Properties

Label 128.4.a.g.1.1
Level 128
Weight 4
Character 128.1
Self dual yes
Analytic conductor 7.552
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 128.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.92820 q^{3} -15.8564 q^{5} +17.8564 q^{7} -2.71281 q^{9} +O(q^{10})\) \(q-4.92820 q^{3} -15.8564 q^{5} +17.8564 q^{7} -2.71281 q^{9} +52.9282 q^{11} -8.43078 q^{13} +78.1436 q^{15} +129.138 q^{17} +50.4974 q^{19} -88.0000 q^{21} -128.708 q^{23} +126.426 q^{25} +146.431 q^{27} -111.282 q^{29} +302.851 q^{31} -260.841 q^{33} -283.138 q^{35} +182.995 q^{37} +41.5486 q^{39} -94.5744 q^{41} +184.641 q^{43} +43.0155 q^{45} -296.841 q^{47} -24.1487 q^{49} -636.420 q^{51} +102.995 q^{53} -839.251 q^{55} -248.862 q^{57} -93.3693 q^{59} +338.974 q^{61} -48.4411 q^{63} +133.682 q^{65} +489.041 q^{67} +634.297 q^{69} +86.9845 q^{71} -154.267 q^{73} -623.051 q^{75} +945.108 q^{77} +449.415 q^{79} -648.395 q^{81} +383.933 q^{83} -2047.67 q^{85} +548.420 q^{87} -517.672 q^{89} -150.543 q^{91} -1492.51 q^{93} -800.708 q^{95} +1739.39 q^{97} -143.584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 4q^{5} + 8q^{7} + 50q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 4q^{5} + 8q^{7} + 50q^{9} + 92q^{11} - 100q^{13} + 184q^{15} + 92q^{17} + 4q^{19} - 176q^{21} - 8q^{23} + 142q^{25} + 376q^{27} - 84q^{29} + 384q^{31} + 88q^{33} - 400q^{35} + 172q^{37} - 776q^{39} - 300q^{41} + 300q^{43} + 668q^{45} + 16q^{47} - 270q^{49} - 968q^{51} + 12q^{53} - 376q^{55} - 664q^{57} - 644q^{59} - 292q^{61} - 568q^{63} - 952q^{65} - 172q^{67} + 1712q^{69} - 408q^{71} + 412q^{73} - 484q^{75} + 560q^{77} + 400q^{79} - 22q^{81} + 948q^{83} - 2488q^{85} + 792q^{87} + 572q^{89} + 752q^{91} - 768q^{93} - 1352q^{95} + 2204q^{97} + 1916q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.92820 −0.948433 −0.474217 0.880408i \(-0.657269\pi\)
−0.474217 + 0.880408i \(0.657269\pi\)
\(4\) 0 0
\(5\) −15.8564 −1.41824 −0.709120 0.705088i \(-0.750908\pi\)
−0.709120 + 0.705088i \(0.750908\pi\)
\(6\) 0 0
\(7\) 17.8564 0.964155 0.482078 0.876128i \(-0.339882\pi\)
0.482078 + 0.876128i \(0.339882\pi\)
\(8\) 0 0
\(9\) −2.71281 −0.100475
\(10\) 0 0
\(11\) 52.9282 1.45077 0.725384 0.688344i \(-0.241662\pi\)
0.725384 + 0.688344i \(0.241662\pi\)
\(12\) 0 0
\(13\) −8.43078 −0.179868 −0.0899338 0.995948i \(-0.528666\pi\)
−0.0899338 + 0.995948i \(0.528666\pi\)
\(14\) 0 0
\(15\) 78.1436 1.34511
\(16\) 0 0
\(17\) 129.138 1.84239 0.921196 0.389098i \(-0.127213\pi\)
0.921196 + 0.389098i \(0.127213\pi\)
\(18\) 0 0
\(19\) 50.4974 0.609732 0.304866 0.952395i \(-0.401388\pi\)
0.304866 + 0.952395i \(0.401388\pi\)
\(20\) 0 0
\(21\) −88.0000 −0.914437
\(22\) 0 0
\(23\) −128.708 −1.16684 −0.583422 0.812169i \(-0.698287\pi\)
−0.583422 + 0.812169i \(0.698287\pi\)
\(24\) 0 0
\(25\) 126.426 1.01141
\(26\) 0 0
\(27\) 146.431 1.04373
\(28\) 0 0
\(29\) −111.282 −0.712571 −0.356285 0.934377i \(-0.615957\pi\)
−0.356285 + 0.934377i \(0.615957\pi\)
\(30\) 0 0
\(31\) 302.851 1.75464 0.877318 0.479910i \(-0.159331\pi\)
0.877318 + 0.479910i \(0.159331\pi\)
\(32\) 0 0
\(33\) −260.841 −1.37596
\(34\) 0 0
\(35\) −283.138 −1.36740
\(36\) 0 0
\(37\) 182.995 0.813086 0.406543 0.913632i \(-0.366734\pi\)
0.406543 + 0.913632i \(0.366734\pi\)
\(38\) 0 0
\(39\) 41.5486 0.170592
\(40\) 0 0
\(41\) −94.5744 −0.360245 −0.180122 0.983644i \(-0.557649\pi\)
−0.180122 + 0.983644i \(0.557649\pi\)
\(42\) 0 0
\(43\) 184.641 0.654825 0.327413 0.944881i \(-0.393823\pi\)
0.327413 + 0.944881i \(0.393823\pi\)
\(44\) 0 0
\(45\) 43.0155 0.142497
\(46\) 0 0
\(47\) −296.841 −0.921249 −0.460624 0.887595i \(-0.652375\pi\)
−0.460624 + 0.887595i \(0.652375\pi\)
\(48\) 0 0
\(49\) −24.1487 −0.0704045
\(50\) 0 0
\(51\) −636.420 −1.74739
\(52\) 0 0
\(53\) 102.995 0.266933 0.133466 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(54\) 0 0
\(55\) −839.251 −2.05754
\(56\) 0 0
\(57\) −248.862 −0.578290
\(58\) 0 0
\(59\) −93.3693 −0.206028 −0.103014 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(60\) 0 0
\(61\) 338.974 0.711495 0.355748 0.934582i \(-0.384226\pi\)
0.355748 + 0.934582i \(0.384226\pi\)
\(62\) 0 0
\(63\) −48.4411 −0.0968731
\(64\) 0 0
\(65\) 133.682 0.255095
\(66\) 0 0
\(67\) 489.041 0.891729 0.445865 0.895100i \(-0.352896\pi\)
0.445865 + 0.895100i \(0.352896\pi\)
\(68\) 0 0
\(69\) 634.297 1.10667
\(70\) 0 0
\(71\) 86.9845 0.145397 0.0726983 0.997354i \(-0.476839\pi\)
0.0726983 + 0.997354i \(0.476839\pi\)
\(72\) 0 0
\(73\) −154.267 −0.247336 −0.123668 0.992324i \(-0.539466\pi\)
−0.123668 + 0.992324i \(0.539466\pi\)
\(74\) 0 0
\(75\) −623.051 −0.959250
\(76\) 0 0
\(77\) 945.108 1.39877
\(78\) 0 0
\(79\) 449.415 0.640040 0.320020 0.947411i \(-0.396310\pi\)
0.320020 + 0.947411i \(0.396310\pi\)
\(80\) 0 0
\(81\) −648.395 −0.889430
\(82\) 0 0
\(83\) 383.933 0.507737 0.253868 0.967239i \(-0.418297\pi\)
0.253868 + 0.967239i \(0.418297\pi\)
\(84\) 0 0
\(85\) −2047.67 −2.61295
\(86\) 0 0
\(87\) 548.420 0.675826
\(88\) 0 0
\(89\) −517.672 −0.616551 −0.308276 0.951297i \(-0.599752\pi\)
−0.308276 + 0.951297i \(0.599752\pi\)
\(90\) 0 0
\(91\) −150.543 −0.173420
\(92\) 0 0
\(93\) −1492.51 −1.66415
\(94\) 0 0
\(95\) −800.708 −0.864746
\(96\) 0 0
\(97\) 1739.39 1.82071 0.910355 0.413829i \(-0.135809\pi\)
0.910355 + 0.413829i \(0.135809\pi\)
\(98\) 0 0
\(99\) −143.584 −0.145765
\(100\) 0 0
\(101\) 1158.99 1.14182 0.570912 0.821011i \(-0.306590\pi\)
0.570912 + 0.821011i \(0.306590\pi\)
\(102\) 0 0
\(103\) 1635.01 1.56410 0.782048 0.623218i \(-0.214175\pi\)
0.782048 + 0.623218i \(0.214175\pi\)
\(104\) 0 0
\(105\) 1395.36 1.29689
\(106\) 0 0
\(107\) 282.589 0.255317 0.127659 0.991818i \(-0.459254\pi\)
0.127659 + 0.991818i \(0.459254\pi\)
\(108\) 0 0
\(109\) −1004.98 −0.883120 −0.441560 0.897232i \(-0.645575\pi\)
−0.441560 + 0.897232i \(0.645575\pi\)
\(110\) 0 0
\(111\) −901.836 −0.771157
\(112\) 0 0
\(113\) −567.108 −0.472115 −0.236057 0.971739i \(-0.575855\pi\)
−0.236057 + 0.971739i \(0.575855\pi\)
\(114\) 0 0
\(115\) 2040.84 1.65486
\(116\) 0 0
\(117\) 22.8711 0.0180721
\(118\) 0 0
\(119\) 2305.95 1.77635
\(120\) 0 0
\(121\) 1470.39 1.10473
\(122\) 0 0
\(123\) 466.082 0.341668
\(124\) 0 0
\(125\) −22.6053 −0.0161750
\(126\) 0 0
\(127\) 496.616 0.346988 0.173494 0.984835i \(-0.444494\pi\)
0.173494 + 0.984835i \(0.444494\pi\)
\(128\) 0 0
\(129\) −909.948 −0.621058
\(130\) 0 0
\(131\) 1658.72 1.10628 0.553142 0.833087i \(-0.313429\pi\)
0.553142 + 0.833087i \(0.313429\pi\)
\(132\) 0 0
\(133\) 901.703 0.587876
\(134\) 0 0
\(135\) −2321.87 −1.48025
\(136\) 0 0
\(137\) −432.872 −0.269947 −0.134974 0.990849i \(-0.543095\pi\)
−0.134974 + 0.990849i \(0.543095\pi\)
\(138\) 0 0
\(139\) −146.148 −0.0891809 −0.0445904 0.999005i \(-0.514198\pi\)
−0.0445904 + 0.999005i \(0.514198\pi\)
\(140\) 0 0
\(141\) 1462.89 0.873743
\(142\) 0 0
\(143\) −446.226 −0.260946
\(144\) 0 0
\(145\) 1764.53 1.01060
\(146\) 0 0
\(147\) 119.010 0.0667740
\(148\) 0 0
\(149\) −3556.29 −1.95532 −0.977659 0.210198i \(-0.932589\pi\)
−0.977659 + 0.210198i \(0.932589\pi\)
\(150\) 0 0
\(151\) 1320.32 0.711562 0.355781 0.934569i \(-0.384215\pi\)
0.355781 + 0.934569i \(0.384215\pi\)
\(152\) 0 0
\(153\) −350.328 −0.185114
\(154\) 0 0
\(155\) −4802.13 −2.48849
\(156\) 0 0
\(157\) −2040.43 −1.03722 −0.518612 0.855010i \(-0.673551\pi\)
−0.518612 + 0.855010i \(0.673551\pi\)
\(158\) 0 0
\(159\) −507.580 −0.253168
\(160\) 0 0
\(161\) −2298.26 −1.12502
\(162\) 0 0
\(163\) −2392.21 −1.14952 −0.574762 0.818321i \(-0.694905\pi\)
−0.574762 + 0.818321i \(0.694905\pi\)
\(164\) 0 0
\(165\) 4136.00 1.95144
\(166\) 0 0
\(167\) −2068.34 −0.958400 −0.479200 0.877706i \(-0.659073\pi\)
−0.479200 + 0.877706i \(0.659073\pi\)
\(168\) 0 0
\(169\) −2125.92 −0.967648
\(170\) 0 0
\(171\) −136.990 −0.0612625
\(172\) 0 0
\(173\) 2441.87 1.07313 0.536566 0.843859i \(-0.319721\pi\)
0.536566 + 0.843859i \(0.319721\pi\)
\(174\) 0 0
\(175\) 2257.51 0.975152
\(176\) 0 0
\(177\) 460.143 0.195404
\(178\) 0 0
\(179\) −2430.84 −1.01502 −0.507512 0.861645i \(-0.669435\pi\)
−0.507512 + 0.861645i \(0.669435\pi\)
\(180\) 0 0
\(181\) −1928.96 −0.792148 −0.396074 0.918219i \(-0.629628\pi\)
−0.396074 + 0.918219i \(0.629628\pi\)
\(182\) 0 0
\(183\) −1670.53 −0.674806
\(184\) 0 0
\(185\) −2901.64 −1.15315
\(186\) 0 0
\(187\) 6835.07 2.67288
\(188\) 0 0
\(189\) 2614.73 1.00631
\(190\) 0 0
\(191\) −4087.34 −1.54843 −0.774214 0.632924i \(-0.781855\pi\)
−0.774214 + 0.632924i \(0.781855\pi\)
\(192\) 0 0
\(193\) −1156.52 −0.431339 −0.215669 0.976466i \(-0.569193\pi\)
−0.215669 + 0.976466i \(0.569193\pi\)
\(194\) 0 0
\(195\) −658.811 −0.241941
\(196\) 0 0
\(197\) 2658.44 0.961452 0.480726 0.876871i \(-0.340373\pi\)
0.480726 + 0.876871i \(0.340373\pi\)
\(198\) 0 0
\(199\) 742.369 0.264448 0.132224 0.991220i \(-0.457788\pi\)
0.132224 + 0.991220i \(0.457788\pi\)
\(200\) 0 0
\(201\) −2410.09 −0.845745
\(202\) 0 0
\(203\) −1987.10 −0.687029
\(204\) 0 0
\(205\) 1499.61 0.510914
\(206\) 0 0
\(207\) 349.160 0.117238
\(208\) 0 0
\(209\) 2672.74 0.884580
\(210\) 0 0
\(211\) −3838.30 −1.25232 −0.626160 0.779694i \(-0.715374\pi\)
−0.626160 + 0.779694i \(0.715374\pi\)
\(212\) 0 0
\(213\) −428.677 −0.137899
\(214\) 0 0
\(215\) −2927.74 −0.928700
\(216\) 0 0
\(217\) 5407.84 1.69174
\(218\) 0 0
\(219\) 760.257 0.234582
\(220\) 0 0
\(221\) −1088.74 −0.331387
\(222\) 0 0
\(223\) 3418.34 1.02650 0.513249 0.858240i \(-0.328442\pi\)
0.513249 + 0.858240i \(0.328442\pi\)
\(224\) 0 0
\(225\) −342.969 −0.101620
\(226\) 0 0
\(227\) 515.729 0.150793 0.0753967 0.997154i \(-0.475978\pi\)
0.0753967 + 0.997154i \(0.475978\pi\)
\(228\) 0 0
\(229\) −1595.26 −0.460340 −0.230170 0.973150i \(-0.573928\pi\)
−0.230170 + 0.973150i \(0.573928\pi\)
\(230\) 0 0
\(231\) −4657.68 −1.32664
\(232\) 0 0
\(233\) 5251.27 1.47649 0.738245 0.674533i \(-0.235655\pi\)
0.738245 + 0.674533i \(0.235655\pi\)
\(234\) 0 0
\(235\) 4706.83 1.30655
\(236\) 0 0
\(237\) −2214.81 −0.607035
\(238\) 0 0
\(239\) 2029.27 0.549216 0.274608 0.961556i \(-0.411452\pi\)
0.274608 + 0.961556i \(0.411452\pi\)
\(240\) 0 0
\(241\) −4615.72 −1.23371 −0.616856 0.787076i \(-0.711594\pi\)
−0.616856 + 0.787076i \(0.711594\pi\)
\(242\) 0 0
\(243\) −758.210 −0.200161
\(244\) 0 0
\(245\) 382.912 0.0998505
\(246\) 0 0
\(247\) −425.733 −0.109671
\(248\) 0 0
\(249\) −1892.10 −0.481554
\(250\) 0 0
\(251\) −2702.52 −0.679607 −0.339804 0.940496i \(-0.610361\pi\)
−0.339804 + 0.940496i \(0.610361\pi\)
\(252\) 0 0
\(253\) −6812.27 −1.69282
\(254\) 0 0
\(255\) 10091.3 2.47821
\(256\) 0 0
\(257\) 991.785 0.240723 0.120362 0.992730i \(-0.461595\pi\)
0.120362 + 0.992730i \(0.461595\pi\)
\(258\) 0 0
\(259\) 3267.63 0.783941
\(260\) 0 0
\(261\) 301.887 0.0715952
\(262\) 0 0
\(263\) 7430.74 1.74220 0.871101 0.491105i \(-0.163407\pi\)
0.871101 + 0.491105i \(0.163407\pi\)
\(264\) 0 0
\(265\) −1633.13 −0.378575
\(266\) 0 0
\(267\) 2551.19 0.584758
\(268\) 0 0
\(269\) −3281.50 −0.743778 −0.371889 0.928277i \(-0.621290\pi\)
−0.371889 + 0.928277i \(0.621290\pi\)
\(270\) 0 0
\(271\) 221.518 0.0496542 0.0248271 0.999692i \(-0.492096\pi\)
0.0248271 + 0.999692i \(0.492096\pi\)
\(272\) 0 0
\(273\) 741.909 0.164477
\(274\) 0 0
\(275\) 6691.48 1.46731
\(276\) 0 0
\(277\) −1106.28 −0.239963 −0.119982 0.992776i \(-0.538284\pi\)
−0.119982 + 0.992776i \(0.538284\pi\)
\(278\) 0 0
\(279\) −821.579 −0.176296
\(280\) 0 0
\(281\) −2372.52 −0.503676 −0.251838 0.967769i \(-0.581035\pi\)
−0.251838 + 0.967769i \(0.581035\pi\)
\(282\) 0 0
\(283\) 2549.89 0.535602 0.267801 0.963474i \(-0.413703\pi\)
0.267801 + 0.963474i \(0.413703\pi\)
\(284\) 0 0
\(285\) 3946.05 0.820154
\(286\) 0 0
\(287\) −1688.76 −0.347332
\(288\) 0 0
\(289\) 11763.7 2.39441
\(290\) 0 0
\(291\) −8572.09 −1.72682
\(292\) 0 0
\(293\) −130.319 −0.0259840 −0.0129920 0.999916i \(-0.504136\pi\)
−0.0129920 + 0.999916i \(0.504136\pi\)
\(294\) 0 0
\(295\) 1480.50 0.292197
\(296\) 0 0
\(297\) 7750.32 1.51421
\(298\) 0 0
\(299\) 1085.11 0.209877
\(300\) 0 0
\(301\) 3297.03 0.631353
\(302\) 0 0
\(303\) −5711.76 −1.08294
\(304\) 0 0
\(305\) −5374.91 −1.00907
\(306\) 0 0
\(307\) −62.1182 −0.0115481 −0.00577406 0.999983i \(-0.501838\pi\)
−0.00577406 + 0.999983i \(0.501838\pi\)
\(308\) 0 0
\(309\) −8057.64 −1.48344
\(310\) 0 0
\(311\) 593.837 0.108275 0.0541373 0.998533i \(-0.482759\pi\)
0.0541373 + 0.998533i \(0.482759\pi\)
\(312\) 0 0
\(313\) 3669.69 0.662694 0.331347 0.943509i \(-0.392497\pi\)
0.331347 + 0.943509i \(0.392497\pi\)
\(314\) 0 0
\(315\) 768.102 0.137389
\(316\) 0 0
\(317\) −374.339 −0.0663249 −0.0331625 0.999450i \(-0.510558\pi\)
−0.0331625 + 0.999450i \(0.510558\pi\)
\(318\) 0 0
\(319\) −5889.96 −1.03378
\(320\) 0 0
\(321\) −1392.66 −0.242151
\(322\) 0 0
\(323\) 6521.16 1.12337
\(324\) 0 0
\(325\) −1065.87 −0.181919
\(326\) 0 0
\(327\) 4952.77 0.837580
\(328\) 0 0
\(329\) −5300.51 −0.888227
\(330\) 0 0
\(331\) −5871.13 −0.974945 −0.487472 0.873138i \(-0.662081\pi\)
−0.487472 + 0.873138i \(0.662081\pi\)
\(332\) 0 0
\(333\) −496.431 −0.0816944
\(334\) 0 0
\(335\) −7754.43 −1.26469
\(336\) 0 0
\(337\) 1047.83 0.169373 0.0846865 0.996408i \(-0.473011\pi\)
0.0846865 + 0.996408i \(0.473011\pi\)
\(338\) 0 0
\(339\) 2794.82 0.447769
\(340\) 0 0
\(341\) 16029.4 2.54557
\(342\) 0 0
\(343\) −6555.96 −1.03204
\(344\) 0 0
\(345\) −10057.7 −1.56953
\(346\) 0 0
\(347\) −3921.88 −0.606737 −0.303368 0.952873i \(-0.598111\pi\)
−0.303368 + 0.952873i \(0.598111\pi\)
\(348\) 0 0
\(349\) 11604.9 1.77993 0.889966 0.456027i \(-0.150728\pi\)
0.889966 + 0.456027i \(0.150728\pi\)
\(350\) 0 0
\(351\) −1234.53 −0.187733
\(352\) 0 0
\(353\) −1530.22 −0.230724 −0.115362 0.993324i \(-0.536803\pi\)
−0.115362 + 0.993324i \(0.536803\pi\)
\(354\) 0 0
\(355\) −1379.26 −0.206207
\(356\) 0 0
\(357\) −11364.2 −1.68475
\(358\) 0 0
\(359\) 10764.0 1.58246 0.791231 0.611518i \(-0.209441\pi\)
0.791231 + 0.611518i \(0.209441\pi\)
\(360\) 0 0
\(361\) −4309.01 −0.628227
\(362\) 0 0
\(363\) −7246.40 −1.04776
\(364\) 0 0
\(365\) 2446.11 0.350782
\(366\) 0 0
\(367\) 10713.0 1.52374 0.761869 0.647731i \(-0.224282\pi\)
0.761869 + 0.647731i \(0.224282\pi\)
\(368\) 0 0
\(369\) 256.563 0.0361954
\(370\) 0 0
\(371\) 1839.12 0.257365
\(372\) 0 0
\(373\) 9186.32 1.27520 0.637600 0.770368i \(-0.279927\pi\)
0.637600 + 0.770368i \(0.279927\pi\)
\(374\) 0 0
\(375\) 111.404 0.0153409
\(376\) 0 0
\(377\) 938.194 0.128168
\(378\) 0 0
\(379\) −14201.0 −1.92468 −0.962342 0.271842i \(-0.912367\pi\)
−0.962342 + 0.271842i \(0.912367\pi\)
\(380\) 0 0
\(381\) −2447.42 −0.329095
\(382\) 0 0
\(383\) −9691.24 −1.29295 −0.646474 0.762936i \(-0.723757\pi\)
−0.646474 + 0.762936i \(0.723757\pi\)
\(384\) 0 0
\(385\) −14986.0 −1.98379
\(386\) 0 0
\(387\) −500.897 −0.0657933
\(388\) 0 0
\(389\) −6358.79 −0.828801 −0.414400 0.910095i \(-0.636009\pi\)
−0.414400 + 0.910095i \(0.636009\pi\)
\(390\) 0 0
\(391\) −16621.1 −2.14978
\(392\) 0 0
\(393\) −8174.52 −1.04924
\(394\) 0 0
\(395\) −7126.11 −0.907731
\(396\) 0 0
\(397\) −13034.2 −1.64778 −0.823888 0.566752i \(-0.808200\pi\)
−0.823888 + 0.566752i \(0.808200\pi\)
\(398\) 0 0
\(399\) −4443.77 −0.557561
\(400\) 0 0
\(401\) 6705.06 0.834999 0.417499 0.908677i \(-0.362907\pi\)
0.417499 + 0.908677i \(0.362907\pi\)
\(402\) 0 0
\(403\) −2553.27 −0.315602
\(404\) 0 0
\(405\) 10281.2 1.26143
\(406\) 0 0
\(407\) 9685.59 1.17960
\(408\) 0 0
\(409\) 10967.8 1.32598 0.662988 0.748630i \(-0.269288\pi\)
0.662988 + 0.748630i \(0.269288\pi\)
\(410\) 0 0
\(411\) 2133.28 0.256027
\(412\) 0 0
\(413\) −1667.24 −0.198643
\(414\) 0 0
\(415\) −6087.80 −0.720093
\(416\) 0 0
\(417\) 720.249 0.0845821
\(418\) 0 0
\(419\) 13686.0 1.59571 0.797856 0.602849i \(-0.205968\pi\)
0.797856 + 0.602849i \(0.205968\pi\)
\(420\) 0 0
\(421\) 8552.92 0.990128 0.495064 0.868857i \(-0.335145\pi\)
0.495064 + 0.868857i \(0.335145\pi\)
\(422\) 0 0
\(423\) 805.274 0.0925621
\(424\) 0 0
\(425\) 16326.4 1.86340
\(426\) 0 0
\(427\) 6052.86 0.685992
\(428\) 0 0
\(429\) 2199.09 0.247490
\(430\) 0 0
\(431\) −12187.6 −1.36208 −0.681039 0.732247i \(-0.738472\pi\)
−0.681039 + 0.732247i \(0.738472\pi\)
\(432\) 0 0
\(433\) −8906.35 −0.988480 −0.494240 0.869326i \(-0.664554\pi\)
−0.494240 + 0.869326i \(0.664554\pi\)
\(434\) 0 0
\(435\) −8695.98 −0.958483
\(436\) 0 0
\(437\) −6499.41 −0.711462
\(438\) 0 0
\(439\) 4655.21 0.506107 0.253054 0.967452i \(-0.418565\pi\)
0.253054 + 0.967452i \(0.418565\pi\)
\(440\) 0 0
\(441\) 65.5110 0.00707386
\(442\) 0 0
\(443\) 10008.8 1.07343 0.536716 0.843763i \(-0.319665\pi\)
0.536716 + 0.843763i \(0.319665\pi\)
\(444\) 0 0
\(445\) 8208.41 0.874418
\(446\) 0 0
\(447\) 17526.1 1.85449
\(448\) 0 0
\(449\) −3373.71 −0.354600 −0.177300 0.984157i \(-0.556736\pi\)
−0.177300 + 0.984157i \(0.556736\pi\)
\(450\) 0 0
\(451\) −5005.65 −0.522632
\(452\) 0 0
\(453\) −6506.79 −0.674869
\(454\) 0 0
\(455\) 2387.08 0.245952
\(456\) 0 0
\(457\) −13600.4 −1.39212 −0.696060 0.717984i \(-0.745065\pi\)
−0.696060 + 0.717984i \(0.745065\pi\)
\(458\) 0 0
\(459\) 18909.8 1.92295
\(460\) 0 0
\(461\) 3559.45 0.359609 0.179805 0.983702i \(-0.442453\pi\)
0.179805 + 0.983702i \(0.442453\pi\)
\(462\) 0 0
\(463\) 8019.14 0.804926 0.402463 0.915436i \(-0.368154\pi\)
0.402463 + 0.915436i \(0.368154\pi\)
\(464\) 0 0
\(465\) 23665.9 2.36017
\(466\) 0 0
\(467\) −2066.14 −0.204731 −0.102365 0.994747i \(-0.532641\pi\)
−0.102365 + 0.994747i \(0.532641\pi\)
\(468\) 0 0
\(469\) 8732.51 0.859765
\(470\) 0 0
\(471\) 10055.7 0.983737
\(472\) 0 0
\(473\) 9772.72 0.950000
\(474\) 0 0
\(475\) 6384.17 0.616686
\(476\) 0 0
\(477\) −279.406 −0.0268199
\(478\) 0 0
\(479\) 3679.26 0.350960 0.175480 0.984483i \(-0.443852\pi\)
0.175480 + 0.984483i \(0.443852\pi\)
\(480\) 0 0
\(481\) −1542.79 −0.146248
\(482\) 0 0
\(483\) 11326.3 1.06700
\(484\) 0 0
\(485\) −27580.5 −2.58220
\(486\) 0 0
\(487\) −10884.4 −1.01277 −0.506386 0.862307i \(-0.669019\pi\)
−0.506386 + 0.862307i \(0.669019\pi\)
\(488\) 0 0
\(489\) 11789.3 1.09025
\(490\) 0 0
\(491\) −14548.9 −1.33723 −0.668616 0.743608i \(-0.733113\pi\)
−0.668616 + 0.743608i \(0.733113\pi\)
\(492\) 0 0
\(493\) −14370.8 −1.31284
\(494\) 0 0
\(495\) 2276.73 0.206730
\(496\) 0 0
\(497\) 1553.23 0.140185
\(498\) 0 0
\(499\) 8461.92 0.759134 0.379567 0.925164i \(-0.376073\pi\)
0.379567 + 0.925164i \(0.376073\pi\)
\(500\) 0 0
\(501\) 10193.2 0.908978
\(502\) 0 0
\(503\) 983.702 0.0871990 0.0435995 0.999049i \(-0.486117\pi\)
0.0435995 + 0.999049i \(0.486117\pi\)
\(504\) 0 0
\(505\) −18377.5 −1.61938
\(506\) 0 0
\(507\) 10477.0 0.917749
\(508\) 0 0
\(509\) −14472.5 −1.26028 −0.630140 0.776481i \(-0.717003\pi\)
−0.630140 + 0.776481i \(0.717003\pi\)
\(510\) 0 0
\(511\) −2754.65 −0.238470
\(512\) 0 0
\(513\) 7394.38 0.636393
\(514\) 0 0
\(515\) −25925.3 −2.21826
\(516\) 0 0
\(517\) −15711.3 −1.33652
\(518\) 0 0
\(519\) −12034.0 −1.01779
\(520\) 0 0
\(521\) −10761.8 −0.904959 −0.452480 0.891775i \(-0.649460\pi\)
−0.452480 + 0.891775i \(0.649460\pi\)
\(522\) 0 0
\(523\) −1516.67 −0.126806 −0.0634029 0.997988i \(-0.520195\pi\)
−0.0634029 + 0.997988i \(0.520195\pi\)
\(524\) 0 0
\(525\) −11125.5 −0.924866
\(526\) 0 0
\(527\) 39109.7 3.23273
\(528\) 0 0
\(529\) 4398.66 0.361524
\(530\) 0 0
\(531\) 253.293 0.0207006
\(532\) 0 0
\(533\) 797.336 0.0647963
\(534\) 0 0
\(535\) −4480.85 −0.362101
\(536\) 0 0
\(537\) 11979.7 0.962682
\(538\) 0 0
\(539\) −1278.15 −0.102141
\(540\) 0 0
\(541\) 22541.9 1.79141 0.895704 0.444652i \(-0.146673\pi\)
0.895704 + 0.444652i \(0.146673\pi\)
\(542\) 0 0
\(543\) 9506.33 0.751299
\(544\) 0 0
\(545\) 15935.4 1.25248
\(546\) 0 0
\(547\) −10202.0 −0.797448 −0.398724 0.917071i \(-0.630547\pi\)
−0.398724 + 0.917071i \(0.630547\pi\)
\(548\) 0 0
\(549\) −919.574 −0.0714872
\(550\) 0 0
\(551\) −5619.46 −0.434477
\(552\) 0 0
\(553\) 8024.94 0.617098
\(554\) 0 0
\(555\) 14299.9 1.09369
\(556\) 0 0
\(557\) 7139.06 0.543073 0.271536 0.962428i \(-0.412468\pi\)
0.271536 + 0.962428i \(0.412468\pi\)
\(558\) 0 0
\(559\) −1556.67 −0.117782
\(560\) 0 0
\(561\) −33684.6 −2.53505
\(562\) 0 0
\(563\) 19558.0 1.46407 0.732034 0.681268i \(-0.238571\pi\)
0.732034 + 0.681268i \(0.238571\pi\)
\(564\) 0 0
\(565\) 8992.29 0.669572
\(566\) 0 0
\(567\) −11578.0 −0.857549
\(568\) 0 0
\(569\) −5835.86 −0.429969 −0.214984 0.976617i \(-0.568970\pi\)
−0.214984 + 0.976617i \(0.568970\pi\)
\(570\) 0 0
\(571\) −9820.65 −0.719757 −0.359879 0.932999i \(-0.617182\pi\)
−0.359879 + 0.932999i \(0.617182\pi\)
\(572\) 0 0
\(573\) 20143.3 1.46858
\(574\) 0 0
\(575\) −16271.9 −1.18015
\(576\) 0 0
\(577\) 451.231 0.0325563 0.0162782 0.999868i \(-0.494818\pi\)
0.0162782 + 0.999868i \(0.494818\pi\)
\(578\) 0 0
\(579\) 5699.58 0.409096
\(580\) 0 0
\(581\) 6855.67 0.489537
\(582\) 0 0
\(583\) 5451.33 0.387257
\(584\) 0 0
\(585\) −362.654 −0.0256306
\(586\) 0 0
\(587\) 24775.7 1.74208 0.871042 0.491209i \(-0.163445\pi\)
0.871042 + 0.491209i \(0.163445\pi\)
\(588\) 0 0
\(589\) 15293.2 1.06986
\(590\) 0 0
\(591\) −13101.3 −0.911873
\(592\) 0 0
\(593\) −4207.64 −0.291378 −0.145689 0.989330i \(-0.546540\pi\)
−0.145689 + 0.989330i \(0.546540\pi\)
\(594\) 0 0
\(595\) −36564.1 −2.51929
\(596\) 0 0
\(597\) −3658.54 −0.250811
\(598\) 0 0
\(599\) −1916.32 −0.130715 −0.0653577 0.997862i \(-0.520819\pi\)
−0.0653577 + 0.997862i \(0.520819\pi\)
\(600\) 0 0
\(601\) −19361.1 −1.31407 −0.657034 0.753861i \(-0.728189\pi\)
−0.657034 + 0.753861i \(0.728189\pi\)
\(602\) 0 0
\(603\) −1326.68 −0.0895961
\(604\) 0 0
\(605\) −23315.2 −1.56677
\(606\) 0 0
\(607\) −6097.02 −0.407694 −0.203847 0.979003i \(-0.565345\pi\)
−0.203847 + 0.979003i \(0.565345\pi\)
\(608\) 0 0
\(609\) 9792.82 0.651601
\(610\) 0 0
\(611\) 2502.60 0.165703
\(612\) 0 0
\(613\) 13837.7 0.911743 0.455872 0.890046i \(-0.349328\pi\)
0.455872 + 0.890046i \(0.349328\pi\)
\(614\) 0 0
\(615\) −7390.38 −0.484567
\(616\) 0 0
\(617\) 11959.1 0.780318 0.390159 0.920747i \(-0.372420\pi\)
0.390159 + 0.920747i \(0.372420\pi\)
\(618\) 0 0
\(619\) −8385.02 −0.544463 −0.272231 0.962232i \(-0.587762\pi\)
−0.272231 + 0.962232i \(0.587762\pi\)
\(620\) 0 0
\(621\) −18846.8 −1.21787
\(622\) 0 0
\(623\) −9243.75 −0.594451
\(624\) 0 0
\(625\) −15444.8 −0.988465
\(626\) 0 0
\(627\) −13171.8 −0.838965
\(628\) 0 0
\(629\) 23631.7 1.49802
\(630\) 0 0
\(631\) −13592.1 −0.857513 −0.428757 0.903420i \(-0.641048\pi\)
−0.428757 + 0.903420i \(0.641048\pi\)
\(632\) 0 0
\(633\) 18915.9 1.18774
\(634\) 0 0
\(635\) −7874.54 −0.492113
\(636\) 0 0
\(637\) 203.593 0.0126635
\(638\) 0 0
\(639\) −235.973 −0.0146087
\(640\) 0 0
\(641\) −18123.4 −1.11674 −0.558371 0.829591i \(-0.688573\pi\)
−0.558371 + 0.829591i \(0.688573\pi\)
\(642\) 0 0
\(643\) −25042.1 −1.53587 −0.767935 0.640528i \(-0.778716\pi\)
−0.767935 + 0.640528i \(0.778716\pi\)
\(644\) 0 0
\(645\) 14428.5 0.880809
\(646\) 0 0
\(647\) 6022.08 0.365923 0.182962 0.983120i \(-0.441432\pi\)
0.182962 + 0.983120i \(0.441432\pi\)
\(648\) 0 0
\(649\) −4941.87 −0.298899
\(650\) 0 0
\(651\) −26650.9 −1.60450
\(652\) 0 0
\(653\) 14093.8 0.844617 0.422308 0.906452i \(-0.361220\pi\)
0.422308 + 0.906452i \(0.361220\pi\)
\(654\) 0 0
\(655\) −26301.4 −1.56898
\(656\) 0 0
\(657\) 418.496 0.0248510
\(658\) 0 0
\(659\) 17240.7 1.01912 0.509562 0.860434i \(-0.329807\pi\)
0.509562 + 0.860434i \(0.329807\pi\)
\(660\) 0 0
\(661\) −20209.5 −1.18920 −0.594598 0.804023i \(-0.702689\pi\)
−0.594598 + 0.804023i \(0.702689\pi\)
\(662\) 0 0
\(663\) 5365.52 0.314298
\(664\) 0 0
\(665\) −14297.8 −0.833749
\(666\) 0 0
\(667\) 14322.8 0.831459
\(668\) 0 0
\(669\) −16846.3 −0.973564
\(670\) 0 0
\(671\) 17941.3 1.03221
\(672\) 0 0
\(673\) 6838.55 0.391689 0.195845 0.980635i \(-0.437255\pi\)
0.195845 + 0.980635i \(0.437255\pi\)
\(674\) 0 0
\(675\) 18512.6 1.05563
\(676\) 0 0
\(677\) 187.425 0.0106401 0.00532003 0.999986i \(-0.498307\pi\)
0.00532003 + 0.999986i \(0.498307\pi\)
\(678\) 0 0
\(679\) 31059.3 1.75545
\(680\) 0 0
\(681\) −2541.62 −0.143018
\(682\) 0 0
\(683\) −23476.0 −1.31520 −0.657601 0.753367i \(-0.728429\pi\)
−0.657601 + 0.753367i \(0.728429\pi\)
\(684\) 0 0
\(685\) 6863.79 0.382850
\(686\) 0 0
\(687\) 7861.77 0.436602
\(688\) 0 0
\(689\) −868.327 −0.0480125
\(690\) 0 0
\(691\) 6309.92 0.347382 0.173691 0.984800i \(-0.444431\pi\)
0.173691 + 0.984800i \(0.444431\pi\)
\(692\) 0 0
\(693\) −2563.90 −0.140540
\(694\) 0 0
\(695\) 2317.39 0.126480
\(696\) 0 0
\(697\) −12213.2 −0.663712
\(698\) 0 0
\(699\) −25879.3 −1.40035
\(700\) 0 0
\(701\) 24920.4 1.34270 0.671348 0.741142i \(-0.265716\pi\)
0.671348 + 0.741142i \(0.265716\pi\)
\(702\) 0 0
\(703\) 9240.77 0.495764
\(704\) 0 0
\(705\) −23196.2 −1.23918
\(706\) 0 0
\(707\) 20695.5 1.10090
\(708\) 0 0
\(709\) −14478.9 −0.766949 −0.383474 0.923551i \(-0.625273\pi\)
−0.383474 + 0.923551i \(0.625273\pi\)
\(710\) 0 0
\(711\) −1219.18 −0.0643077
\(712\) 0 0
\(713\) −38979.3 −2.04738
\(714\) 0 0
\(715\) 7075.54 0.370084
\(716\) 0 0
\(717\) −10000.7 −0.520895
\(718\) 0 0
\(719\) −6411.30 −0.332547 −0.166273 0.986080i \(-0.553173\pi\)
−0.166273 + 0.986080i \(0.553173\pi\)
\(720\) 0 0
\(721\) 29195.3 1.50803
\(722\) 0 0
\(723\) 22747.2 1.17009
\(724\) 0 0
\(725\) −14068.9 −0.720698
\(726\) 0 0
\(727\) −20626.3 −1.05225 −0.526126 0.850407i \(-0.676356\pi\)
−0.526126 + 0.850407i \(0.676356\pi\)
\(728\) 0 0
\(729\) 21243.3 1.07927
\(730\) 0 0
\(731\) 23844.3 1.20645
\(732\) 0 0
\(733\) −5732.29 −0.288850 −0.144425 0.989516i \(-0.546133\pi\)
−0.144425 + 0.989516i \(0.546133\pi\)
\(734\) 0 0
\(735\) −1887.07 −0.0947015
\(736\) 0 0
\(737\) 25884.1 1.29369
\(738\) 0 0
\(739\) 36336.0 1.80872 0.904359 0.426773i \(-0.140350\pi\)
0.904359 + 0.426773i \(0.140350\pi\)
\(740\) 0 0
\(741\) 2098.10 0.104016
\(742\) 0 0
\(743\) 4363.83 0.215469 0.107734 0.994180i \(-0.465640\pi\)
0.107734 + 0.994180i \(0.465640\pi\)
\(744\) 0 0
\(745\) 56389.9 2.77311
\(746\) 0 0
\(747\) −1041.54 −0.0510146
\(748\) 0 0
\(749\) 5046.03 0.246166
\(750\) 0 0
\(751\) −16162.7 −0.785334 −0.392667 0.919681i \(-0.628448\pi\)
−0.392667 + 0.919681i \(0.628448\pi\)
\(752\) 0 0
\(753\) 13318.6 0.644562
\(754\) 0 0
\(755\) −20935.5 −1.00917
\(756\) 0 0
\(757\) −4411.92 −0.211828 −0.105914 0.994375i \(-0.533777\pi\)
−0.105914 + 0.994375i \(0.533777\pi\)
\(758\) 0 0
\(759\) 33572.2 1.60553
\(760\) 0 0
\(761\) 28281.8 1.34719 0.673597 0.739099i \(-0.264749\pi\)
0.673597 + 0.739099i \(0.264749\pi\)
\(762\) 0 0
\(763\) −17945.4 −0.851465
\(764\) 0 0
\(765\) 5554.95 0.262535
\(766\) 0 0
\(767\) 787.176 0.0370577
\(768\) 0 0
\(769\) −15016.8 −0.704186 −0.352093 0.935965i \(-0.614530\pi\)
−0.352093 + 0.935965i \(0.614530\pi\)
\(770\) 0 0
\(771\) −4887.72 −0.228310
\(772\) 0 0
\(773\) −8674.11 −0.403604 −0.201802 0.979426i \(-0.564680\pi\)
−0.201802 + 0.979426i \(0.564680\pi\)
\(774\) 0 0
\(775\) 38288.2 1.77465
\(776\) 0 0
\(777\) −16103.5 −0.743515
\(778\) 0 0
\(779\) −4775.76 −0.219653
\(780\) 0 0
\(781\) 4603.94 0.210937
\(782\) 0 0
\(783\) −16295.1 −0.743729
\(784\) 0 0
\(785\) 32353.9 1.47103
\(786\) 0 0
\(787\) 23660.2 1.07166 0.535829 0.844327i \(-0.319999\pi\)
0.535829 + 0.844327i \(0.319999\pi\)
\(788\) 0 0
\(789\) −36620.2 −1.65236
\(790\) 0 0
\(791\) −10126.5 −0.455192
\(792\) 0 0
\(793\) −2857.82 −0.127975
\(794\) 0 0
\(795\) 8048.39 0.359053
\(796\) 0 0
\(797\) −35432.5 −1.57476 −0.787380 0.616468i \(-0.788563\pi\)
−0.787380 + 0.616468i \(0.788563\pi\)
\(798\) 0 0
\(799\) −38333.6 −1.69730
\(800\) 0 0
\(801\) 1404.35 0.0619477
\(802\) 0 0
\(803\) −8165.05 −0.358827
\(804\) 0 0
\(805\) 36442.1 1.59555
\(806\) 0 0
\(807\) 16171.9 0.705424
\(808\) 0 0
\(809\) −14005.5 −0.608664 −0.304332 0.952566i \(-0.598433\pi\)
−0.304332 + 0.952566i \(0.598433\pi\)
\(810\) 0 0
\(811\) 22580.8 0.977706 0.488853 0.872366i \(-0.337416\pi\)
0.488853 + 0.872366i \(0.337416\pi\)
\(812\) 0 0
\(813\) −1091.69 −0.0470937
\(814\) 0 0
\(815\) 37931.8 1.63030
\(816\) 0 0
\(817\) 9323.90 0.399268
\(818\) 0 0
\(819\) 408.396 0.0174243
\(820\) 0 0
\(821\) −21308.9 −0.905827 −0.452914 0.891554i \(-0.649615\pi\)
−0.452914 + 0.891554i \(0.649615\pi\)
\(822\) 0 0
\(823\) 13571.4 0.574810 0.287405 0.957809i \(-0.407207\pi\)
0.287405 + 0.957809i \(0.407207\pi\)
\(824\) 0 0
\(825\) −32977.0 −1.39165
\(826\) 0 0
\(827\) 19295.2 0.811318 0.405659 0.914025i \(-0.367042\pi\)
0.405659 + 0.914025i \(0.367042\pi\)
\(828\) 0 0
\(829\) −28752.7 −1.20461 −0.602306 0.798265i \(-0.705751\pi\)
−0.602306 + 0.798265i \(0.705751\pi\)
\(830\) 0 0
\(831\) 5451.96 0.227589
\(832\) 0 0
\(833\) −3118.53 −0.129713
\(834\) 0 0
\(835\) 32796.4 1.35924
\(836\) 0 0
\(837\) 44346.7 1.83136
\(838\) 0 0
\(839\) −566.927 −0.0233284 −0.0116642 0.999932i \(-0.503713\pi\)
−0.0116642 + 0.999932i \(0.503713\pi\)
\(840\) 0 0
\(841\) −12005.3 −0.492243
\(842\) 0 0
\(843\) 11692.3 0.477703
\(844\) 0 0
\(845\) 33709.5 1.37236
\(846\) 0 0
\(847\) 26256.0 1.06513
\(848\) 0 0
\(849\) −12566.4 −0.507983
\(850\) 0 0
\(851\) −23552.8 −0.948744
\(852\) 0 0
\(853\) −34698.1 −1.39278 −0.696389 0.717665i \(-0.745211\pi\)
−0.696389 + 0.717665i \(0.745211\pi\)
\(854\) 0 0
\(855\) 2172.17 0.0868850
\(856\) 0 0
\(857\) 37362.0 1.48922 0.744610 0.667500i \(-0.232636\pi\)
0.744610 + 0.667500i \(0.232636\pi\)
\(858\) 0 0
\(859\) −29572.2 −1.17461 −0.587305 0.809366i \(-0.699811\pi\)
−0.587305 + 0.809366i \(0.699811\pi\)
\(860\) 0 0
\(861\) 8322.54 0.329421
\(862\) 0 0
\(863\) −11636.5 −0.458994 −0.229497 0.973309i \(-0.573708\pi\)
−0.229497 + 0.973309i \(0.573708\pi\)
\(864\) 0 0
\(865\) −38719.2 −1.52196
\(866\) 0 0
\(867\) −57974.1 −2.27094
\(868\) 0 0
\(869\) 23786.7 0.928550
\(870\) 0 0
\(871\) −4123.00 −0.160393
\(872\) 0 0
\(873\) −4718.65 −0.182935
\(874\) 0 0
\(875\) −403.649 −0.0155952
\(876\) 0 0
\(877\) −19878.2 −0.765380 −0.382690 0.923877i \(-0.625002\pi\)
−0.382690 + 0.923877i \(0.625002\pi\)
\(878\) 0 0
\(879\) 642.238 0.0246441
\(880\) 0 0
\(881\) −36460.3 −1.39430 −0.697150 0.716926i \(-0.745549\pi\)
−0.697150 + 0.716926i \(0.745549\pi\)
\(882\) 0 0
\(883\) 28119.9 1.07170 0.535849 0.844314i \(-0.319991\pi\)
0.535849 + 0.844314i \(0.319991\pi\)
\(884\) 0 0
\(885\) −7296.21 −0.277129
\(886\) 0 0
\(887\) −41888.4 −1.58565 −0.792826 0.609448i \(-0.791391\pi\)
−0.792826 + 0.609448i \(0.791391\pi\)
\(888\) 0 0
\(889\) 8867.77 0.334551
\(890\) 0 0
\(891\) −34318.4 −1.29036
\(892\) 0 0
\(893\) −14989.7 −0.561715
\(894\) 0 0
\(895\) 38544.3 1.43955
\(896\) 0 0
\(897\) −5347.62 −0.199055
\(898\) 0 0
\(899\) −33701.9 −1.25030
\(900\) 0 0
\(901\) 13300.6 0.491795
\(902\) 0 0
\(903\) −16248.4 −0.598796
\(904\) 0 0
\(905\) 30586.4 1.12346
\(906\) 0 0
\(907\) −12575.3 −0.460371 −0.230185 0.973147i \(-0.573933\pi\)
−0.230185 + 0.973147i \(0.573933\pi\)
\(908\) 0 0
\(909\) −3144.14 −0.114724
\(910\) 0 0
\(911\) −39814.7 −1.44799 −0.723995 0.689805i \(-0.757696\pi\)
−0.723995 + 0.689805i \(0.757696\pi\)
\(912\) 0 0
\(913\) 20320.9 0.736609
\(914\) 0 0
\(915\) 26488.7 0.957036
\(916\) 0 0
\(917\) 29618.8 1.06663
\(918\) 0 0
\(919\) 8987.93 0.322616 0.161308 0.986904i \(-0.448429\pi\)
0.161308 + 0.986904i \(0.448429\pi\)
\(920\) 0 0
\(921\) 306.131 0.0109526
\(922\) 0 0
\(923\) −733.348 −0.0261521
\(924\) 0 0
\(925\) 23135.2 0.822359
\(926\) 0 0
\(927\) −4435.46 −0.157152
\(928\) 0 0
\(929\) 234.252 0.00827293 0.00413647 0.999991i \(-0.498683\pi\)
0.00413647 + 0.999991i \(0.498683\pi\)
\(930\) 0 0
\(931\) −1219.45 −0.0429279
\(932\) 0 0
\(933\) −2926.55 −0.102691
\(934\) 0 0
\(935\) −108380. −3.79079
\(936\) 0 0
\(937\) 17624.6 0.614483 0.307241 0.951632i \(-0.400594\pi\)
0.307241 + 0.951632i \(0.400594\pi\)
\(938\) 0 0
\(939\) −18085.0 −0.628521
\(940\) 0 0
\(941\) 30532.1 1.05772 0.528861 0.848708i \(-0.322619\pi\)
0.528861 + 0.848708i \(0.322619\pi\)
\(942\) 0 0
\(943\) 12172.4 0.420349
\(944\) 0 0
\(945\) −41460.2 −1.42720
\(946\) 0 0
\(947\) −42092.8 −1.44438 −0.722192 0.691692i \(-0.756865\pi\)
−0.722192 + 0.691692i \(0.756865\pi\)
\(948\) 0 0
\(949\) 1300.59 0.0444877
\(950\) 0 0
\(951\) 1844.82 0.0629048
\(952\) 0 0
\(953\) −48641.0 −1.65334 −0.826672 0.562683i \(-0.809769\pi\)
−0.826672 + 0.562683i \(0.809769\pi\)
\(954\) 0 0
\(955\) 64810.6 2.19604
\(956\) 0 0
\(957\) 29026.9 0.980467
\(958\) 0 0
\(959\) −7729.54 −0.260271
\(960\) 0 0
\(961\) 61927.9 2.07874
\(962\) 0 0
\(963\) −766.612 −0.0256529
\(964\) 0 0
\(965\) 18338.3 0.611742
\(966\) 0 0
\(967\) −53000.5 −1.76254 −0.881272 0.472609i \(-0.843312\pi\)
−0.881272 + 0.472609i \(0.843312\pi\)
\(968\) 0 0
\(969\) −32137.6 −1.06544
\(970\) 0 0
\(971\) 26286.3 0.868761 0.434381 0.900729i \(-0.356967\pi\)
0.434381 + 0.900729i \(0.356967\pi\)
\(972\) 0 0
\(973\) −2609.68 −0.0859842
\(974\) 0 0
\(975\) 5252.81 0.172538
\(976\) 0 0
\(977\) −30508.0 −0.999014 −0.499507 0.866310i \(-0.666486\pi\)
−0.499507 + 0.866310i \(0.666486\pi\)
\(978\) 0 0
\(979\) −27399.4 −0.894473
\(980\) 0 0
\(981\) 2726.34 0.0887311
\(982\) 0 0
\(983\) 37044.3 1.20196 0.600982 0.799263i \(-0.294776\pi\)
0.600982 + 0.799263i \(0.294776\pi\)
\(984\) 0 0
\(985\) −42153.3 −1.36357
\(986\) 0 0
\(987\) 26122.0 0.842424
\(988\) 0 0
\(989\) −23764.7 −0.764079
\(990\) 0 0
\(991\) 47803.5 1.53232 0.766160 0.642650i \(-0.222165\pi\)
0.766160 + 0.642650i \(0.222165\pi\)
\(992\) 0 0
\(993\) 28934.1 0.924670
\(994\) 0 0
\(995\) −11771.3 −0.375051
\(996\) 0 0
\(997\) 13478.1 0.428140 0.214070 0.976818i \(-0.431328\pi\)
0.214070 + 0.976818i \(0.431328\pi\)
\(998\) 0 0
\(999\) 26796.1 0.848639
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 128.4.a.g.1.1 yes 2
3.2 odd 2 1152.4.a.t.1.2 2
4.3 odd 2 128.4.a.e.1.2 2
8.3 odd 2 128.4.a.h.1.1 yes 2
8.5 even 2 128.4.a.f.1.2 yes 2
12.11 even 2 1152.4.a.s.1.2 2
16.3 odd 4 256.4.b.i.129.3 4
16.5 even 4 256.4.b.h.129.3 4
16.11 odd 4 256.4.b.i.129.2 4
16.13 even 4 256.4.b.h.129.2 4
24.5 odd 2 1152.4.a.r.1.1 2
24.11 even 2 1152.4.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.e.1.2 2 4.3 odd 2
128.4.a.f.1.2 yes 2 8.5 even 2
128.4.a.g.1.1 yes 2 1.1 even 1 trivial
128.4.a.h.1.1 yes 2 8.3 odd 2
256.4.b.h.129.2 4 16.13 even 4
256.4.b.h.129.3 4 16.5 even 4
256.4.b.i.129.2 4 16.11 odd 4
256.4.b.i.129.3 4 16.3 odd 4
1152.4.a.q.1.1 2 24.11 even 2
1152.4.a.r.1.1 2 24.5 odd 2
1152.4.a.s.1.2 2 12.11 even 2
1152.4.a.t.1.2 2 3.2 odd 2