Properties

Label 4851.2.a.bp.1.2
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.7354300205\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 4851.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +O(q^{10})\) \(q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +0.637062 q^{10} -1.00000 q^{11} -3.80451 q^{13} +3.83255 q^{16} +0.334898 q^{17} -8.13941 q^{19} -7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} -0.637062 q^{26} -0.195488 q^{29} +9.94392 q^{31} +1.97196 q^{32} +0.0560785 q^{34} -4.47431 q^{37} -1.36294 q^{38} -2.53039 q^{40} -6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} -12.1394 q^{47} +1.58647 q^{50} +7.50235 q^{52} -7.94392 q^{53} -3.80451 q^{55} -0.0327344 q^{58} +3.74843 q^{59} -6.00000 q^{61} +1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} -0.660406 q^{68} -4.66980 q^{71} -4.19549 q^{73} -0.749219 q^{74} +16.0506 q^{76} +3.33020 q^{79} +14.5810 q^{80} -1.05138 q^{82} -13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} -9.88784 q^{89} -3.28352 q^{92} -2.03273 q^{94} -30.9665 q^{95} -0.0560785 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.167449 0.118404 0.0592022 0.998246i \(-0.481144\pi\)
0.0592022 + 0.998246i \(0.481144\pi\)
\(3\) 0 0
\(4\) −1.97196 −0.985980
\(5\) 3.80451 1.70143 0.850715 0.525628i \(-0.176170\pi\)
0.850715 + 0.525628i \(0.176170\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.665102 −0.235149
\(9\) 0 0
\(10\) 0.637062 0.201457
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.80451 −1.05518 −0.527591 0.849499i \(-0.676905\pi\)
−0.527591 + 0.849499i \(0.676905\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.83255 0.958138
\(17\) 0.334898 0.0812248 0.0406124 0.999175i \(-0.487069\pi\)
0.0406124 + 0.999175i \(0.487069\pi\)
\(18\) 0 0
\(19\) −8.13941 −1.86731 −0.933654 0.358175i \(-0.883399\pi\)
−0.933654 + 0.358175i \(0.883399\pi\)
\(20\) −7.50235 −1.67758
\(21\) 0 0
\(22\) −0.167449 −0.0357003
\(23\) 1.66510 0.347198 0.173599 0.984816i \(-0.444460\pi\)
0.173599 + 0.984816i \(0.444460\pi\)
\(24\) 0 0
\(25\) 9.47431 1.89486
\(26\) −0.637062 −0.124938
\(27\) 0 0
\(28\) 0 0
\(29\) −0.195488 −0.0363013 −0.0181506 0.999835i \(-0.505778\pi\)
−0.0181506 + 0.999835i \(0.505778\pi\)
\(30\) 0 0
\(31\) 9.94392 1.78598 0.892991 0.450075i \(-0.148603\pi\)
0.892991 + 0.450075i \(0.148603\pi\)
\(32\) 1.97196 0.348597
\(33\) 0 0
\(34\) 0.0560785 0.00961738
\(35\) 0 0
\(36\) 0 0
\(37\) −4.47431 −0.735572 −0.367786 0.929911i \(-0.619884\pi\)
−0.367786 + 0.929911i \(0.619884\pi\)
\(38\) −1.36294 −0.221098
\(39\) 0 0
\(40\) −2.53039 −0.400089
\(41\) −6.27882 −0.980587 −0.490293 0.871557i \(-0.663110\pi\)
−0.490293 + 0.871557i \(0.663110\pi\)
\(42\) 0 0
\(43\) 2.33490 0.356069 0.178034 0.984024i \(-0.443026\pi\)
0.178034 + 0.984024i \(0.443026\pi\)
\(44\) 1.97196 0.297284
\(45\) 0 0
\(46\) 0.278820 0.0411098
\(47\) −12.1394 −1.77071 −0.885357 0.464911i \(-0.846086\pi\)
−0.885357 + 0.464911i \(0.846086\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.58647 0.224360
\(51\) 0 0
\(52\) 7.50235 1.04039
\(53\) −7.94392 −1.09118 −0.545591 0.838052i \(-0.683695\pi\)
−0.545591 + 0.838052i \(0.683695\pi\)
\(54\) 0 0
\(55\) −3.80451 −0.513000
\(56\) 0 0
\(57\) 0 0
\(58\) −0.0327344 −0.00429823
\(59\) 3.74843 0.488004 0.244002 0.969775i \(-0.421540\pi\)
0.244002 + 0.969775i \(0.421540\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 1.66510 0.211468
\(63\) 0 0
\(64\) −7.33490 −0.916862
\(65\) −14.4743 −1.79532
\(66\) 0 0
\(67\) −0.139410 −0.0170316 −0.00851582 0.999964i \(-0.502711\pi\)
−0.00851582 + 0.999964i \(0.502711\pi\)
\(68\) −0.660406 −0.0800860
\(69\) 0 0
\(70\) 0 0
\(71\) −4.66980 −0.554203 −0.277101 0.960841i \(-0.589374\pi\)
−0.277101 + 0.960841i \(0.589374\pi\)
\(72\) 0 0
\(73\) −4.19549 −0.491045 −0.245522 0.969391i \(-0.578959\pi\)
−0.245522 + 0.969391i \(0.578959\pi\)
\(74\) −0.749219 −0.0870950
\(75\) 0 0
\(76\) 16.0506 1.84113
\(77\) 0 0
\(78\) 0 0
\(79\) 3.33020 0.374677 0.187339 0.982295i \(-0.440014\pi\)
0.187339 + 0.982295i \(0.440014\pi\)
\(80\) 14.5810 1.63020
\(81\) 0 0
\(82\) −1.05138 −0.116106
\(83\) −13.9439 −1.53054 −0.765272 0.643707i \(-0.777396\pi\)
−0.765272 + 0.643707i \(0.777396\pi\)
\(84\) 0 0
\(85\) 1.27412 0.138198
\(86\) 0.390977 0.0421601
\(87\) 0 0
\(88\) 0.665102 0.0709001
\(89\) −9.88784 −1.04811 −0.524055 0.851685i \(-0.675581\pi\)
−0.524055 + 0.851685i \(0.675581\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.28352 −0.342330
\(93\) 0 0
\(94\) −2.03273 −0.209661
\(95\) −30.9665 −3.17709
\(96\) 0 0
\(97\) −0.0560785 −0.00569391 −0.00284695 0.999996i \(-0.500906\pi\)
−0.00284695 + 0.999996i \(0.500906\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −18.6830 −1.86830
\(101\) −18.8831 −1.87894 −0.939472 0.342626i \(-0.888683\pi\)
−0.939472 + 0.342626i \(0.888683\pi\)
\(102\) 0 0
\(103\) 8.27882 0.815736 0.407868 0.913041i \(-0.366272\pi\)
0.407868 + 0.913041i \(0.366272\pi\)
\(104\) 2.53039 0.248125
\(105\) 0 0
\(106\) −1.33020 −0.129201
\(107\) 8.13941 0.786866 0.393433 0.919353i \(-0.371287\pi\)
0.393433 + 0.919353i \(0.371287\pi\)
\(108\) 0 0
\(109\) 11.5529 1.10657 0.553286 0.832992i \(-0.313374\pi\)
0.553286 + 0.832992i \(0.313374\pi\)
\(110\) −0.637062 −0.0607415
\(111\) 0 0
\(112\) 0 0
\(113\) 1.33020 0.125135 0.0625675 0.998041i \(-0.480071\pi\)
0.0625675 + 0.998041i \(0.480071\pi\)
\(114\) 0 0
\(115\) 6.33490 0.590732
\(116\) 0.385496 0.0357924
\(117\) 0 0
\(118\) 0.627672 0.0577819
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.00470 −0.0909608
\(123\) 0 0
\(124\) −19.6090 −1.76094
\(125\) 17.0226 1.52254
\(126\) 0 0
\(127\) 14.6137 1.29676 0.648379 0.761318i \(-0.275447\pi\)
0.648379 + 0.761318i \(0.275447\pi\)
\(128\) −5.17214 −0.457157
\(129\) 0 0
\(130\) −2.42371 −0.212574
\(131\) 20.5576 1.79613 0.898065 0.439863i \(-0.144973\pi\)
0.898065 + 0.439863i \(0.144973\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.0233441 −0.00201662
\(135\) 0 0
\(136\) −0.222741 −0.0190999
\(137\) 16.2227 1.38600 0.693001 0.720936i \(-0.256288\pi\)
0.693001 + 0.720936i \(0.256288\pi\)
\(138\) 0 0
\(139\) −22.5482 −1.91252 −0.956259 0.292522i \(-0.905506\pi\)
−0.956259 + 0.292522i \(0.905506\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.781954 −0.0656201
\(143\) 3.80451 0.318149
\(144\) 0 0
\(145\) −0.743738 −0.0617641
\(146\) −0.702531 −0.0581419
\(147\) 0 0
\(148\) 8.82316 0.725259
\(149\) −8.19549 −0.671401 −0.335700 0.941969i \(-0.608973\pi\)
−0.335700 + 0.941969i \(0.608973\pi\)
\(150\) 0 0
\(151\) −13.2741 −1.08023 −0.540116 0.841590i \(-0.681620\pi\)
−0.540116 + 0.841590i \(0.681620\pi\)
\(152\) 5.41353 0.439096
\(153\) 0 0
\(154\) 0 0
\(155\) 37.8318 3.03872
\(156\) 0 0
\(157\) 16.9392 1.35190 0.675949 0.736949i \(-0.263734\pi\)
0.675949 + 0.736949i \(0.263734\pi\)
\(158\) 0.557640 0.0443634
\(159\) 0 0
\(160\) 7.50235 0.593113
\(161\) 0 0
\(162\) 0 0
\(163\) −6.79982 −0.532603 −0.266301 0.963890i \(-0.585802\pi\)
−0.266301 + 0.963890i \(0.585802\pi\)
\(164\) 12.3816 0.966839
\(165\) 0 0
\(166\) −2.33490 −0.181223
\(167\) −18.2227 −1.41012 −0.705059 0.709149i \(-0.749080\pi\)
−0.705059 + 0.709149i \(0.749080\pi\)
\(168\) 0 0
\(169\) 1.47431 0.113408
\(170\) 0.213351 0.0163633
\(171\) 0 0
\(172\) −4.60433 −0.351077
\(173\) −1.72118 −0.130859 −0.0654294 0.997857i \(-0.520842\pi\)
−0.0654294 + 0.997857i \(0.520842\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.83255 −0.288889
\(177\) 0 0
\(178\) −1.65571 −0.124101
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −0.725875 −0.0539539 −0.0269769 0.999636i \(-0.508588\pi\)
−0.0269769 + 0.999636i \(0.508588\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.10746 −0.0816432
\(185\) −17.0226 −1.25152
\(186\) 0 0
\(187\) −0.334898 −0.0244902
\(188\) 23.9384 1.74589
\(189\) 0 0
\(190\) −5.18531 −0.376182
\(191\) −5.27412 −0.381622 −0.190811 0.981627i \(-0.561112\pi\)
−0.190811 + 0.981627i \(0.561112\pi\)
\(192\) 0 0
\(193\) −19.8318 −1.42752 −0.713761 0.700390i \(-0.753010\pi\)
−0.713761 + 0.700390i \(0.753010\pi\)
\(194\) −0.00939029 −0.000674184 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.66980 −0.190215 −0.0951076 0.995467i \(-0.530320\pi\)
−0.0951076 + 0.995467i \(0.530320\pi\)
\(198\) 0 0
\(199\) −13.5529 −0.960743 −0.480371 0.877065i \(-0.659498\pi\)
−0.480371 + 0.877065i \(0.659498\pi\)
\(200\) −6.30138 −0.445575
\(201\) 0 0
\(202\) −3.16197 −0.222475
\(203\) 0 0
\(204\) 0 0
\(205\) −23.8878 −1.66840
\(206\) 1.38628 0.0965868
\(207\) 0 0
\(208\) −14.5810 −1.01101
\(209\) 8.13941 0.563015
\(210\) 0 0
\(211\) −4.27882 −0.294566 −0.147283 0.989094i \(-0.547053\pi\)
−0.147283 + 0.989094i \(0.547053\pi\)
\(212\) 15.6651 1.07588
\(213\) 0 0
\(214\) 1.36294 0.0931685
\(215\) 8.88315 0.605826
\(216\) 0 0
\(217\) 0 0
\(218\) 1.93453 0.131023
\(219\) 0 0
\(220\) 7.50235 0.505808
\(221\) −1.27412 −0.0857069
\(222\) 0 0
\(223\) 10.2694 0.687692 0.343846 0.939026i \(-0.388270\pi\)
0.343846 + 0.939026i \(0.388270\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.222741 0.0148165
\(227\) −0.390977 −0.0259500 −0.0129750 0.999916i \(-0.504130\pi\)
−0.0129750 + 0.999916i \(0.504130\pi\)
\(228\) 0 0
\(229\) −7.94392 −0.524949 −0.262475 0.964939i \(-0.584539\pi\)
−0.262475 + 0.964939i \(0.584539\pi\)
\(230\) 1.06077 0.0699453
\(231\) 0 0
\(232\) 0.130020 0.00853621
\(233\) −26.5576 −1.73985 −0.869924 0.493185i \(-0.835833\pi\)
−0.869924 + 0.493185i \(0.835833\pi\)
\(234\) 0 0
\(235\) −46.1845 −3.01275
\(236\) −7.39176 −0.481163
\(237\) 0 0
\(238\) 0 0
\(239\) −10.7998 −0.698582 −0.349291 0.937014i \(-0.613578\pi\)
−0.349291 + 0.937014i \(0.613578\pi\)
\(240\) 0 0
\(241\) 12.0833 0.778356 0.389178 0.921163i \(-0.372759\pi\)
0.389178 + 0.921163i \(0.372759\pi\)
\(242\) 0.167449 0.0107640
\(243\) 0 0
\(244\) 11.8318 0.757451
\(245\) 0 0
\(246\) 0 0
\(247\) 30.9665 1.97035
\(248\) −6.61372 −0.419972
\(249\) 0 0
\(250\) 2.85041 0.180276
\(251\) 4.80921 0.303554 0.151777 0.988415i \(-0.451500\pi\)
0.151777 + 0.988415i \(0.451500\pi\)
\(252\) 0 0
\(253\) −1.66510 −0.104684
\(254\) 2.44706 0.153542
\(255\) 0 0
\(256\) 13.8037 0.862733
\(257\) −16.7531 −1.04503 −0.522516 0.852630i \(-0.675006\pi\)
−0.522516 + 0.852630i \(0.675006\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 28.5428 1.77015
\(261\) 0 0
\(262\) 3.44236 0.212670
\(263\) 12.1394 0.748548 0.374274 0.927318i \(-0.377892\pi\)
0.374274 + 0.927318i \(0.377892\pi\)
\(264\) 0 0
\(265\) −30.2227 −1.85657
\(266\) 0 0
\(267\) 0 0
\(268\) 0.274911 0.0167929
\(269\) −25.2180 −1.53757 −0.768786 0.639506i \(-0.779139\pi\)
−0.768786 + 0.639506i \(0.779139\pi\)
\(270\) 0 0
\(271\) −4.13941 −0.251451 −0.125726 0.992065i \(-0.540126\pi\)
−0.125726 + 0.992065i \(0.540126\pi\)
\(272\) 1.28352 0.0778245
\(273\) 0 0
\(274\) 2.71648 0.164109
\(275\) −9.47431 −0.571322
\(276\) 0 0
\(277\) −18.2788 −1.09827 −0.549134 0.835734i \(-0.685042\pi\)
−0.549134 + 0.835734i \(0.685042\pi\)
\(278\) −3.77569 −0.226451
\(279\) 0 0
\(280\) 0 0
\(281\) −6.74374 −0.402298 −0.201149 0.979561i \(-0.564467\pi\)
−0.201149 + 0.979561i \(0.564467\pi\)
\(282\) 0 0
\(283\) −23.3575 −1.38846 −0.694228 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(284\) 9.20866 0.546433
\(285\) 0 0
\(286\) 0.637062 0.0376703
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8878 −0.993403
\(290\) −0.124538 −0.00731314
\(291\) 0 0
\(292\) 8.27334 0.484161
\(293\) −14.1667 −0.827625 −0.413813 0.910362i \(-0.635803\pi\)
−0.413813 + 0.910362i \(0.635803\pi\)
\(294\) 0 0
\(295\) 14.2610 0.830305
\(296\) 2.97587 0.172969
\(297\) 0 0
\(298\) −1.37233 −0.0794968
\(299\) −6.33490 −0.366357
\(300\) 0 0
\(301\) 0 0
\(302\) −2.22274 −0.127904
\(303\) 0 0
\(304\) −31.1947 −1.78914
\(305\) −22.8271 −1.30707
\(306\) 0 0
\(307\) 12.5576 0.716702 0.358351 0.933587i \(-0.383339\pi\)
0.358351 + 0.933587i \(0.383339\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6.33490 0.359798
\(311\) 9.33959 0.529600 0.264800 0.964303i \(-0.414694\pi\)
0.264800 + 0.964303i \(0.414694\pi\)
\(312\) 0 0
\(313\) 2.99530 0.169305 0.0846523 0.996411i \(-0.473022\pi\)
0.0846523 + 0.996411i \(0.473022\pi\)
\(314\) 2.83646 0.160071
\(315\) 0 0
\(316\) −6.56703 −0.369424
\(317\) −9.93453 −0.557979 −0.278989 0.960294i \(-0.589999\pi\)
−0.278989 + 0.960294i \(0.589999\pi\)
\(318\) 0 0
\(319\) 0.195488 0.0109453
\(320\) −27.9057 −1.55998
\(321\) 0 0
\(322\) 0 0
\(323\) −2.72588 −0.151672
\(324\) 0 0
\(325\) −36.0451 −1.99942
\(326\) −1.13862 −0.0630625
\(327\) 0 0
\(328\) 4.17605 0.230584
\(329\) 0 0
\(330\) 0 0
\(331\) 22.5482 1.23936 0.619682 0.784853i \(-0.287262\pi\)
0.619682 + 0.784853i \(0.287262\pi\)
\(332\) 27.4969 1.50909
\(333\) 0 0
\(334\) −3.05138 −0.166964
\(335\) −0.530387 −0.0289781
\(336\) 0 0
\(337\) 13.2835 0.723599 0.361800 0.932256i \(-0.382162\pi\)
0.361800 + 0.932256i \(0.382162\pi\)
\(338\) 0.246872 0.0134281
\(339\) 0 0
\(340\) −2.51252 −0.136261
\(341\) −9.94392 −0.538494
\(342\) 0 0
\(343\) 0 0
\(344\) −1.55294 −0.0837292
\(345\) 0 0
\(346\) −0.288210 −0.0154943
\(347\) 27.7757 1.49108 0.745538 0.666463i \(-0.232192\pi\)
0.745538 + 0.666463i \(0.232192\pi\)
\(348\) 0 0
\(349\) 2.85589 0.152873 0.0764363 0.997074i \(-0.475646\pi\)
0.0764363 + 0.997074i \(0.475646\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.97196 −0.105106
\(353\) 24.6410 1.31151 0.655753 0.754975i \(-0.272351\pi\)
0.655753 + 0.754975i \(0.272351\pi\)
\(354\) 0 0
\(355\) −17.7663 −0.942937
\(356\) 19.4984 1.03342
\(357\) 0 0
\(358\) 2.00939 0.106200
\(359\) 11.8878 0.627416 0.313708 0.949519i \(-0.398429\pi\)
0.313708 + 0.949519i \(0.398429\pi\)
\(360\) 0 0
\(361\) 47.2500 2.48684
\(362\) −0.121547 −0.00638838
\(363\) 0 0
\(364\) 0 0
\(365\) −15.9618 −0.835478
\(366\) 0 0
\(367\) −3.60902 −0.188389 −0.0941947 0.995554i \(-0.530028\pi\)
−0.0941947 + 0.995554i \(0.530028\pi\)
\(368\) 6.38159 0.332663
\(369\) 0 0
\(370\) −2.85041 −0.148186
\(371\) 0 0
\(372\) 0 0
\(373\) 12.3349 0.638677 0.319338 0.947641i \(-0.396539\pi\)
0.319338 + 0.947641i \(0.396539\pi\)
\(374\) −0.0560785 −0.00289975
\(375\) 0 0
\(376\) 8.07394 0.416382
\(377\) 0.743738 0.0383045
\(378\) 0 0
\(379\) 23.3575 1.19979 0.599896 0.800078i \(-0.295209\pi\)
0.599896 + 0.800078i \(0.295209\pi\)
\(380\) 61.0647 3.13255
\(381\) 0 0
\(382\) −0.883148 −0.0451858
\(383\) −15.2180 −0.777606 −0.388803 0.921321i \(-0.627111\pi\)
−0.388803 + 0.921321i \(0.627111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.32081 −0.169025
\(387\) 0 0
\(388\) 0.110585 0.00561408
\(389\) −3.73057 −0.189147 −0.0945737 0.995518i \(-0.530149\pi\)
−0.0945737 + 0.995518i \(0.530149\pi\)
\(390\) 0 0
\(391\) 0.557640 0.0282011
\(392\) 0 0
\(393\) 0 0
\(394\) −0.447055 −0.0225223
\(395\) 12.6698 0.637487
\(396\) 0 0
\(397\) −20.8925 −1.04857 −0.524283 0.851544i \(-0.675667\pi\)
−0.524283 + 0.851544i \(0.675667\pi\)
\(398\) −2.26943 −0.113756
\(399\) 0 0
\(400\) 36.3108 1.81554
\(401\) −23.2741 −1.16225 −0.581127 0.813813i \(-0.697388\pi\)
−0.581127 + 0.813813i \(0.697388\pi\)
\(402\) 0 0
\(403\) −37.8318 −1.88453
\(404\) 37.2368 1.85260
\(405\) 0 0
\(406\) 0 0
\(407\) 4.47431 0.221783
\(408\) 0 0
\(409\) 23.8972 1.18164 0.590821 0.806803i \(-0.298804\pi\)
0.590821 + 0.806803i \(0.298804\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −16.3255 −0.804300
\(413\) 0 0
\(414\) 0 0
\(415\) −53.0498 −2.60411
\(416\) −7.50235 −0.367833
\(417\) 0 0
\(418\) 1.36294 0.0666635
\(419\) −30.2967 −1.48009 −0.740045 0.672557i \(-0.765196\pi\)
−0.740045 + 0.672557i \(0.765196\pi\)
\(420\) 0 0
\(421\) −15.5257 −0.756676 −0.378338 0.925668i \(-0.623504\pi\)
−0.378338 + 0.925668i \(0.623504\pi\)
\(422\) −0.716485 −0.0348779
\(423\) 0 0
\(424\) 5.28352 0.256590
\(425\) 3.17293 0.153910
\(426\) 0 0
\(427\) 0 0
\(428\) −16.0506 −0.775835
\(429\) 0 0
\(430\) 1.48748 0.0717325
\(431\) −3.07864 −0.148293 −0.0741463 0.997247i \(-0.523623\pi\)
−0.0741463 + 0.997247i \(0.523623\pi\)
\(432\) 0 0
\(433\) −16.9392 −0.814047 −0.407024 0.913418i \(-0.633433\pi\)
−0.407024 + 0.913418i \(0.633433\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −22.7820 −1.09106
\(437\) −13.5529 −0.648325
\(438\) 0 0
\(439\) 11.0786 0.528754 0.264377 0.964419i \(-0.414834\pi\)
0.264377 + 0.964419i \(0.414834\pi\)
\(440\) 2.53039 0.120631
\(441\) 0 0
\(442\) −0.213351 −0.0101481
\(443\) 14.5482 0.691208 0.345604 0.938380i \(-0.387674\pi\)
0.345604 + 0.938380i \(0.387674\pi\)
\(444\) 0 0
\(445\) −37.6184 −1.78328
\(446\) 1.71961 0.0814258
\(447\) 0 0
\(448\) 0 0
\(449\) −27.4875 −1.29721 −0.648607 0.761123i \(-0.724648\pi\)
−0.648607 + 0.761123i \(0.724648\pi\)
\(450\) 0 0
\(451\) 6.27882 0.295658
\(452\) −2.62311 −0.123381
\(453\) 0 0
\(454\) −0.0654688 −0.00307260
\(455\) 0 0
\(456\) 0 0
\(457\) −7.94392 −0.371601 −0.185800 0.982587i \(-0.559488\pi\)
−0.185800 + 0.982587i \(0.559488\pi\)
\(458\) −1.33020 −0.0621563
\(459\) 0 0
\(460\) −12.4922 −0.582450
\(461\) 8.26943 0.385146 0.192573 0.981283i \(-0.438317\pi\)
0.192573 + 0.981283i \(0.438317\pi\)
\(462\) 0 0
\(463\) 9.73904 0.452612 0.226306 0.974056i \(-0.427335\pi\)
0.226306 + 0.974056i \(0.427335\pi\)
\(464\) −0.749219 −0.0347816
\(465\) 0 0
\(466\) −4.44706 −0.206006
\(467\) −40.6970 −1.88323 −0.941617 0.336685i \(-0.890694\pi\)
−0.941617 + 0.336685i \(0.890694\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −7.73356 −0.356723
\(471\) 0 0
\(472\) −2.49309 −0.114754
\(473\) −2.33490 −0.107359
\(474\) 0 0
\(475\) −77.1153 −3.53829
\(476\) 0 0
\(477\) 0 0
\(478\) −1.80842 −0.0827152
\(479\) 37.8318 1.72858 0.864289 0.502996i \(-0.167769\pi\)
0.864289 + 0.502996i \(0.167769\pi\)
\(480\) 0 0
\(481\) 17.0226 0.776162
\(482\) 2.02334 0.0921608
\(483\) 0 0
\(484\) −1.97196 −0.0896346
\(485\) −0.213351 −0.00968778
\(486\) 0 0
\(487\) 40.5576 1.83784 0.918921 0.394442i \(-0.129062\pi\)
0.918921 + 0.394442i \(0.129062\pi\)
\(488\) 3.99061 0.180646
\(489\) 0 0
\(490\) 0 0
\(491\) −31.0786 −1.40256 −0.701280 0.712886i \(-0.747388\pi\)
−0.701280 + 0.712886i \(0.747388\pi\)
\(492\) 0 0
\(493\) −0.0654688 −0.00294856
\(494\) 5.18531 0.233298
\(495\) 0 0
\(496\) 38.1106 1.71122
\(497\) 0 0
\(498\) 0 0
\(499\) 15.3575 0.687494 0.343747 0.939062i \(-0.388304\pi\)
0.343747 + 0.939062i \(0.388304\pi\)
\(500\) −33.5678 −1.50120
\(501\) 0 0
\(502\) 0.805298 0.0359422
\(503\) 14.8925 0.664025 0.332013 0.943275i \(-0.392272\pi\)
0.332013 + 0.943275i \(0.392272\pi\)
\(504\) 0 0
\(505\) −71.8412 −3.19689
\(506\) −0.278820 −0.0123951
\(507\) 0 0
\(508\) −28.8177 −1.27858
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12.6557 0.559309
\(513\) 0 0
\(514\) −2.80530 −0.123736
\(515\) 31.4969 1.38792
\(516\) 0 0
\(517\) 12.1394 0.533891
\(518\) 0 0
\(519\) 0 0
\(520\) 9.62689 0.422167
\(521\) 27.6924 1.21322 0.606612 0.794998i \(-0.292528\pi\)
0.606612 + 0.794998i \(0.292528\pi\)
\(522\) 0 0
\(523\) −18.4088 −0.804962 −0.402481 0.915428i \(-0.631852\pi\)
−0.402481 + 0.915428i \(0.631852\pi\)
\(524\) −40.5389 −1.77095
\(525\) 0 0
\(526\) 2.03273 0.0886314
\(527\) 3.33020 0.145066
\(528\) 0 0
\(529\) −20.2274 −0.879454
\(530\) −5.06077 −0.219826
\(531\) 0 0
\(532\) 0 0
\(533\) 23.8878 1.03470
\(534\) 0 0
\(535\) 30.9665 1.33880
\(536\) 0.0927218 0.00400497
\(537\) 0 0
\(538\) −4.22274 −0.182055
\(539\) 0 0
\(540\) 0 0
\(541\) −4.26943 −0.183557 −0.0917786 0.995779i \(-0.529255\pi\)
−0.0917786 + 0.995779i \(0.529255\pi\)
\(542\) −0.693141 −0.0297729
\(543\) 0 0
\(544\) 0.660406 0.0283147
\(545\) 43.9533 1.88275
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −31.9906 −1.36657
\(549\) 0 0
\(550\) −1.58647 −0.0676471
\(551\) 1.59116 0.0677857
\(552\) 0 0
\(553\) 0 0
\(554\) −3.06077 −0.130040
\(555\) 0 0
\(556\) 44.4643 1.88570
\(557\) 12.4743 0.528553 0.264277 0.964447i \(-0.414867\pi\)
0.264277 + 0.964447i \(0.414867\pi\)
\(558\) 0 0
\(559\) −8.88315 −0.375717
\(560\) 0 0
\(561\) 0 0
\(562\) −1.12923 −0.0476338
\(563\) −32.7710 −1.38113 −0.690566 0.723269i \(-0.742639\pi\)
−0.690566 + 0.723269i \(0.742639\pi\)
\(564\) 0 0
\(565\) 5.06077 0.212908
\(566\) −3.91119 −0.164399
\(567\) 0 0
\(568\) 3.10589 0.130320
\(569\) 29.2180 1.22488 0.612442 0.790515i \(-0.290187\pi\)
0.612442 + 0.790515i \(0.290187\pi\)
\(570\) 0 0
\(571\) 32.4455 1.35780 0.678901 0.734230i \(-0.262457\pi\)
0.678901 + 0.734230i \(0.262457\pi\)
\(572\) −7.50235 −0.313689
\(573\) 0 0
\(574\) 0 0
\(575\) 15.7757 0.657892
\(576\) 0 0
\(577\) 14.8831 0.619594 0.309797 0.950803i \(-0.399739\pi\)
0.309797 + 0.950803i \(0.399739\pi\)
\(578\) −2.82786 −0.117623
\(579\) 0 0
\(580\) 1.46662 0.0608982
\(581\) 0 0
\(582\) 0 0
\(583\) 7.94392 0.329004
\(584\) 2.79043 0.115469
\(585\) 0 0
\(586\) −2.37220 −0.0979945
\(587\) −4.53039 −0.186989 −0.0934945 0.995620i \(-0.529804\pi\)
−0.0934945 + 0.995620i \(0.529804\pi\)
\(588\) 0 0
\(589\) −80.9377 −3.33498
\(590\) 2.38799 0.0983118
\(591\) 0 0
\(592\) −17.1480 −0.704779
\(593\) 8.28821 0.340356 0.170178 0.985413i \(-0.445566\pi\)
0.170178 + 0.985413i \(0.445566\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.1612 0.661988
\(597\) 0 0
\(598\) −1.06077 −0.0433783
\(599\) −1.99061 −0.0813341 −0.0406671 0.999173i \(-0.512948\pi\)
−0.0406671 + 0.999173i \(0.512948\pi\)
\(600\) 0 0
\(601\) 8.47431 0.345674 0.172837 0.984950i \(-0.444707\pi\)
0.172837 + 0.984950i \(0.444707\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 26.1761 1.06509
\(605\) 3.80451 0.154675
\(606\) 0 0
\(607\) 22.9665 0.932181 0.466090 0.884737i \(-0.345662\pi\)
0.466090 + 0.884737i \(0.345662\pi\)
\(608\) −16.0506 −0.650938
\(609\) 0 0
\(610\) −3.82237 −0.154763
\(611\) 46.1845 1.86843
\(612\) 0 0
\(613\) 3.55294 0.143502 0.0717510 0.997423i \(-0.477141\pi\)
0.0717510 + 0.997423i \(0.477141\pi\)
\(614\) 2.10277 0.0848608
\(615\) 0 0
\(616\) 0 0
\(617\) −22.6698 −0.912652 −0.456326 0.889813i \(-0.650835\pi\)
−0.456326 + 0.889813i \(0.650835\pi\)
\(618\) 0 0
\(619\) −8.71648 −0.350345 −0.175173 0.984538i \(-0.556048\pi\)
−0.175173 + 0.984538i \(0.556048\pi\)
\(620\) −74.6028 −2.99612
\(621\) 0 0
\(622\) 1.56391 0.0627070
\(623\) 0 0
\(624\) 0 0
\(625\) 17.3910 0.695639
\(626\) 0.501561 0.0200464
\(627\) 0 0
\(628\) −33.4035 −1.33294
\(629\) −1.49844 −0.0597467
\(630\) 0 0
\(631\) −11.8878 −0.473248 −0.236624 0.971601i \(-0.576041\pi\)
−0.236624 + 0.971601i \(0.576041\pi\)
\(632\) −2.21492 −0.0881049
\(633\) 0 0
\(634\) −1.66353 −0.0660672
\(635\) 55.5981 2.20634
\(636\) 0 0
\(637\) 0 0
\(638\) 0.0327344 0.00129597
\(639\) 0 0
\(640\) −19.6775 −0.777821
\(641\) 4.89254 0.193244 0.0966218 0.995321i \(-0.469196\pi\)
0.0966218 + 0.995321i \(0.469196\pi\)
\(642\) 0 0
\(643\) −22.7804 −0.898371 −0.449185 0.893439i \(-0.648286\pi\)
−0.449185 + 0.893439i \(0.648286\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.456446 −0.0179586
\(647\) 31.2453 1.22838 0.614190 0.789158i \(-0.289483\pi\)
0.614190 + 0.789158i \(0.289483\pi\)
\(648\) 0 0
\(649\) −3.74843 −0.147139
\(650\) −6.03573 −0.236741
\(651\) 0 0
\(652\) 13.4090 0.525136
\(653\) −28.2882 −1.10700 −0.553502 0.832848i \(-0.686709\pi\)
−0.553502 + 0.832848i \(0.686709\pi\)
\(654\) 0 0
\(655\) 78.2118 3.05599
\(656\) −24.0639 −0.939537
\(657\) 0 0
\(658\) 0 0
\(659\) −30.2967 −1.18019 −0.590096 0.807333i \(-0.700910\pi\)
−0.590096 + 0.807333i \(0.700910\pi\)
\(660\) 0 0
\(661\) 35.7196 1.38933 0.694666 0.719333i \(-0.255552\pi\)
0.694666 + 0.719333i \(0.255552\pi\)
\(662\) 3.77569 0.146746
\(663\) 0 0
\(664\) 9.27412 0.359906
\(665\) 0 0
\(666\) 0 0
\(667\) −0.325508 −0.0126037
\(668\) 35.9345 1.39035
\(669\) 0 0
\(670\) −0.0888128 −0.00343114
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 34.9377 1.34675 0.673374 0.739302i \(-0.264844\pi\)
0.673374 + 0.739302i \(0.264844\pi\)
\(674\) 2.22431 0.0856774
\(675\) 0 0
\(676\) −2.90728 −0.111818
\(677\) 20.0094 0.769023 0.384512 0.923120i \(-0.374370\pi\)
0.384512 + 0.923120i \(0.374370\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.847422 −0.0324972
\(681\) 0 0
\(682\) −1.66510 −0.0637600
\(683\) −25.0498 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(684\) 0 0
\(685\) 61.7196 2.35818
\(686\) 0 0
\(687\) 0 0
\(688\) 8.94862 0.341163
\(689\) 30.2227 1.15139
\(690\) 0 0
\(691\) 38.8271 1.47705 0.738526 0.674225i \(-0.235522\pi\)
0.738526 + 0.674225i \(0.235522\pi\)
\(692\) 3.39410 0.129024
\(693\) 0 0
\(694\) 4.65102 0.176550
\(695\) −85.7851 −3.25401
\(696\) 0 0
\(697\) −2.10277 −0.0796480
\(698\) 0.478217 0.0181008
\(699\) 0 0
\(700\) 0 0
\(701\) 41.7757 1.57785 0.788923 0.614492i \(-0.210639\pi\)
0.788923 + 0.614492i \(0.210639\pi\)
\(702\) 0 0
\(703\) 36.4182 1.37354
\(704\) 7.33490 0.276444
\(705\) 0 0
\(706\) 4.12611 0.155288
\(707\) 0 0
\(708\) 0 0
\(709\) −13.4041 −0.503403 −0.251702 0.967805i \(-0.580990\pi\)
−0.251702 + 0.967805i \(0.580990\pi\)
\(710\) −2.97495 −0.111648
\(711\) 0 0
\(712\) 6.57642 0.246462
\(713\) 16.5576 0.620088
\(714\) 0 0
\(715\) 14.4743 0.541308
\(716\) −23.6635 −0.884348
\(717\) 0 0
\(718\) 1.99061 0.0742889
\(719\) 5.47900 0.204332 0.102166 0.994767i \(-0.467423\pi\)
0.102166 + 0.994767i \(0.467423\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.91197 0.294453
\(723\) 0 0
\(724\) 1.43140 0.0531975
\(725\) −1.85212 −0.0687859
\(726\) 0 0
\(727\) 32.8831 1.21957 0.609784 0.792567i \(-0.291256\pi\)
0.609784 + 0.792567i \(0.291256\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.67279 −0.0989243
\(731\) 0.781954 0.0289216
\(732\) 0 0
\(733\) 35.8972 1.32589 0.662947 0.748666i \(-0.269305\pi\)
0.662947 + 0.748666i \(0.269305\pi\)
\(734\) −0.604328 −0.0223062
\(735\) 0 0
\(736\) 3.28352 0.121032
\(737\) 0.139410 0.00513523
\(738\) 0 0
\(739\) 29.6651 1.09125 0.545624 0.838030i \(-0.316293\pi\)
0.545624 + 0.838030i \(0.316293\pi\)
\(740\) 33.5678 1.23398
\(741\) 0 0
\(742\) 0 0
\(743\) 18.7998 0.689698 0.344849 0.938658i \(-0.387930\pi\)
0.344849 + 0.938658i \(0.387930\pi\)
\(744\) 0 0
\(745\) −31.1798 −1.14234
\(746\) 2.06547 0.0756222
\(747\) 0 0
\(748\) 0.660406 0.0241469
\(749\) 0 0
\(750\) 0 0
\(751\) −9.59116 −0.349986 −0.174993 0.984570i \(-0.555990\pi\)
−0.174993 + 0.984570i \(0.555990\pi\)
\(752\) −46.5249 −1.69659
\(753\) 0 0
\(754\) 0.124538 0.00453542
\(755\) −50.5016 −1.83794
\(756\) 0 0
\(757\) 2.74374 0.0997229 0.0498614 0.998756i \(-0.484122\pi\)
0.0498614 + 0.998756i \(0.484122\pi\)
\(758\) 3.91119 0.142061
\(759\) 0 0
\(760\) 20.5959 0.747090
\(761\) 6.39098 0.231673 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10.4004 0.376272
\(765\) 0 0
\(766\) −2.54825 −0.0920720
\(767\) −14.2610 −0.514933
\(768\) 0 0
\(769\) −17.7951 −0.641708 −0.320854 0.947129i \(-0.603970\pi\)
−0.320854 + 0.947129i \(0.603970\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 39.1075 1.40751
\(773\) −31.5257 −1.13390 −0.566950 0.823752i \(-0.691877\pi\)
−0.566950 + 0.823752i \(0.691877\pi\)
\(774\) 0 0
\(775\) 94.2118 3.38419
\(776\) 0.0372979 0.00133892
\(777\) 0 0
\(778\) −0.624681 −0.0223959
\(779\) 51.1059 1.83106
\(780\) 0 0
\(781\) 4.66980 0.167098
\(782\) 0.0933763 0.00333913
\(783\) 0 0
\(784\) 0 0
\(785\) 64.4455 2.30016
\(786\) 0 0
\(787\) −31.8606 −1.13571 −0.567854 0.823130i \(-0.692226\pi\)
−0.567854 + 0.823130i \(0.692226\pi\)
\(788\) 5.26473 0.187548
\(789\) 0 0
\(790\) 2.12155 0.0754813
\(791\) 0 0
\(792\) 0 0
\(793\) 22.8271 0.810613
\(794\) −3.49844 −0.124155
\(795\) 0 0
\(796\) 26.7259 0.947274
\(797\) 43.8590 1.55357 0.776783 0.629768i \(-0.216850\pi\)
0.776783 + 0.629768i \(0.216850\pi\)
\(798\) 0 0
\(799\) −4.06547 −0.143826
\(800\) 18.6830 0.660543
\(801\) 0 0
\(802\) −3.89723 −0.137616
\(803\) 4.19549 0.148056
\(804\) 0 0
\(805\) 0 0
\(806\) −6.33490 −0.223137
\(807\) 0 0
\(808\) 12.5592 0.441832
\(809\) −6.74374 −0.237097 −0.118549 0.992948i \(-0.537824\pi\)
−0.118549 + 0.992948i \(0.537824\pi\)
\(810\) 0 0
\(811\) −3.97275 −0.139502 −0.0697510 0.997564i \(-0.522220\pi\)
−0.0697510 + 0.997564i \(0.522220\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.749219 0.0262601
\(815\) −25.8700 −0.906186
\(816\) 0 0
\(817\) −19.0047 −0.664890
\(818\) 4.00157 0.139912
\(819\) 0 0
\(820\) 47.1059 1.64501
\(821\) −49.5896 −1.73069 −0.865344 0.501178i \(-0.832900\pi\)
−0.865344 + 0.501178i \(0.832900\pi\)
\(822\) 0 0
\(823\) 23.1908 0.808380 0.404190 0.914675i \(-0.367553\pi\)
0.404190 + 0.914675i \(0.367553\pi\)
\(824\) −5.50626 −0.191820
\(825\) 0 0
\(826\) 0 0
\(827\) −27.7484 −0.964908 −0.482454 0.875921i \(-0.660254\pi\)
−0.482454 + 0.875921i \(0.660254\pi\)
\(828\) 0 0
\(829\) −0.269430 −0.00935768 −0.00467884 0.999989i \(-0.501489\pi\)
−0.00467884 + 0.999989i \(0.501489\pi\)
\(830\) −8.88315 −0.308339
\(831\) 0 0
\(832\) 27.9057 0.967456
\(833\) 0 0
\(834\) 0 0
\(835\) −69.3286 −2.39922
\(836\) −16.0506 −0.555122
\(837\) 0 0
\(838\) −5.07316 −0.175249
\(839\) −27.3575 −0.944484 −0.472242 0.881469i \(-0.656555\pi\)
−0.472242 + 0.881469i \(0.656555\pi\)
\(840\) 0 0
\(841\) −28.9618 −0.998682
\(842\) −2.59976 −0.0895938
\(843\) 0 0
\(844\) 8.43767 0.290436
\(845\) 5.60902 0.192956
\(846\) 0 0
\(847\) 0 0
\(848\) −30.4455 −1.04550
\(849\) 0 0
\(850\) 0.531305 0.0182236
\(851\) −7.45018 −0.255389
\(852\) 0 0
\(853\) 28.5482 0.977473 0.488737 0.872431i \(-0.337458\pi\)
0.488737 + 0.872431i \(0.337458\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.41353 −0.185031
\(857\) −39.8972 −1.36286 −0.681432 0.731882i \(-0.738642\pi\)
−0.681432 + 0.731882i \(0.738642\pi\)
\(858\) 0 0
\(859\) −20.5576 −0.701418 −0.350709 0.936485i \(-0.614059\pi\)
−0.350709 + 0.936485i \(0.614059\pi\)
\(860\) −17.5172 −0.597332
\(861\) 0 0
\(862\) −0.515515 −0.0175585
\(863\) 9.66510 0.329004 0.164502 0.986377i \(-0.447398\pi\)
0.164502 + 0.986377i \(0.447398\pi\)
\(864\) 0 0
\(865\) −6.54825 −0.222647
\(866\) −2.83646 −0.0963868
\(867\) 0 0
\(868\) 0 0
\(869\) −3.33020 −0.112969
\(870\) 0 0
\(871\) 0.530387 0.0179715
\(872\) −7.68388 −0.260209
\(873\) 0 0
\(874\) −2.26943 −0.0767646
\(875\) 0 0
\(876\) 0 0
\(877\) −25.7212 −0.868543 −0.434271 0.900782i \(-0.642994\pi\)
−0.434271 + 0.900782i \(0.642994\pi\)
\(878\) 1.85511 0.0626069
\(879\) 0 0
\(880\) −14.5810 −0.491525
\(881\) −14.6316 −0.492950 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(882\) 0 0
\(883\) 18.6877 0.628890 0.314445 0.949276i \(-0.398182\pi\)
0.314445 + 0.949276i \(0.398182\pi\)
\(884\) 2.51252 0.0845053
\(885\) 0 0
\(886\) 2.43609 0.0818421
\(887\) 38.6137 1.29652 0.648261 0.761418i \(-0.275497\pi\)
0.648261 + 0.761418i \(0.275497\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −6.29917 −0.211149
\(891\) 0 0
\(892\) −20.2509 −0.678051
\(893\) 98.8076 3.30647
\(894\) 0 0
\(895\) 45.6541 1.52605
\(896\) 0 0
\(897\) 0 0
\(898\) −4.60276 −0.153596
\(899\) −1.94392 −0.0648334
\(900\) 0 0
\(901\) −2.66041 −0.0886310
\(902\) 1.05138 0.0350072
\(903\) 0 0
\(904\) −0.884720 −0.0294254
\(905\) −2.76160 −0.0917987
\(906\) 0 0
\(907\) 49.0965 1.63022 0.815111 0.579304i \(-0.196676\pi\)
0.815111 + 0.579304i \(0.196676\pi\)
\(908\) 0.770991 0.0255862
\(909\) 0 0
\(910\) 0 0
\(911\) −29.3753 −0.973248 −0.486624 0.873612i \(-0.661772\pi\)
−0.486624 + 0.873612i \(0.661772\pi\)
\(912\) 0 0
\(913\) 13.9439 0.461476
\(914\) −1.33020 −0.0439992
\(915\) 0 0
\(916\) 15.6651 0.517590
\(917\) 0 0
\(918\) 0 0
\(919\) −27.0047 −0.890803 −0.445401 0.895331i \(-0.646939\pi\)
−0.445401 + 0.895331i \(0.646939\pi\)
\(920\) −4.21335 −0.138910
\(921\) 0 0
\(922\) 1.38471 0.0456030
\(923\) 17.7663 0.584785
\(924\) 0 0
\(925\) −42.3910 −1.39381
\(926\) 1.63079 0.0535912
\(927\) 0 0
\(928\) −0.385496 −0.0126545
\(929\) 57.8496 1.89798 0.948992 0.315299i \(-0.102105\pi\)
0.948992 + 0.315299i \(0.102105\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 52.3706 1.71546
\(933\) 0 0
\(934\) −6.81469 −0.222983
\(935\) −1.27412 −0.0416683
\(936\) 0 0
\(937\) −25.2180 −0.823838 −0.411919 0.911221i \(-0.635141\pi\)
−0.411919 + 0.911221i \(0.635141\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 91.0741 2.97051
\(941\) −34.2788 −1.11746 −0.558729 0.829350i \(-0.688711\pi\)
−0.558729 + 0.829350i \(0.688711\pi\)
\(942\) 0 0
\(943\) −10.4549 −0.340458
\(944\) 14.3661 0.467575
\(945\) 0 0
\(946\) −0.390977 −0.0127118
\(947\) 18.1573 0.590032 0.295016 0.955492i \(-0.404675\pi\)
0.295016 + 0.955492i \(0.404675\pi\)
\(948\) 0 0
\(949\) 15.9618 0.518141
\(950\) −12.9129 −0.418950
\(951\) 0 0
\(952\) 0 0
\(953\) 19.8045 0.641531 0.320766 0.947159i \(-0.396060\pi\)
0.320766 + 0.947159i \(0.396060\pi\)
\(954\) 0 0
\(955\) −20.0655 −0.649303
\(956\) 21.2968 0.688788
\(957\) 0 0
\(958\) 6.33490 0.204671
\(959\) 0 0
\(960\) 0 0
\(961\) 67.8816 2.18973
\(962\) 2.85041 0.0919010
\(963\) 0 0
\(964\) −23.8279 −0.767444
\(965\) −75.4502 −2.42883
\(966\) 0 0
\(967\) −0.278820 −0.00896624 −0.00448312 0.999990i \(-0.501427\pi\)
−0.00448312 + 0.999990i \(0.501427\pi\)
\(968\) −0.665102 −0.0213772
\(969\) 0 0
\(970\) −0.0357255 −0.00114708
\(971\) 25.3481 0.813458 0.406729 0.913549i \(-0.366669\pi\)
0.406729 + 0.913549i \(0.366669\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 6.79134 0.217609
\(975\) 0 0
\(976\) −22.9953 −0.736062
\(977\) −6.71648 −0.214879 −0.107440 0.994212i \(-0.534265\pi\)
−0.107440 + 0.994212i \(0.534265\pi\)
\(978\) 0 0
\(979\) 9.88784 0.316017
\(980\) 0 0
\(981\) 0 0
\(982\) −5.20409 −0.166069
\(983\) 9.99061 0.318651 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(984\) 0 0
\(985\) −10.1573 −0.323638
\(986\) −0.0109627 −0.000349123 0
\(987\) 0 0
\(988\) −61.0647 −1.94273
\(989\) 3.88784 0.123626
\(990\) 0 0
\(991\) −26.7064 −0.848358 −0.424179 0.905578i \(-0.639437\pi\)
−0.424179 + 0.905578i \(0.639437\pi\)
\(992\) 19.6090 0.622587
\(993\) 0 0
\(994\) 0 0
\(995\) −51.5623 −1.63464
\(996\) 0 0
\(997\) −54.3333 −1.72075 −0.860377 0.509658i \(-0.829772\pi\)
−0.860377 + 0.509658i \(0.829772\pi\)
\(998\) 2.57159 0.0814024
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4851.2.a.bp.1.2 3
3.2 odd 2 1617.2.a.s.1.2 3
7.6 odd 2 693.2.a.m.1.2 3
21.20 even 2 231.2.a.d.1.2 3
77.76 even 2 7623.2.a.cb.1.2 3
84.83 odd 2 3696.2.a.bp.1.3 3
105.104 even 2 5775.2.a.bw.1.2 3
231.230 odd 2 2541.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 21.20 even 2
693.2.a.m.1.2 3 7.6 odd 2
1617.2.a.s.1.2 3 3.2 odd 2
2541.2.a.bi.1.2 3 231.230 odd 2
3696.2.a.bp.1.3 3 84.83 odd 2
4851.2.a.bp.1.2 3 1.1 even 1 trivial
5775.2.a.bw.1.2 3 105.104 even 2
7623.2.a.cb.1.2 3 77.76 even 2