Properties

Label 8001.2.a.z.1.22
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.22
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +O(q^{10})\) \(q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +2.30815 q^{10} +4.25741 q^{11} +3.54385 q^{13} -0.959867 q^{14} -0.679190 q^{16} +0.302404 q^{17} -6.75953 q^{19} -2.59380 q^{20} +4.08655 q^{22} -6.51256 q^{23} +0.782373 q^{25} +3.40162 q^{26} +1.07866 q^{28} -3.90770 q^{29} -5.48700 q^{31} +5.25827 q^{32} +0.290267 q^{34} -2.40466 q^{35} -6.55479 q^{37} -6.48825 q^{38} -7.10600 q^{40} -2.17456 q^{41} +9.47361 q^{43} -4.59228 q^{44} -6.25119 q^{46} -1.43429 q^{47} +1.00000 q^{49} +0.750974 q^{50} -3.82259 q^{52} -10.5740 q^{53} +10.2376 q^{55} +2.95510 q^{56} -3.75087 q^{58} +7.97145 q^{59} +1.05914 q^{61} -5.26679 q^{62} +6.40561 q^{64} +8.52173 q^{65} -4.72133 q^{67} -0.326189 q^{68} -2.30815 q^{70} -6.25244 q^{71} -11.4772 q^{73} -6.29172 q^{74} +7.29120 q^{76} -4.25741 q^{77} +4.55627 q^{79} -1.63322 q^{80} -2.08729 q^{82} -10.1365 q^{83} +0.727177 q^{85} +9.09340 q^{86} -12.5811 q^{88} -8.24382 q^{89} -3.54385 q^{91} +7.02481 q^{92} -1.37672 q^{94} -16.2543 q^{95} +1.74407 q^{97} +0.959867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959867 0.678728 0.339364 0.940655i \(-0.389788\pi\)
0.339364 + 0.940655i \(0.389788\pi\)
\(3\) 0 0
\(4\) −1.07866 −0.539328
\(5\) 2.40466 1.07540 0.537698 0.843138i \(-0.319294\pi\)
0.537698 + 0.843138i \(0.319294\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.95510 −1.04479
\(9\) 0 0
\(10\) 2.30815 0.729901
\(11\) 4.25741 1.28366 0.641829 0.766848i \(-0.278176\pi\)
0.641829 + 0.766848i \(0.278176\pi\)
\(12\) 0 0
\(13\) 3.54385 0.982886 0.491443 0.870910i \(-0.336470\pi\)
0.491443 + 0.870910i \(0.336470\pi\)
\(14\) −0.959867 −0.256535
\(15\) 0 0
\(16\) −0.679190 −0.169797
\(17\) 0.302404 0.0733436 0.0366718 0.999327i \(-0.488324\pi\)
0.0366718 + 0.999327i \(0.488324\pi\)
\(18\) 0 0
\(19\) −6.75953 −1.55074 −0.775371 0.631506i \(-0.782437\pi\)
−0.775371 + 0.631506i \(0.782437\pi\)
\(20\) −2.59380 −0.579991
\(21\) 0 0
\(22\) 4.08655 0.871255
\(23\) −6.51256 −1.35796 −0.678981 0.734155i \(-0.737578\pi\)
−0.678981 + 0.734155i \(0.737578\pi\)
\(24\) 0 0
\(25\) 0.782373 0.156475
\(26\) 3.40162 0.667113
\(27\) 0 0
\(28\) 1.07866 0.203847
\(29\) −3.90770 −0.725641 −0.362820 0.931859i \(-0.618186\pi\)
−0.362820 + 0.931859i \(0.618186\pi\)
\(30\) 0 0
\(31\) −5.48700 −0.985495 −0.492748 0.870172i \(-0.664007\pi\)
−0.492748 + 0.870172i \(0.664007\pi\)
\(32\) 5.25827 0.929539
\(33\) 0 0
\(34\) 0.290267 0.0497804
\(35\) −2.40466 −0.406461
\(36\) 0 0
\(37\) −6.55479 −1.07760 −0.538800 0.842434i \(-0.681122\pi\)
−0.538800 + 0.842434i \(0.681122\pi\)
\(38\) −6.48825 −1.05253
\(39\) 0 0
\(40\) −7.10600 −1.12356
\(41\) −2.17456 −0.339609 −0.169805 0.985478i \(-0.554314\pi\)
−0.169805 + 0.985478i \(0.554314\pi\)
\(42\) 0 0
\(43\) 9.47361 1.44471 0.722356 0.691521i \(-0.243059\pi\)
0.722356 + 0.691521i \(0.243059\pi\)
\(44\) −4.59228 −0.692313
\(45\) 0 0
\(46\) −6.25119 −0.921688
\(47\) −1.43429 −0.209212 −0.104606 0.994514i \(-0.533358\pi\)
−0.104606 + 0.994514i \(0.533358\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.750974 0.106204
\(51\) 0 0
\(52\) −3.82259 −0.530098
\(53\) −10.5740 −1.45246 −0.726228 0.687454i \(-0.758728\pi\)
−0.726228 + 0.687454i \(0.758728\pi\)
\(54\) 0 0
\(55\) 10.2376 1.38044
\(56\) 2.95510 0.394892
\(57\) 0 0
\(58\) −3.75087 −0.492513
\(59\) 7.97145 1.03779 0.518897 0.854837i \(-0.326343\pi\)
0.518897 + 0.854837i \(0.326343\pi\)
\(60\) 0 0
\(61\) 1.05914 0.135608 0.0678042 0.997699i \(-0.478401\pi\)
0.0678042 + 0.997699i \(0.478401\pi\)
\(62\) −5.26679 −0.668883
\(63\) 0 0
\(64\) 6.40561 0.800702
\(65\) 8.52173 1.05699
\(66\) 0 0
\(67\) −4.72133 −0.576802 −0.288401 0.957510i \(-0.593124\pi\)
−0.288401 + 0.957510i \(0.593124\pi\)
\(68\) −0.326189 −0.0395563
\(69\) 0 0
\(70\) −2.30815 −0.275877
\(71\) −6.25244 −0.742028 −0.371014 0.928627i \(-0.620990\pi\)
−0.371014 + 0.928627i \(0.620990\pi\)
\(72\) 0 0
\(73\) −11.4772 −1.34330 −0.671652 0.740867i \(-0.734415\pi\)
−0.671652 + 0.740867i \(0.734415\pi\)
\(74\) −6.29172 −0.731398
\(75\) 0 0
\(76\) 7.29120 0.836358
\(77\) −4.25741 −0.485177
\(78\) 0 0
\(79\) 4.55627 0.512620 0.256310 0.966595i \(-0.417493\pi\)
0.256310 + 0.966595i \(0.417493\pi\)
\(80\) −1.63322 −0.182599
\(81\) 0 0
\(82\) −2.08729 −0.230502
\(83\) −10.1365 −1.11263 −0.556315 0.830972i \(-0.687785\pi\)
−0.556315 + 0.830972i \(0.687785\pi\)
\(84\) 0 0
\(85\) 0.727177 0.0788734
\(86\) 9.09340 0.980567
\(87\) 0 0
\(88\) −12.5811 −1.34115
\(89\) −8.24382 −0.873844 −0.436922 0.899499i \(-0.643931\pi\)
−0.436922 + 0.899499i \(0.643931\pi\)
\(90\) 0 0
\(91\) −3.54385 −0.371496
\(92\) 7.02481 0.732387
\(93\) 0 0
\(94\) −1.37672 −0.141998
\(95\) −16.2543 −1.66766
\(96\) 0 0
\(97\) 1.74407 0.177084 0.0885419 0.996072i \(-0.471779\pi\)
0.0885419 + 0.996072i \(0.471779\pi\)
\(98\) 0.959867 0.0969612
\(99\) 0 0
\(100\) −0.843912 −0.0843912
\(101\) 3.61578 0.359783 0.179892 0.983686i \(-0.442425\pi\)
0.179892 + 0.983686i \(0.442425\pi\)
\(102\) 0 0
\(103\) 5.92884 0.584186 0.292093 0.956390i \(-0.405648\pi\)
0.292093 + 0.956390i \(0.405648\pi\)
\(104\) −10.4724 −1.02690
\(105\) 0 0
\(106\) −10.1497 −0.985823
\(107\) −6.45523 −0.624050 −0.312025 0.950074i \(-0.601007\pi\)
−0.312025 + 0.950074i \(0.601007\pi\)
\(108\) 0 0
\(109\) 20.5285 1.96628 0.983138 0.182865i \(-0.0585371\pi\)
0.983138 + 0.182865i \(0.0585371\pi\)
\(110\) 9.82675 0.936943
\(111\) 0 0
\(112\) 0.679190 0.0641774
\(113\) 15.2547 1.43504 0.717520 0.696538i \(-0.245277\pi\)
0.717520 + 0.696538i \(0.245277\pi\)
\(114\) 0 0
\(115\) −15.6605 −1.46035
\(116\) 4.21506 0.391358
\(117\) 0 0
\(118\) 7.65153 0.704380
\(119\) −0.302404 −0.0277213
\(120\) 0 0
\(121\) 7.12556 0.647778
\(122\) 1.01663 0.0920412
\(123\) 0 0
\(124\) 5.91859 0.531505
\(125\) −10.1419 −0.907123
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −4.36800 −0.386080
\(129\) 0 0
\(130\) 8.17973 0.717410
\(131\) 5.94653 0.519551 0.259775 0.965669i \(-0.416352\pi\)
0.259775 + 0.965669i \(0.416352\pi\)
\(132\) 0 0
\(133\) 6.75953 0.586125
\(134\) −4.53185 −0.391492
\(135\) 0 0
\(136\) −0.893633 −0.0766284
\(137\) −1.37243 −0.117255 −0.0586273 0.998280i \(-0.518672\pi\)
−0.0586273 + 0.998280i \(0.518672\pi\)
\(138\) 0 0
\(139\) 3.13422 0.265841 0.132921 0.991127i \(-0.457564\pi\)
0.132921 + 0.991127i \(0.457564\pi\)
\(140\) 2.59380 0.219216
\(141\) 0 0
\(142\) −6.00151 −0.503635
\(143\) 15.0876 1.26169
\(144\) 0 0
\(145\) −9.39666 −0.780351
\(146\) −11.0166 −0.911738
\(147\) 0 0
\(148\) 7.07036 0.581180
\(149\) −13.3703 −1.09534 −0.547670 0.836694i \(-0.684485\pi\)
−0.547670 + 0.836694i \(0.684485\pi\)
\(150\) 0 0
\(151\) −10.2915 −0.837513 −0.418757 0.908098i \(-0.637534\pi\)
−0.418757 + 0.908098i \(0.637534\pi\)
\(152\) 19.9751 1.62019
\(153\) 0 0
\(154\) −4.08655 −0.329303
\(155\) −13.1944 −1.05980
\(156\) 0 0
\(157\) 7.70342 0.614800 0.307400 0.951580i \(-0.400541\pi\)
0.307400 + 0.951580i \(0.400541\pi\)
\(158\) 4.37341 0.347930
\(159\) 0 0
\(160\) 12.6443 0.999622
\(161\) 6.51256 0.513262
\(162\) 0 0
\(163\) 6.54115 0.512342 0.256171 0.966631i \(-0.417539\pi\)
0.256171 + 0.966631i \(0.417539\pi\)
\(164\) 2.34560 0.183161
\(165\) 0 0
\(166\) −9.72973 −0.755173
\(167\) −2.06466 −0.159769 −0.0798843 0.996804i \(-0.525455\pi\)
−0.0798843 + 0.996804i \(0.525455\pi\)
\(168\) 0 0
\(169\) −0.441156 −0.0339351
\(170\) 0.697993 0.0535336
\(171\) 0 0
\(172\) −10.2188 −0.779174
\(173\) 11.0221 0.837991 0.418996 0.907988i \(-0.362382\pi\)
0.418996 + 0.907988i \(0.362382\pi\)
\(174\) 0 0
\(175\) −0.782373 −0.0591419
\(176\) −2.89159 −0.217962
\(177\) 0 0
\(178\) −7.91297 −0.593102
\(179\) −23.6676 −1.76900 −0.884501 0.466537i \(-0.845501\pi\)
−0.884501 + 0.466537i \(0.845501\pi\)
\(180\) 0 0
\(181\) −15.1437 −1.12563 −0.562813 0.826584i \(-0.690281\pi\)
−0.562813 + 0.826584i \(0.690281\pi\)
\(182\) −3.40162 −0.252145
\(183\) 0 0
\(184\) 19.2453 1.41878
\(185\) −15.7620 −1.15885
\(186\) 0 0
\(187\) 1.28746 0.0941482
\(188\) 1.54710 0.112834
\(189\) 0 0
\(190\) −15.6020 −1.13189
\(191\) −11.4808 −0.830722 −0.415361 0.909657i \(-0.636345\pi\)
−0.415361 + 0.909657i \(0.636345\pi\)
\(192\) 0 0
\(193\) −8.41906 −0.606017 −0.303009 0.952988i \(-0.597991\pi\)
−0.303009 + 0.952988i \(0.597991\pi\)
\(194\) 1.67408 0.120192
\(195\) 0 0
\(196\) −1.07866 −0.0770468
\(197\) −6.10162 −0.434723 −0.217361 0.976091i \(-0.569745\pi\)
−0.217361 + 0.976091i \(0.569745\pi\)
\(198\) 0 0
\(199\) −10.7103 −0.759231 −0.379616 0.925144i \(-0.623944\pi\)
−0.379616 + 0.925144i \(0.623944\pi\)
\(200\) −2.31199 −0.163482
\(201\) 0 0
\(202\) 3.47066 0.244195
\(203\) 3.90770 0.274266
\(204\) 0 0
\(205\) −5.22907 −0.365214
\(206\) 5.69089 0.396503
\(207\) 0 0
\(208\) −2.40694 −0.166891
\(209\) −28.7781 −1.99062
\(210\) 0 0
\(211\) −13.8349 −0.952436 −0.476218 0.879327i \(-0.657993\pi\)
−0.476218 + 0.879327i \(0.657993\pi\)
\(212\) 11.4057 0.783350
\(213\) 0 0
\(214\) −6.19616 −0.423561
\(215\) 22.7808 1.55364
\(216\) 0 0
\(217\) 5.48700 0.372482
\(218\) 19.7046 1.33457
\(219\) 0 0
\(220\) −11.0429 −0.744510
\(221\) 1.07167 0.0720884
\(222\) 0 0
\(223\) −5.00800 −0.335360 −0.167680 0.985841i \(-0.553628\pi\)
−0.167680 + 0.985841i \(0.553628\pi\)
\(224\) −5.25827 −0.351333
\(225\) 0 0
\(226\) 14.6425 0.974002
\(227\) 23.4386 1.55567 0.777837 0.628467i \(-0.216317\pi\)
0.777837 + 0.628467i \(0.216317\pi\)
\(228\) 0 0
\(229\) −3.95298 −0.261220 −0.130610 0.991434i \(-0.541694\pi\)
−0.130610 + 0.991434i \(0.541694\pi\)
\(230\) −15.0320 −0.991179
\(231\) 0 0
\(232\) 11.5476 0.758139
\(233\) −1.27357 −0.0834346 −0.0417173 0.999129i \(-0.513283\pi\)
−0.0417173 + 0.999129i \(0.513283\pi\)
\(234\) 0 0
\(235\) −3.44897 −0.224986
\(236\) −8.59845 −0.559712
\(237\) 0 0
\(238\) −0.290267 −0.0188152
\(239\) −8.13559 −0.526248 −0.263124 0.964762i \(-0.584753\pi\)
−0.263124 + 0.964762i \(0.584753\pi\)
\(240\) 0 0
\(241\) 19.4833 1.25503 0.627513 0.778606i \(-0.284073\pi\)
0.627513 + 0.778606i \(0.284073\pi\)
\(242\) 6.83959 0.439665
\(243\) 0 0
\(244\) −1.14244 −0.0731374
\(245\) 2.40466 0.153628
\(246\) 0 0
\(247\) −23.9547 −1.52420
\(248\) 16.2146 1.02963
\(249\) 0 0
\(250\) −9.73491 −0.615690
\(251\) −2.25176 −0.142130 −0.0710650 0.997472i \(-0.522640\pi\)
−0.0710650 + 0.997472i \(0.522640\pi\)
\(252\) 0 0
\(253\) −27.7267 −1.74316
\(254\) −0.959867 −0.0602274
\(255\) 0 0
\(256\) −17.0039 −1.06275
\(257\) −15.5072 −0.967310 −0.483655 0.875259i \(-0.660691\pi\)
−0.483655 + 0.875259i \(0.660691\pi\)
\(258\) 0 0
\(259\) 6.55479 0.407295
\(260\) −9.19202 −0.570065
\(261\) 0 0
\(262\) 5.70787 0.352634
\(263\) −3.83687 −0.236591 −0.118296 0.992978i \(-0.537743\pi\)
−0.118296 + 0.992978i \(0.537743\pi\)
\(264\) 0 0
\(265\) −25.4269 −1.56196
\(266\) 6.48825 0.397820
\(267\) 0 0
\(268\) 5.09269 0.311086
\(269\) −15.6381 −0.953470 −0.476735 0.879047i \(-0.658180\pi\)
−0.476735 + 0.879047i \(0.658180\pi\)
\(270\) 0 0
\(271\) 23.4561 1.42485 0.712427 0.701746i \(-0.247596\pi\)
0.712427 + 0.701746i \(0.247596\pi\)
\(272\) −0.205389 −0.0124536
\(273\) 0 0
\(274\) −1.31735 −0.0795840
\(275\) 3.33089 0.200860
\(276\) 0 0
\(277\) 26.8299 1.61205 0.806025 0.591881i \(-0.201615\pi\)
0.806025 + 0.591881i \(0.201615\pi\)
\(278\) 3.00844 0.180434
\(279\) 0 0
\(280\) 7.10600 0.424665
\(281\) 20.9721 1.25109 0.625545 0.780188i \(-0.284877\pi\)
0.625545 + 0.780188i \(0.284877\pi\)
\(282\) 0 0
\(283\) −16.5527 −0.983959 −0.491979 0.870607i \(-0.663726\pi\)
−0.491979 + 0.870607i \(0.663726\pi\)
\(284\) 6.74423 0.400196
\(285\) 0 0
\(286\) 14.4821 0.856344
\(287\) 2.17456 0.128360
\(288\) 0 0
\(289\) −16.9086 −0.994621
\(290\) −9.01955 −0.529646
\(291\) 0 0
\(292\) 12.3799 0.724481
\(293\) 16.6113 0.970443 0.485221 0.874391i \(-0.338739\pi\)
0.485221 + 0.874391i \(0.338739\pi\)
\(294\) 0 0
\(295\) 19.1686 1.11604
\(296\) 19.3701 1.12586
\(297\) 0 0
\(298\) −12.8337 −0.743439
\(299\) −23.0795 −1.33472
\(300\) 0 0
\(301\) −9.47361 −0.546050
\(302\) −9.87850 −0.568444
\(303\) 0 0
\(304\) 4.59100 0.263312
\(305\) 2.54686 0.145833
\(306\) 0 0
\(307\) −25.6493 −1.46388 −0.731942 0.681367i \(-0.761386\pi\)
−0.731942 + 0.681367i \(0.761386\pi\)
\(308\) 4.59228 0.261670
\(309\) 0 0
\(310\) −12.6648 −0.719314
\(311\) −6.28790 −0.356554 −0.178277 0.983980i \(-0.557052\pi\)
−0.178277 + 0.983980i \(0.557052\pi\)
\(312\) 0 0
\(313\) −24.0220 −1.35781 −0.678903 0.734228i \(-0.737544\pi\)
−0.678903 + 0.734228i \(0.737544\pi\)
\(314\) 7.39425 0.417282
\(315\) 0 0
\(316\) −4.91464 −0.276470
\(317\) 11.7159 0.658031 0.329016 0.944324i \(-0.393283\pi\)
0.329016 + 0.944324i \(0.393283\pi\)
\(318\) 0 0
\(319\) −16.6367 −0.931475
\(320\) 15.4033 0.861071
\(321\) 0 0
\(322\) 6.25119 0.348365
\(323\) −2.04411 −0.113737
\(324\) 0 0
\(325\) 2.77261 0.153797
\(326\) 6.27863 0.347741
\(327\) 0 0
\(328\) 6.42604 0.354819
\(329\) 1.43429 0.0790748
\(330\) 0 0
\(331\) −14.4126 −0.792187 −0.396094 0.918210i \(-0.629634\pi\)
−0.396094 + 0.918210i \(0.629634\pi\)
\(332\) 10.9338 0.600072
\(333\) 0 0
\(334\) −1.98180 −0.108439
\(335\) −11.3532 −0.620290
\(336\) 0 0
\(337\) −14.1181 −0.769061 −0.384531 0.923112i \(-0.625637\pi\)
−0.384531 + 0.923112i \(0.625637\pi\)
\(338\) −0.423451 −0.0230327
\(339\) 0 0
\(340\) −0.784373 −0.0425386
\(341\) −23.3604 −1.26504
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −27.9955 −1.50941
\(345\) 0 0
\(346\) 10.5797 0.568768
\(347\) 28.8833 1.55053 0.775267 0.631633i \(-0.217615\pi\)
0.775267 + 0.631633i \(0.217615\pi\)
\(348\) 0 0
\(349\) 24.3596 1.30394 0.651971 0.758244i \(-0.273943\pi\)
0.651971 + 0.758244i \(0.273943\pi\)
\(350\) −0.750974 −0.0401413
\(351\) 0 0
\(352\) 22.3866 1.19321
\(353\) −22.1309 −1.17791 −0.588955 0.808166i \(-0.700461\pi\)
−0.588955 + 0.808166i \(0.700461\pi\)
\(354\) 0 0
\(355\) −15.0350 −0.797973
\(356\) 8.89225 0.471288
\(357\) 0 0
\(358\) −22.7178 −1.20067
\(359\) −34.4554 −1.81848 −0.909242 0.416268i \(-0.863338\pi\)
−0.909242 + 0.416268i \(0.863338\pi\)
\(360\) 0 0
\(361\) 26.6912 1.40480
\(362\) −14.5360 −0.763994
\(363\) 0 0
\(364\) 3.82259 0.200358
\(365\) −27.5987 −1.44458
\(366\) 0 0
\(367\) −29.6470 −1.54756 −0.773780 0.633455i \(-0.781636\pi\)
−0.773780 + 0.633455i \(0.781636\pi\)
\(368\) 4.42326 0.230579
\(369\) 0 0
\(370\) −15.1294 −0.786542
\(371\) 10.5740 0.548977
\(372\) 0 0
\(373\) −24.6924 −1.27852 −0.639262 0.768989i \(-0.720760\pi\)
−0.639262 + 0.768989i \(0.720760\pi\)
\(374\) 1.23579 0.0639010
\(375\) 0 0
\(376\) 4.23846 0.218582
\(377\) −13.8483 −0.713222
\(378\) 0 0
\(379\) −37.1608 −1.90882 −0.954412 0.298493i \(-0.903516\pi\)
−0.954412 + 0.298493i \(0.903516\pi\)
\(380\) 17.5328 0.899416
\(381\) 0 0
\(382\) −11.0200 −0.563834
\(383\) −16.4370 −0.839894 −0.419947 0.907549i \(-0.637951\pi\)
−0.419947 + 0.907549i \(0.637951\pi\)
\(384\) 0 0
\(385\) −10.2376 −0.521757
\(386\) −8.08118 −0.411321
\(387\) 0 0
\(388\) −1.88125 −0.0955062
\(389\) −3.32747 −0.168709 −0.0843547 0.996436i \(-0.526883\pi\)
−0.0843547 + 0.996436i \(0.526883\pi\)
\(390\) 0 0
\(391\) −1.96942 −0.0995979
\(392\) −2.95510 −0.149255
\(393\) 0 0
\(394\) −5.85675 −0.295059
\(395\) 10.9563 0.551269
\(396\) 0 0
\(397\) 4.88603 0.245223 0.122611 0.992455i \(-0.460873\pi\)
0.122611 + 0.992455i \(0.460873\pi\)
\(398\) −10.2804 −0.515312
\(399\) 0 0
\(400\) −0.531380 −0.0265690
\(401\) 18.0905 0.903396 0.451698 0.892171i \(-0.350818\pi\)
0.451698 + 0.892171i \(0.350818\pi\)
\(402\) 0 0
\(403\) −19.4451 −0.968629
\(404\) −3.90018 −0.194041
\(405\) 0 0
\(406\) 3.75087 0.186152
\(407\) −27.9064 −1.38327
\(408\) 0 0
\(409\) 13.1826 0.651837 0.325918 0.945398i \(-0.394327\pi\)
0.325918 + 0.945398i \(0.394327\pi\)
\(410\) −5.01921 −0.247881
\(411\) 0 0
\(412\) −6.39518 −0.315068
\(413\) −7.97145 −0.392249
\(414\) 0 0
\(415\) −24.3749 −1.19652
\(416\) 18.6345 0.913631
\(417\) 0 0
\(418\) −27.6231 −1.35109
\(419\) 33.2567 1.62470 0.812348 0.583173i \(-0.198189\pi\)
0.812348 + 0.583173i \(0.198189\pi\)
\(420\) 0 0
\(421\) −4.43830 −0.216309 −0.108155 0.994134i \(-0.534494\pi\)
−0.108155 + 0.994134i \(0.534494\pi\)
\(422\) −13.2797 −0.646445
\(423\) 0 0
\(424\) 31.2473 1.51750
\(425\) 0.236592 0.0114764
\(426\) 0 0
\(427\) −1.05914 −0.0512552
\(428\) 6.96297 0.336568
\(429\) 0 0
\(430\) 21.8665 1.05450
\(431\) 19.7379 0.950743 0.475371 0.879785i \(-0.342314\pi\)
0.475371 + 0.879785i \(0.342314\pi\)
\(432\) 0 0
\(433\) 29.6295 1.42390 0.711952 0.702228i \(-0.247811\pi\)
0.711952 + 0.702228i \(0.247811\pi\)
\(434\) 5.26679 0.252814
\(435\) 0 0
\(436\) −22.1432 −1.06047
\(437\) 44.0218 2.10585
\(438\) 0 0
\(439\) −41.2017 −1.96645 −0.983224 0.182403i \(-0.941612\pi\)
−0.983224 + 0.182403i \(0.941612\pi\)
\(440\) −30.2532 −1.44226
\(441\) 0 0
\(442\) 1.02866 0.0489285
\(443\) 24.9988 1.18773 0.593863 0.804566i \(-0.297602\pi\)
0.593863 + 0.804566i \(0.297602\pi\)
\(444\) 0 0
\(445\) −19.8236 −0.939727
\(446\) −4.80701 −0.227619
\(447\) 0 0
\(448\) −6.40561 −0.302637
\(449\) −4.39785 −0.207548 −0.103774 0.994601i \(-0.533092\pi\)
−0.103774 + 0.994601i \(0.533092\pi\)
\(450\) 0 0
\(451\) −9.25800 −0.435942
\(452\) −16.4545 −0.773957
\(453\) 0 0
\(454\) 22.4979 1.05588
\(455\) −8.52173 −0.399505
\(456\) 0 0
\(457\) 13.3749 0.625653 0.312826 0.949810i \(-0.398724\pi\)
0.312826 + 0.949810i \(0.398724\pi\)
\(458\) −3.79433 −0.177298
\(459\) 0 0
\(460\) 16.8923 0.787606
\(461\) 10.2563 0.477685 0.238842 0.971058i \(-0.423232\pi\)
0.238842 + 0.971058i \(0.423232\pi\)
\(462\) 0 0
\(463\) −16.2905 −0.757084 −0.378542 0.925584i \(-0.623575\pi\)
−0.378542 + 0.925584i \(0.623575\pi\)
\(464\) 2.65407 0.123212
\(465\) 0 0
\(466\) −1.22246 −0.0566294
\(467\) 8.99794 0.416375 0.208188 0.978089i \(-0.433244\pi\)
0.208188 + 0.978089i \(0.433244\pi\)
\(468\) 0 0
\(469\) 4.72133 0.218011
\(470\) −3.31055 −0.152704
\(471\) 0 0
\(472\) −23.5564 −1.08427
\(473\) 40.3331 1.85452
\(474\) 0 0
\(475\) −5.28847 −0.242652
\(476\) 0.326189 0.0149509
\(477\) 0 0
\(478\) −7.80908 −0.357179
\(479\) 14.2639 0.651733 0.325866 0.945416i \(-0.394344\pi\)
0.325866 + 0.945416i \(0.394344\pi\)
\(480\) 0 0
\(481\) −23.2292 −1.05916
\(482\) 18.7013 0.851822
\(483\) 0 0
\(484\) −7.68603 −0.349365
\(485\) 4.19390 0.190435
\(486\) 0 0
\(487\) −10.7755 −0.488284 −0.244142 0.969740i \(-0.578506\pi\)
−0.244142 + 0.969740i \(0.578506\pi\)
\(488\) −3.12985 −0.141682
\(489\) 0 0
\(490\) 2.30815 0.104272
\(491\) 6.80357 0.307041 0.153520 0.988145i \(-0.450939\pi\)
0.153520 + 0.988145i \(0.450939\pi\)
\(492\) 0 0
\(493\) −1.18170 −0.0532211
\(494\) −22.9933 −1.03452
\(495\) 0 0
\(496\) 3.72672 0.167335
\(497\) 6.25244 0.280460
\(498\) 0 0
\(499\) −30.2923 −1.35607 −0.678035 0.735030i \(-0.737168\pi\)
−0.678035 + 0.735030i \(0.737168\pi\)
\(500\) 10.9397 0.489237
\(501\) 0 0
\(502\) −2.16139 −0.0964677
\(503\) −14.4114 −0.642571 −0.321286 0.946982i \(-0.604115\pi\)
−0.321286 + 0.946982i \(0.604115\pi\)
\(504\) 0 0
\(505\) 8.69470 0.386909
\(506\) −26.6139 −1.18313
\(507\) 0 0
\(508\) 1.07866 0.0478576
\(509\) 0.467272 0.0207115 0.0103557 0.999946i \(-0.496704\pi\)
0.0103557 + 0.999946i \(0.496704\pi\)
\(510\) 0 0
\(511\) 11.4772 0.507721
\(512\) −7.58551 −0.335235
\(513\) 0 0
\(514\) −14.8848 −0.656541
\(515\) 14.2568 0.628230
\(516\) 0 0
\(517\) −6.10635 −0.268557
\(518\) 6.29172 0.276442
\(519\) 0 0
\(520\) −25.1826 −1.10433
\(521\) 34.7244 1.52130 0.760651 0.649160i \(-0.224880\pi\)
0.760651 + 0.649160i \(0.224880\pi\)
\(522\) 0 0
\(523\) −9.13868 −0.399607 −0.199803 0.979836i \(-0.564030\pi\)
−0.199803 + 0.979836i \(0.564030\pi\)
\(524\) −6.41426 −0.280208
\(525\) 0 0
\(526\) −3.68288 −0.160581
\(527\) −1.65929 −0.0722798
\(528\) 0 0
\(529\) 19.4134 0.844063
\(530\) −24.4065 −1.06015
\(531\) 0 0
\(532\) −7.29120 −0.316114
\(533\) −7.70631 −0.333797
\(534\) 0 0
\(535\) −15.5226 −0.671101
\(536\) 13.9520 0.602635
\(537\) 0 0
\(538\) −15.0105 −0.647147
\(539\) 4.25741 0.183380
\(540\) 0 0
\(541\) 8.80281 0.378462 0.189231 0.981933i \(-0.439400\pi\)
0.189231 + 0.981933i \(0.439400\pi\)
\(542\) 22.5147 0.967089
\(543\) 0 0
\(544\) 1.59012 0.0681758
\(545\) 49.3641 2.11452
\(546\) 0 0
\(547\) 2.47099 0.105652 0.0528258 0.998604i \(-0.483177\pi\)
0.0528258 + 0.998604i \(0.483177\pi\)
\(548\) 1.48038 0.0632386
\(549\) 0 0
\(550\) 3.19721 0.136329
\(551\) 26.4142 1.12528
\(552\) 0 0
\(553\) −4.55627 −0.193752
\(554\) 25.7531 1.09414
\(555\) 0 0
\(556\) −3.38075 −0.143376
\(557\) −32.2057 −1.36460 −0.682299 0.731073i \(-0.739020\pi\)
−0.682299 + 0.731073i \(0.739020\pi\)
\(558\) 0 0
\(559\) 33.5730 1.41999
\(560\) 1.63322 0.0690161
\(561\) 0 0
\(562\) 20.1304 0.849150
\(563\) −13.9684 −0.588696 −0.294348 0.955698i \(-0.595102\pi\)
−0.294348 + 0.955698i \(0.595102\pi\)
\(564\) 0 0
\(565\) 36.6823 1.54323
\(566\) −15.8884 −0.667841
\(567\) 0 0
\(568\) 18.4766 0.775260
\(569\) 18.2780 0.766252 0.383126 0.923696i \(-0.374847\pi\)
0.383126 + 0.923696i \(0.374847\pi\)
\(570\) 0 0
\(571\) 40.3925 1.69037 0.845187 0.534470i \(-0.179489\pi\)
0.845187 + 0.534470i \(0.179489\pi\)
\(572\) −16.2743 −0.680464
\(573\) 0 0
\(574\) 2.08729 0.0871217
\(575\) −5.09525 −0.212487
\(576\) 0 0
\(577\) 4.42495 0.184213 0.0921065 0.995749i \(-0.470640\pi\)
0.0921065 + 0.995749i \(0.470640\pi\)
\(578\) −16.2300 −0.675077
\(579\) 0 0
\(580\) 10.1358 0.420865
\(581\) 10.1365 0.420534
\(582\) 0 0
\(583\) −45.0180 −1.86446
\(584\) 33.9162 1.40346
\(585\) 0 0
\(586\) 15.9446 0.658667
\(587\) 22.3504 0.922500 0.461250 0.887270i \(-0.347401\pi\)
0.461250 + 0.887270i \(0.347401\pi\)
\(588\) 0 0
\(589\) 37.0896 1.52825
\(590\) 18.3993 0.757487
\(591\) 0 0
\(592\) 4.45194 0.182974
\(593\) 39.1966 1.60961 0.804805 0.593539i \(-0.202270\pi\)
0.804805 + 0.593539i \(0.202270\pi\)
\(594\) 0 0
\(595\) −0.727177 −0.0298113
\(596\) 14.4220 0.590748
\(597\) 0 0
\(598\) −22.1533 −0.905914
\(599\) 20.9632 0.856534 0.428267 0.903652i \(-0.359124\pi\)
0.428267 + 0.903652i \(0.359124\pi\)
\(600\) 0 0
\(601\) 39.3899 1.60675 0.803375 0.595474i \(-0.203036\pi\)
0.803375 + 0.595474i \(0.203036\pi\)
\(602\) −9.09340 −0.370620
\(603\) 0 0
\(604\) 11.1010 0.451694
\(605\) 17.1345 0.696618
\(606\) 0 0
\(607\) −6.62814 −0.269028 −0.134514 0.990912i \(-0.542947\pi\)
−0.134514 + 0.990912i \(0.542947\pi\)
\(608\) −35.5434 −1.44148
\(609\) 0 0
\(610\) 2.44464 0.0989807
\(611\) −5.08289 −0.205632
\(612\) 0 0
\(613\) −22.9071 −0.925210 −0.462605 0.886565i \(-0.653085\pi\)
−0.462605 + 0.886565i \(0.653085\pi\)
\(614\) −24.6199 −0.993579
\(615\) 0 0
\(616\) 12.5811 0.506906
\(617\) 40.6908 1.63815 0.819076 0.573685i \(-0.194487\pi\)
0.819076 + 0.573685i \(0.194487\pi\)
\(618\) 0 0
\(619\) −24.2607 −0.975118 −0.487559 0.873090i \(-0.662113\pi\)
−0.487559 + 0.873090i \(0.662113\pi\)
\(620\) 14.2322 0.571578
\(621\) 0 0
\(622\) −6.03554 −0.242003
\(623\) 8.24382 0.330282
\(624\) 0 0
\(625\) −28.2998 −1.13199
\(626\) −23.0579 −0.921581
\(627\) 0 0
\(628\) −8.30933 −0.331579
\(629\) −1.98219 −0.0790351
\(630\) 0 0
\(631\) −36.8846 −1.46835 −0.734177 0.678959i \(-0.762432\pi\)
−0.734177 + 0.678959i \(0.762432\pi\)
\(632\) −13.4642 −0.535578
\(633\) 0 0
\(634\) 11.2457 0.446624
\(635\) −2.40466 −0.0954259
\(636\) 0 0
\(637\) 3.54385 0.140412
\(638\) −15.9690 −0.632218
\(639\) 0 0
\(640\) −10.5035 −0.415189
\(641\) 37.0112 1.46186 0.730928 0.682455i \(-0.239088\pi\)
0.730928 + 0.682455i \(0.239088\pi\)
\(642\) 0 0
\(643\) −19.7735 −0.779790 −0.389895 0.920859i \(-0.627489\pi\)
−0.389895 + 0.920859i \(0.627489\pi\)
\(644\) −7.02481 −0.276816
\(645\) 0 0
\(646\) −1.96207 −0.0771966
\(647\) 8.97670 0.352910 0.176455 0.984309i \(-0.443537\pi\)
0.176455 + 0.984309i \(0.443537\pi\)
\(648\) 0 0
\(649\) 33.9378 1.33217
\(650\) 2.66134 0.104386
\(651\) 0 0
\(652\) −7.05565 −0.276321
\(653\) −11.3033 −0.442331 −0.221166 0.975236i \(-0.570986\pi\)
−0.221166 + 0.975236i \(0.570986\pi\)
\(654\) 0 0
\(655\) 14.2994 0.558722
\(656\) 1.47694 0.0576648
\(657\) 0 0
\(658\) 1.37672 0.0536703
\(659\) −44.1617 −1.72029 −0.860147 0.510046i \(-0.829629\pi\)
−0.860147 + 0.510046i \(0.829629\pi\)
\(660\) 0 0
\(661\) 35.8877 1.39587 0.697936 0.716160i \(-0.254102\pi\)
0.697936 + 0.716160i \(0.254102\pi\)
\(662\) −13.8342 −0.537680
\(663\) 0 0
\(664\) 29.9545 1.16246
\(665\) 16.2543 0.630316
\(666\) 0 0
\(667\) 25.4491 0.985393
\(668\) 2.22706 0.0861676
\(669\) 0 0
\(670\) −10.8975 −0.421009
\(671\) 4.50918 0.174075
\(672\) 0 0
\(673\) 30.0012 1.15646 0.578230 0.815873i \(-0.303743\pi\)
0.578230 + 0.815873i \(0.303743\pi\)
\(674\) −13.5515 −0.521984
\(675\) 0 0
\(676\) 0.475856 0.0183022
\(677\) 12.7361 0.489488 0.244744 0.969588i \(-0.421296\pi\)
0.244744 + 0.969588i \(0.421296\pi\)
\(678\) 0 0
\(679\) −1.74407 −0.0669314
\(680\) −2.14888 −0.0824058
\(681\) 0 0
\(682\) −22.4229 −0.858618
\(683\) −21.9682 −0.840591 −0.420295 0.907387i \(-0.638074\pi\)
−0.420295 + 0.907387i \(0.638074\pi\)
\(684\) 0 0
\(685\) −3.30022 −0.126095
\(686\) −0.959867 −0.0366479
\(687\) 0 0
\(688\) −6.43438 −0.245308
\(689\) −37.4728 −1.42760
\(690\) 0 0
\(691\) 34.3489 1.30669 0.653347 0.757058i \(-0.273364\pi\)
0.653347 + 0.757058i \(0.273364\pi\)
\(692\) −11.8890 −0.451952
\(693\) 0 0
\(694\) 27.7241 1.05239
\(695\) 7.53673 0.285884
\(696\) 0 0
\(697\) −0.657595 −0.0249082
\(698\) 23.3820 0.885022
\(699\) 0 0
\(700\) 0.843912 0.0318969
\(701\) −44.3643 −1.67562 −0.837809 0.545963i \(-0.816164\pi\)
−0.837809 + 0.545963i \(0.816164\pi\)
\(702\) 0 0
\(703\) 44.3073 1.67108
\(704\) 27.2713 1.02783
\(705\) 0 0
\(706\) −21.2427 −0.799481
\(707\) −3.61578 −0.135985
\(708\) 0 0
\(709\) 26.5449 0.996913 0.498457 0.866915i \(-0.333900\pi\)
0.498457 + 0.866915i \(0.333900\pi\)
\(710\) −14.4316 −0.541607
\(711\) 0 0
\(712\) 24.3613 0.912979
\(713\) 35.7345 1.33827
\(714\) 0 0
\(715\) 36.2805 1.35681
\(716\) 25.5292 0.954073
\(717\) 0 0
\(718\) −33.0725 −1.23426
\(719\) −29.8919 −1.11478 −0.557389 0.830251i \(-0.688197\pi\)
−0.557389 + 0.830251i \(0.688197\pi\)
\(720\) 0 0
\(721\) −5.92884 −0.220801
\(722\) 25.6200 0.953478
\(723\) 0 0
\(724\) 16.3349 0.607081
\(725\) −3.05728 −0.113544
\(726\) 0 0
\(727\) −42.0562 −1.55978 −0.779889 0.625918i \(-0.784725\pi\)
−0.779889 + 0.625918i \(0.784725\pi\)
\(728\) 10.4724 0.388134
\(729\) 0 0
\(730\) −26.4911 −0.980478
\(731\) 2.86485 0.105960
\(732\) 0 0
\(733\) 11.2662 0.416128 0.208064 0.978115i \(-0.433284\pi\)
0.208064 + 0.978115i \(0.433284\pi\)
\(734\) −28.4571 −1.05037
\(735\) 0 0
\(736\) −34.2448 −1.26228
\(737\) −20.1006 −0.740417
\(738\) 0 0
\(739\) 30.3213 1.11539 0.557694 0.830046i \(-0.311686\pi\)
0.557694 + 0.830046i \(0.311686\pi\)
\(740\) 17.0018 0.624998
\(741\) 0 0
\(742\) 10.1497 0.372606
\(743\) 39.3851 1.44490 0.722450 0.691423i \(-0.243016\pi\)
0.722450 + 0.691423i \(0.243016\pi\)
\(744\) 0 0
\(745\) −32.1511 −1.17792
\(746\) −23.7014 −0.867771
\(747\) 0 0
\(748\) −1.38872 −0.0507767
\(749\) 6.45523 0.235869
\(750\) 0 0
\(751\) 1.95873 0.0714751 0.0357375 0.999361i \(-0.488622\pi\)
0.0357375 + 0.999361i \(0.488622\pi\)
\(752\) 0.974153 0.0355237
\(753\) 0 0
\(754\) −13.2925 −0.484084
\(755\) −24.7476 −0.900658
\(756\) 0 0
\(757\) 29.2998 1.06492 0.532460 0.846455i \(-0.321268\pi\)
0.532460 + 0.846455i \(0.321268\pi\)
\(758\) −35.6694 −1.29557
\(759\) 0 0
\(760\) 48.0332 1.74235
\(761\) 2.27331 0.0824075 0.0412038 0.999151i \(-0.486881\pi\)
0.0412038 + 0.999151i \(0.486881\pi\)
\(762\) 0 0
\(763\) −20.5285 −0.743183
\(764\) 12.3838 0.448031
\(765\) 0 0
\(766\) −15.7774 −0.570060
\(767\) 28.2496 1.02003
\(768\) 0 0
\(769\) −37.8564 −1.36514 −0.682568 0.730822i \(-0.739137\pi\)
−0.682568 + 0.730822i \(0.739137\pi\)
\(770\) −9.82675 −0.354131
\(771\) 0 0
\(772\) 9.08127 0.326842
\(773\) −31.5316 −1.13411 −0.567056 0.823679i \(-0.691918\pi\)
−0.567056 + 0.823679i \(0.691918\pi\)
\(774\) 0 0
\(775\) −4.29289 −0.154205
\(776\) −5.15391 −0.185015
\(777\) 0 0
\(778\) −3.19393 −0.114508
\(779\) 14.6990 0.526646
\(780\) 0 0
\(781\) −26.6192 −0.952510
\(782\) −1.89038 −0.0675999
\(783\) 0 0
\(784\) −0.679190 −0.0242568
\(785\) 18.5241 0.661152
\(786\) 0 0
\(787\) −5.10263 −0.181889 −0.0909446 0.995856i \(-0.528989\pi\)
−0.0909446 + 0.995856i \(0.528989\pi\)
\(788\) 6.58155 0.234458
\(789\) 0 0
\(790\) 10.5165 0.374162
\(791\) −15.2547 −0.542394
\(792\) 0 0
\(793\) 3.75341 0.133288
\(794\) 4.68994 0.166440
\(795\) 0 0
\(796\) 11.5527 0.409475
\(797\) 20.3107 0.719441 0.359720 0.933060i \(-0.382872\pi\)
0.359720 + 0.933060i \(0.382872\pi\)
\(798\) 0 0
\(799\) −0.433733 −0.0153444
\(800\) 4.11393 0.145449
\(801\) 0 0
\(802\) 17.3645 0.613161
\(803\) −48.8631 −1.72434
\(804\) 0 0
\(805\) 15.6605 0.551959
\(806\) −18.6647 −0.657436
\(807\) 0 0
\(808\) −10.6850 −0.375896
\(809\) 41.8654 1.47191 0.735955 0.677030i \(-0.236733\pi\)
0.735955 + 0.677030i \(0.236733\pi\)
\(810\) 0 0
\(811\) 1.27580 0.0447992 0.0223996 0.999749i \(-0.492869\pi\)
0.0223996 + 0.999749i \(0.492869\pi\)
\(812\) −4.21506 −0.147920
\(813\) 0 0
\(814\) −26.7865 −0.938865
\(815\) 15.7292 0.550970
\(816\) 0 0
\(817\) −64.0371 −2.24038
\(818\) 12.6535 0.442420
\(819\) 0 0
\(820\) 5.64037 0.196970
\(821\) −41.2734 −1.44045 −0.720226 0.693740i \(-0.755962\pi\)
−0.720226 + 0.693740i \(0.755962\pi\)
\(822\) 0 0
\(823\) 37.3455 1.30178 0.650891 0.759171i \(-0.274396\pi\)
0.650891 + 0.759171i \(0.274396\pi\)
\(824\) −17.5203 −0.610349
\(825\) 0 0
\(826\) −7.65153 −0.266231
\(827\) 5.74511 0.199777 0.0998886 0.994999i \(-0.468151\pi\)
0.0998886 + 0.994999i \(0.468151\pi\)
\(828\) 0 0
\(829\) 46.6365 1.61975 0.809876 0.586601i \(-0.199534\pi\)
0.809876 + 0.586601i \(0.199534\pi\)
\(830\) −23.3966 −0.812110
\(831\) 0 0
\(832\) 22.7005 0.786999
\(833\) 0.302404 0.0104777
\(834\) 0 0
\(835\) −4.96481 −0.171814
\(836\) 31.0417 1.07360
\(837\) 0 0
\(838\) 31.9220 1.10273
\(839\) −31.6074 −1.09121 −0.545605 0.838043i \(-0.683700\pi\)
−0.545605 + 0.838043i \(0.683700\pi\)
\(840\) 0 0
\(841\) −13.7299 −0.473446
\(842\) −4.26017 −0.146815
\(843\) 0 0
\(844\) 14.9231 0.513675
\(845\) −1.06083 −0.0364937
\(846\) 0 0
\(847\) −7.12556 −0.244837
\(848\) 7.18178 0.246623
\(849\) 0 0
\(850\) 0.227097 0.00778937
\(851\) 42.6885 1.46334
\(852\) 0 0
\(853\) −35.7198 −1.22302 −0.611512 0.791235i \(-0.709438\pi\)
−0.611512 + 0.791235i \(0.709438\pi\)
\(854\) −1.01663 −0.0347883
\(855\) 0 0
\(856\) 19.0758 0.651999
\(857\) 11.5668 0.395113 0.197556 0.980292i \(-0.436699\pi\)
0.197556 + 0.980292i \(0.436699\pi\)
\(858\) 0 0
\(859\) −34.8509 −1.18910 −0.594549 0.804059i \(-0.702669\pi\)
−0.594549 + 0.804059i \(0.702669\pi\)
\(860\) −24.5726 −0.837920
\(861\) 0 0
\(862\) 18.9458 0.645296
\(863\) −11.4883 −0.391068 −0.195534 0.980697i \(-0.562644\pi\)
−0.195534 + 0.980697i \(0.562644\pi\)
\(864\) 0 0
\(865\) 26.5043 0.901172
\(866\) 28.4404 0.966444
\(867\) 0 0
\(868\) −5.91859 −0.200890
\(869\) 19.3979 0.658029
\(870\) 0 0
\(871\) −16.7317 −0.566931
\(872\) −60.6638 −2.05434
\(873\) 0 0
\(874\) 42.2551 1.42930
\(875\) 10.1419 0.342860
\(876\) 0 0
\(877\) −7.92650 −0.267659 −0.133829 0.991004i \(-0.542727\pi\)
−0.133829 + 0.991004i \(0.542727\pi\)
\(878\) −39.5481 −1.33468
\(879\) 0 0
\(880\) −6.95328 −0.234395
\(881\) 31.4350 1.05907 0.529536 0.848287i \(-0.322366\pi\)
0.529536 + 0.848287i \(0.322366\pi\)
\(882\) 0 0
\(883\) 1.77775 0.0598260 0.0299130 0.999553i \(-0.490477\pi\)
0.0299130 + 0.999553i \(0.490477\pi\)
\(884\) −1.15596 −0.0388793
\(885\) 0 0
\(886\) 23.9955 0.806144
\(887\) −10.3134 −0.346292 −0.173146 0.984896i \(-0.555393\pi\)
−0.173146 + 0.984896i \(0.555393\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −19.0280 −0.637819
\(891\) 0 0
\(892\) 5.40191 0.180869
\(893\) 9.69510 0.324434
\(894\) 0 0
\(895\) −56.9126 −1.90238
\(896\) 4.36800 0.145925
\(897\) 0 0
\(898\) −4.22135 −0.140868
\(899\) 21.4415 0.715115
\(900\) 0 0
\(901\) −3.19763 −0.106528
\(902\) −8.88644 −0.295886
\(903\) 0 0
\(904\) −45.0791 −1.49931
\(905\) −36.4155 −1.21049
\(906\) 0 0
\(907\) 2.28463 0.0758598 0.0379299 0.999280i \(-0.487924\pi\)
0.0379299 + 0.999280i \(0.487924\pi\)
\(908\) −25.2822 −0.839018
\(909\) 0 0
\(910\) −8.17973 −0.271155
\(911\) 5.90967 0.195796 0.0978980 0.995196i \(-0.468788\pi\)
0.0978980 + 0.995196i \(0.468788\pi\)
\(912\) 0 0
\(913\) −43.1554 −1.42824
\(914\) 12.8381 0.424648
\(915\) 0 0
\(916\) 4.26390 0.140883
\(917\) −5.94653 −0.196372
\(918\) 0 0
\(919\) −45.1535 −1.48947 −0.744737 0.667358i \(-0.767425\pi\)
−0.744737 + 0.667358i \(0.767425\pi\)
\(920\) 46.2783 1.52575
\(921\) 0 0
\(922\) 9.84471 0.324218
\(923\) −22.1577 −0.729329
\(924\) 0 0
\(925\) −5.12829 −0.168617
\(926\) −15.6367 −0.513855
\(927\) 0 0
\(928\) −20.5477 −0.674511
\(929\) −43.7874 −1.43662 −0.718309 0.695724i \(-0.755084\pi\)
−0.718309 + 0.695724i \(0.755084\pi\)
\(930\) 0 0
\(931\) −6.75953 −0.221535
\(932\) 1.37375 0.0449986
\(933\) 0 0
\(934\) 8.63683 0.282606
\(935\) 3.09589 0.101246
\(936\) 0 0
\(937\) 39.0370 1.27528 0.637642 0.770333i \(-0.279910\pi\)
0.637642 + 0.770333i \(0.279910\pi\)
\(938\) 4.53185 0.147970
\(939\) 0 0
\(940\) 3.72025 0.121341
\(941\) −15.5982 −0.508488 −0.254244 0.967140i \(-0.581827\pi\)
−0.254244 + 0.967140i \(0.581827\pi\)
\(942\) 0 0
\(943\) 14.1620 0.461177
\(944\) −5.41413 −0.176215
\(945\) 0 0
\(946\) 38.7144 1.25871
\(947\) 51.9065 1.68673 0.843367 0.537338i \(-0.180570\pi\)
0.843367 + 0.537338i \(0.180570\pi\)
\(948\) 0 0
\(949\) −40.6734 −1.32031
\(950\) −5.07623 −0.164695
\(951\) 0 0
\(952\) 0.893633 0.0289628
\(953\) 9.02726 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(954\) 0 0
\(955\) −27.6074 −0.893354
\(956\) 8.77550 0.283820
\(957\) 0 0
\(958\) 13.6914 0.442349
\(959\) 1.37243 0.0443180
\(960\) 0 0
\(961\) −0.892778 −0.0287993
\(962\) −22.2969 −0.718881
\(963\) 0 0
\(964\) −21.0157 −0.676871
\(965\) −20.2449 −0.651708
\(966\) 0 0
\(967\) 6.81098 0.219027 0.109513 0.993985i \(-0.465071\pi\)
0.109513 + 0.993985i \(0.465071\pi\)
\(968\) −21.0567 −0.676789
\(969\) 0 0
\(970\) 4.02558 0.129254
\(971\) −24.2046 −0.776764 −0.388382 0.921499i \(-0.626966\pi\)
−0.388382 + 0.921499i \(0.626966\pi\)
\(972\) 0 0
\(973\) −3.13422 −0.100479
\(974\) −10.3430 −0.331412
\(975\) 0 0
\(976\) −0.719354 −0.0230260
\(977\) 25.4883 0.815443 0.407722 0.913106i \(-0.366323\pi\)
0.407722 + 0.913106i \(0.366323\pi\)
\(978\) 0 0
\(979\) −35.0974 −1.12172
\(980\) −2.59380 −0.0828558
\(981\) 0 0
\(982\) 6.53052 0.208397
\(983\) 21.3210 0.680035 0.340017 0.940419i \(-0.389567\pi\)
0.340017 + 0.940419i \(0.389567\pi\)
\(984\) 0 0
\(985\) −14.6723 −0.467499
\(986\) −1.13428 −0.0361227
\(987\) 0 0
\(988\) 25.8389 0.822045
\(989\) −61.6975 −1.96187
\(990\) 0 0
\(991\) 54.5238 1.73200 0.866002 0.500040i \(-0.166681\pi\)
0.866002 + 0.500040i \(0.166681\pi\)
\(992\) −28.8521 −0.916056
\(993\) 0 0
\(994\) 6.00151 0.190356
\(995\) −25.7545 −0.816473
\(996\) 0 0
\(997\) −54.8655 −1.73761 −0.868804 0.495155i \(-0.835111\pi\)
−0.868804 + 0.495155i \(0.835111\pi\)
\(998\) −29.0766 −0.920403
\(999\) 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 8001.2.a.z.1.22 yes 32
3.2 odd 2 inner 8001.2.a.z.1.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.11 32 3.2 odd 2 inner
8001.2.a.z.1.22 yes 32 1.1 even 1 trivial