Properties

Label 2842.2.a.x.1.3
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1019601.1
Defining polynomial: \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.20194\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.515089 q^{3} +1.00000 q^{4} -3.68685 q^{5} -0.515089 q^{6} +1.00000 q^{8} -2.73468 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.515089 q^{3} +1.00000 q^{4} -3.68685 q^{5} -0.515089 q^{6} +1.00000 q^{8} -2.73468 q^{9} -3.68685 q^{10} +6.45172 q^{11} -0.515089 q^{12} -2.84856 q^{13} +1.89906 q^{15} +1.00000 q^{16} +1.53541 q^{17} -2.73468 q^{18} +3.91898 q^{19} -3.68685 q^{20} +6.45172 q^{22} +0.380892 q^{23} -0.515089 q^{24} +8.59290 q^{25} -2.84856 q^{26} +2.95387 q^{27} -1.00000 q^{29} +1.89906 q^{30} -7.98405 q^{31} +1.00000 q^{32} -3.32321 q^{33} +1.53541 q^{34} -2.73468 q^{36} -7.46896 q^{37} +3.91898 q^{38} +1.46726 q^{39} -3.68685 q^{40} +4.98713 q^{41} -11.5673 q^{43} +6.45172 q^{44} +10.0824 q^{45} +0.380892 q^{46} -7.85862 q^{47} -0.515089 q^{48} +8.59290 q^{50} -0.790874 q^{51} -2.84856 q^{52} -1.22186 q^{53} +2.95387 q^{54} -23.7865 q^{55} -2.01862 q^{57} -1.00000 q^{58} +0.696710 q^{59} +1.89906 q^{60} +3.10100 q^{61} -7.98405 q^{62} +1.00000 q^{64} +10.5022 q^{65} -3.32321 q^{66} -7.22494 q^{67} +1.53541 q^{68} -0.196193 q^{69} -13.2927 q^{71} -2.73468 q^{72} -14.9513 q^{73} -7.46896 q^{74} -4.42611 q^{75} +3.91898 q^{76} +1.46726 q^{78} +5.13590 q^{79} -3.68685 q^{80} +6.68254 q^{81} +4.98713 q^{82} -12.8154 q^{83} -5.66084 q^{85} -11.5673 q^{86} +0.515089 q^{87} +6.45172 q^{88} -9.39630 q^{89} +10.0824 q^{90} +0.380892 q^{92} +4.11250 q^{93} -7.85862 q^{94} -14.4487 q^{95} -0.515089 q^{96} +2.56729 q^{97} -17.6434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9} - 7 q^{10} - 3 q^{12} - 10 q^{13} - 10 q^{15} + 5 q^{16} - 8 q^{17} + 8 q^{18} - 2 q^{19} - 7 q^{20} + q^{23} - 3 q^{24} + 12 q^{25} - 10 q^{26} - 15 q^{27} - 5 q^{29} - 10 q^{30} - 11 q^{31} + 5 q^{32} - 9 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 2 q^{38} + 18 q^{39} - 7 q^{40} - 23 q^{41} - 3 q^{43} - 4 q^{45} + q^{46} - 16 q^{47} - 3 q^{48} + 12 q^{50} + 7 q^{51} - 10 q^{52} + 7 q^{53} - 15 q^{54} - 6 q^{55} - 34 q^{57} - 5 q^{58} + 9 q^{59} - 10 q^{60} - 15 q^{61} - 11 q^{62} + 5 q^{64} + 5 q^{65} - 9 q^{66} - 4 q^{67} - 8 q^{68} - 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} + 34 q^{75} - 2 q^{76} + 18 q^{78} - 13 q^{79} - 7 q^{80} + 17 q^{81} - 23 q^{82} - 28 q^{83} - 7 q^{85} - 3 q^{86} + 3 q^{87} - 17 q^{89} - 4 q^{90} + q^{92} + 17 q^{93} - 16 q^{94} - 9 q^{95} - 3 q^{96} - 42 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.515089 −0.297387 −0.148693 0.988883i \(-0.547507\pi\)
−0.148693 + 0.988883i \(0.547507\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.68685 −1.64881 −0.824406 0.565999i \(-0.808491\pi\)
−0.824406 + 0.565999i \(0.808491\pi\)
\(6\) −0.515089 −0.210284
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.73468 −0.911561
\(10\) −3.68685 −1.16589
\(11\) 6.45172 1.94527 0.972633 0.232348i \(-0.0746407\pi\)
0.972633 + 0.232348i \(0.0746407\pi\)
\(12\) −0.515089 −0.148693
\(13\) −2.84856 −0.790048 −0.395024 0.918671i \(-0.629264\pi\)
−0.395024 + 0.918671i \(0.629264\pi\)
\(14\) 0 0
\(15\) 1.89906 0.490335
\(16\) 1.00000 0.250000
\(17\) 1.53541 0.372392 0.186196 0.982513i \(-0.440384\pi\)
0.186196 + 0.982513i \(0.440384\pi\)
\(18\) −2.73468 −0.644571
\(19\) 3.91898 0.899075 0.449537 0.893261i \(-0.351589\pi\)
0.449537 + 0.893261i \(0.351589\pi\)
\(20\) −3.68685 −0.824406
\(21\) 0 0
\(22\) 6.45172 1.37551
\(23\) 0.380892 0.0794214 0.0397107 0.999211i \(-0.487356\pi\)
0.0397107 + 0.999211i \(0.487356\pi\)
\(24\) −0.515089 −0.105142
\(25\) 8.59290 1.71858
\(26\) −2.84856 −0.558648
\(27\) 2.95387 0.568473
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 1.89906 0.346719
\(31\) −7.98405 −1.43398 −0.716989 0.697085i \(-0.754480\pi\)
−0.716989 + 0.697085i \(0.754480\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.32321 −0.578496
\(34\) 1.53541 0.263321
\(35\) 0 0
\(36\) −2.73468 −0.455781
\(37\) −7.46896 −1.22789 −0.613945 0.789349i \(-0.710418\pi\)
−0.613945 + 0.789349i \(0.710418\pi\)
\(38\) 3.91898 0.635742
\(39\) 1.46726 0.234950
\(40\) −3.68685 −0.582943
\(41\) 4.98713 0.778859 0.389429 0.921056i \(-0.372672\pi\)
0.389429 + 0.921056i \(0.372672\pi\)
\(42\) 0 0
\(43\) −11.5673 −1.76400 −0.881998 0.471254i \(-0.843802\pi\)
−0.881998 + 0.471254i \(0.843802\pi\)
\(44\) 6.45172 0.972633
\(45\) 10.0824 1.50299
\(46\) 0.380892 0.0561594
\(47\) −7.85862 −1.14630 −0.573149 0.819451i \(-0.694278\pi\)
−0.573149 + 0.819451i \(0.694278\pi\)
\(48\) −0.515089 −0.0743467
\(49\) 0 0
\(50\) 8.59290 1.21522
\(51\) −0.790874 −0.110745
\(52\) −2.84856 −0.395024
\(53\) −1.22186 −0.167836 −0.0839179 0.996473i \(-0.526743\pi\)
−0.0839179 + 0.996473i \(0.526743\pi\)
\(54\) 2.95387 0.401971
\(55\) −23.7865 −3.20738
\(56\) 0 0
\(57\) −2.01862 −0.267373
\(58\) −1.00000 −0.131306
\(59\) 0.696710 0.0907039 0.0453519 0.998971i \(-0.485559\pi\)
0.0453519 + 0.998971i \(0.485559\pi\)
\(60\) 1.89906 0.245167
\(61\) 3.10100 0.397043 0.198521 0.980097i \(-0.436386\pi\)
0.198521 + 0.980097i \(0.436386\pi\)
\(62\) −7.98405 −1.01398
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.5022 1.30264
\(66\) −3.32321 −0.409059
\(67\) −7.22494 −0.882667 −0.441333 0.897343i \(-0.645494\pi\)
−0.441333 + 0.897343i \(0.645494\pi\)
\(68\) 1.53541 0.186196
\(69\) −0.196193 −0.0236189
\(70\) 0 0
\(71\) −13.2927 −1.57755 −0.788776 0.614681i \(-0.789285\pi\)
−0.788776 + 0.614681i \(0.789285\pi\)
\(72\) −2.73468 −0.322286
\(73\) −14.9513 −1.74991 −0.874956 0.484202i \(-0.839110\pi\)
−0.874956 + 0.484202i \(0.839110\pi\)
\(74\) −7.46896 −0.868249
\(75\) −4.42611 −0.511083
\(76\) 3.91898 0.449537
\(77\) 0 0
\(78\) 1.46726 0.166135
\(79\) 5.13590 0.577834 0.288917 0.957354i \(-0.406705\pi\)
0.288917 + 0.957354i \(0.406705\pi\)
\(80\) −3.68685 −0.412203
\(81\) 6.68254 0.742505
\(82\) 4.98713 0.550736
\(83\) −12.8154 −1.40667 −0.703334 0.710859i \(-0.748306\pi\)
−0.703334 + 0.710859i \(0.748306\pi\)
\(84\) 0 0
\(85\) −5.66084 −0.614005
\(86\) −11.5673 −1.24733
\(87\) 0.515089 0.0552233
\(88\) 6.45172 0.687755
\(89\) −9.39630 −0.996006 −0.498003 0.867175i \(-0.665933\pi\)
−0.498003 + 0.867175i \(0.665933\pi\)
\(90\) 10.0824 1.06278
\(91\) 0 0
\(92\) 0.380892 0.0397107
\(93\) 4.11250 0.426446
\(94\) −7.85862 −0.810555
\(95\) −14.4487 −1.48241
\(96\) −0.515089 −0.0525711
\(97\) 2.56729 0.260669 0.130334 0.991470i \(-0.458395\pi\)
0.130334 + 0.991470i \(0.458395\pi\)
\(98\) 0 0
\(99\) −17.6434 −1.77323
\(100\) 8.59290 0.859290
\(101\) 10.0344 0.998456 0.499228 0.866471i \(-0.333617\pi\)
0.499228 + 0.866471i \(0.333617\pi\)
\(102\) −0.790874 −0.0783082
\(103\) 9.34353 0.920646 0.460323 0.887752i \(-0.347734\pi\)
0.460323 + 0.887752i \(0.347734\pi\)
\(104\) −2.84856 −0.279324
\(105\) 0 0
\(106\) −1.22186 −0.118678
\(107\) −11.7990 −1.14065 −0.570327 0.821418i \(-0.693183\pi\)
−0.570327 + 0.821418i \(0.693183\pi\)
\(108\) 2.95387 0.284237
\(109\) −6.05447 −0.579913 −0.289957 0.957040i \(-0.593641\pi\)
−0.289957 + 0.957040i \(0.593641\pi\)
\(110\) −23.7865 −2.26796
\(111\) 3.84718 0.365158
\(112\) 0 0
\(113\) −3.50810 −0.330015 −0.165007 0.986292i \(-0.552765\pi\)
−0.165007 + 0.986292i \(0.552765\pi\)
\(114\) −2.01862 −0.189061
\(115\) −1.40429 −0.130951
\(116\) −1.00000 −0.0928477
\(117\) 7.78990 0.720177
\(118\) 0.696710 0.0641373
\(119\) 0 0
\(120\) 1.89906 0.173360
\(121\) 30.6246 2.78406
\(122\) 3.10100 0.280752
\(123\) −2.56882 −0.231622
\(124\) −7.98405 −0.716989
\(125\) −13.2465 −1.18480
\(126\) 0 0
\(127\) 2.64032 0.234291 0.117145 0.993115i \(-0.462626\pi\)
0.117145 + 0.993115i \(0.462626\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.95819 0.524589
\(130\) 10.5022 0.921105
\(131\) 10.7569 0.939832 0.469916 0.882711i \(-0.344284\pi\)
0.469916 + 0.882711i \(0.344284\pi\)
\(132\) −3.32321 −0.289248
\(133\) 0 0
\(134\) −7.22494 −0.624140
\(135\) −10.8905 −0.937305
\(136\) 1.53541 0.131661
\(137\) 8.47238 0.723845 0.361922 0.932208i \(-0.382121\pi\)
0.361922 + 0.932208i \(0.382121\pi\)
\(138\) −0.196193 −0.0167011
\(139\) 2.40081 0.203634 0.101817 0.994803i \(-0.467534\pi\)
0.101817 + 0.994803i \(0.467534\pi\)
\(140\) 0 0
\(141\) 4.04789 0.340894
\(142\) −13.2927 −1.11550
\(143\) −18.3781 −1.53685
\(144\) −2.73468 −0.227890
\(145\) 3.68685 0.306177
\(146\) −14.9513 −1.23738
\(147\) 0 0
\(148\) −7.46896 −0.613945
\(149\) −10.3750 −0.849954 −0.424977 0.905204i \(-0.639718\pi\)
−0.424977 + 0.905204i \(0.639718\pi\)
\(150\) −4.42611 −0.361390
\(151\) 6.75687 0.549866 0.274933 0.961463i \(-0.411344\pi\)
0.274933 + 0.961463i \(0.411344\pi\)
\(152\) 3.91898 0.317871
\(153\) −4.19887 −0.339458
\(154\) 0 0
\(155\) 29.4360 2.36436
\(156\) 1.46726 0.117475
\(157\) 3.89720 0.311030 0.155515 0.987834i \(-0.450296\pi\)
0.155515 + 0.987834i \(0.450296\pi\)
\(158\) 5.13590 0.408590
\(159\) 0.629368 0.0499121
\(160\) −3.68685 −0.291471
\(161\) 0 0
\(162\) 6.68254 0.525030
\(163\) 0.389316 0.0304935 0.0152468 0.999884i \(-0.495147\pi\)
0.0152468 + 0.999884i \(0.495147\pi\)
\(164\) 4.98713 0.389429
\(165\) 12.2522 0.953832
\(166\) −12.8154 −0.994665
\(167\) −10.3042 −0.797362 −0.398681 0.917090i \(-0.630532\pi\)
−0.398681 + 0.917090i \(0.630532\pi\)
\(168\) 0 0
\(169\) −4.88572 −0.375825
\(170\) −5.66084 −0.434167
\(171\) −10.7172 −0.819562
\(172\) −11.5673 −0.881998
\(173\) −23.0407 −1.75175 −0.875876 0.482536i \(-0.839716\pi\)
−0.875876 + 0.482536i \(0.839716\pi\)
\(174\) 0.515089 0.0390488
\(175\) 0 0
\(176\) 6.45172 0.486316
\(177\) −0.358868 −0.0269741
\(178\) −9.39630 −0.704283
\(179\) −6.69801 −0.500632 −0.250316 0.968164i \(-0.580535\pi\)
−0.250316 + 0.968164i \(0.580535\pi\)
\(180\) 10.0824 0.751496
\(181\) −13.2453 −0.984513 −0.492256 0.870450i \(-0.663828\pi\)
−0.492256 + 0.870450i \(0.663828\pi\)
\(182\) 0 0
\(183\) −1.59729 −0.118075
\(184\) 0.380892 0.0280797
\(185\) 27.5370 2.02456
\(186\) 4.11250 0.301543
\(187\) 9.90604 0.724402
\(188\) −7.85862 −0.573149
\(189\) 0 0
\(190\) −14.4487 −1.04822
\(191\) −20.3346 −1.47136 −0.735681 0.677328i \(-0.763138\pi\)
−0.735681 + 0.677328i \(0.763138\pi\)
\(192\) −0.515089 −0.0371734
\(193\) 13.2040 0.950448 0.475224 0.879865i \(-0.342367\pi\)
0.475224 + 0.879865i \(0.342367\pi\)
\(194\) 2.56729 0.184321
\(195\) −5.40958 −0.387388
\(196\) 0 0
\(197\) −9.01237 −0.642104 −0.321052 0.947061i \(-0.604037\pi\)
−0.321052 + 0.947061i \(0.604037\pi\)
\(198\) −17.6434 −1.25386
\(199\) 5.86360 0.415659 0.207830 0.978165i \(-0.433360\pi\)
0.207830 + 0.978165i \(0.433360\pi\)
\(200\) 8.59290 0.607610
\(201\) 3.72149 0.262494
\(202\) 10.0344 0.706015
\(203\) 0 0
\(204\) −0.790874 −0.0553723
\(205\) −18.3868 −1.28419
\(206\) 9.34353 0.650995
\(207\) −1.04162 −0.0723975
\(208\) −2.84856 −0.197512
\(209\) 25.2841 1.74894
\(210\) 0 0
\(211\) 20.9741 1.44391 0.721957 0.691938i \(-0.243243\pi\)
0.721957 + 0.691938i \(0.243243\pi\)
\(212\) −1.22186 −0.0839179
\(213\) 6.84692 0.469143
\(214\) −11.7990 −0.806564
\(215\) 42.6469 2.90850
\(216\) 2.95387 0.200986
\(217\) 0 0
\(218\) −6.05447 −0.410060
\(219\) 7.70123 0.520401
\(220\) −23.7865 −1.60369
\(221\) −4.37371 −0.294208
\(222\) 3.84718 0.258206
\(223\) 1.10629 0.0740825 0.0370412 0.999314i \(-0.488207\pi\)
0.0370412 + 0.999314i \(0.488207\pi\)
\(224\) 0 0
\(225\) −23.4989 −1.56659
\(226\) −3.50810 −0.233356
\(227\) 0.869629 0.0577193 0.0288596 0.999583i \(-0.490812\pi\)
0.0288596 + 0.999583i \(0.490812\pi\)
\(228\) −2.01862 −0.133687
\(229\) 7.00356 0.462809 0.231404 0.972858i \(-0.425668\pi\)
0.231404 + 0.972858i \(0.425668\pi\)
\(230\) −1.40429 −0.0925963
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 0.411278 0.0269437 0.0134719 0.999909i \(-0.495712\pi\)
0.0134719 + 0.999909i \(0.495712\pi\)
\(234\) 7.78990 0.509242
\(235\) 28.9736 1.89003
\(236\) 0.696710 0.0453519
\(237\) −2.64545 −0.171840
\(238\) 0 0
\(239\) −12.6415 −0.817713 −0.408857 0.912599i \(-0.634072\pi\)
−0.408857 + 0.912599i \(0.634072\pi\)
\(240\) 1.89906 0.122584
\(241\) −20.9092 −1.34688 −0.673439 0.739242i \(-0.735184\pi\)
−0.673439 + 0.739242i \(0.735184\pi\)
\(242\) 30.6246 1.96863
\(243\) −12.3037 −0.789284
\(244\) 3.10100 0.198521
\(245\) 0 0
\(246\) −2.56882 −0.163782
\(247\) −11.1634 −0.710312
\(248\) −7.98405 −0.506988
\(249\) 6.60105 0.418325
\(250\) −13.2465 −0.837782
\(251\) 23.2297 1.46624 0.733121 0.680098i \(-0.238063\pi\)
0.733121 + 0.680098i \(0.238063\pi\)
\(252\) 0 0
\(253\) 2.45741 0.154496
\(254\) 2.64032 0.165669
\(255\) 2.91584 0.182597
\(256\) 1.00000 0.0625000
\(257\) 15.5148 0.967784 0.483892 0.875128i \(-0.339223\pi\)
0.483892 + 0.875128i \(0.339223\pi\)
\(258\) 5.95819 0.370940
\(259\) 0 0
\(260\) 10.5022 0.651320
\(261\) 2.73468 0.169273
\(262\) 10.7569 0.664562
\(263\) −2.94512 −0.181604 −0.0908021 0.995869i \(-0.528943\pi\)
−0.0908021 + 0.995869i \(0.528943\pi\)
\(264\) −3.32321 −0.204529
\(265\) 4.50483 0.276729
\(266\) 0 0
\(267\) 4.83993 0.296199
\(268\) −7.22494 −0.441333
\(269\) 6.76917 0.412724 0.206362 0.978476i \(-0.433838\pi\)
0.206362 + 0.978476i \(0.433838\pi\)
\(270\) −10.8905 −0.662775
\(271\) −16.8757 −1.02513 −0.512564 0.858649i \(-0.671304\pi\)
−0.512564 + 0.858649i \(0.671304\pi\)
\(272\) 1.53541 0.0930980
\(273\) 0 0
\(274\) 8.47238 0.511835
\(275\) 55.4389 3.34309
\(276\) −0.196193 −0.0118094
\(277\) 26.3155 1.58115 0.790573 0.612368i \(-0.209783\pi\)
0.790573 + 0.612368i \(0.209783\pi\)
\(278\) 2.40081 0.143991
\(279\) 21.8338 1.30716
\(280\) 0 0
\(281\) 0.662563 0.0395252 0.0197626 0.999805i \(-0.493709\pi\)
0.0197626 + 0.999805i \(0.493709\pi\)
\(282\) 4.04789 0.241048
\(283\) 24.9296 1.48191 0.740955 0.671555i \(-0.234373\pi\)
0.740955 + 0.671555i \(0.234373\pi\)
\(284\) −13.2927 −0.788776
\(285\) 7.44237 0.440848
\(286\) −18.3781 −1.08672
\(287\) 0 0
\(288\) −2.73468 −0.161143
\(289\) −14.6425 −0.861324
\(290\) 3.68685 0.216500
\(291\) −1.32238 −0.0775195
\(292\) −14.9513 −0.874956
\(293\) −9.98069 −0.583078 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(294\) 0 0
\(295\) −2.56867 −0.149554
\(296\) −7.46896 −0.434125
\(297\) 19.0575 1.10583
\(298\) −10.3750 −0.601008
\(299\) −1.08499 −0.0627467
\(300\) −4.42611 −0.255541
\(301\) 0 0
\(302\) 6.75687 0.388814
\(303\) −5.16859 −0.296928
\(304\) 3.91898 0.224769
\(305\) −11.4329 −0.654649
\(306\) −4.19887 −0.240033
\(307\) 2.78572 0.158990 0.0794948 0.996835i \(-0.474669\pi\)
0.0794948 + 0.996835i \(0.474669\pi\)
\(308\) 0 0
\(309\) −4.81275 −0.273788
\(310\) 29.4360 1.67185
\(311\) −15.2581 −0.865209 −0.432605 0.901584i \(-0.642405\pi\)
−0.432605 + 0.901584i \(0.642405\pi\)
\(312\) 1.46726 0.0830673
\(313\) −34.3024 −1.93889 −0.969443 0.245318i \(-0.921108\pi\)
−0.969443 + 0.245318i \(0.921108\pi\)
\(314\) 3.89720 0.219931
\(315\) 0 0
\(316\) 5.13590 0.288917
\(317\) −23.5203 −1.32103 −0.660516 0.750812i \(-0.729662\pi\)
−0.660516 + 0.750812i \(0.729662\pi\)
\(318\) 0.629368 0.0352932
\(319\) −6.45172 −0.361227
\(320\) −3.68685 −0.206101
\(321\) 6.07754 0.339215
\(322\) 0 0
\(323\) 6.01725 0.334808
\(324\) 6.68254 0.371252
\(325\) −24.4774 −1.35776
\(326\) 0.389316 0.0215622
\(327\) 3.11859 0.172459
\(328\) 4.98713 0.275368
\(329\) 0 0
\(330\) 12.2522 0.674461
\(331\) −20.9170 −1.14970 −0.574851 0.818258i \(-0.694940\pi\)
−0.574851 + 0.818258i \(0.694940\pi\)
\(332\) −12.8154 −0.703334
\(333\) 20.4252 1.11930
\(334\) −10.3042 −0.563820
\(335\) 26.6373 1.45535
\(336\) 0 0
\(337\) 35.1825 1.91651 0.958256 0.285910i \(-0.0922959\pi\)
0.958256 + 0.285910i \(0.0922959\pi\)
\(338\) −4.88572 −0.265748
\(339\) 1.80699 0.0981420
\(340\) −5.66084 −0.307002
\(341\) −51.5108 −2.78947
\(342\) −10.7172 −0.579518
\(343\) 0 0
\(344\) −11.5673 −0.623667
\(345\) 0.723336 0.0389431
\(346\) −23.0407 −1.23868
\(347\) −14.8661 −0.798053 −0.399026 0.916939i \(-0.630652\pi\)
−0.399026 + 0.916939i \(0.630652\pi\)
\(348\) 0.515089 0.0276117
\(349\) 20.1456 1.07837 0.539185 0.842187i \(-0.318732\pi\)
0.539185 + 0.842187i \(0.318732\pi\)
\(350\) 0 0
\(351\) −8.41428 −0.449121
\(352\) 6.45172 0.343878
\(353\) 9.55582 0.508605 0.254302 0.967125i \(-0.418154\pi\)
0.254302 + 0.967125i \(0.418154\pi\)
\(354\) −0.358868 −0.0190736
\(355\) 49.0082 2.60109
\(356\) −9.39630 −0.498003
\(357\) 0 0
\(358\) −6.69801 −0.354001
\(359\) 12.6878 0.669637 0.334818 0.942283i \(-0.391325\pi\)
0.334818 + 0.942283i \(0.391325\pi\)
\(360\) 10.0824 0.531388
\(361\) −3.64162 −0.191664
\(362\) −13.2453 −0.696156
\(363\) −15.7744 −0.827942
\(364\) 0 0
\(365\) 55.1231 2.88528
\(366\) −1.59729 −0.0834918
\(367\) 33.0253 1.72391 0.861953 0.506989i \(-0.169241\pi\)
0.861953 + 0.506989i \(0.169241\pi\)
\(368\) 0.380892 0.0198554
\(369\) −13.6382 −0.709977
\(370\) 27.5370 1.43158
\(371\) 0 0
\(372\) 4.11250 0.213223
\(373\) −19.9474 −1.03284 −0.516420 0.856336i \(-0.672735\pi\)
−0.516420 + 0.856336i \(0.672735\pi\)
\(374\) 9.90604 0.512229
\(375\) 6.82313 0.352345
\(376\) −7.85862 −0.405277
\(377\) 2.84856 0.146708
\(378\) 0 0
\(379\) 27.8725 1.43171 0.715857 0.698247i \(-0.246036\pi\)
0.715857 + 0.698247i \(0.246036\pi\)
\(380\) −14.4487 −0.741203
\(381\) −1.36000 −0.0696750
\(382\) −20.3346 −1.04041
\(383\) −21.1598 −1.08122 −0.540609 0.841274i \(-0.681806\pi\)
−0.540609 + 0.841274i \(0.681806\pi\)
\(384\) −0.515089 −0.0262855
\(385\) 0 0
\(386\) 13.2040 0.672068
\(387\) 31.6329 1.60799
\(388\) 2.56729 0.130334
\(389\) 8.79804 0.446078 0.223039 0.974810i \(-0.428402\pi\)
0.223039 + 0.974810i \(0.428402\pi\)
\(390\) −5.40958 −0.273925
\(391\) 0.584826 0.0295759
\(392\) 0 0
\(393\) −5.54075 −0.279494
\(394\) −9.01237 −0.454036
\(395\) −18.9353 −0.952739
\(396\) −17.6434 −0.886614
\(397\) 7.58413 0.380637 0.190318 0.981722i \(-0.439048\pi\)
0.190318 + 0.981722i \(0.439048\pi\)
\(398\) 5.86360 0.293916
\(399\) 0 0
\(400\) 8.59290 0.429645
\(401\) 11.8271 0.590616 0.295308 0.955402i \(-0.404578\pi\)
0.295308 + 0.955402i \(0.404578\pi\)
\(402\) 3.72149 0.185611
\(403\) 22.7430 1.13291
\(404\) 10.0344 0.499228
\(405\) −24.6376 −1.22425
\(406\) 0 0
\(407\) −48.1876 −2.38857
\(408\) −0.790874 −0.0391541
\(409\) −11.8506 −0.585975 −0.292987 0.956116i \(-0.594649\pi\)
−0.292987 + 0.956116i \(0.594649\pi\)
\(410\) −18.3868 −0.908060
\(411\) −4.36403 −0.215262
\(412\) 9.34353 0.460323
\(413\) 0 0
\(414\) −1.04162 −0.0511928
\(415\) 47.2484 2.31933
\(416\) −2.84856 −0.139662
\(417\) −1.23663 −0.0605581
\(418\) 25.2841 1.23669
\(419\) −0.933145 −0.0455871 −0.0227935 0.999740i \(-0.507256\pi\)
−0.0227935 + 0.999740i \(0.507256\pi\)
\(420\) 0 0
\(421\) −3.17478 −0.154729 −0.0773647 0.997003i \(-0.524651\pi\)
−0.0773647 + 0.997003i \(0.524651\pi\)
\(422\) 20.9741 1.02100
\(423\) 21.4908 1.04492
\(424\) −1.22186 −0.0593389
\(425\) 13.1936 0.639986
\(426\) 6.84692 0.331734
\(427\) 0 0
\(428\) −11.7990 −0.570327
\(429\) 9.46635 0.457040
\(430\) 42.6469 2.05662
\(431\) −25.2882 −1.21809 −0.609044 0.793137i \(-0.708447\pi\)
−0.609044 + 0.793137i \(0.708447\pi\)
\(432\) 2.95387 0.142118
\(433\) −25.3900 −1.22017 −0.610084 0.792337i \(-0.708864\pi\)
−0.610084 + 0.792337i \(0.708864\pi\)
\(434\) 0 0
\(435\) −1.89906 −0.0910529
\(436\) −6.05447 −0.289957
\(437\) 1.49271 0.0714058
\(438\) 7.70123 0.367979
\(439\) 9.95941 0.475337 0.237668 0.971346i \(-0.423617\pi\)
0.237668 + 0.971346i \(0.423617\pi\)
\(440\) −23.7865 −1.13398
\(441\) 0 0
\(442\) −4.37371 −0.208036
\(443\) −9.69053 −0.460411 −0.230206 0.973142i \(-0.573940\pi\)
−0.230206 + 0.973142i \(0.573940\pi\)
\(444\) 3.84718 0.182579
\(445\) 34.6428 1.64223
\(446\) 1.10629 0.0523842
\(447\) 5.34405 0.252765
\(448\) 0 0
\(449\) 13.7314 0.648025 0.324013 0.946053i \(-0.394968\pi\)
0.324013 + 0.946053i \(0.394968\pi\)
\(450\) −23.4989 −1.10775
\(451\) 32.1755 1.51509
\(452\) −3.50810 −0.165007
\(453\) −3.48039 −0.163523
\(454\) 0.869629 0.0408137
\(455\) 0 0
\(456\) −2.01862 −0.0945306
\(457\) 12.3389 0.577188 0.288594 0.957452i \(-0.406812\pi\)
0.288594 + 0.957452i \(0.406812\pi\)
\(458\) 7.00356 0.327255
\(459\) 4.53541 0.211695
\(460\) −1.40429 −0.0654755
\(461\) 11.8322 0.551081 0.275540 0.961290i \(-0.411143\pi\)
0.275540 + 0.961290i \(0.411143\pi\)
\(462\) 0 0
\(463\) −30.3189 −1.40904 −0.704519 0.709685i \(-0.748837\pi\)
−0.704519 + 0.709685i \(0.748837\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −15.1622 −0.703129
\(466\) 0.411278 0.0190521
\(467\) 8.31097 0.384586 0.192293 0.981338i \(-0.438408\pi\)
0.192293 + 0.981338i \(0.438408\pi\)
\(468\) 7.78990 0.360088
\(469\) 0 0
\(470\) 28.9736 1.33645
\(471\) −2.00740 −0.0924962
\(472\) 0.696710 0.0320687
\(473\) −74.6289 −3.43144
\(474\) −2.64545 −0.121509
\(475\) 33.6754 1.54513
\(476\) 0 0
\(477\) 3.34141 0.152992
\(478\) −12.6415 −0.578210
\(479\) −31.3096 −1.43057 −0.715286 0.698832i \(-0.753703\pi\)
−0.715286 + 0.698832i \(0.753703\pi\)
\(480\) 1.89906 0.0866798
\(481\) 21.2758 0.970091
\(482\) −20.9092 −0.952387
\(483\) 0 0
\(484\) 30.6246 1.39203
\(485\) −9.46523 −0.429794
\(486\) −12.3037 −0.558108
\(487\) −21.7094 −0.983748 −0.491874 0.870666i \(-0.663688\pi\)
−0.491874 + 0.870666i \(0.663688\pi\)
\(488\) 3.10100 0.140376
\(489\) −0.200532 −0.00906838
\(490\) 0 0
\(491\) −28.6142 −1.29134 −0.645670 0.763616i \(-0.723422\pi\)
−0.645670 + 0.763616i \(0.723422\pi\)
\(492\) −2.56882 −0.115811
\(493\) −1.53541 −0.0691515
\(494\) −11.1634 −0.502266
\(495\) 65.0487 2.92372
\(496\) −7.98405 −0.358494
\(497\) 0 0
\(498\) 6.60105 0.295800
\(499\) −22.3287 −0.999572 −0.499786 0.866149i \(-0.666588\pi\)
−0.499786 + 0.866149i \(0.666588\pi\)
\(500\) −13.2465 −0.592401
\(501\) 5.30757 0.237125
\(502\) 23.2297 1.03679
\(503\) 13.7215 0.611810 0.305905 0.952062i \(-0.401041\pi\)
0.305905 + 0.952062i \(0.401041\pi\)
\(504\) 0 0
\(505\) −36.9952 −1.64627
\(506\) 2.45741 0.109245
\(507\) 2.51658 0.111765
\(508\) 2.64032 0.117145
\(509\) 6.80605 0.301673 0.150836 0.988559i \(-0.451803\pi\)
0.150836 + 0.988559i \(0.451803\pi\)
\(510\) 2.91584 0.129115
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 11.5762 0.511100
\(514\) 15.5148 0.684327
\(515\) −34.4482 −1.51797
\(516\) 5.95819 0.262294
\(517\) −50.7016 −2.22985
\(518\) 0 0
\(519\) 11.8680 0.520948
\(520\) 10.5022 0.460553
\(521\) −5.55924 −0.243555 −0.121777 0.992557i \(-0.538859\pi\)
−0.121777 + 0.992557i \(0.538859\pi\)
\(522\) 2.73468 0.119694
\(523\) 12.0800 0.528221 0.264111 0.964492i \(-0.414922\pi\)
0.264111 + 0.964492i \(0.414922\pi\)
\(524\) 10.7569 0.469916
\(525\) 0 0
\(526\) −2.94512 −0.128414
\(527\) −12.2588 −0.534002
\(528\) −3.32321 −0.144624
\(529\) −22.8549 −0.993692
\(530\) 4.50483 0.195677
\(531\) −1.90528 −0.0826821
\(532\) 0 0
\(533\) −14.2061 −0.615335
\(534\) 4.83993 0.209444
\(535\) 43.5012 1.88072
\(536\) −7.22494 −0.312070
\(537\) 3.45007 0.148881
\(538\) 6.76917 0.291840
\(539\) 0 0
\(540\) −10.8905 −0.468652
\(541\) 28.3762 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(542\) −16.8757 −0.724875
\(543\) 6.82249 0.292781
\(544\) 1.53541 0.0658303
\(545\) 22.3220 0.956167
\(546\) 0 0
\(547\) −19.5396 −0.835455 −0.417727 0.908572i \(-0.637173\pi\)
−0.417727 + 0.908572i \(0.637173\pi\)
\(548\) 8.47238 0.361922
\(549\) −8.48026 −0.361929
\(550\) 55.4389 2.36392
\(551\) −3.91898 −0.166954
\(552\) −0.196193 −0.00835054
\(553\) 0 0
\(554\) 26.3155 1.11804
\(555\) −14.1840 −0.602077
\(556\) 2.40081 0.101817
\(557\) 16.2974 0.690543 0.345272 0.938503i \(-0.387787\pi\)
0.345272 + 0.938503i \(0.387787\pi\)
\(558\) 21.8338 0.924301
\(559\) 32.9501 1.39364
\(560\) 0 0
\(561\) −5.10250 −0.215428
\(562\) 0.662563 0.0279485
\(563\) 13.8524 0.583810 0.291905 0.956447i \(-0.405711\pi\)
0.291905 + 0.956447i \(0.405711\pi\)
\(564\) 4.04789 0.170447
\(565\) 12.9339 0.544132
\(566\) 24.9296 1.04787
\(567\) 0 0
\(568\) −13.2927 −0.557749
\(569\) −40.4417 −1.69541 −0.847703 0.530472i \(-0.822015\pi\)
−0.847703 + 0.530472i \(0.822015\pi\)
\(570\) 7.44237 0.311726
\(571\) 18.8558 0.789091 0.394546 0.918876i \(-0.370902\pi\)
0.394546 + 0.918876i \(0.370902\pi\)
\(572\) −18.3781 −0.768426
\(573\) 10.4741 0.437564
\(574\) 0 0
\(575\) 3.27296 0.136492
\(576\) −2.73468 −0.113945
\(577\) 18.4743 0.769095 0.384548 0.923105i \(-0.374357\pi\)
0.384548 + 0.923105i \(0.374357\pi\)
\(578\) −14.6425 −0.609048
\(579\) −6.80126 −0.282651
\(580\) 3.68685 0.153088
\(581\) 0 0
\(582\) −1.32238 −0.0548146
\(583\) −7.88311 −0.326485
\(584\) −14.9513 −0.618688
\(585\) −28.7202 −1.18744
\(586\) −9.98069 −0.412299
\(587\) −29.5731 −1.22061 −0.610306 0.792166i \(-0.708954\pi\)
−0.610306 + 0.792166i \(0.708954\pi\)
\(588\) 0 0
\(589\) −31.2893 −1.28925
\(590\) −2.56867 −0.105750
\(591\) 4.64217 0.190953
\(592\) −7.46896 −0.306972
\(593\) −35.5469 −1.45974 −0.729869 0.683587i \(-0.760419\pi\)
−0.729869 + 0.683587i \(0.760419\pi\)
\(594\) 19.0575 0.781941
\(595\) 0 0
\(596\) −10.3750 −0.424977
\(597\) −3.02027 −0.123612
\(598\) −1.08499 −0.0443686
\(599\) −1.11873 −0.0457099 −0.0228550 0.999739i \(-0.507276\pi\)
−0.0228550 + 0.999739i \(0.507276\pi\)
\(600\) −4.42611 −0.180695
\(601\) 5.46816 0.223051 0.111525 0.993762i \(-0.464426\pi\)
0.111525 + 0.993762i \(0.464426\pi\)
\(602\) 0 0
\(603\) 19.7579 0.804605
\(604\) 6.75687 0.274933
\(605\) −112.909 −4.59039
\(606\) −5.16859 −0.209959
\(607\) 24.0318 0.975421 0.487710 0.873006i \(-0.337832\pi\)
0.487710 + 0.873006i \(0.337832\pi\)
\(608\) 3.91898 0.158936
\(609\) 0 0
\(610\) −11.4329 −0.462906
\(611\) 22.3857 0.905630
\(612\) −4.19887 −0.169729
\(613\) 11.9080 0.480958 0.240479 0.970654i \(-0.422695\pi\)
0.240479 + 0.970654i \(0.422695\pi\)
\(614\) 2.78572 0.112423
\(615\) 9.47085 0.381902
\(616\) 0 0
\(617\) −40.3723 −1.62533 −0.812664 0.582733i \(-0.801983\pi\)
−0.812664 + 0.582733i \(0.801983\pi\)
\(618\) −4.81275 −0.193597
\(619\) 25.7432 1.03471 0.517353 0.855772i \(-0.326918\pi\)
0.517353 + 0.855772i \(0.326918\pi\)
\(620\) 29.4360 1.18218
\(621\) 1.12511 0.0451489
\(622\) −15.2581 −0.611795
\(623\) 0 0
\(624\) 1.46726 0.0587374
\(625\) 5.87342 0.234937
\(626\) −34.3024 −1.37100
\(627\) −13.0236 −0.520112
\(628\) 3.89720 0.155515
\(629\) −11.4679 −0.457257
\(630\) 0 0
\(631\) 26.6745 1.06189 0.530947 0.847405i \(-0.321836\pi\)
0.530947 + 0.847405i \(0.321836\pi\)
\(632\) 5.13590 0.204295
\(633\) −10.8035 −0.429401
\(634\) −23.5203 −0.934110
\(635\) −9.73449 −0.386301
\(636\) 0.629368 0.0249561
\(637\) 0 0
\(638\) −6.45172 −0.255426
\(639\) 36.3513 1.43803
\(640\) −3.68685 −0.145736
\(641\) 23.3821 0.923538 0.461769 0.887000i \(-0.347215\pi\)
0.461769 + 0.887000i \(0.347215\pi\)
\(642\) 6.07754 0.239861
\(643\) 5.26893 0.207786 0.103893 0.994588i \(-0.466870\pi\)
0.103893 + 0.994588i \(0.466870\pi\)
\(644\) 0 0
\(645\) −21.9670 −0.864948
\(646\) 6.01725 0.236745
\(647\) −10.4615 −0.411286 −0.205643 0.978627i \(-0.565929\pi\)
−0.205643 + 0.978627i \(0.565929\pi\)
\(648\) 6.68254 0.262515
\(649\) 4.49497 0.176443
\(650\) −24.4774 −0.960081
\(651\) 0 0
\(652\) 0.389316 0.0152468
\(653\) 12.6550 0.495229 0.247615 0.968859i \(-0.420353\pi\)
0.247615 + 0.968859i \(0.420353\pi\)
\(654\) 3.11859 0.121947
\(655\) −39.6590 −1.54961
\(656\) 4.98713 0.194715
\(657\) 40.8870 1.59515
\(658\) 0 0
\(659\) 27.0561 1.05396 0.526978 0.849879i \(-0.323325\pi\)
0.526978 + 0.849879i \(0.323325\pi\)
\(660\) 12.2522 0.476916
\(661\) 21.1950 0.824389 0.412194 0.911096i \(-0.364762\pi\)
0.412194 + 0.911096i \(0.364762\pi\)
\(662\) −20.9170 −0.812961
\(663\) 2.25285 0.0874934
\(664\) −12.8154 −0.497333
\(665\) 0 0
\(666\) 20.4252 0.791462
\(667\) −0.380892 −0.0147482
\(668\) −10.3042 −0.398681
\(669\) −0.569837 −0.0220312
\(670\) 26.6373 1.02909
\(671\) 20.0068 0.772354
\(672\) 0 0
\(673\) 6.95037 0.267917 0.133959 0.990987i \(-0.457231\pi\)
0.133959 + 0.990987i \(0.457231\pi\)
\(674\) 35.1825 1.35518
\(675\) 25.3823 0.976966
\(676\) −4.88572 −0.187912
\(677\) 45.9370 1.76550 0.882752 0.469840i \(-0.155688\pi\)
0.882752 + 0.469840i \(0.155688\pi\)
\(678\) 1.80699 0.0693969
\(679\) 0 0
\(680\) −5.66084 −0.217083
\(681\) −0.447936 −0.0171650
\(682\) −51.5108 −1.97245
\(683\) 1.50183 0.0574657 0.0287329 0.999587i \(-0.490853\pi\)
0.0287329 + 0.999587i \(0.490853\pi\)
\(684\) −10.7172 −0.409781
\(685\) −31.2364 −1.19348
\(686\) 0 0
\(687\) −3.60746 −0.137633
\(688\) −11.5673 −0.440999
\(689\) 3.48055 0.132598
\(690\) 0.723336 0.0275369
\(691\) 32.0262 1.21833 0.609167 0.793042i \(-0.291504\pi\)
0.609167 + 0.793042i \(0.291504\pi\)
\(692\) −23.0407 −0.875876
\(693\) 0 0
\(694\) −14.8661 −0.564308
\(695\) −8.85144 −0.335754
\(696\) 0.515089 0.0195244
\(697\) 7.65730 0.290041
\(698\) 20.1456 0.762523
\(699\) −0.211845 −0.00801271
\(700\) 0 0
\(701\) 20.1243 0.760083 0.380042 0.924969i \(-0.375910\pi\)
0.380042 + 0.924969i \(0.375910\pi\)
\(702\) −8.41428 −0.317576
\(703\) −29.2707 −1.10396
\(704\) 6.45172 0.243158
\(705\) −14.9240 −0.562070
\(706\) 9.55582 0.359638
\(707\) 0 0
\(708\) −0.358868 −0.0134871
\(709\) −4.06107 −0.152517 −0.0762584 0.997088i \(-0.524297\pi\)
−0.0762584 + 0.997088i \(0.524297\pi\)
\(710\) 49.0082 1.83925
\(711\) −14.0451 −0.526731
\(712\) −9.39630 −0.352141
\(713\) −3.04106 −0.113889
\(714\) 0 0
\(715\) 67.7573 2.53398
\(716\) −6.69801 −0.250316
\(717\) 6.51152 0.243177
\(718\) 12.6878 0.473505
\(719\) −1.51190 −0.0563842 −0.0281921 0.999603i \(-0.508975\pi\)
−0.0281921 + 0.999603i \(0.508975\pi\)
\(720\) 10.0824 0.375748
\(721\) 0 0
\(722\) −3.64162 −0.135527
\(723\) 10.7701 0.400544
\(724\) −13.2453 −0.492256
\(725\) −8.59290 −0.319132
\(726\) −15.7744 −0.585444
\(727\) 28.1717 1.04483 0.522414 0.852692i \(-0.325031\pi\)
0.522414 + 0.852692i \(0.325031\pi\)
\(728\) 0 0
\(729\) −13.7101 −0.507782
\(730\) 55.1231 2.04020
\(731\) −17.7606 −0.656898
\(732\) −1.59729 −0.0590376
\(733\) −6.31545 −0.233266 −0.116633 0.993175i \(-0.537210\pi\)
−0.116633 + 0.993175i \(0.537210\pi\)
\(734\) 33.0253 1.21899
\(735\) 0 0
\(736\) 0.380892 0.0140399
\(737\) −46.6133 −1.71702
\(738\) −13.6382 −0.502030
\(739\) 8.64546 0.318029 0.159014 0.987276i \(-0.449168\pi\)
0.159014 + 0.987276i \(0.449168\pi\)
\(740\) 27.5370 1.01228
\(741\) 5.75016 0.211237
\(742\) 0 0
\(743\) 18.3123 0.671814 0.335907 0.941895i \(-0.390957\pi\)
0.335907 + 0.941895i \(0.390957\pi\)
\(744\) 4.11250 0.150771
\(745\) 38.2511 1.40141
\(746\) −19.9474 −0.730327
\(747\) 35.0460 1.28226
\(748\) 9.90604 0.362201
\(749\) 0 0
\(750\) 6.82313 0.249145
\(751\) −2.30974 −0.0842835 −0.0421417 0.999112i \(-0.513418\pi\)
−0.0421417 + 0.999112i \(0.513418\pi\)
\(752\) −7.85862 −0.286574
\(753\) −11.9653 −0.436041
\(754\) 2.84856 0.103738
\(755\) −24.9116 −0.906626
\(756\) 0 0
\(757\) 53.8978 1.95895 0.979474 0.201573i \(-0.0646052\pi\)
0.979474 + 0.201573i \(0.0646052\pi\)
\(758\) 27.8725 1.01237
\(759\) −1.26578 −0.0459450
\(760\) −14.4487 −0.524109
\(761\) 30.9134 1.12061 0.560305 0.828286i \(-0.310684\pi\)
0.560305 + 0.828286i \(0.310684\pi\)
\(762\) −1.36000 −0.0492677
\(763\) 0 0
\(764\) −20.3346 −0.735681
\(765\) 15.4806 0.559703
\(766\) −21.1598 −0.764536
\(767\) −1.98462 −0.0716604
\(768\) −0.515089 −0.0185867
\(769\) −8.51208 −0.306953 −0.153477 0.988152i \(-0.549047\pi\)
−0.153477 + 0.988152i \(0.549047\pi\)
\(770\) 0 0
\(771\) −7.99148 −0.287806
\(772\) 13.2040 0.475224
\(773\) −2.51529 −0.0904685 −0.0452343 0.998976i \(-0.514403\pi\)
−0.0452343 + 0.998976i \(0.514403\pi\)
\(774\) 31.6329 1.13702
\(775\) −68.6061 −2.46441
\(776\) 2.56729 0.0921604
\(777\) 0 0
\(778\) 8.79804 0.315425
\(779\) 19.5444 0.700252
\(780\) −5.40958 −0.193694
\(781\) −85.7606 −3.06876
\(782\) 0.584826 0.0209133
\(783\) −2.95387 −0.105563
\(784\) 0 0
\(785\) −14.3684 −0.512830
\(786\) −5.54075 −0.197632
\(787\) −5.98304 −0.213273 −0.106636 0.994298i \(-0.534008\pi\)
−0.106636 + 0.994298i \(0.534008\pi\)
\(788\) −9.01237 −0.321052
\(789\) 1.51700 0.0540067
\(790\) −18.9353 −0.673688
\(791\) 0 0
\(792\) −17.6434 −0.626931
\(793\) −8.83338 −0.313683
\(794\) 7.58413 0.269151
\(795\) −2.32039 −0.0822957
\(796\) 5.86360 0.207830
\(797\) 0.207213 0.00733988 0.00366994 0.999993i \(-0.498832\pi\)
0.00366994 + 0.999993i \(0.498832\pi\)
\(798\) 0 0
\(799\) −12.0662 −0.426872
\(800\) 8.59290 0.303805
\(801\) 25.6959 0.907920
\(802\) 11.8271 0.417629
\(803\) −96.4613 −3.40405
\(804\) 3.72149 0.131247
\(805\) 0 0
\(806\) 22.7430 0.801089
\(807\) −3.48673 −0.122739
\(808\) 10.0344 0.353007
\(809\) 31.8360 1.11929 0.559646 0.828732i \(-0.310937\pi\)
0.559646 + 0.828732i \(0.310937\pi\)
\(810\) −24.6376 −0.865676
\(811\) 19.8976 0.698699 0.349350 0.936992i \(-0.386403\pi\)
0.349350 + 0.936992i \(0.386403\pi\)
\(812\) 0 0
\(813\) 8.69250 0.304859
\(814\) −48.1876 −1.68898
\(815\) −1.43535 −0.0502781
\(816\) −0.790874 −0.0276861
\(817\) −45.3319 −1.58596
\(818\) −11.8506 −0.414347
\(819\) 0 0
\(820\) −18.3868 −0.642096
\(821\) −24.3901 −0.851221 −0.425610 0.904907i \(-0.639941\pi\)
−0.425610 + 0.904907i \(0.639941\pi\)
\(822\) −4.36403 −0.152213
\(823\) −50.1588 −1.74843 −0.874213 0.485542i \(-0.838622\pi\)
−0.874213 + 0.485542i \(0.838622\pi\)
\(824\) 9.34353 0.325497
\(825\) −28.5560 −0.994192
\(826\) 0 0
\(827\) −12.5847 −0.437612 −0.218806 0.975768i \(-0.570216\pi\)
−0.218806 + 0.975768i \(0.570216\pi\)
\(828\) −1.04162 −0.0361987
\(829\) 50.0977 1.73996 0.869982 0.493083i \(-0.164130\pi\)
0.869982 + 0.493083i \(0.164130\pi\)
\(830\) 47.2484 1.64002
\(831\) −13.5548 −0.470212
\(832\) −2.84856 −0.0987560
\(833\) 0 0
\(834\) −1.23663 −0.0428210
\(835\) 37.9900 1.31470
\(836\) 25.2841 0.874470
\(837\) −23.5839 −0.815178
\(838\) −0.933145 −0.0322349
\(839\) 26.7756 0.924397 0.462199 0.886776i \(-0.347061\pi\)
0.462199 + 0.886776i \(0.347061\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −3.17478 −0.109410
\(843\) −0.341279 −0.0117543
\(844\) 20.9741 0.721957
\(845\) 18.0129 0.619664
\(846\) 21.4908 0.738870
\(847\) 0 0
\(848\) −1.22186 −0.0419589
\(849\) −12.8410 −0.440701
\(850\) 13.1936 0.452538
\(851\) −2.84487 −0.0975208
\(852\) 6.84692 0.234572
\(853\) 16.8716 0.577673 0.288837 0.957378i \(-0.406732\pi\)
0.288837 + 0.957378i \(0.406732\pi\)
\(854\) 0 0
\(855\) 39.5126 1.35130
\(856\) −11.7990 −0.403282
\(857\) −8.11457 −0.277188 −0.138594 0.990349i \(-0.544258\pi\)
−0.138594 + 0.990349i \(0.544258\pi\)
\(858\) 9.46635 0.323176
\(859\) −49.5252 −1.68978 −0.844889 0.534942i \(-0.820333\pi\)
−0.844889 + 0.534942i \(0.820333\pi\)
\(860\) 42.6469 1.45425
\(861\) 0 0
\(862\) −25.2882 −0.861318
\(863\) 36.6653 1.24810 0.624051 0.781383i \(-0.285486\pi\)
0.624051 + 0.781383i \(0.285486\pi\)
\(864\) 2.95387 0.100493
\(865\) 84.9477 2.88831
\(866\) −25.3900 −0.862789
\(867\) 7.54220 0.256146
\(868\) 0 0
\(869\) 33.1354 1.12404
\(870\) −1.89906 −0.0643841
\(871\) 20.5807 0.697349
\(872\) −6.05447 −0.205030
\(873\) −7.02073 −0.237616
\(874\) 1.49271 0.0504915
\(875\) 0 0
\(876\) 7.70123 0.260200
\(877\) 2.99441 0.101114 0.0505570 0.998721i \(-0.483900\pi\)
0.0505570 + 0.998721i \(0.483900\pi\)
\(878\) 9.95941 0.336114
\(879\) 5.14095 0.173400
\(880\) −23.7865 −0.801844
\(881\) −14.1339 −0.476183 −0.238092 0.971243i \(-0.576522\pi\)
−0.238092 + 0.971243i \(0.576522\pi\)
\(882\) 0 0
\(883\) −15.9154 −0.535596 −0.267798 0.963475i \(-0.586296\pi\)
−0.267798 + 0.963475i \(0.586296\pi\)
\(884\) −4.37371 −0.147104
\(885\) 1.32309 0.0444753
\(886\) −9.69053 −0.325560
\(887\) −2.14593 −0.0720532 −0.0360266 0.999351i \(-0.511470\pi\)
−0.0360266 + 0.999351i \(0.511470\pi\)
\(888\) 3.84718 0.129103
\(889\) 0 0
\(890\) 34.6428 1.16123
\(891\) 43.1139 1.44437
\(892\) 1.10629 0.0370412
\(893\) −30.7978 −1.03061
\(894\) 5.34405 0.178732
\(895\) 24.6946 0.825449
\(896\) 0 0
\(897\) 0.558868 0.0186600
\(898\) 13.7314 0.458223
\(899\) 7.98405 0.266283
\(900\) −23.4989 −0.783295
\(901\) −1.87606 −0.0625007
\(902\) 32.1755 1.07133
\(903\) 0 0
\(904\) −3.50810 −0.116678
\(905\) 48.8334 1.62328
\(906\) −3.48039 −0.115628
\(907\) 19.4259 0.645025 0.322513 0.946565i \(-0.395473\pi\)
0.322513 + 0.946565i \(0.395473\pi\)
\(908\) 0.869629 0.0288596
\(909\) −27.4408 −0.910153
\(910\) 0 0
\(911\) −1.60092 −0.0530408 −0.0265204 0.999648i \(-0.508443\pi\)
−0.0265204 + 0.999648i \(0.508443\pi\)
\(912\) −2.01862 −0.0668433
\(913\) −82.6811 −2.73634
\(914\) 12.3389 0.408134
\(915\) 5.88899 0.194684
\(916\) 7.00356 0.231404
\(917\) 0 0
\(918\) 4.53541 0.149691
\(919\) −50.8625 −1.67780 −0.838899 0.544287i \(-0.816800\pi\)
−0.838899 + 0.544287i \(0.816800\pi\)
\(920\) −1.40429 −0.0462982
\(921\) −1.43490 −0.0472814
\(922\) 11.8322 0.389673
\(923\) 37.8650 1.24634
\(924\) 0 0
\(925\) −64.1800 −2.11023
\(926\) −30.3189 −0.996340
\(927\) −25.5516 −0.839225
\(928\) −1.00000 −0.0328266
\(929\) 27.9717 0.917723 0.458861 0.888508i \(-0.348257\pi\)
0.458861 + 0.888508i \(0.348257\pi\)
\(930\) −15.1622 −0.497187
\(931\) 0 0
\(932\) 0.411278 0.0134719
\(933\) 7.85930 0.257302
\(934\) 8.31097 0.271943
\(935\) −36.5221 −1.19440
\(936\) 7.78990 0.254621
\(937\) −42.4694 −1.38741 −0.693707 0.720258i \(-0.744024\pi\)
−0.693707 + 0.720258i \(0.744024\pi\)
\(938\) 0 0
\(939\) 17.6688 0.576599
\(940\) 28.9736 0.945014
\(941\) −34.4273 −1.12230 −0.561149 0.827715i \(-0.689640\pi\)
−0.561149 + 0.827715i \(0.689640\pi\)
\(942\) −2.00740 −0.0654047
\(943\) 1.89956 0.0618581
\(944\) 0.696710 0.0226760
\(945\) 0 0
\(946\) −74.6289 −2.42639
\(947\) −28.0358 −0.911040 −0.455520 0.890226i \(-0.650547\pi\)
−0.455520 + 0.890226i \(0.650547\pi\)
\(948\) −2.64545 −0.0859201
\(949\) 42.5895 1.38251
\(950\) 33.6754 1.09257
\(951\) 12.1150 0.392857
\(952\) 0 0
\(953\) 50.4431 1.63401 0.817007 0.576628i \(-0.195632\pi\)
0.817007 + 0.576628i \(0.195632\pi\)
\(954\) 3.34141 0.108182
\(955\) 74.9708 2.42600
\(956\) −12.6415 −0.408857
\(957\) 3.32321 0.107424
\(958\) −31.3096 −1.01157
\(959\) 0 0
\(960\) 1.89906 0.0612919
\(961\) 32.7451 1.05629
\(962\) 21.2758 0.685958
\(963\) 32.2666 1.03978
\(964\) −20.9092 −0.673439
\(965\) −48.6814 −1.56711
\(966\) 0 0
\(967\) 40.9091 1.31555 0.657774 0.753215i \(-0.271498\pi\)
0.657774 + 0.753215i \(0.271498\pi\)
\(968\) 30.6246 0.984313
\(969\) −3.09942 −0.0995676
\(970\) −9.46523 −0.303910
\(971\) −43.1138 −1.38359 −0.691794 0.722095i \(-0.743179\pi\)
−0.691794 + 0.722095i \(0.743179\pi\)
\(972\) −12.3037 −0.394642
\(973\) 0 0
\(974\) −21.7094 −0.695615
\(975\) 12.6080 0.403780
\(976\) 3.10100 0.0992607
\(977\) 49.1685 1.57304 0.786519 0.617566i \(-0.211881\pi\)
0.786519 + 0.617566i \(0.211881\pi\)
\(978\) −0.200532 −0.00641231
\(979\) −60.6223 −1.93750
\(980\) 0 0
\(981\) 16.5571 0.528626
\(982\) −28.6142 −0.913115
\(983\) −5.41718 −0.172781 −0.0863907 0.996261i \(-0.527533\pi\)
−0.0863907 + 0.996261i \(0.527533\pi\)
\(984\) −2.56882 −0.0818908
\(985\) 33.2273 1.05871
\(986\) −1.53541 −0.0488975
\(987\) 0 0
\(988\) −11.1634 −0.355156
\(989\) −4.40589 −0.140099
\(990\) 65.0487 2.06738
\(991\) 6.28876 0.199769 0.0998845 0.994999i \(-0.468153\pi\)
0.0998845 + 0.994999i \(0.468153\pi\)
\(992\) −7.98405 −0.253494
\(993\) 10.7741 0.341906
\(994\) 0 0
\(995\) −21.6182 −0.685344
\(996\) 6.60105 0.209162
\(997\) −7.11434 −0.225313 −0.112657 0.993634i \(-0.535936\pi\)
−0.112657 + 0.993634i \(0.535936\pi\)
\(998\) −22.3287 −0.706804
\(999\) −22.0624 −0.698022
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 2842.2.a.x.1.3 5
7.3 odd 6 406.2.e.a.233.3 10
7.5 odd 6 406.2.e.a.291.3 yes 10
7.6 odd 2 2842.2.a.z.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.3 10 7.3 odd 6
406.2.e.a.291.3 yes 10 7.5 odd 6
2842.2.a.x.1.3 5 1.1 even 1 trivial
2842.2.a.z.1.3 5 7.6 odd 2