Properties

Label 4025.2.a.p.1.4
Level 4025
Weight 2
Character 4025.1
Self dual yes
Analytic conductor 32.140
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2147108.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.50216\)
Character \(\chi\) = 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.50216 q^{2} +3.04067 q^{3} +0.256481 q^{4} +4.56757 q^{6} -1.00000 q^{7} -2.61904 q^{8} +6.24568 q^{9} +O(q^{10})\) \(q+1.50216 q^{2} +3.04067 q^{3} +0.256481 q^{4} +4.56757 q^{6} -1.00000 q^{7} -2.61904 q^{8} +6.24568 q^{9} -0.542019 q^{11} +0.779873 q^{12} +1.21662 q^{13} -1.50216 q^{14} -4.44718 q^{16} +3.66299 q^{17} +9.38200 q^{18} +5.00432 q^{19} -3.04067 q^{21} -0.814198 q^{22} +1.00000 q^{23} -7.96365 q^{24} +1.82756 q^{26} +9.86904 q^{27} -0.256481 q^{28} +3.72840 q^{29} +9.04067 q^{31} -1.44228 q^{32} -1.64810 q^{33} +5.50239 q^{34} +1.60189 q^{36} +9.08566 q^{37} +7.51728 q^{38} +3.69934 q^{39} -4.86472 q^{41} -4.56757 q^{42} -3.64810 q^{43} -0.139017 q^{44} +1.50216 q^{46} -10.6393 q^{47} -13.5224 q^{48} +1.00000 q^{49} +11.1379 q^{51} +0.312039 q^{52} -7.48704 q^{53} +14.8249 q^{54} +2.61904 q^{56} +15.2165 q^{57} +5.60065 q^{58} +5.18027 q^{59} +3.26161 q^{61} +13.5805 q^{62} -6.24568 q^{63} +6.72782 q^{64} -2.47571 q^{66} +12.0377 q^{67} +0.939485 q^{68} +3.04067 q^{69} -5.29796 q^{71} -16.3577 q^{72} -8.19070 q^{73} +13.6481 q^{74} +1.28351 q^{76} +0.542019 q^{77} +5.55700 q^{78} +10.0291 q^{79} +11.2715 q^{81} -7.30758 q^{82} -7.64810 q^{83} -0.779873 q^{84} -5.48003 q^{86} +11.3368 q^{87} +1.41957 q^{88} -4.74703 q^{89} -1.21662 q^{91} +0.256481 q^{92} +27.4897 q^{93} -15.9819 q^{94} -4.38551 q^{96} +10.9894 q^{97} +1.50216 q^{98} -3.38527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} + O(q^{10}) \) \( 5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} - 4q^{11} - 3q^{12} + 6q^{13} + 2q^{14} + 10q^{16} + 12q^{17} + 19q^{18} + 6q^{19} - 14q^{22} + 5q^{23} - 36q^{24} + q^{26} - 12q^{28} - 4q^{29} + 30q^{31} - 8q^{32} + 22q^{33} + 6q^{34} - q^{36} - 4q^{37} + 40q^{38} + 16q^{39} + 6q^{41} + 3q^{42} + 12q^{43} - 26q^{44} - 2q^{46} - 10q^{47} - 25q^{48} + 5q^{49} - 4q^{51} + 21q^{52} - 16q^{53} + 33q^{54} + 3q^{56} - 6q^{57} - 13q^{58} + 22q^{59} - 18q^{61} - 15q^{62} - 11q^{63} + 25q^{64} + 4q^{66} + 2q^{67} - 12q^{68} + 4q^{71} + 41q^{72} + 2q^{73} + 38q^{74} + 10q^{76} + 4q^{77} - 41q^{78} + 30q^{79} - 3q^{81} + 7q^{82} - 8q^{83} + 3q^{84} + 8q^{86} + 12q^{87} - 4q^{88} - 20q^{89} - 6q^{91} + 12q^{92} + 26q^{93} - 25q^{94} - q^{96} + 12q^{97} - 2q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50216 1.06219 0.531093 0.847313i \(-0.321781\pi\)
0.531093 + 0.847313i \(0.321781\pi\)
\(3\) 3.04067 1.75553 0.877766 0.479090i \(-0.159033\pi\)
0.877766 + 0.479090i \(0.159033\pi\)
\(4\) 0.256481 0.128240
\(5\) 0 0
\(6\) 4.56757 1.86470
\(7\) −1.00000 −0.377964
\(8\) −2.61904 −0.925971
\(9\) 6.24568 2.08189
\(10\) 0 0
\(11\) −0.542019 −0.163425 −0.0817124 0.996656i \(-0.526039\pi\)
−0.0817124 + 0.996656i \(0.526039\pi\)
\(12\) 0.779873 0.225130
\(13\) 1.21662 0.337430 0.168715 0.985665i \(-0.446038\pi\)
0.168715 + 0.985665i \(0.446038\pi\)
\(14\) −1.50216 −0.401469
\(15\) 0 0
\(16\) −4.44718 −1.11179
\(17\) 3.66299 0.888405 0.444203 0.895926i \(-0.353487\pi\)
0.444203 + 0.895926i \(0.353487\pi\)
\(18\) 9.38200 2.21136
\(19\) 5.00432 1.14807 0.574035 0.818831i \(-0.305378\pi\)
0.574035 + 0.818831i \(0.305378\pi\)
\(20\) 0 0
\(21\) −3.04067 −0.663529
\(22\) −0.814198 −0.173588
\(23\) 1.00000 0.208514
\(24\) −7.96365 −1.62557
\(25\) 0 0
\(26\) 1.82756 0.358413
\(27\) 9.86904 1.89930
\(28\) −0.256481 −0.0484703
\(29\) 3.72840 0.692346 0.346173 0.938171i \(-0.387481\pi\)
0.346173 + 0.938171i \(0.387481\pi\)
\(30\) 0 0
\(31\) 9.04067 1.62375 0.811876 0.583830i \(-0.198446\pi\)
0.811876 + 0.583830i \(0.198446\pi\)
\(32\) −1.44228 −0.254962
\(33\) −1.64810 −0.286897
\(34\) 5.50239 0.943652
\(35\) 0 0
\(36\) 1.60189 0.266982
\(37\) 9.08566 1.49367 0.746837 0.665008i \(-0.231572\pi\)
0.746837 + 0.665008i \(0.231572\pi\)
\(38\) 7.51728 1.21946
\(39\) 3.69934 0.592369
\(40\) 0 0
\(41\) −4.86472 −0.759742 −0.379871 0.925040i \(-0.624032\pi\)
−0.379871 + 0.925040i \(0.624032\pi\)
\(42\) −4.56757 −0.704791
\(43\) −3.64810 −0.556330 −0.278165 0.960533i \(-0.589726\pi\)
−0.278165 + 0.960533i \(0.589726\pi\)
\(44\) −0.139017 −0.0209576
\(45\) 0 0
\(46\) 1.50216 0.221481
\(47\) −10.6393 −1.55190 −0.775950 0.630794i \(-0.782729\pi\)
−0.775950 + 0.630794i \(0.782729\pi\)
\(48\) −13.5224 −1.95179
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 11.1379 1.55962
\(52\) 0.312039 0.0432721
\(53\) −7.48704 −1.02842 −0.514212 0.857663i \(-0.671916\pi\)
−0.514212 + 0.857663i \(0.671916\pi\)
\(54\) 14.8249 2.01741
\(55\) 0 0
\(56\) 2.61904 0.349984
\(57\) 15.2165 2.01547
\(58\) 5.60065 0.735401
\(59\) 5.18027 0.674413 0.337207 0.941431i \(-0.390518\pi\)
0.337207 + 0.941431i \(0.390518\pi\)
\(60\) 0 0
\(61\) 3.26161 0.417606 0.208803 0.977958i \(-0.433043\pi\)
0.208803 + 0.977958i \(0.433043\pi\)
\(62\) 13.5805 1.72473
\(63\) −6.24568 −0.786881
\(64\) 6.72782 0.840978
\(65\) 0 0
\(66\) −2.47571 −0.304739
\(67\) 12.0377 1.47064 0.735319 0.677721i \(-0.237032\pi\)
0.735319 + 0.677721i \(0.237032\pi\)
\(68\) 0.939485 0.113929
\(69\) 3.04067 0.366054
\(70\) 0 0
\(71\) −5.29796 −0.628752 −0.314376 0.949298i \(-0.601795\pi\)
−0.314376 + 0.949298i \(0.601795\pi\)
\(72\) −16.3577 −1.92777
\(73\) −8.19070 −0.958649 −0.479324 0.877638i \(-0.659118\pi\)
−0.479324 + 0.877638i \(0.659118\pi\)
\(74\) 13.6481 1.58656
\(75\) 0 0
\(76\) 1.28351 0.147229
\(77\) 0.542019 0.0617688
\(78\) 5.55700 0.629206
\(79\) 10.0291 1.12836 0.564179 0.825653i \(-0.309193\pi\)
0.564179 + 0.825653i \(0.309193\pi\)
\(80\) 0 0
\(81\) 11.2715 1.25238
\(82\) −7.30758 −0.806987
\(83\) −7.64810 −0.839488 −0.419744 0.907643i \(-0.637880\pi\)
−0.419744 + 0.907643i \(0.637880\pi\)
\(84\) −0.779873 −0.0850911
\(85\) 0 0
\(86\) −5.48003 −0.590926
\(87\) 11.3368 1.21544
\(88\) 1.41957 0.151327
\(89\) −4.74703 −0.503184 −0.251592 0.967833i \(-0.580954\pi\)
−0.251592 + 0.967833i \(0.580954\pi\)
\(90\) 0 0
\(91\) −1.21662 −0.127536
\(92\) 0.256481 0.0267399
\(93\) 27.4897 2.85055
\(94\) −15.9819 −1.64841
\(95\) 0 0
\(96\) −4.38551 −0.447594
\(97\) 10.9894 1.11581 0.557904 0.829906i \(-0.311606\pi\)
0.557904 + 0.829906i \(0.311606\pi\)
\(98\) 1.50216 0.151741
\(99\) −3.38527 −0.340233
\(100\) 0 0
\(101\) 3.59862 0.358076 0.179038 0.983842i \(-0.442701\pi\)
0.179038 + 0.983842i \(0.442701\pi\)
\(102\) 16.7310 1.65661
\(103\) −9.48704 −0.934786 −0.467393 0.884050i \(-0.654807\pi\)
−0.467393 + 0.884050i \(0.654807\pi\)
\(104\) −3.18638 −0.312450
\(105\) 0 0
\(106\) −11.2467 −1.09238
\(107\) −6.60294 −0.638330 −0.319165 0.947699i \(-0.603402\pi\)
−0.319165 + 0.947699i \(0.603402\pi\)
\(108\) 2.53122 0.243566
\(109\) 1.10726 0.106057 0.0530283 0.998593i \(-0.483113\pi\)
0.0530283 + 0.998593i \(0.483113\pi\)
\(110\) 0 0
\(111\) 27.6265 2.62219
\(112\) 4.44718 0.420219
\(113\) −0.433241 −0.0407559 −0.0203779 0.999792i \(-0.506487\pi\)
−0.0203779 + 0.999792i \(0.506487\pi\)
\(114\) 22.8576 2.14081
\(115\) 0 0
\(116\) 0.956262 0.0887867
\(117\) 7.59862 0.702493
\(118\) 7.78158 0.716353
\(119\) −3.66299 −0.335786
\(120\) 0 0
\(121\) −10.7062 −0.973292
\(122\) 4.89945 0.443576
\(123\) −14.7920 −1.33375
\(124\) 2.31876 0.208230
\(125\) 0 0
\(126\) −9.38200 −0.835815
\(127\) −8.84149 −0.784556 −0.392278 0.919847i \(-0.628313\pi\)
−0.392278 + 0.919847i \(0.628313\pi\)
\(128\) 12.9908 1.14824
\(129\) −11.0927 −0.976655
\(130\) 0 0
\(131\) −2.54931 −0.222735 −0.111367 0.993779i \(-0.535523\pi\)
−0.111367 + 0.993779i \(0.535523\pi\)
\(132\) −0.422706 −0.0367918
\(133\) −5.00432 −0.433929
\(134\) 18.0825 1.56209
\(135\) 0 0
\(136\) −9.59352 −0.822638
\(137\) −13.6265 −1.16419 −0.582095 0.813121i \(-0.697767\pi\)
−0.582095 + 0.813121i \(0.697767\pi\)
\(138\) 4.56757 0.388817
\(139\) −18.0466 −1.53069 −0.765347 0.643618i \(-0.777432\pi\)
−0.765347 + 0.643618i \(0.777432\pi\)
\(140\) 0 0
\(141\) −32.3506 −2.72441
\(142\) −7.95838 −0.667852
\(143\) −0.659431 −0.0551444
\(144\) −27.7756 −2.31464
\(145\) 0 0
\(146\) −12.3037 −1.01826
\(147\) 3.04067 0.250790
\(148\) 2.33029 0.191549
\(149\) 13.8178 1.13200 0.565999 0.824406i \(-0.308491\pi\)
0.565999 + 0.824406i \(0.308491\pi\)
\(150\) 0 0
\(151\) 16.1412 1.31355 0.656777 0.754085i \(-0.271919\pi\)
0.656777 + 0.754085i \(0.271919\pi\)
\(152\) −13.1065 −1.06308
\(153\) 22.8778 1.84956
\(154\) 0.814198 0.0656100
\(155\) 0 0
\(156\) 0.948809 0.0759655
\(157\) 15.3111 1.22196 0.610979 0.791647i \(-0.290776\pi\)
0.610979 + 0.791647i \(0.290776\pi\)
\(158\) 15.0652 1.19853
\(159\) −22.7656 −1.80543
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 16.9315 1.33027
\(163\) 3.94876 0.309291 0.154645 0.987970i \(-0.450577\pi\)
0.154645 + 0.987970i \(0.450577\pi\)
\(164\) −1.24771 −0.0974295
\(165\) 0 0
\(166\) −11.4887 −0.891693
\(167\) −1.87752 −0.145287 −0.0726433 0.997358i \(-0.523143\pi\)
−0.0726433 + 0.997358i \(0.523143\pi\)
\(168\) 7.96365 0.614409
\(169\) −11.5198 −0.886141
\(170\) 0 0
\(171\) 31.2554 2.39016
\(172\) −0.935667 −0.0713439
\(173\) −10.9246 −0.830582 −0.415291 0.909689i \(-0.636320\pi\)
−0.415291 + 0.909689i \(0.636320\pi\)
\(174\) 17.0297 1.29102
\(175\) 0 0
\(176\) 2.41045 0.181695
\(177\) 15.7515 1.18395
\(178\) −7.13079 −0.534475
\(179\) 4.74253 0.354474 0.177237 0.984168i \(-0.443284\pi\)
0.177237 + 0.984168i \(0.443284\pi\)
\(180\) 0 0
\(181\) 2.82837 0.210231 0.105115 0.994460i \(-0.466479\pi\)
0.105115 + 0.994460i \(0.466479\pi\)
\(182\) −1.82756 −0.135468
\(183\) 9.91748 0.733121
\(184\) −2.61904 −0.193078
\(185\) 0 0
\(186\) 41.2939 3.02782
\(187\) −1.98541 −0.145187
\(188\) −2.72877 −0.199016
\(189\) −9.86904 −0.717867
\(190\) 0 0
\(191\) −3.56838 −0.258199 −0.129099 0.991632i \(-0.541209\pi\)
−0.129099 + 0.991632i \(0.541209\pi\)
\(192\) 20.4571 1.47636
\(193\) −20.9158 −1.50555 −0.752777 0.658276i \(-0.771286\pi\)
−0.752777 + 0.658276i \(0.771286\pi\)
\(194\) 16.5079 1.18520
\(195\) 0 0
\(196\) 0.256481 0.0183200
\(197\) −5.54576 −0.395119 −0.197559 0.980291i \(-0.563302\pi\)
−0.197559 + 0.980291i \(0.563302\pi\)
\(198\) −5.08522 −0.361391
\(199\) −7.73808 −0.548538 −0.274269 0.961653i \(-0.588436\pi\)
−0.274269 + 0.961653i \(0.588436\pi\)
\(200\) 0 0
\(201\) 36.6027 2.58175
\(202\) 5.40570 0.380344
\(203\) −3.72840 −0.261682
\(204\) 2.85667 0.200007
\(205\) 0 0
\(206\) −14.2510 −0.992917
\(207\) 6.24568 0.434105
\(208\) −5.41053 −0.375153
\(209\) −2.71243 −0.187623
\(210\) 0 0
\(211\) 9.43918 0.649820 0.324910 0.945745i \(-0.394666\pi\)
0.324910 + 0.945745i \(0.394666\pi\)
\(212\) −1.92028 −0.131885
\(213\) −16.1094 −1.10379
\(214\) −9.91866 −0.678026
\(215\) 0 0
\(216\) −25.8474 −1.75870
\(217\) −9.04067 −0.613721
\(218\) 1.66329 0.112652
\(219\) −24.9052 −1.68294
\(220\) 0 0
\(221\) 4.45647 0.299774
\(222\) 41.4994 2.78526
\(223\) −11.1010 −0.743379 −0.371689 0.928357i \(-0.621221\pi\)
−0.371689 + 0.928357i \(0.621221\pi\)
\(224\) 1.44228 0.0963665
\(225\) 0 0
\(226\) −0.650797 −0.0432903
\(227\) −20.9827 −1.39267 −0.696336 0.717716i \(-0.745188\pi\)
−0.696336 + 0.717716i \(0.745188\pi\)
\(228\) 3.90273 0.258465
\(229\) 4.63545 0.306319 0.153159 0.988201i \(-0.451055\pi\)
0.153159 + 0.988201i \(0.451055\pi\)
\(230\) 0 0
\(231\) 1.64810 0.108437
\(232\) −9.76484 −0.641093
\(233\) −16.1703 −1.05935 −0.529675 0.848201i \(-0.677686\pi\)
−0.529675 + 0.848201i \(0.677686\pi\)
\(234\) 11.4143 0.746178
\(235\) 0 0
\(236\) 1.32864 0.0864869
\(237\) 30.4951 1.98087
\(238\) −5.50239 −0.356667
\(239\) 1.26610 0.0818973 0.0409486 0.999161i \(-0.486962\pi\)
0.0409486 + 0.999161i \(0.486962\pi\)
\(240\) 0 0
\(241\) −2.09623 −0.135030 −0.0675150 0.997718i \(-0.521507\pi\)
−0.0675150 + 0.997718i \(0.521507\pi\)
\(242\) −16.0824 −1.03382
\(243\) 4.66568 0.299304
\(244\) 0.836539 0.0535539
\(245\) 0 0
\(246\) −22.2200 −1.41669
\(247\) 6.08835 0.387393
\(248\) −23.6779 −1.50355
\(249\) −23.2554 −1.47375
\(250\) 0 0
\(251\) 11.5365 0.728179 0.364089 0.931364i \(-0.381380\pi\)
0.364089 + 0.931364i \(0.381380\pi\)
\(252\) −1.60189 −0.100910
\(253\) −0.542019 −0.0340764
\(254\) −13.2813 −0.833345
\(255\) 0 0
\(256\) 6.05863 0.378664
\(257\) −14.6546 −0.914131 −0.457066 0.889433i \(-0.651100\pi\)
−0.457066 + 0.889433i \(0.651100\pi\)
\(258\) −16.6630 −1.03739
\(259\) −9.08566 −0.564555
\(260\) 0 0
\(261\) 23.2864 1.44139
\(262\) −3.82947 −0.236586
\(263\) 22.7133 1.40056 0.700282 0.713866i \(-0.253058\pi\)
0.700282 + 0.713866i \(0.253058\pi\)
\(264\) 4.31645 0.265659
\(265\) 0 0
\(266\) −7.51728 −0.460914
\(267\) −14.4341 −0.883355
\(268\) 3.08743 0.188595
\(269\) 25.0855 1.52949 0.764746 0.644332i \(-0.222865\pi\)
0.764746 + 0.644332i \(0.222865\pi\)
\(270\) 0 0
\(271\) −28.0232 −1.70229 −0.851145 0.524930i \(-0.824091\pi\)
−0.851145 + 0.524930i \(0.824091\pi\)
\(272\) −16.2900 −0.987724
\(273\) −3.69934 −0.223894
\(274\) −20.4692 −1.23659
\(275\) 0 0
\(276\) 0.779873 0.0469428
\(277\) 16.2692 0.977524 0.488762 0.872417i \(-0.337449\pi\)
0.488762 + 0.872417i \(0.337449\pi\)
\(278\) −27.1089 −1.62588
\(279\) 56.4651 3.38048
\(280\) 0 0
\(281\) −13.4694 −0.803518 −0.401759 0.915745i \(-0.631601\pi\)
−0.401759 + 0.915745i \(0.631601\pi\)
\(282\) −48.5957 −2.89383
\(283\) −18.0516 −1.07305 −0.536527 0.843883i \(-0.680264\pi\)
−0.536527 + 0.843883i \(0.680264\pi\)
\(284\) −1.35882 −0.0806314
\(285\) 0 0
\(286\) −0.990570 −0.0585736
\(287\) 4.86472 0.287155
\(288\) −9.00803 −0.530803
\(289\) −3.58251 −0.210736
\(290\) 0 0
\(291\) 33.4152 1.95884
\(292\) −2.10075 −0.122937
\(293\) −23.1521 −1.35256 −0.676280 0.736644i \(-0.736409\pi\)
−0.676280 + 0.736644i \(0.736409\pi\)
\(294\) 4.56757 0.266386
\(295\) 0 0
\(296\) −23.7957 −1.38310
\(297\) −5.34920 −0.310392
\(298\) 20.7565 1.20239
\(299\) 1.21662 0.0703590
\(300\) 0 0
\(301\) 3.64810 0.210273
\(302\) 24.2467 1.39524
\(303\) 10.9422 0.628614
\(304\) −22.2551 −1.27642
\(305\) 0 0
\(306\) 34.3662 1.96458
\(307\) 1.14301 0.0652353 0.0326176 0.999468i \(-0.489616\pi\)
0.0326176 + 0.999468i \(0.489616\pi\)
\(308\) 0.139017 0.00792124
\(309\) −28.8470 −1.64105
\(310\) 0 0
\(311\) −19.3196 −1.09551 −0.547757 0.836638i \(-0.684518\pi\)
−0.547757 + 0.836638i \(0.684518\pi\)
\(312\) −9.68874 −0.548517
\(313\) 17.4140 0.984299 0.492150 0.870511i \(-0.336211\pi\)
0.492150 + 0.870511i \(0.336211\pi\)
\(314\) 22.9997 1.29795
\(315\) 0 0
\(316\) 2.57226 0.144701
\(317\) −1.42612 −0.0800989 −0.0400495 0.999198i \(-0.512752\pi\)
−0.0400495 + 0.999198i \(0.512752\pi\)
\(318\) −34.1976 −1.91770
\(319\) −2.02086 −0.113147
\(320\) 0 0
\(321\) −20.0774 −1.12061
\(322\) −1.50216 −0.0837120
\(323\) 18.3308 1.01995
\(324\) 2.89091 0.160606
\(325\) 0 0
\(326\) 5.93166 0.328524
\(327\) 3.36682 0.186186
\(328\) 12.7409 0.703499
\(329\) 10.6393 0.586563
\(330\) 0 0
\(331\) 3.91690 0.215292 0.107646 0.994189i \(-0.465669\pi\)
0.107646 + 0.994189i \(0.465669\pi\)
\(332\) −1.96159 −0.107656
\(333\) 56.7461 3.10967
\(334\) −2.82033 −0.154321
\(335\) 0 0
\(336\) 13.5224 0.737708
\(337\) −7.73646 −0.421432 −0.210716 0.977547i \(-0.567579\pi\)
−0.210716 + 0.977547i \(0.567579\pi\)
\(338\) −17.3046 −0.941247
\(339\) −1.31734 −0.0715482
\(340\) 0 0
\(341\) −4.90021 −0.265361
\(342\) 46.9505 2.53879
\(343\) −1.00000 −0.0539949
\(344\) 9.55453 0.515146
\(345\) 0 0
\(346\) −16.4105 −0.882233
\(347\) 32.0316 1.71955 0.859773 0.510677i \(-0.170605\pi\)
0.859773 + 0.510677i \(0.170605\pi\)
\(348\) 2.90768 0.155868
\(349\) 29.0447 1.55473 0.777363 0.629052i \(-0.216557\pi\)
0.777363 + 0.629052i \(0.216557\pi\)
\(350\) 0 0
\(351\) 12.0069 0.640880
\(352\) 0.781744 0.0416671
\(353\) −14.1412 −0.752661 −0.376331 0.926485i \(-0.622814\pi\)
−0.376331 + 0.926485i \(0.622814\pi\)
\(354\) 23.6612 1.25758
\(355\) 0 0
\(356\) −1.21752 −0.0645284
\(357\) −11.1379 −0.589482
\(358\) 7.12404 0.376517
\(359\) 7.25851 0.383089 0.191545 0.981484i \(-0.438650\pi\)
0.191545 + 0.981484i \(0.438650\pi\)
\(360\) 0 0
\(361\) 6.04319 0.318063
\(362\) 4.24866 0.223304
\(363\) −32.5541 −1.70865
\(364\) −0.312039 −0.0163553
\(365\) 0 0
\(366\) 14.8976 0.778711
\(367\) 15.7667 0.823015 0.411507 0.911406i \(-0.365002\pi\)
0.411507 + 0.911406i \(0.365002\pi\)
\(368\) −4.44718 −0.231825
\(369\) −30.3835 −1.58170
\(370\) 0 0
\(371\) 7.48704 0.388708
\(372\) 7.05057 0.365555
\(373\) −4.16807 −0.215815 −0.107907 0.994161i \(-0.534415\pi\)
−0.107907 + 0.994161i \(0.534415\pi\)
\(374\) −2.98240 −0.154216
\(375\) 0 0
\(376\) 27.8648 1.43702
\(377\) 4.53605 0.233618
\(378\) −14.8249 −0.762509
\(379\) 18.6572 0.958356 0.479178 0.877718i \(-0.340935\pi\)
0.479178 + 0.877718i \(0.340935\pi\)
\(380\) 0 0
\(381\) −26.8841 −1.37731
\(382\) −5.36027 −0.274255
\(383\) 30.2829 1.54738 0.773692 0.633562i \(-0.218408\pi\)
0.773692 + 0.633562i \(0.218408\pi\)
\(384\) 39.5008 2.01577
\(385\) 0 0
\(386\) −31.4189 −1.59918
\(387\) −22.7849 −1.15822
\(388\) 2.81857 0.143091
\(389\) −14.0727 −0.713515 −0.356757 0.934197i \(-0.616118\pi\)
−0.356757 + 0.934197i \(0.616118\pi\)
\(390\) 0 0
\(391\) 3.66299 0.185245
\(392\) −2.61904 −0.132282
\(393\) −7.75163 −0.391018
\(394\) −8.33061 −0.419690
\(395\) 0 0
\(396\) −0.868257 −0.0436316
\(397\) −30.2423 −1.51782 −0.758908 0.651198i \(-0.774267\pi\)
−0.758908 + 0.651198i \(0.774267\pi\)
\(398\) −11.6238 −0.582649
\(399\) −15.2165 −0.761777
\(400\) 0 0
\(401\) −0.668630 −0.0333898 −0.0166949 0.999861i \(-0.505314\pi\)
−0.0166949 + 0.999861i \(0.505314\pi\)
\(402\) 54.9830 2.74230
\(403\) 10.9991 0.547902
\(404\) 0.922976 0.0459198
\(405\) 0 0
\(406\) −5.60065 −0.277955
\(407\) −4.92460 −0.244103
\(408\) −29.1707 −1.44417
\(409\) −9.05602 −0.447791 −0.223896 0.974613i \(-0.571878\pi\)
−0.223896 + 0.974613i \(0.571878\pi\)
\(410\) 0 0
\(411\) −41.4337 −2.04377
\(412\) −2.43324 −0.119877
\(413\) −5.18027 −0.254904
\(414\) 9.38200 0.461100
\(415\) 0 0
\(416\) −1.75471 −0.0860318
\(417\) −54.8738 −2.68718
\(418\) −4.07451 −0.199291
\(419\) −19.9592 −0.975068 −0.487534 0.873104i \(-0.662104\pi\)
−0.487534 + 0.873104i \(0.662104\pi\)
\(420\) 0 0
\(421\) −6.86296 −0.334480 −0.167240 0.985916i \(-0.553486\pi\)
−0.167240 + 0.985916i \(0.553486\pi\)
\(422\) 14.1791 0.690230
\(423\) −66.4496 −3.23089
\(424\) 19.6089 0.952291
\(425\) 0 0
\(426\) −24.1988 −1.17244
\(427\) −3.26161 −0.157840
\(428\) −1.69352 −0.0818596
\(429\) −2.00511 −0.0968078
\(430\) 0 0
\(431\) 4.52160 0.217798 0.108899 0.994053i \(-0.465268\pi\)
0.108899 + 0.994053i \(0.465268\pi\)
\(432\) −43.8894 −2.11163
\(433\) −14.5170 −0.697641 −0.348820 0.937190i \(-0.613418\pi\)
−0.348820 + 0.937190i \(0.613418\pi\)
\(434\) −13.5805 −0.651886
\(435\) 0 0
\(436\) 0.283992 0.0136007
\(437\) 5.00432 0.239389
\(438\) −37.4116 −1.78759
\(439\) 25.5058 1.21732 0.608662 0.793430i \(-0.291707\pi\)
0.608662 + 0.793430i \(0.291707\pi\)
\(440\) 0 0
\(441\) 6.24568 0.297413
\(442\) 6.69432 0.318416
\(443\) −13.9434 −0.662469 −0.331235 0.943548i \(-0.607465\pi\)
−0.331235 + 0.943548i \(0.607465\pi\)
\(444\) 7.08566 0.336270
\(445\) 0 0
\(446\) −16.6755 −0.789607
\(447\) 42.0154 1.98726
\(448\) −6.72782 −0.317860
\(449\) −22.5307 −1.06329 −0.531644 0.846968i \(-0.678426\pi\)
−0.531644 + 0.846968i \(0.678426\pi\)
\(450\) 0 0
\(451\) 2.63677 0.124161
\(452\) −0.111118 −0.00522654
\(453\) 49.0801 2.30599
\(454\) −31.5194 −1.47928
\(455\) 0 0
\(456\) −39.8526 −1.86627
\(457\) 12.9589 0.606191 0.303096 0.952960i \(-0.401980\pi\)
0.303096 + 0.952960i \(0.401980\pi\)
\(458\) 6.96317 0.325368
\(459\) 36.1502 1.68735
\(460\) 0 0
\(461\) 14.9315 0.695428 0.347714 0.937601i \(-0.386958\pi\)
0.347714 + 0.937601i \(0.386958\pi\)
\(462\) 2.47571 0.115180
\(463\) −17.4941 −0.813018 −0.406509 0.913647i \(-0.633254\pi\)
−0.406509 + 0.913647i \(0.633254\pi\)
\(464\) −16.5809 −0.769747
\(465\) 0 0
\(466\) −24.2903 −1.12523
\(467\) 17.6281 0.815732 0.407866 0.913042i \(-0.366273\pi\)
0.407866 + 0.913042i \(0.366273\pi\)
\(468\) 1.94890 0.0900878
\(469\) −12.0377 −0.555849
\(470\) 0 0
\(471\) 46.5560 2.14519
\(472\) −13.5673 −0.624487
\(473\) 1.97734 0.0909181
\(474\) 45.8084 2.10405
\(475\) 0 0
\(476\) −0.939485 −0.0430612
\(477\) −46.7616 −2.14107
\(478\) 1.90188 0.0869902
\(479\) −11.7457 −0.536673 −0.268336 0.963325i \(-0.586474\pi\)
−0.268336 + 0.963325i \(0.586474\pi\)
\(480\) 0 0
\(481\) 11.0538 0.504010
\(482\) −3.14887 −0.143427
\(483\) −3.04067 −0.138355
\(484\) −2.74594 −0.124815
\(485\) 0 0
\(486\) 7.00860 0.317916
\(487\) −22.9380 −1.03942 −0.519710 0.854342i \(-0.673960\pi\)
−0.519710 + 0.854342i \(0.673960\pi\)
\(488\) −8.54229 −0.386691
\(489\) 12.0069 0.542970
\(490\) 0 0
\(491\) 11.5682 0.522067 0.261034 0.965330i \(-0.415937\pi\)
0.261034 + 0.965330i \(0.415937\pi\)
\(492\) −3.79386 −0.171041
\(493\) 13.6571 0.615084
\(494\) 9.14567 0.411483
\(495\) 0 0
\(496\) −40.2055 −1.80528
\(497\) 5.29796 0.237646
\(498\) −34.9332 −1.56540
\(499\) 25.0063 1.11943 0.559717 0.828684i \(-0.310910\pi\)
0.559717 + 0.828684i \(0.310910\pi\)
\(500\) 0 0
\(501\) −5.70891 −0.255055
\(502\) 17.3297 0.773462
\(503\) −16.1238 −0.718925 −0.359463 0.933160i \(-0.617040\pi\)
−0.359463 + 0.933160i \(0.617040\pi\)
\(504\) 16.3577 0.728630
\(505\) 0 0
\(506\) −0.814198 −0.0361955
\(507\) −35.0280 −1.55565
\(508\) −2.26767 −0.100612
\(509\) −1.46604 −0.0649809 −0.0324905 0.999472i \(-0.510344\pi\)
−0.0324905 + 0.999472i \(0.510344\pi\)
\(510\) 0 0
\(511\) 8.19070 0.362335
\(512\) −16.8806 −0.746025
\(513\) 49.3878 2.18052
\(514\) −22.0136 −0.970978
\(515\) 0 0
\(516\) −2.84505 −0.125247
\(517\) 5.76669 0.253619
\(518\) −13.6481 −0.599663
\(519\) −33.2181 −1.45811
\(520\) 0 0
\(521\) −31.1921 −1.36655 −0.683276 0.730160i \(-0.739446\pi\)
−0.683276 + 0.730160i \(0.739446\pi\)
\(522\) 34.9798 1.53103
\(523\) 25.0064 1.09345 0.546727 0.837311i \(-0.315874\pi\)
0.546727 + 0.837311i \(0.315874\pi\)
\(524\) −0.653850 −0.0285636
\(525\) 0 0
\(526\) 34.1190 1.48766
\(527\) 33.1159 1.44255
\(528\) 7.32940 0.318971
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 32.3543 1.40406
\(532\) −1.28351 −0.0556472
\(533\) −5.91852 −0.256360
\(534\) −21.6824 −0.938288
\(535\) 0 0
\(536\) −31.5272 −1.36177
\(537\) 14.4205 0.622290
\(538\) 37.6824 1.62460
\(539\) −0.542019 −0.0233464
\(540\) 0 0
\(541\) −29.8577 −1.28368 −0.641842 0.766837i \(-0.721829\pi\)
−0.641842 + 0.766837i \(0.721829\pi\)
\(542\) −42.0954 −1.80815
\(543\) 8.60013 0.369067
\(544\) −5.28306 −0.226510
\(545\) 0 0
\(546\) −5.55700 −0.237818
\(547\) −29.1669 −1.24708 −0.623542 0.781789i \(-0.714307\pi\)
−0.623542 + 0.781789i \(0.714307\pi\)
\(548\) −3.49493 −0.149296
\(549\) 20.3710 0.869411
\(550\) 0 0
\(551\) 18.6581 0.794862
\(552\) −7.96365 −0.338955
\(553\) −10.0291 −0.426479
\(554\) 24.4390 1.03831
\(555\) 0 0
\(556\) −4.62860 −0.196297
\(557\) −40.2543 −1.70563 −0.852814 0.522214i \(-0.825106\pi\)
−0.852814 + 0.522214i \(0.825106\pi\)
\(558\) 84.8196 3.59070
\(559\) −4.43835 −0.187722
\(560\) 0 0
\(561\) −6.03697 −0.254881
\(562\) −20.2332 −0.853486
\(563\) 22.5130 0.948808 0.474404 0.880307i \(-0.342664\pi\)
0.474404 + 0.880307i \(0.342664\pi\)
\(564\) −8.29729 −0.349379
\(565\) 0 0
\(566\) −27.1163 −1.13978
\(567\) −11.2715 −0.473357
\(568\) 13.8756 0.582207
\(569\) 6.81862 0.285852 0.142926 0.989733i \(-0.454349\pi\)
0.142926 + 0.989733i \(0.454349\pi\)
\(570\) 0 0
\(571\) −0.873495 −0.0365546 −0.0182773 0.999833i \(-0.505818\pi\)
−0.0182773 + 0.999833i \(0.505818\pi\)
\(572\) −0.169131 −0.00707173
\(573\) −10.8503 −0.453276
\(574\) 7.30758 0.305013
\(575\) 0 0
\(576\) 42.0198 1.75083
\(577\) 16.4150 0.683366 0.341683 0.939815i \(-0.389003\pi\)
0.341683 + 0.939815i \(0.389003\pi\)
\(578\) −5.38150 −0.223841
\(579\) −63.5981 −2.64305
\(580\) 0 0
\(581\) 7.64810 0.317297
\(582\) 50.1950 2.08065
\(583\) 4.05812 0.168070
\(584\) 21.4518 0.887681
\(585\) 0 0
\(586\) −34.7781 −1.43667
\(587\) 9.97331 0.411643 0.205821 0.978590i \(-0.434013\pi\)
0.205821 + 0.978590i \(0.434013\pi\)
\(588\) 0.779873 0.0321614
\(589\) 45.2424 1.86418
\(590\) 0 0
\(591\) −16.8628 −0.693644
\(592\) −40.4055 −1.66066
\(593\) 20.5516 0.843951 0.421976 0.906607i \(-0.361337\pi\)
0.421976 + 0.906607i \(0.361337\pi\)
\(594\) −8.03535 −0.329694
\(595\) 0 0
\(596\) 3.54400 0.145168
\(597\) −23.5289 −0.962975
\(598\) 1.82756 0.0747344
\(599\) 15.5468 0.635224 0.317612 0.948221i \(-0.397119\pi\)
0.317612 + 0.948221i \(0.397119\pi\)
\(600\) 0 0
\(601\) −29.3814 −1.19849 −0.599246 0.800565i \(-0.704533\pi\)
−0.599246 + 0.800565i \(0.704533\pi\)
\(602\) 5.48003 0.223349
\(603\) 75.1836 3.06171
\(604\) 4.13991 0.168450
\(605\) 0 0
\(606\) 16.4369 0.667705
\(607\) −31.1108 −1.26275 −0.631374 0.775478i \(-0.717509\pi\)
−0.631374 + 0.775478i \(0.717509\pi\)
\(608\) −7.21764 −0.292714
\(609\) −11.3368 −0.459392
\(610\) 0 0
\(611\) −12.9440 −0.523657
\(612\) 5.86772 0.237189
\(613\) 27.7365 1.12026 0.560132 0.828403i \(-0.310750\pi\)
0.560132 + 0.828403i \(0.310750\pi\)
\(614\) 1.71699 0.0692920
\(615\) 0 0
\(616\) −1.41957 −0.0571961
\(617\) 20.6041 0.829490 0.414745 0.909938i \(-0.363871\pi\)
0.414745 + 0.909938i \(0.363871\pi\)
\(618\) −43.3327 −1.74310
\(619\) −9.83894 −0.395460 −0.197730 0.980256i \(-0.563357\pi\)
−0.197730 + 0.980256i \(0.563357\pi\)
\(620\) 0 0
\(621\) 9.86904 0.396031
\(622\) −29.0211 −1.16364
\(623\) 4.74703 0.190186
\(624\) −16.4516 −0.658593
\(625\) 0 0
\(626\) 26.1586 1.04551
\(627\) −8.24762 −0.329378
\(628\) 3.92700 0.156704
\(629\) 33.2807 1.32699
\(630\) 0 0
\(631\) 2.65314 0.105620 0.0528098 0.998605i \(-0.483182\pi\)
0.0528098 + 0.998605i \(0.483182\pi\)
\(632\) −26.2665 −1.04483
\(633\) 28.7014 1.14078
\(634\) −2.14226 −0.0850800
\(635\) 0 0
\(636\) −5.83894 −0.231529
\(637\) 1.21662 0.0482043
\(638\) −3.03566 −0.120183
\(639\) −33.0894 −1.30899
\(640\) 0 0
\(641\) −46.9550 −1.85461 −0.927306 0.374304i \(-0.877882\pi\)
−0.927306 + 0.374304i \(0.877882\pi\)
\(642\) −30.1594 −1.19030
\(643\) 8.01924 0.316248 0.158124 0.987419i \(-0.449455\pi\)
0.158124 + 0.987419i \(0.449455\pi\)
\(644\) −0.256481 −0.0101067
\(645\) 0 0
\(646\) 27.5357 1.08338
\(647\) −20.3950 −0.801810 −0.400905 0.916120i \(-0.631304\pi\)
−0.400905 + 0.916120i \(0.631304\pi\)
\(648\) −29.5204 −1.15967
\(649\) −2.80780 −0.110216
\(650\) 0 0
\(651\) −27.4897 −1.07741
\(652\) 1.01278 0.0396635
\(653\) 30.9134 1.20973 0.604867 0.796326i \(-0.293226\pi\)
0.604867 + 0.796326i \(0.293226\pi\)
\(654\) 5.05750 0.197764
\(655\) 0 0
\(656\) 21.6343 0.844677
\(657\) −51.1565 −1.99580
\(658\) 15.9819 0.623039
\(659\) 8.74476 0.340647 0.170324 0.985388i \(-0.445519\pi\)
0.170324 + 0.985388i \(0.445519\pi\)
\(660\) 0 0
\(661\) 36.1030 1.40424 0.702121 0.712057i \(-0.252236\pi\)
0.702121 + 0.712057i \(0.252236\pi\)
\(662\) 5.88380 0.228680
\(663\) 13.5506 0.526264
\(664\) 20.0307 0.777342
\(665\) 0 0
\(666\) 85.2416 3.30305
\(667\) 3.72840 0.144364
\(668\) −0.481546 −0.0186316
\(669\) −33.7545 −1.30502
\(670\) 0 0
\(671\) −1.76785 −0.0682472
\(672\) 4.38551 0.169175
\(673\) 37.5762 1.44845 0.724227 0.689561i \(-0.242197\pi\)
0.724227 + 0.689561i \(0.242197\pi\)
\(674\) −11.6214 −0.447639
\(675\) 0 0
\(676\) −2.95461 −0.113639
\(677\) 21.5389 0.827807 0.413904 0.910321i \(-0.364165\pi\)
0.413904 + 0.910321i \(0.364165\pi\)
\(678\) −1.97886 −0.0759976
\(679\) −10.9894 −0.421736
\(680\) 0 0
\(681\) −63.8015 −2.44488
\(682\) −7.36090 −0.281863
\(683\) −20.0545 −0.767364 −0.383682 0.923465i \(-0.625344\pi\)
−0.383682 + 0.923465i \(0.625344\pi\)
\(684\) 8.01639 0.306514
\(685\) 0 0
\(686\) −1.50216 −0.0573527
\(687\) 14.0949 0.537753
\(688\) 16.2238 0.618525
\(689\) −9.10888 −0.347021
\(690\) 0 0
\(691\) 36.3245 1.38185 0.690925 0.722927i \(-0.257204\pi\)
0.690925 + 0.722927i \(0.257204\pi\)
\(692\) −2.80195 −0.106514
\(693\) 3.38527 0.128596
\(694\) 48.1165 1.82648
\(695\) 0 0
\(696\) −29.6917 −1.12546
\(697\) −17.8194 −0.674959
\(698\) 43.6297 1.65141
\(699\) −49.1685 −1.85972
\(700\) 0 0
\(701\) 28.3217 1.06969 0.534847 0.844949i \(-0.320369\pi\)
0.534847 + 0.844949i \(0.320369\pi\)
\(702\) 18.0362 0.680734
\(703\) 45.4675 1.71484
\(704\) −3.64660 −0.137437
\(705\) 0 0
\(706\) −21.2424 −0.799466
\(707\) −3.59862 −0.135340
\(708\) 4.03995 0.151831
\(709\) 21.4326 0.804919 0.402460 0.915438i \(-0.368155\pi\)
0.402460 + 0.915438i \(0.368155\pi\)
\(710\) 0 0
\(711\) 62.6383 2.34912
\(712\) 12.4327 0.465934
\(713\) 9.04067 0.338576
\(714\) −16.7310 −0.626140
\(715\) 0 0
\(716\) 1.21637 0.0454578
\(717\) 3.84980 0.143773
\(718\) 10.9034 0.406912
\(719\) 46.7125 1.74208 0.871040 0.491212i \(-0.163446\pi\)
0.871040 + 0.491212i \(0.163446\pi\)
\(720\) 0 0
\(721\) 9.48704 0.353316
\(722\) 9.07783 0.337842
\(723\) −6.37394 −0.237050
\(724\) 0.725421 0.0269601
\(725\) 0 0
\(726\) −48.9014 −1.81490
\(727\) −17.3705 −0.644237 −0.322119 0.946699i \(-0.604395\pi\)
−0.322119 + 0.946699i \(0.604395\pi\)
\(728\) 3.18638 0.118095
\(729\) −19.6276 −0.726947
\(730\) 0 0
\(731\) −13.3630 −0.494247
\(732\) 2.54364 0.0940156
\(733\) −15.4702 −0.571404 −0.285702 0.958318i \(-0.592227\pi\)
−0.285702 + 0.958318i \(0.592227\pi\)
\(734\) 23.6841 0.874195
\(735\) 0 0
\(736\) −1.44228 −0.0531632
\(737\) −6.52465 −0.240339
\(738\) −45.6408 −1.68006
\(739\) −17.5944 −0.647221 −0.323611 0.946190i \(-0.604897\pi\)
−0.323611 + 0.946190i \(0.604897\pi\)
\(740\) 0 0
\(741\) 18.5127 0.680080
\(742\) 11.2467 0.412880
\(743\) 32.0161 1.17456 0.587278 0.809385i \(-0.300199\pi\)
0.587278 + 0.809385i \(0.300199\pi\)
\(744\) −71.9967 −2.63953
\(745\) 0 0
\(746\) −6.26111 −0.229236
\(747\) −47.7676 −1.74772
\(748\) −0.509219 −0.0186189
\(749\) 6.60294 0.241266
\(750\) 0 0
\(751\) 38.1097 1.39064 0.695321 0.718700i \(-0.255262\pi\)
0.695321 + 0.718700i \(0.255262\pi\)
\(752\) 47.3148 1.72539
\(753\) 35.0788 1.27834
\(754\) 6.81386 0.248146
\(755\) 0 0
\(756\) −2.53122 −0.0920594
\(757\) −19.5846 −0.711814 −0.355907 0.934521i \(-0.615828\pi\)
−0.355907 + 0.934521i \(0.615828\pi\)
\(758\) 28.0261 1.01795
\(759\) −1.64810 −0.0598223
\(760\) 0 0
\(761\) −1.99465 −0.0723058 −0.0361529 0.999346i \(-0.511510\pi\)
−0.0361529 + 0.999346i \(0.511510\pi\)
\(762\) −40.3841 −1.46296
\(763\) −1.10726 −0.0400856
\(764\) −0.915220 −0.0331115
\(765\) 0 0
\(766\) 45.4897 1.64361
\(767\) 6.30242 0.227567
\(768\) 18.4223 0.664757
\(769\) 11.5731 0.417338 0.208669 0.977986i \(-0.433087\pi\)
0.208669 + 0.977986i \(0.433087\pi\)
\(770\) 0 0
\(771\) −44.5599 −1.60479
\(772\) −5.36450 −0.193073
\(773\) 47.3961 1.70472 0.852360 0.522955i \(-0.175170\pi\)
0.852360 + 0.522955i \(0.175170\pi\)
\(774\) −34.2265 −1.23025
\(775\) 0 0
\(776\) −28.7818 −1.03321
\(777\) −27.6265 −0.991095
\(778\) −21.1394 −0.757886
\(779\) −24.3446 −0.872236
\(780\) 0 0
\(781\) 2.87159 0.102754
\(782\) 5.50239 0.196765
\(783\) 36.7957 1.31497
\(784\) −4.44718 −0.158828
\(785\) 0 0
\(786\) −11.6442 −0.415334
\(787\) 30.1964 1.07639 0.538193 0.842822i \(-0.319107\pi\)
0.538193 + 0.842822i \(0.319107\pi\)
\(788\) −1.42238 −0.0506701
\(789\) 69.0638 2.45874
\(790\) 0 0
\(791\) 0.433241 0.0154043
\(792\) 8.86618 0.315046
\(793\) 3.96814 0.140913
\(794\) −45.4287 −1.61220
\(795\) 0 0
\(796\) −1.98467 −0.0703446
\(797\) 41.0916 1.45554 0.727770 0.685821i \(-0.240557\pi\)
0.727770 + 0.685821i \(0.240557\pi\)
\(798\) −22.8576 −0.809149
\(799\) −38.9716 −1.37872
\(800\) 0 0
\(801\) −29.6484 −1.04757
\(802\) −1.00439 −0.0354662
\(803\) 4.43951 0.156667
\(804\) 9.38787 0.331085
\(805\) 0 0
\(806\) 16.5223 0.581975
\(807\) 76.2768 2.68507
\(808\) −9.42494 −0.331568
\(809\) 17.9154 0.629870 0.314935 0.949113i \(-0.398017\pi\)
0.314935 + 0.949113i \(0.398017\pi\)
\(810\) 0 0
\(811\) 10.1577 0.356686 0.178343 0.983968i \(-0.442926\pi\)
0.178343 + 0.983968i \(0.442926\pi\)
\(812\) −0.956262 −0.0335582
\(813\) −85.2095 −2.98843
\(814\) −7.39753 −0.259283
\(815\) 0 0
\(816\) −49.5324 −1.73398
\(817\) −18.2563 −0.638705
\(818\) −13.6036 −0.475638
\(819\) −7.59862 −0.265517
\(820\) 0 0
\(821\) −15.2289 −0.531491 −0.265745 0.964043i \(-0.585618\pi\)
−0.265745 + 0.964043i \(0.585618\pi\)
\(822\) −62.2400 −2.17087
\(823\) 20.0090 0.697468 0.348734 0.937222i \(-0.386612\pi\)
0.348734 + 0.937222i \(0.386612\pi\)
\(824\) 24.8470 0.865585
\(825\) 0 0
\(826\) −7.78158 −0.270756
\(827\) −18.0259 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(828\) 1.60189 0.0556697
\(829\) 14.2616 0.495327 0.247664 0.968846i \(-0.420337\pi\)
0.247664 + 0.968846i \(0.420337\pi\)
\(830\) 0 0
\(831\) 49.4694 1.71607
\(832\) 8.18520 0.283771
\(833\) 3.66299 0.126915
\(834\) −82.4291 −2.85429
\(835\) 0 0
\(836\) −0.695686 −0.0240608
\(837\) 89.2227 3.08399
\(838\) −29.9818 −1.03570
\(839\) 51.5877 1.78101 0.890503 0.454977i \(-0.150353\pi\)
0.890503 + 0.454977i \(0.150353\pi\)
\(840\) 0 0
\(841\) −15.0990 −0.520656
\(842\) −10.3093 −0.355280
\(843\) −40.9561 −1.41060
\(844\) 2.42097 0.0833331
\(845\) 0 0
\(846\) −99.8178 −3.43181
\(847\) 10.7062 0.367870
\(848\) 33.2962 1.14340
\(849\) −54.8889 −1.88378
\(850\) 0 0
\(851\) 9.08566 0.311452
\(852\) −4.13174 −0.141551
\(853\) −48.2610 −1.65243 −0.826213 0.563358i \(-0.809509\pi\)
−0.826213 + 0.563358i \(0.809509\pi\)
\(854\) −4.89945 −0.167656
\(855\) 0 0
\(856\) 17.2934 0.591075
\(857\) 1.37665 0.0470256 0.0235128 0.999724i \(-0.492515\pi\)
0.0235128 + 0.999724i \(0.492515\pi\)
\(858\) −3.01200 −0.102828
\(859\) −36.3622 −1.24066 −0.620331 0.784340i \(-0.713002\pi\)
−0.620331 + 0.784340i \(0.713002\pi\)
\(860\) 0 0
\(861\) 14.7920 0.504110
\(862\) 6.79215 0.231342
\(863\) −44.8731 −1.52750 −0.763749 0.645514i \(-0.776643\pi\)
−0.763749 + 0.645514i \(0.776643\pi\)
\(864\) −14.2339 −0.484248
\(865\) 0 0
\(866\) −21.8068 −0.741025
\(867\) −10.8932 −0.369954
\(868\) −2.31876 −0.0787037
\(869\) −5.43594 −0.184402
\(870\) 0 0
\(871\) 14.6453 0.496237
\(872\) −2.89997 −0.0982054
\(873\) 68.6364 2.32299
\(874\) 7.51728 0.254276
\(875\) 0 0
\(876\) −6.38770 −0.215820
\(877\) −42.2559 −1.42688 −0.713441 0.700716i \(-0.752864\pi\)
−0.713441 + 0.700716i \(0.752864\pi\)
\(878\) 38.3137 1.29303
\(879\) −70.3979 −2.37446
\(880\) 0 0
\(881\) 51.1432 1.72306 0.861529 0.507708i \(-0.169507\pi\)
0.861529 + 0.507708i \(0.169507\pi\)
\(882\) 9.38200 0.315908
\(883\) −52.6889 −1.77312 −0.886562 0.462610i \(-0.846913\pi\)
−0.886562 + 0.462610i \(0.846913\pi\)
\(884\) 1.14300 0.0384432
\(885\) 0 0
\(886\) −20.9451 −0.703666
\(887\) 42.9102 1.44078 0.720391 0.693568i \(-0.243962\pi\)
0.720391 + 0.693568i \(0.243962\pi\)
\(888\) −72.3550 −2.42807
\(889\) 8.84149 0.296534
\(890\) 0 0
\(891\) −6.10934 −0.204671
\(892\) −2.84719 −0.0953311
\(893\) −53.2424 −1.78169
\(894\) 63.1137 2.11084
\(895\) 0 0
\(896\) −12.9908 −0.433993
\(897\) 3.69934 0.123517
\(898\) −33.8446 −1.12941
\(899\) 33.7072 1.12420
\(900\) 0 0
\(901\) −27.4249 −0.913657
\(902\) 3.96085 0.131882
\(903\) 11.0927 0.369141
\(904\) 1.13468 0.0377388
\(905\) 0 0
\(906\) 73.7261 2.44939
\(907\) 24.5213 0.814217 0.407108 0.913380i \(-0.366537\pi\)
0.407108 + 0.913380i \(0.366537\pi\)
\(908\) −5.38166 −0.178597
\(909\) 22.4758 0.745476
\(910\) 0 0
\(911\) −31.6347 −1.04810 −0.524052 0.851686i \(-0.675580\pi\)
−0.524052 + 0.851686i \(0.675580\pi\)
\(912\) −67.6704 −2.24079
\(913\) 4.14541 0.137193
\(914\) 19.4663 0.643888
\(915\) 0 0
\(916\) 1.18890 0.0392824
\(917\) 2.54931 0.0841858
\(918\) 54.3033 1.79228
\(919\) −43.7293 −1.44250 −0.721249 0.692676i \(-0.756431\pi\)
−0.721249 + 0.692676i \(0.756431\pi\)
\(920\) 0 0
\(921\) 3.47553 0.114523
\(922\) 22.4294 0.738674
\(923\) −6.44561 −0.212160
\(924\) 0.422706 0.0139060
\(925\) 0 0
\(926\) −26.2788 −0.863576
\(927\) −59.2530 −1.94612
\(928\) −5.37741 −0.176522
\(929\) 3.63411 0.119231 0.0596157 0.998221i \(-0.481012\pi\)
0.0596157 + 0.998221i \(0.481012\pi\)
\(930\) 0 0
\(931\) 5.00432 0.164010
\(932\) −4.14736 −0.135851
\(933\) −58.7445 −1.92321
\(934\) 26.4802 0.866460
\(935\) 0 0
\(936\) −19.9011 −0.650488
\(937\) 10.3649 0.338606 0.169303 0.985564i \(-0.445848\pi\)
0.169303 + 0.985564i \(0.445848\pi\)
\(938\) −18.0825 −0.590415
\(939\) 52.9503 1.72797
\(940\) 0 0
\(941\) −29.8720 −0.973800 −0.486900 0.873458i \(-0.661872\pi\)
−0.486900 + 0.873458i \(0.661872\pi\)
\(942\) 69.9345 2.27859
\(943\) −4.86472 −0.158417
\(944\) −23.0376 −0.749809
\(945\) 0 0
\(946\) 2.97028 0.0965720
\(947\) −7.17017 −0.232999 −0.116500 0.993191i \(-0.537167\pi\)
−0.116500 + 0.993191i \(0.537167\pi\)
\(948\) 7.82139 0.254027
\(949\) −9.96497 −0.323477
\(950\) 0 0
\(951\) −4.33636 −0.140616
\(952\) 9.59352 0.310928
\(953\) −16.0495 −0.519894 −0.259947 0.965623i \(-0.583705\pi\)
−0.259947 + 0.965623i \(0.583705\pi\)
\(954\) −70.2434 −2.27421
\(955\) 0 0
\(956\) 0.324730 0.0105025
\(957\) −6.14478 −0.198632
\(958\) −17.6438 −0.570046
\(959\) 13.6265 0.440022
\(960\) 0 0
\(961\) 50.7337 1.63657
\(962\) 16.6046 0.535352
\(963\) −41.2398 −1.32893
\(964\) −0.537642 −0.0173163
\(965\) 0 0
\(966\) −4.56757 −0.146959
\(967\) −33.5360 −1.07845 −0.539223 0.842163i \(-0.681282\pi\)
−0.539223 + 0.842163i \(0.681282\pi\)
\(968\) 28.0400 0.901241
\(969\) 55.7378 1.79056
\(970\) 0 0
\(971\) −17.5189 −0.562208 −0.281104 0.959677i \(-0.590701\pi\)
−0.281104 + 0.959677i \(0.590701\pi\)
\(972\) 1.19666 0.0383828
\(973\) 18.0466 0.578548
\(974\) −34.4565 −1.10406
\(975\) 0 0
\(976\) −14.5050 −0.464292
\(977\) −50.1902 −1.60573 −0.802863 0.596163i \(-0.796691\pi\)
−0.802863 + 0.596163i \(0.796691\pi\)
\(978\) 18.0362 0.576735
\(979\) 2.57298 0.0822327
\(980\) 0 0
\(981\) 6.91561 0.220798
\(982\) 17.3773 0.554533
\(983\) 22.7924 0.726966 0.363483 0.931601i \(-0.381587\pi\)
0.363483 + 0.931601i \(0.381587\pi\)
\(984\) 38.7409 1.23502
\(985\) 0 0
\(986\) 20.5151 0.653334
\(987\) 32.3506 1.02973
\(988\) 1.56154 0.0496793
\(989\) −3.64810 −0.116003
\(990\) 0 0
\(991\) −0.965443 −0.0306683 −0.0153342 0.999882i \(-0.504881\pi\)
−0.0153342 + 0.999882i \(0.504881\pi\)
\(992\) −13.0392 −0.413995
\(993\) 11.9100 0.377952
\(994\) 7.95838 0.252424
\(995\) 0 0
\(996\) −5.96455 −0.188994
\(997\) 4.22619 0.133845 0.0669224 0.997758i \(-0.478682\pi\)
0.0669224 + 0.997758i \(0.478682\pi\)
\(998\) 37.5634 1.18905
\(999\) 89.6667 2.83693
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 4025.2.a.p.1.4 5
5.4 even 2 161.2.a.d.1.2 5
15.14 odd 2 1449.2.a.r.1.4 5
20.19 odd 2 2576.2.a.bd.1.5 5
35.34 odd 2 1127.2.a.h.1.2 5
115.114 odd 2 3703.2.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.d.1.2 5 5.4 even 2
1127.2.a.h.1.2 5 35.34 odd 2
1449.2.a.r.1.4 5 15.14 odd 2
2576.2.a.bd.1.5 5 20.19 odd 2
3703.2.a.j.1.2 5 115.114 odd 2
4025.2.a.p.1.4 5 1.1 even 1 trivial