \(0.341682\pi\)
\(464\) 20.6532 0.958803
\(465\) 0 0
\(466\) −23.6902 −1.09743
\(467\) −33.6714 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(468\) 0 0
\(469\) 6.17828 0.285286
\(470\) −5.17112 −0.238526
\(471\) 0 0
\(472\) 8.71178 0.400992
\(473\) 0 0
\(474\) 0 0
\(475\) −42.6429 −1.95659
\(476\) −4.64501 −0.212904
\(477\) 0 0
\(478\) 13.7231 0.627680
\(479\) −14.1708 −0.647479 −0.323740 0.946146i \(-0.604940\pi\)
−0.323740 + 0.946146i \(0.604940\pi\)
\(480\) 0 0
\(481\) 0.287493 0.0131085
\(482\) −37.0522 −1.68768
\(483\) 0 0
\(484\) 0 0
\(485\) −51.5111 −2.33900
\(486\) 0 0
\(487\) 27.7415 1.25709 0.628543 0.777775i \(-0.283652\pi\)
0.628543 + 0.777775i \(0.283652\pi\)
\(488\) −35.2686 −1.59653
\(489\) 0 0
\(490\) −8.55150 −0.386317
\(491\) 3.44295 0.155378 0.0776891 0.996978i \(-0.475246\pi\)
0.0776891 + 0.996978i \(0.475246\pi\)
\(492\) 0 0
\(493\) 5.20086 0.234235
\(494\) −9.79837 −0.440850
\(495\) 0 0
\(496\) 12.6275 0.566992
\(497\) −5.41687 −0.242980
\(498\) 0 0
\(499\) −2.58923 −0.115910 −0.0579550 0.998319i \(-0.518458\pi\)
−0.0579550 + 0.998319i \(0.518458\pi\)
\(500\) 28.5797 1.27812
\(501\) 0 0
\(502\) 68.1140 3.04008
\(503\) 22.9026 1.02118 0.510589 0.859825i \(-0.329427\pi\)
0.510589 + 0.859825i \(0.329427\pi\)
\(504\) 0 0
\(505\) −29.8818 −1.32972
\(506\) 0 0
\(507\) 0 0
\(508\) −32.5407 −1.44376
\(509\) 24.0985 1.06815 0.534074 0.845438i \(-0.320660\pi\)
0.534074 + 0.845438i \(0.320660\pi\)
\(510\) 0 0
\(511\) −6.70198 −0.296478
\(512\) 42.5884 1.88216
\(513\) 0 0
\(514\) −70.2229 −3.09740
\(515\) −3.23432 −0.142521
\(516\) 0 0
\(517\) 0 0
\(518\) −1.08477 −0.0476619
\(519\) 0 0
\(520\) 11.6550 0.511105
\(521\) 33.5057 1.46791 0.733956 0.679197i \(-0.237672\pi\)
0.733956 + 0.679197i \(0.237672\pi\)
\(522\) 0 0
\(523\) −31.1574 −1.36242 −0.681208 0.732090i \(-0.738545\pi\)
−0.681208 + 0.732090i \(0.738545\pi\)
\(524\) 19.6275 0.857432
\(525\) 0 0
\(526\) 34.9791 1.52516
\(527\) 3.17983 0.138516
\(528\) 0 0
\(529\) 21.4500 0.932608
\(530\) 83.9968 3.64859
\(531\) 0 0
\(532\) 24.8191 1.07605
\(533\) −3.85919 −0.167160
\(534\) 0 0
\(535\) −22.9571 −0.992522
\(536\) 31.7721 1.37235
\(537\) 0 0
\(538\) −59.6793 −2.57296
\(539\) 0 0
\(540\) 0 0
\(541\) 22.6071 0.971957 0.485979 0.873971i \(-0.338463\pi\)
0.485979 + 0.873971i \(0.338463\pi\)
\(542\) −18.3765 −0.789340
\(543\) 0 0
\(544\) −0.971245 −0.0416418
\(545\) −14.3005 −0.612567
\(546\) 0 0
\(547\) −27.4442 −1.17343 −0.586715 0.809794i \(-0.699579\pi\)
−0.586715 + 0.809794i \(0.699579\pi\)
\(548\) 89.3654 3.81750
\(549\) 0 0
\(550\) 0 0
\(551\) −27.7891 −1.18386
\(552\) 0 0
\(553\) −2.65375 −0.112849
\(554\) 47.3341 2.01103
\(555\) 0 0
\(556\) −80.9344 −3.43238
\(557\) 30.2504 1.28175 0.640875 0.767645i \(-0.278572\pi\)
0.640875 + 0.767645i \(0.278572\pi\)
\(558\) 0 0
\(559\) −5.69300 −0.240788
\(560\) −15.6550 −0.661544
\(561\) 0 0
\(562\) −4.68834 −0.197766
\(563\) −7.46234 −0.314500 −0.157250 0.987559i \(-0.550263\pi\)
−0.157250 + 0.987559i \(0.550263\pi\)
\(564\) 0 0
\(565\) 65.2640 2.74568
\(566\) −17.8882 −0.751896
\(567\) 0 0
\(568\) −27.8565 −1.16883
\(569\) −35.6483 −1.49446 −0.747228 0.664568i \(-0.768616\pi\)
−0.747228 + 0.664568i \(0.768616\pi\)
\(570\) 0 0
\(571\) 25.8902 1.08347 0.541737 0.840548i \(-0.317767\pi\)
0.541737 + 0.840548i \(0.317767\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 14.5615 0.607785
\(575\) 46.7911 1.95132
\(576\) 0 0
\(577\) 8.16382 0.339864 0.169932 0.985456i \(-0.445645\pi\)
0.169932 + 0.985456i \(0.445645\pi\)
\(578\) 38.7447 1.61157
\(579\) 0 0
\(580\) 64.7654 2.68924
\(581\) 6.69658 0.277821
\(582\) 0 0
\(583\) 0 0
\(584\) −34.4653 −1.42619
\(585\) 0 0
\(586\) −8.06860 −0.333311
\(587\) 12.4634 0.514419 0.257210 0.966356i \(-0.417197\pi\)
0.257210 + 0.966356i \(0.417197\pi\)
\(588\) 0 0
\(589\) −16.9904 −0.700079
\(590\) 14.4867 0.596409
\(591\) 0 0
\(592\) −1.98585 −0.0816180
\(593\) 23.6707 0.972037 0.486019 0.873948i \(-0.338449\pi\)
0.486019 + 0.873948i \(0.338449\pi\)
\(594\) 0 0
\(595\) −3.94221 −0.161615
\(596\) −12.9165 −0.529080
\(597\) 0 0
\(598\) 10.7515 0.439663
\(599\) −38.9809 −1.59272 −0.796358 0.604826i \(-0.793243\pi\)
−0.796358 + 0.604826i \(0.793243\pi\)
\(600\) 0 0
\(601\) 30.5510 1.24620 0.623100 0.782142i \(-0.285873\pi\)
0.623100 + 0.782142i \(0.285873\pi\)
\(602\) 21.4808 0.875492
\(603\) 0 0
\(604\) 36.4690 1.48390
\(605\) 0 0
\(606\) 0 0
\(607\) 37.6173 1.52684 0.763420 0.645903i \(-0.223519\pi\)
0.763420 + 0.645903i \(0.223519\pi\)
\(608\) 5.18954 0.210464
\(609\) 0 0
\(610\) −58.6477 −2.37458
\(611\) −0.395326 −0.0159932
\(612\) 0 0
\(613\) 17.6272 0.711956 0.355978 0.934494i \(-0.384148\pi\)
0.355978 + 0.934494i \(0.384148\pi\)
\(614\) −77.9786 −3.14696
\(615\) 0 0
\(616\) 0 0
\(617\) 44.4849 1.79089 0.895447 0.445168i \(-0.146856\pi\)
0.895447 + 0.445168i \(0.146856\pi\)
\(618\) 0 0
\(619\) −6.20424 −0.249369 −0.124685 0.992196i \(-0.539792\pi\)
−0.124685 + 0.992196i \(0.539792\pi\)
\(620\) 39.5979 1.59029
\(621\) 0 0
\(622\) −22.2950 −0.893949
\(623\) −0.698213 −0.0279733
\(624\) 0 0
\(625\) −10.8356 −0.433424
\(626\) −35.7652 −1.42947
\(627\) 0 0
\(628\) 4.65531 0.185767
\(629\) −0.500073 −0.0199392
\(630\) 0 0
\(631\) 44.8057 1.78369 0.891844 0.452344i \(-0.149412\pi\)
0.891844 + 0.452344i \(0.149412\pi\)
\(632\) −13.6471 −0.542851
\(633\) 0 0
\(634\) −45.4307 −1.80428
\(635\) −27.6172 −1.09595
\(636\) 0 0
\(637\) −0.653752 −0.0259026
\(638\) 0 0
\(639\) 0 0
\(640\) 65.1386 2.57483
\(641\) 10.2756 0.405863 0.202932 0.979193i \(-0.434953\pi\)
0.202932 + 0.979193i \(0.434953\pi\)
\(642\) 0 0
\(643\) −16.4446 −0.648511 −0.324255 0.945970i \(-0.605114\pi\)
−0.324255 + 0.945970i \(0.605114\pi\)
\(644\) −27.2335 −1.07315
\(645\) 0 0
\(646\) 17.0436 0.670570
\(647\) 26.9686 1.06025 0.530123 0.847920i \(-0.322146\pi\)
0.530123 + 0.847920i \(0.322146\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 11.3178 0.443921
\(651\) 0 0
\(652\) −20.6986 −0.810621
\(653\) −7.10602 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(654\) 0 0
\(655\) 16.6578 0.650875
\(656\) 26.6573 1.04079
\(657\) 0 0
\(658\) 1.49164 0.0581502
\(659\) 32.6279 1.27100 0.635502 0.772099i \(-0.280793\pi\)
0.635502 + 0.772099i \(0.280793\pi\)
\(660\) 0 0
\(661\) −33.8165 −1.31531 −0.657654 0.753320i \(-0.728451\pi\)
−0.657654 + 0.753320i \(0.728451\pi\)
\(662\) 15.9759 0.620919
\(663\) 0 0
\(664\) 34.4375 1.33644
\(665\) 21.0639 0.816824
\(666\) 0 0
\(667\) 30.4924 1.18067
\(668\) 80.4980 3.11456
\(669\) 0 0
\(670\) 52.8335 2.04114
\(671\) 0 0
\(672\) 0 0
\(673\) 23.7496 0.915478 0.457739 0.889087i \(-0.348659\pi\)
0.457739 + 0.889087i \(0.348659\pi\)
\(674\) −15.4339 −0.594491
\(675\) 0 0
\(676\) −51.3562 −1.97524
\(677\) 45.6311 1.75375 0.876874 0.480721i \(-0.159625\pi\)
0.876874 + 0.480721i \(0.159625\pi\)
\(678\) 0 0
\(679\) 14.8587 0.570224
\(680\) −20.2730 −0.777435
\(681\) 0 0
\(682\) 0 0
\(683\) 28.8727 1.10478 0.552392 0.833585i \(-0.313715\pi\)
0.552392 + 0.833585i \(0.313715\pi\)
\(684\) 0 0
\(685\) 75.8442 2.89786
\(686\) 2.46673 0.0941803
\(687\) 0 0
\(688\) 39.3243 1.49923
\(689\) 6.42145 0.244638
\(690\) 0 0
\(691\) −27.0635 −1.02954 −0.514772 0.857327i \(-0.672123\pi\)
−0.514772 + 0.857327i \(0.672123\pi\)
\(692\) 58.7994 2.23522
\(693\) 0 0
\(694\) 56.0334 2.12700
\(695\) −68.6889 −2.60552
\(696\) 0 0
\(697\) 6.71279 0.254265
\(698\) −73.2473 −2.77245
\(699\) 0 0
\(700\) −28.6678 −1.08354
\(701\) −38.5156 −1.45471 −0.727357 0.686259i \(-0.759252\pi\)
−0.727357 + 0.686259i \(0.759252\pi\)
\(702\) 0 0
\(703\) 2.67198 0.100776
\(704\) 0 0
\(705\) 0 0
\(706\) −16.8356 −0.633616
\(707\) 8.61959 0.324173
\(708\) 0 0
\(709\) −1.89464 −0.0711548 −0.0355774 0.999367i \(-0.511327\pi\)
−0.0355774 + 0.999367i \(0.511327\pi\)
\(710\) −46.3223 −1.73845
\(711\) 0 0
\(712\) −3.59060 −0.134564
\(713\) 18.6432 0.698194
\(714\) 0 0
\(715\) 0 0
\(716\) 19.0522 0.712013
\(717\) 0 0
\(718\) −17.9077 −0.668311
\(719\) −14.5772 −0.543639 −0.271819 0.962348i \(-0.587625\pi\)
−0.271819 + 0.962348i \(0.587625\pi\)
\(720\) 0 0
\(721\) 0.932958 0.0347452
\(722\) −44.1990 −1.64492
\(723\) 0 0
\(724\) 40.4721 1.50413
\(725\) 32.0984 1.19210
\(726\) 0 0
\(727\) 4.04780 0.150125 0.0750623 0.997179i \(-0.476084\pi\)
0.0750623 + 0.997179i \(0.476084\pi\)
\(728\) −3.36196 −0.124602
\(729\) 0 0
\(730\) −57.3120 −2.12121
\(731\) 9.90257 0.366260
\(732\) 0 0
\(733\) −23.5012 −0.868036 −0.434018 0.900904i \(-0.642905\pi\)
−0.434018 + 0.900904i \(0.642905\pi\)
\(734\) 89.6327 3.30840
\(735\) 0 0
\(736\) −5.69437 −0.209897
\(737\) 0 0
\(738\) 0 0
\(739\) −32.6786 −1.20210 −0.601051 0.799210i \(-0.705251\pi\)
−0.601051 + 0.799210i \(0.705251\pi\)
\(740\) −6.22732 −0.228921
\(741\) 0 0
\(742\) −24.2294 −0.889489
\(743\) 18.1058 0.664237 0.332119 0.943238i \(-0.392237\pi\)
0.332119 + 0.943238i \(0.392237\pi\)
\(744\) 0 0
\(745\) −10.9622 −0.401624
\(746\) 35.2529 1.29070
\(747\) 0 0
\(748\) 0 0
\(749\) 6.62212 0.241967
\(750\) 0 0
\(751\) 9.14357 0.333654 0.166827 0.985986i \(-0.446648\pi\)
0.166827 + 0.985986i \(0.446648\pi\)
\(752\) 2.73071 0.0995787
\(753\) 0 0
\(754\) 7.37548 0.268599
\(755\) 30.9512 1.12643
\(756\) 0 0
\(757\) 47.3509 1.72100 0.860499 0.509452i \(-0.170152\pi\)
0.860499 + 0.509452i \(0.170152\pi\)
\(758\) 6.27851 0.228046
\(759\) 0 0
\(760\) 108.323 3.92927
\(761\) −3.44828 −0.125000 −0.0625000 0.998045i \(-0.519907\pi\)
−0.0625000 + 0.998045i \(0.519907\pi\)
\(762\) 0 0
\(763\) 4.12507 0.149338
\(764\) −40.6146 −1.46939
\(765\) 0 0
\(766\) −57.0720 −2.06210
\(767\) 1.10749 0.0399892
\(768\) 0 0
\(769\) 34.9787 1.26137 0.630683 0.776041i \(-0.282775\pi\)
0.630683 + 0.776041i \(0.282775\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.1408 0.544928
\(773\) −25.3794 −0.912832 −0.456416 0.889766i \(-0.650867\pi\)
−0.456416 + 0.889766i \(0.650867\pi\)
\(774\) 0 0
\(775\) 19.6251 0.704956
\(776\) 76.4117 2.74302
\(777\) 0 0
\(778\) 74.6469 2.67622
\(779\) −35.8677 −1.28509
\(780\) 0 0
\(781\) 0 0
\(782\) −18.7015 −0.668765
\(783\) 0 0
\(784\) 4.51578 0.161278
\(785\) 3.95095 0.141016
\(786\) 0 0
\(787\) 29.7552 1.06066 0.530329 0.847792i \(-0.322068\pi\)
0.530329 + 0.847792i \(0.322068\pi\)
\(788\) −24.1811 −0.861415
\(789\) 0 0
\(790\) −22.6936 −0.807400
\(791\) −18.8258 −0.669369
\(792\) 0 0
\(793\) −4.48355 −0.159215
\(794\) 55.9308 1.98491
\(795\) 0 0
\(796\) 46.9103 1.66269
\(797\) 6.24330 0.221149 0.110575 0.993868i \(-0.464731\pi\)
0.110575 + 0.993868i \(0.464731\pi\)
\(798\) 0 0
\(799\) 0.687641 0.0243270
\(800\) −5.99428 −0.211930
\(801\) 0 0
\(802\) −40.0069 −1.41269
\(803\) 0 0
\(804\) 0 0
\(805\) −23.1130 −0.814626
\(806\) 4.50941 0.158837
\(807\) 0 0
\(808\) 44.3268 1.55941
\(809\) −39.1860 −1.37771 −0.688854 0.724900i \(-0.741886\pi\)
−0.688854 + 0.724900i \(0.741886\pi\)
\(810\) 0 0
\(811\) −12.3809 −0.434753 −0.217376 0.976088i \(-0.569750\pi\)
−0.217376 + 0.976088i \(0.569750\pi\)
\(812\) −18.6820 −0.655609
\(813\) 0 0
\(814\) 0 0
\(815\) −17.5669 −0.615341
\(816\) 0 0
\(817\) −52.9112 −1.85113
\(818\) 86.4871 3.02395
\(819\) 0 0
\(820\) 83.5933 2.91920
\(821\) −47.6987 −1.66470 −0.832349 0.554252i \(-0.813004\pi\)
−0.832349 + 0.554252i \(0.813004\pi\)
\(822\) 0 0
\(823\) −14.5207 −0.506161 −0.253080 0.967445i \(-0.581444\pi\)
−0.253080 + 0.967445i \(0.581444\pi\)
\(824\) 4.79779 0.167139
\(825\) 0 0
\(826\) −4.17878 −0.145398
\(827\) 30.9372 1.07579 0.537896 0.843011i \(-0.319219\pi\)
0.537896 + 0.843011i \(0.319219\pi\)
\(828\) 0 0
\(829\) 0.0858057 0.00298015 0.00149008 0.999999i \(-0.499526\pi\)
0.00149008 + 0.999999i \(0.499526\pi\)
\(830\) 57.2658 1.98772
\(831\) 0 0
\(832\) 4.52705 0.156947
\(833\) 1.13715 0.0394000
\(834\) 0 0
\(835\) 68.3185 2.36426
\(836\) 0 0
\(837\) 0 0
\(838\) 69.7179 2.40837
\(839\) −10.6905 −0.369078 −0.184539 0.982825i \(-0.559079\pi\)
−0.184539 + 0.982825i \(0.559079\pi\)
\(840\) 0 0
\(841\) −8.08244 −0.278705
\(842\) 34.0278 1.17267
\(843\) 0 0
\(844\) 35.6391 1.22675
\(845\) −43.5859 −1.49940
\(846\) 0 0
\(847\) 0 0
\(848\) −44.3561 −1.52319
\(849\) 0 0
\(850\) −19.6865 −0.675242
\(851\) −2.93190 −0.100504
\(852\) 0 0
\(853\) 21.2003 0.725884 0.362942 0.931812i \(-0.381772\pi\)
0.362942 + 0.931812i \(0.381772\pi\)
\(854\) 16.9173 0.578898
\(855\) 0 0
\(856\) 34.0546 1.16396
\(857\) 9.45359 0.322929 0.161464 0.986879i \(-0.448378\pi\)
0.161464 + 0.986879i \(0.448378\pi\)
\(858\) 0 0
\(859\) −38.8261 −1.32473 −0.662365 0.749181i \(-0.730447\pi\)
−0.662365 + 0.749181i \(0.730447\pi\)
\(860\) 123.315 4.20501
\(861\) 0 0
\(862\) 69.4879 2.36677
\(863\) 37.6046 1.28008 0.640038 0.768343i \(-0.278919\pi\)
0.640038 + 0.768343i \(0.278919\pi\)
\(864\) 0 0
\(865\) 49.9029 1.69675
\(866\) −34.9766 −1.18855
\(867\) 0 0
\(868\) −11.4223 −0.387697
\(869\) 0 0
\(870\) 0 0
\(871\) 4.03906 0.136858
\(872\) 21.2134 0.718377
\(873\) 0 0
\(874\) 99.9257 3.38004
\(875\) −6.99666 −0.236530
\(876\) 0 0
\(877\) 22.0086 0.743178 0.371589 0.928397i \(-0.378813\pi\)
0.371589 + 0.928397i \(0.378813\pi\)
\(878\) −69.1142 −2.33249
\(879\) 0 0
\(880\) 0 0
\(881\) 6.92969 0.233467 0.116734 0.993163i \(-0.462758\pi\)
0.116734 + 0.993163i \(0.462758\pi\)
\(882\) 0 0
\(883\) 42.3388 1.42481 0.712407 0.701767i \(-0.247605\pi\)
0.712407 + 0.701767i \(0.247605\pi\)
\(884\) −3.03668 −0.102135
\(885\) 0 0
\(886\) 42.7657 1.43674
\(887\) −8.05647 −0.270510 −0.135255 0.990811i \(-0.543185\pi\)
−0.135255 + 0.990811i \(0.543185\pi\)
\(888\) 0 0
\(889\) 7.96635 0.267183
\(890\) −5.97077 −0.200141
\(891\) 0 0
\(892\) 42.2072 1.41320
\(893\) −3.67419 −0.122952
\(894\) 0 0
\(895\) 16.1695 0.540488
\(896\) −18.7896 −0.627717
\(897\) 0 0
\(898\) −72.8216 −2.43009
\(899\) 12.7891 0.426541
\(900\) 0 0
\(901\) −11.1697 −0.372115
\(902\) 0 0
\(903\) 0 0
\(904\) −96.8128 −3.21995
\(905\) 34.3485 1.14178
\(906\) 0 0
\(907\) 2.89429 0.0961033 0.0480517 0.998845i \(-0.484699\pi\)
0.0480517 + 0.998845i \(0.484699\pi\)
\(908\) 54.2006 1.79871
\(909\) 0 0
\(910\) −5.59056 −0.185325
\(911\) 20.2500 0.670912 0.335456 0.942056i \(-0.391110\pi\)
0.335456 + 0.942056i \(0.391110\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −23.9205 −0.791220
\(915\) 0 0
\(916\) −10.1965 −0.336902
\(917\) −4.80505 −0.158677
\(918\) 0 0
\(919\) 18.7261 0.617718 0.308859 0.951108i \(-0.400053\pi\)
0.308859 + 0.951108i \(0.400053\pi\)
\(920\) −118.860 −3.91869
\(921\) 0 0
\(922\) −47.4146 −1.56152
\(923\) −3.54128 −0.116563
\(924\) 0 0
\(925\) −3.08632 −0.101478
\(926\) −50.6486 −1.66442
\(927\) 0 0
\(928\) −3.90630 −0.128230
\(929\) 45.2701 1.48526 0.742631 0.669700i \(-0.233577\pi\)
0.742631 + 0.669700i \(0.233577\pi\)
\(930\) 0 0
\(931\) −6.07602 −0.199134
\(932\) 39.2296 1.28501
\(933\) 0 0
\(934\) 83.0584 2.71775
\(935\) 0 0
\(936\) 0 0
\(937\) 29.4599 0.962413 0.481207 0.876607i \(-0.340199\pi\)
0.481207 + 0.876607i \(0.340199\pi\)
\(938\) −15.2401 −0.497609
\(939\) 0 0
\(940\) 8.56308 0.279297
\(941\) −58.2346 −1.89839 −0.949197 0.314683i \(-0.898102\pi\)
−0.949197 + 0.314683i \(0.898102\pi\)
\(942\) 0 0
\(943\) 39.3568 1.28163
\(944\) −7.64998 −0.248986
\(945\) 0 0
\(946\) 0 0
\(947\) −45.3642 −1.47414 −0.737069 0.675818i \(-0.763791\pi\)
−0.737069 + 0.675818i \(0.763791\pi\)
\(948\) 0 0
\(949\) −4.38143 −0.142227
\(950\) 105.189 3.41277
\(951\) 0 0
\(952\) 5.84788 0.189531
\(953\) 41.3375 1.33905 0.669527 0.742788i \(-0.266497\pi\)
0.669527 + 0.742788i \(0.266497\pi\)
\(954\) 0 0
\(955\) −34.4695 −1.11541
\(956\) −22.7247 −0.734968
\(957\) 0 0
\(958\) 34.9555 1.12936
\(959\) −21.8777 −0.706469
\(960\) 0 0
\(961\) −23.1807 −0.747763
\(962\) −0.709167 −0.0228645
\(963\) 0 0
\(964\) 61.3563 1.97615
\(965\) 12.8499 0.413654
\(966\) 0 0
\(967\) −6.52818 −0.209932 −0.104966 0.994476i \(-0.533473\pi\)
−0.104966 + 0.994476i \(0.533473\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 127.064 4.07978
\(971\) −28.8587 −0.926119 −0.463060 0.886327i \(-0.653248\pi\)
−0.463060 + 0.886327i \(0.653248\pi\)
\(972\) 0 0
\(973\) 19.8137 0.635199
\(974\) −68.4308 −2.19266
\(975\) 0 0
\(976\) 30.9700 0.991327
\(977\) 8.98453 0.287441 0.143720 0.989618i \(-0.454093\pi\)
0.143720 + 0.989618i \(0.454093\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 14.1608 0.452350
\(981\) 0 0
\(982\) −8.49284 −0.271017
\(983\) −36.8819 −1.17635 −0.588175 0.808734i \(-0.700153\pi\)
−0.588175 + 0.808734i \(0.700153\pi\)
\(984\) 0 0
\(985\) −20.5224 −0.653899
\(986\) −12.8291 −0.408562
\(987\) 0 0
\(988\) 16.2255 0.516203
\(989\) 58.0583 1.84615
\(990\) 0 0
\(991\) −2.98352 −0.0947746 −0.0473873 0.998877i \(-0.515089\pi\)
−0.0473873 + 0.998877i \(0.515089\pi\)
\(992\) −2.38833 −0.0758297
\(993\) 0 0
\(994\) 13.3620 0.423815
\(995\) 39.8127 1.26215
\(996\) 0 0
\(997\) 62.8553 1.99065 0.995324 0.0965930i \(-0.0307945\pi\)
0.995324 + 0.0965930i \(0.0307945\pi\)
\(998\) 6.38694 0.202175
\(999\) 0 0
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 7623.2.a.ch.1.1 4
3.2 odd 2 847.2.a.l.1.4 4
11.5 even 5 693.2.m.g.190.2 8
11.9 even 5 693.2.m.g.631.2 8
11.10 odd 2 7623.2.a.co.1.4 4
21.20 even 2 5929.2.a.bi.1.4 4
33.2 even 10 847.2.f.q.323.2 8
33.5 odd 10 77.2.f.a.36.1 yes 8
33.8 even 10 847.2.f.s.372.1 8
33.14 odd 10 847.2.f.p.372.2 8
33.17 even 10 847.2.f.q.729.2 8
33.20 odd 10 77.2.f.a.15.1 8
33.26 odd 10 847.2.f.p.148.2 8
33.29 even 10 847.2.f.s.148.1 8
33.32 even 2 847.2.a.k.1.1 4
231.5 even 30 539.2.q.b.410.2 16
231.20 even 10 539.2.f.d.246.1 8
231.38 even 30 539.2.q.b.520.1 16
231.53 odd 30 539.2.q.c.422.2 16
231.86 odd 30 539.2.q.c.312.1 16
231.104 even 10 539.2.f.d.344.1 8
231.137 odd 30 539.2.q.c.520.1 16
231.152 even 30 539.2.q.b.312.1 16
231.170 odd 30 539.2.q.c.410.2 16
231.185 even 30 539.2.q.b.422.2 16
231.230 odd 2 5929.2.a.bb.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.15.1 8 33.20 odd 10
77.2.f.a.36.1 yes 8 33.5 odd 10
539.2.f.d.246.1 8 231.20 even 10
539.2.f.d.344.1 8 231.104 even 10
539.2.q.b.312.1 16 231.152 even 30
539.2.q.b.410.2 16 231.5 even 30
539.2.q.b.422.2 16 231.185 even 30
539.2.q.b.520.1 16 231.38 even 30
539.2.q.c.312.1 16 231.86 odd 30
539.2.q.c.410.2 16 231.170 odd 30
539.2.q.c.422.2 16 231.53 odd 30
539.2.q.c.520.1 16 231.137 odd 30
693.2.m.g.190.2 8 11.5 even 5
693.2.m.g.631.2 8 11.9 even 5
847.2.a.k.1.1 4 33.32 even 2
847.2.a.l.1.4 4 3.2 odd 2
847.2.f.p.148.2 8 33.26 odd 10
847.2.f.p.372.2 8 33.14 odd 10
847.2.f.q.323.2 8 33.2 even 10
847.2.f.q.729.2 8 33.17 even 10
847.2.f.s.148.1 8 33.29 even 10
847.2.f.s.372.1 8 33.8 even 10
5929.2.a.bb.1.1 4 231.230 odd 2
5929.2.a.bi.1.4 4 21.20 even 2
7623.2.a.ch.1.1 4 1.1 even 1 trivial
7623.2.a.co.1.4 4 11.10 odd 2