Properties

Label 4005.2.a.o.1.3
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.89340\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.584976 q^{2} -1.65780 q^{4} -1.00000 q^{5} -2.74591 q^{7} +2.13973 q^{8} +O(q^{10})\) \(q-0.584976 q^{2} -1.65780 q^{4} -1.00000 q^{5} -2.74591 q^{7} +2.13973 q^{8} +0.584976 q^{10} +0.429723 q^{11} -6.91659 q^{13} +1.60629 q^{14} +2.06392 q^{16} -5.29153 q^{17} -1.63823 q^{19} +1.65780 q^{20} -0.251378 q^{22} +2.26995 q^{23} +1.00000 q^{25} +4.04604 q^{26} +4.55219 q^{28} -3.93691 q^{29} -2.28028 q^{31} -5.48679 q^{32} +3.09541 q^{34} +2.74591 q^{35} -7.74257 q^{37} +0.958324 q^{38} -2.13973 q^{40} +5.18947 q^{41} -6.95847 q^{43} -0.712397 q^{44} -1.32787 q^{46} -8.34712 q^{47} +0.540044 q^{49} -0.584976 q^{50} +11.4664 q^{52} -5.51251 q^{53} -0.429723 q^{55} -5.87550 q^{56} +2.30300 q^{58} +6.35400 q^{59} +9.00292 q^{61} +1.33391 q^{62} -0.918199 q^{64} +6.91659 q^{65} -8.39214 q^{67} +8.77231 q^{68} -1.60629 q^{70} -12.7822 q^{71} +5.08311 q^{73} +4.52921 q^{74} +2.71586 q^{76} -1.17998 q^{77} -1.33370 q^{79} -2.06392 q^{80} -3.03571 q^{82} +6.64852 q^{83} +5.29153 q^{85} +4.07053 q^{86} +0.919490 q^{88} -1.00000 q^{89} +18.9924 q^{91} -3.76314 q^{92} +4.88286 q^{94} +1.63823 q^{95} +0.828069 q^{97} -0.315913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.584976 −0.413640 −0.206820 0.978379i \(-0.566311\pi\)
−0.206820 + 0.978379i \(0.566311\pi\)
\(3\) 0 0
\(4\) −1.65780 −0.828902
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.74591 −1.03786 −0.518929 0.854817i \(-0.673669\pi\)
−0.518929 + 0.854817i \(0.673669\pi\)
\(8\) 2.13973 0.756507
\(9\) 0 0
\(10\) 0.584976 0.184986
\(11\) 0.429723 0.129566 0.0647832 0.997899i \(-0.479364\pi\)
0.0647832 + 0.997899i \(0.479364\pi\)
\(12\) 0 0
\(13\) −6.91659 −1.91832 −0.959159 0.282868i \(-0.908714\pi\)
−0.959159 + 0.282868i \(0.908714\pi\)
\(14\) 1.60629 0.429300
\(15\) 0 0
\(16\) 2.06392 0.515980
\(17\) −5.29153 −1.28338 −0.641692 0.766963i \(-0.721767\pi\)
−0.641692 + 0.766963i \(0.721767\pi\)
\(18\) 0 0
\(19\) −1.63823 −0.375835 −0.187918 0.982185i \(-0.560174\pi\)
−0.187918 + 0.982185i \(0.560174\pi\)
\(20\) 1.65780 0.370696
\(21\) 0 0
\(22\) −0.251378 −0.0535939
\(23\) 2.26995 0.473318 0.236659 0.971593i \(-0.423948\pi\)
0.236659 + 0.971593i \(0.423948\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.04604 0.793493
\(27\) 0 0
\(28\) 4.55219 0.860282
\(29\) −3.93691 −0.731066 −0.365533 0.930798i \(-0.619113\pi\)
−0.365533 + 0.930798i \(0.619113\pi\)
\(30\) 0 0
\(31\) −2.28028 −0.409550 −0.204775 0.978809i \(-0.565646\pi\)
−0.204775 + 0.978809i \(0.565646\pi\)
\(32\) −5.48679 −0.969937
\(33\) 0 0
\(34\) 3.09541 0.530859
\(35\) 2.74591 0.464144
\(36\) 0 0
\(37\) −7.74257 −1.27287 −0.636435 0.771330i \(-0.719592\pi\)
−0.636435 + 0.771330i \(0.719592\pi\)
\(38\) 0.958324 0.155461
\(39\) 0 0
\(40\) −2.13973 −0.338320
\(41\) 5.18947 0.810459 0.405230 0.914215i \(-0.367192\pi\)
0.405230 + 0.914215i \(0.367192\pi\)
\(42\) 0 0
\(43\) −6.95847 −1.06116 −0.530578 0.847636i \(-0.678025\pi\)
−0.530578 + 0.847636i \(0.678025\pi\)
\(44\) −0.712397 −0.107398
\(45\) 0 0
\(46\) −1.32787 −0.195783
\(47\) −8.34712 −1.21755 −0.608776 0.793342i \(-0.708339\pi\)
−0.608776 + 0.793342i \(0.708339\pi\)
\(48\) 0 0
\(49\) 0.540044 0.0771492
\(50\) −0.584976 −0.0827280
\(51\) 0 0
\(52\) 11.4664 1.59010
\(53\) −5.51251 −0.757201 −0.378600 0.925560i \(-0.623595\pi\)
−0.378600 + 0.925560i \(0.623595\pi\)
\(54\) 0 0
\(55\) −0.429723 −0.0579439
\(56\) −5.87550 −0.785147
\(57\) 0 0
\(58\) 2.30300 0.302398
\(59\) 6.35400 0.827220 0.413610 0.910454i \(-0.364268\pi\)
0.413610 + 0.910454i \(0.364268\pi\)
\(60\) 0 0
\(61\) 9.00292 1.15271 0.576353 0.817201i \(-0.304475\pi\)
0.576353 + 0.817201i \(0.304475\pi\)
\(62\) 1.33391 0.169406
\(63\) 0 0
\(64\) −0.918199 −0.114775
\(65\) 6.91659 0.857898
\(66\) 0 0
\(67\) −8.39214 −1.02526 −0.512632 0.858609i \(-0.671329\pi\)
−0.512632 + 0.858609i \(0.671329\pi\)
\(68\) 8.77231 1.06380
\(69\) 0 0
\(70\) −1.60629 −0.191989
\(71\) −12.7822 −1.51696 −0.758482 0.651695i \(-0.774058\pi\)
−0.758482 + 0.651695i \(0.774058\pi\)
\(72\) 0 0
\(73\) 5.08311 0.594933 0.297466 0.954732i \(-0.403858\pi\)
0.297466 + 0.954732i \(0.403858\pi\)
\(74\) 4.52921 0.526510
\(75\) 0 0
\(76\) 2.71586 0.311531
\(77\) −1.17998 −0.134472
\(78\) 0 0
\(79\) −1.33370 −0.150053 −0.0750266 0.997182i \(-0.523904\pi\)
−0.0750266 + 0.997182i \(0.523904\pi\)
\(80\) −2.06392 −0.230753
\(81\) 0 0
\(82\) −3.03571 −0.335238
\(83\) 6.64852 0.729770 0.364885 0.931053i \(-0.381108\pi\)
0.364885 + 0.931053i \(0.381108\pi\)
\(84\) 0 0
\(85\) 5.29153 0.573947
\(86\) 4.07053 0.438937
\(87\) 0 0
\(88\) 0.919490 0.0980180
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) 18.9924 1.99094
\(92\) −3.76314 −0.392334
\(93\) 0 0
\(94\) 4.88286 0.503629
\(95\) 1.63823 0.168079
\(96\) 0 0
\(97\) 0.828069 0.0840776 0.0420388 0.999116i \(-0.486615\pi\)
0.0420388 + 0.999116i \(0.486615\pi\)
\(98\) −0.315913 −0.0319120
\(99\) 0 0
\(100\) −1.65780 −0.165780
\(101\) 1.43539 0.142826 0.0714132 0.997447i \(-0.477249\pi\)
0.0714132 + 0.997447i \(0.477249\pi\)
\(102\) 0 0
\(103\) −6.02978 −0.594132 −0.297066 0.954857i \(-0.596008\pi\)
−0.297066 + 0.954857i \(0.596008\pi\)
\(104\) −14.7996 −1.45122
\(105\) 0 0
\(106\) 3.22468 0.313209
\(107\) 11.0707 1.07024 0.535121 0.844775i \(-0.320266\pi\)
0.535121 + 0.844775i \(0.320266\pi\)
\(108\) 0 0
\(109\) 8.99213 0.861290 0.430645 0.902521i \(-0.358286\pi\)
0.430645 + 0.902521i \(0.358286\pi\)
\(110\) 0.251378 0.0239679
\(111\) 0 0
\(112\) −5.66735 −0.535514
\(113\) 5.54942 0.522045 0.261023 0.965333i \(-0.415940\pi\)
0.261023 + 0.965333i \(0.415940\pi\)
\(114\) 0 0
\(115\) −2.26995 −0.211674
\(116\) 6.52663 0.605982
\(117\) 0 0
\(118\) −3.71694 −0.342172
\(119\) 14.5301 1.33197
\(120\) 0 0
\(121\) −10.8153 −0.983213
\(122\) −5.26649 −0.476806
\(123\) 0 0
\(124\) 3.78025 0.339477
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.6399 −1.47655 −0.738277 0.674497i \(-0.764360\pi\)
−0.738277 + 0.674497i \(0.764360\pi\)
\(128\) 11.5107 1.01741
\(129\) 0 0
\(130\) −4.04604 −0.354861
\(131\) −0.840072 −0.0733974 −0.0366987 0.999326i \(-0.511684\pi\)
−0.0366987 + 0.999326i \(0.511684\pi\)
\(132\) 0 0
\(133\) 4.49843 0.390064
\(134\) 4.90920 0.424090
\(135\) 0 0
\(136\) −11.3224 −0.970889
\(137\) 15.0870 1.28897 0.644485 0.764617i \(-0.277072\pi\)
0.644485 + 0.764617i \(0.277072\pi\)
\(138\) 0 0
\(139\) −4.93675 −0.418730 −0.209365 0.977838i \(-0.567140\pi\)
−0.209365 + 0.977838i \(0.567140\pi\)
\(140\) −4.55219 −0.384730
\(141\) 0 0
\(142\) 7.47725 0.627477
\(143\) −2.97222 −0.248550
\(144\) 0 0
\(145\) 3.93691 0.326943
\(146\) −2.97349 −0.246088
\(147\) 0 0
\(148\) 12.8357 1.05508
\(149\) −22.4744 −1.84117 −0.920587 0.390537i \(-0.872289\pi\)
−0.920587 + 0.390537i \(0.872289\pi\)
\(150\) 0 0
\(151\) 8.83321 0.718837 0.359418 0.933177i \(-0.382975\pi\)
0.359418 + 0.933177i \(0.382975\pi\)
\(152\) −3.50536 −0.284322
\(153\) 0 0
\(154\) 0.690261 0.0556229
\(155\) 2.28028 0.183156
\(156\) 0 0
\(157\) −4.37859 −0.349450 −0.174725 0.984617i \(-0.555904\pi\)
−0.174725 + 0.984617i \(0.555904\pi\)
\(158\) 0.780183 0.0620681
\(159\) 0 0
\(160\) 5.48679 0.433769
\(161\) −6.23310 −0.491237
\(162\) 0 0
\(163\) −17.3126 −1.35603 −0.678014 0.735049i \(-0.737159\pi\)
−0.678014 + 0.735049i \(0.737159\pi\)
\(164\) −8.60312 −0.671791
\(165\) 0 0
\(166\) −3.88922 −0.301862
\(167\) −4.89427 −0.378730 −0.189365 0.981907i \(-0.560643\pi\)
−0.189365 + 0.981907i \(0.560643\pi\)
\(168\) 0 0
\(169\) 34.8393 2.67994
\(170\) −3.09541 −0.237407
\(171\) 0 0
\(172\) 11.5358 0.879594
\(173\) 2.08921 0.158840 0.0794198 0.996841i \(-0.474693\pi\)
0.0794198 + 0.996841i \(0.474693\pi\)
\(174\) 0 0
\(175\) −2.74591 −0.207572
\(176\) 0.886914 0.0668537
\(177\) 0 0
\(178\) 0.584976 0.0438458
\(179\) 17.4654 1.30542 0.652712 0.757606i \(-0.273631\pi\)
0.652712 + 0.757606i \(0.273631\pi\)
\(180\) 0 0
\(181\) 12.0475 0.895485 0.447743 0.894162i \(-0.352228\pi\)
0.447743 + 0.894162i \(0.352228\pi\)
\(182\) −11.1101 −0.823533
\(183\) 0 0
\(184\) 4.85708 0.358069
\(185\) 7.74257 0.569245
\(186\) 0 0
\(187\) −2.27389 −0.166283
\(188\) 13.8379 1.00923
\(189\) 0 0
\(190\) −0.958324 −0.0695241
\(191\) 7.90966 0.572323 0.286161 0.958181i \(-0.407621\pi\)
0.286161 + 0.958181i \(0.407621\pi\)
\(192\) 0 0
\(193\) 2.39653 0.172506 0.0862530 0.996273i \(-0.472511\pi\)
0.0862530 + 0.996273i \(0.472511\pi\)
\(194\) −0.484400 −0.0347779
\(195\) 0 0
\(196\) −0.895288 −0.0639491
\(197\) 16.9355 1.20660 0.603302 0.797513i \(-0.293851\pi\)
0.603302 + 0.797513i \(0.293851\pi\)
\(198\) 0 0
\(199\) −25.0231 −1.77384 −0.886919 0.461926i \(-0.847159\pi\)
−0.886919 + 0.461926i \(0.847159\pi\)
\(200\) 2.13973 0.151301
\(201\) 0 0
\(202\) −0.839666 −0.0590787
\(203\) 10.8104 0.758743
\(204\) 0 0
\(205\) −5.18947 −0.362448
\(206\) 3.52728 0.245757
\(207\) 0 0
\(208\) −14.2753 −0.989814
\(209\) −0.703985 −0.0486957
\(210\) 0 0
\(211\) 8.90186 0.612829 0.306414 0.951898i \(-0.400871\pi\)
0.306414 + 0.951898i \(0.400871\pi\)
\(212\) 9.13865 0.627645
\(213\) 0 0
\(214\) −6.47607 −0.442695
\(215\) 6.95847 0.474564
\(216\) 0 0
\(217\) 6.26145 0.425055
\(218\) −5.26018 −0.356264
\(219\) 0 0
\(220\) 0.712397 0.0480298
\(221\) 36.5993 2.46194
\(222\) 0 0
\(223\) 28.1197 1.88303 0.941516 0.336968i \(-0.109401\pi\)
0.941516 + 0.336968i \(0.109401\pi\)
\(224\) 15.0663 1.00666
\(225\) 0 0
\(226\) −3.24627 −0.215939
\(227\) 16.0209 1.06334 0.531671 0.846951i \(-0.321564\pi\)
0.531671 + 0.846951i \(0.321564\pi\)
\(228\) 0 0
\(229\) 6.71542 0.443767 0.221884 0.975073i \(-0.428779\pi\)
0.221884 + 0.975073i \(0.428779\pi\)
\(230\) 1.32787 0.0875570
\(231\) 0 0
\(232\) −8.42391 −0.553057
\(233\) 9.20337 0.602933 0.301466 0.953477i \(-0.402524\pi\)
0.301466 + 0.953477i \(0.402524\pi\)
\(234\) 0 0
\(235\) 8.34712 0.544506
\(236\) −10.5337 −0.685684
\(237\) 0 0
\(238\) −8.49974 −0.550956
\(239\) 1.01200 0.0654610 0.0327305 0.999464i \(-0.489580\pi\)
0.0327305 + 0.999464i \(0.489580\pi\)
\(240\) 0 0
\(241\) 6.14399 0.395769 0.197885 0.980225i \(-0.436593\pi\)
0.197885 + 0.980225i \(0.436593\pi\)
\(242\) 6.32671 0.406696
\(243\) 0 0
\(244\) −14.9251 −0.955480
\(245\) −0.540044 −0.0345022
\(246\) 0 0
\(247\) 11.3310 0.720972
\(248\) −4.87917 −0.309828
\(249\) 0 0
\(250\) 0.584976 0.0369971
\(251\) −15.4001 −0.972049 −0.486024 0.873945i \(-0.661553\pi\)
−0.486024 + 0.873945i \(0.661553\pi\)
\(252\) 0 0
\(253\) 0.975453 0.0613262
\(254\) 9.73395 0.610762
\(255\) 0 0
\(256\) −4.89709 −0.306068
\(257\) −1.91078 −0.119191 −0.0595957 0.998223i \(-0.518981\pi\)
−0.0595957 + 0.998223i \(0.518981\pi\)
\(258\) 0 0
\(259\) 21.2604 1.32106
\(260\) −11.4664 −0.711113
\(261\) 0 0
\(262\) 0.491421 0.0303601
\(263\) 21.3509 1.31655 0.658277 0.752776i \(-0.271286\pi\)
0.658277 + 0.752776i \(0.271286\pi\)
\(264\) 0 0
\(265\) 5.51251 0.338631
\(266\) −2.63147 −0.161346
\(267\) 0 0
\(268\) 13.9125 0.849843
\(269\) 3.71865 0.226730 0.113365 0.993553i \(-0.463837\pi\)
0.113365 + 0.993553i \(0.463837\pi\)
\(270\) 0 0
\(271\) −29.0720 −1.76600 −0.882999 0.469375i \(-0.844479\pi\)
−0.882999 + 0.469375i \(0.844479\pi\)
\(272\) −10.9213 −0.662200
\(273\) 0 0
\(274\) −8.82553 −0.533170
\(275\) 0.429723 0.0259133
\(276\) 0 0
\(277\) −22.1766 −1.33246 −0.666230 0.745746i \(-0.732093\pi\)
−0.666230 + 0.745746i \(0.732093\pi\)
\(278\) 2.88788 0.173203
\(279\) 0 0
\(280\) 5.87550 0.351128
\(281\) 27.5045 1.64078 0.820391 0.571803i \(-0.193756\pi\)
0.820391 + 0.571803i \(0.193756\pi\)
\(282\) 0 0
\(283\) −11.7574 −0.698902 −0.349451 0.936955i \(-0.613632\pi\)
−0.349451 + 0.936955i \(0.613632\pi\)
\(284\) 21.1903 1.25741
\(285\) 0 0
\(286\) 1.73868 0.102810
\(287\) −14.2498 −0.841141
\(288\) 0 0
\(289\) 11.0002 0.647073
\(290\) −2.30300 −0.135237
\(291\) 0 0
\(292\) −8.42680 −0.493141
\(293\) 12.7263 0.743476 0.371738 0.928338i \(-0.378762\pi\)
0.371738 + 0.928338i \(0.378762\pi\)
\(294\) 0 0
\(295\) −6.35400 −0.369944
\(296\) −16.5670 −0.962935
\(297\) 0 0
\(298\) 13.1470 0.761584
\(299\) −15.7004 −0.907975
\(300\) 0 0
\(301\) 19.1074 1.10133
\(302\) −5.16721 −0.297340
\(303\) 0 0
\(304\) −3.38117 −0.193924
\(305\) −9.00292 −0.515506
\(306\) 0 0
\(307\) −4.59039 −0.261987 −0.130994 0.991383i \(-0.541817\pi\)
−0.130994 + 0.991383i \(0.541817\pi\)
\(308\) 1.95618 0.111464
\(309\) 0 0
\(310\) −1.33391 −0.0757609
\(311\) 15.6135 0.885361 0.442680 0.896679i \(-0.354028\pi\)
0.442680 + 0.896679i \(0.354028\pi\)
\(312\) 0 0
\(313\) 22.3598 1.26385 0.631925 0.775029i \(-0.282265\pi\)
0.631925 + 0.775029i \(0.282265\pi\)
\(314\) 2.56137 0.144547
\(315\) 0 0
\(316\) 2.21102 0.124379
\(317\) −19.3949 −1.08932 −0.544662 0.838655i \(-0.683342\pi\)
−0.544662 + 0.838655i \(0.683342\pi\)
\(318\) 0 0
\(319\) −1.69178 −0.0947217
\(320\) 0.918199 0.0513289
\(321\) 0 0
\(322\) 3.64621 0.203195
\(323\) 8.66873 0.482341
\(324\) 0 0
\(325\) −6.91659 −0.383664
\(326\) 10.1274 0.560908
\(327\) 0 0
\(328\) 11.1040 0.613118
\(329\) 22.9205 1.26365
\(330\) 0 0
\(331\) 25.2472 1.38771 0.693855 0.720115i \(-0.255911\pi\)
0.693855 + 0.720115i \(0.255911\pi\)
\(332\) −11.0219 −0.604908
\(333\) 0 0
\(334\) 2.86303 0.156658
\(335\) 8.39214 0.458512
\(336\) 0 0
\(337\) 4.05916 0.221116 0.110558 0.993870i \(-0.464736\pi\)
0.110558 + 0.993870i \(0.464736\pi\)
\(338\) −20.3801 −1.10853
\(339\) 0 0
\(340\) −8.77231 −0.475745
\(341\) −0.979889 −0.0530640
\(342\) 0 0
\(343\) 17.7385 0.957788
\(344\) −14.8892 −0.802773
\(345\) 0 0
\(346\) −1.22214 −0.0657025
\(347\) −25.2031 −1.35297 −0.676487 0.736454i \(-0.736499\pi\)
−0.676487 + 0.736454i \(0.736499\pi\)
\(348\) 0 0
\(349\) −11.8501 −0.634321 −0.317161 0.948372i \(-0.602729\pi\)
−0.317161 + 0.948372i \(0.602729\pi\)
\(350\) 1.60629 0.0858600
\(351\) 0 0
\(352\) −2.35780 −0.125671
\(353\) −13.9770 −0.743921 −0.371961 0.928249i \(-0.621314\pi\)
−0.371961 + 0.928249i \(0.621314\pi\)
\(354\) 0 0
\(355\) 12.7822 0.678406
\(356\) 1.65780 0.0878634
\(357\) 0 0
\(358\) −10.2168 −0.539976
\(359\) −12.6401 −0.667120 −0.333560 0.942729i \(-0.608250\pi\)
−0.333560 + 0.942729i \(0.608250\pi\)
\(360\) 0 0
\(361\) −16.3162 −0.858748
\(362\) −7.04751 −0.370409
\(363\) 0 0
\(364\) −31.4856 −1.65029
\(365\) −5.08311 −0.266062
\(366\) 0 0
\(367\) −18.5135 −0.966398 −0.483199 0.875510i \(-0.660525\pi\)
−0.483199 + 0.875510i \(0.660525\pi\)
\(368\) 4.68500 0.244223
\(369\) 0 0
\(370\) −4.52921 −0.235463
\(371\) 15.1369 0.785867
\(372\) 0 0
\(373\) 29.1116 1.50734 0.753670 0.657253i \(-0.228282\pi\)
0.753670 + 0.657253i \(0.228282\pi\)
\(374\) 1.33017 0.0687815
\(375\) 0 0
\(376\) −17.8605 −0.921087
\(377\) 27.2300 1.40242
\(378\) 0 0
\(379\) 16.6575 0.855637 0.427818 0.903865i \(-0.359282\pi\)
0.427818 + 0.903865i \(0.359282\pi\)
\(380\) −2.71586 −0.139321
\(381\) 0 0
\(382\) −4.62696 −0.236736
\(383\) −6.49778 −0.332021 −0.166010 0.986124i \(-0.553089\pi\)
−0.166010 + 0.986124i \(0.553089\pi\)
\(384\) 0 0
\(385\) 1.17998 0.0601375
\(386\) −1.40191 −0.0713554
\(387\) 0 0
\(388\) −1.37277 −0.0696921
\(389\) −26.9389 −1.36585 −0.682927 0.730486i \(-0.739294\pi\)
−0.682927 + 0.730486i \(0.739294\pi\)
\(390\) 0 0
\(391\) −12.0115 −0.607449
\(392\) 1.15555 0.0583639
\(393\) 0 0
\(394\) −9.90686 −0.499100
\(395\) 1.33370 0.0671058
\(396\) 0 0
\(397\) 26.0878 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(398\) 14.6379 0.733730
\(399\) 0 0
\(400\) 2.06392 0.103196
\(401\) −37.7689 −1.88609 −0.943045 0.332666i \(-0.892052\pi\)
−0.943045 + 0.332666i \(0.892052\pi\)
\(402\) 0 0
\(403\) 15.7718 0.785647
\(404\) −2.37959 −0.118389
\(405\) 0 0
\(406\) −6.32383 −0.313847
\(407\) −3.32716 −0.164921
\(408\) 0 0
\(409\) −32.8776 −1.62569 −0.812846 0.582479i \(-0.802083\pi\)
−0.812846 + 0.582479i \(0.802083\pi\)
\(410\) 3.03571 0.149923
\(411\) 0 0
\(412\) 9.99620 0.492477
\(413\) −17.4475 −0.858537
\(414\) 0 0
\(415\) −6.64852 −0.326363
\(416\) 37.9499 1.86065
\(417\) 0 0
\(418\) 0.411814 0.0201425
\(419\) 17.5007 0.854964 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(420\) 0 0
\(421\) 2.78590 0.135777 0.0678883 0.997693i \(-0.478374\pi\)
0.0678883 + 0.997693i \(0.478374\pi\)
\(422\) −5.20737 −0.253491
\(423\) 0 0
\(424\) −11.7953 −0.572828
\(425\) −5.29153 −0.256677
\(426\) 0 0
\(427\) −24.7213 −1.19635
\(428\) −18.3530 −0.887126
\(429\) 0 0
\(430\) −4.07053 −0.196299
\(431\) −14.3931 −0.693290 −0.346645 0.937996i \(-0.612679\pi\)
−0.346645 + 0.937996i \(0.612679\pi\)
\(432\) 0 0
\(433\) 6.91601 0.332362 0.166181 0.986095i \(-0.446856\pi\)
0.166181 + 0.986095i \(0.446856\pi\)
\(434\) −3.66280 −0.175820
\(435\) 0 0
\(436\) −14.9072 −0.713925
\(437\) −3.71870 −0.177890
\(438\) 0 0
\(439\) 33.3144 1.59001 0.795005 0.606602i \(-0.207468\pi\)
0.795005 + 0.606602i \(0.207468\pi\)
\(440\) −0.919490 −0.0438350
\(441\) 0 0
\(442\) −21.4097 −1.01836
\(443\) 17.6967 0.840794 0.420397 0.907340i \(-0.361891\pi\)
0.420397 + 0.907340i \(0.361891\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −16.4493 −0.778898
\(447\) 0 0
\(448\) 2.52129 0.119120
\(449\) −42.0230 −1.98319 −0.991593 0.129394i \(-0.958697\pi\)
−0.991593 + 0.129394i \(0.958697\pi\)
\(450\) 0 0
\(451\) 2.23004 0.105008
\(452\) −9.19984 −0.432724
\(453\) 0 0
\(454\) −9.37182 −0.439841
\(455\) −18.9924 −0.890376
\(456\) 0 0
\(457\) −16.9157 −0.791283 −0.395641 0.918405i \(-0.629478\pi\)
−0.395641 + 0.918405i \(0.629478\pi\)
\(458\) −3.92836 −0.183560
\(459\) 0 0
\(460\) 3.76314 0.175457
\(461\) 9.77419 0.455229 0.227615 0.973751i \(-0.426907\pi\)
0.227615 + 0.973751i \(0.426907\pi\)
\(462\) 0 0
\(463\) 40.4822 1.88137 0.940684 0.339283i \(-0.110184\pi\)
0.940684 + 0.339283i \(0.110184\pi\)
\(464\) −8.12547 −0.377216
\(465\) 0 0
\(466\) −5.38375 −0.249397
\(467\) 21.7726 1.00752 0.503759 0.863844i \(-0.331950\pi\)
0.503759 + 0.863844i \(0.331950\pi\)
\(468\) 0 0
\(469\) 23.0441 1.06408
\(470\) −4.88286 −0.225230
\(471\) 0 0
\(472\) 13.5958 0.625798
\(473\) −2.99022 −0.137490
\(474\) 0 0
\(475\) −1.63823 −0.0751671
\(476\) −24.0880 −1.10407
\(477\) 0 0
\(478\) −0.591996 −0.0270773
\(479\) 14.3976 0.657842 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(480\) 0 0
\(481\) 53.5522 2.44177
\(482\) −3.59408 −0.163706
\(483\) 0 0
\(484\) 17.9297 0.814987
\(485\) −0.828069 −0.0376007
\(486\) 0 0
\(487\) −29.7669 −1.34886 −0.674432 0.738337i \(-0.735612\pi\)
−0.674432 + 0.738337i \(0.735612\pi\)
\(488\) 19.2638 0.872031
\(489\) 0 0
\(490\) 0.315913 0.0142715
\(491\) 32.2121 1.45371 0.726857 0.686789i \(-0.240980\pi\)
0.726857 + 0.686789i \(0.240980\pi\)
\(492\) 0 0
\(493\) 20.8323 0.938238
\(494\) −6.62834 −0.298223
\(495\) 0 0
\(496\) −4.70631 −0.211320
\(497\) 35.0987 1.57439
\(498\) 0 0
\(499\) 20.7011 0.926707 0.463353 0.886174i \(-0.346646\pi\)
0.463353 + 0.886174i \(0.346646\pi\)
\(500\) 1.65780 0.0741392
\(501\) 0 0
\(502\) 9.00871 0.402078
\(503\) −39.1454 −1.74541 −0.872703 0.488251i \(-0.837635\pi\)
−0.872703 + 0.488251i \(0.837635\pi\)
\(504\) 0 0
\(505\) −1.43539 −0.0638739
\(506\) −0.570616 −0.0253670
\(507\) 0 0
\(508\) 27.5857 1.22392
\(509\) 8.05887 0.357203 0.178602 0.983921i \(-0.442843\pi\)
0.178602 + 0.983921i \(0.442843\pi\)
\(510\) 0 0
\(511\) −13.9578 −0.617456
\(512\) −20.1567 −0.890811
\(513\) 0 0
\(514\) 1.11776 0.0493023
\(515\) 6.02978 0.265704
\(516\) 0 0
\(517\) −3.58695 −0.157754
\(518\) −12.4368 −0.546443
\(519\) 0 0
\(520\) 14.7996 0.649006
\(521\) −5.83455 −0.255616 −0.127808 0.991799i \(-0.540794\pi\)
−0.127808 + 0.991799i \(0.540794\pi\)
\(522\) 0 0
\(523\) −33.8865 −1.48175 −0.740877 0.671640i \(-0.765590\pi\)
−0.740877 + 0.671640i \(0.765590\pi\)
\(524\) 1.39267 0.0608392
\(525\) 0 0
\(526\) −12.4898 −0.544580
\(527\) 12.0662 0.525610
\(528\) 0 0
\(529\) −17.8473 −0.775970
\(530\) −3.22468 −0.140071
\(531\) 0 0
\(532\) −7.45752 −0.323325
\(533\) −35.8935 −1.55472
\(534\) 0 0
\(535\) −11.0707 −0.478627
\(536\) −17.9569 −0.775619
\(537\) 0 0
\(538\) −2.17532 −0.0937847
\(539\) 0.232070 0.00999595
\(540\) 0 0
\(541\) −22.7941 −0.979995 −0.489997 0.871724i \(-0.663002\pi\)
−0.489997 + 0.871724i \(0.663002\pi\)
\(542\) 17.0064 0.730488
\(543\) 0 0
\(544\) 29.0335 1.24480
\(545\) −8.99213 −0.385181
\(546\) 0 0
\(547\) 21.1319 0.903533 0.451767 0.892136i \(-0.350794\pi\)
0.451767 + 0.892136i \(0.350794\pi\)
\(548\) −25.0113 −1.06843
\(549\) 0 0
\(550\) −0.251378 −0.0107188
\(551\) 6.44956 0.274761
\(552\) 0 0
\(553\) 3.66223 0.155734
\(554\) 12.9727 0.551159
\(555\) 0 0
\(556\) 8.18417 0.347086
\(557\) −19.8251 −0.840017 −0.420008 0.907520i \(-0.637973\pi\)
−0.420008 + 0.907520i \(0.637973\pi\)
\(558\) 0 0
\(559\) 48.1289 2.03564
\(560\) 5.66735 0.239489
\(561\) 0 0
\(562\) −16.0895 −0.678693
\(563\) 10.7352 0.452433 0.226217 0.974077i \(-0.427364\pi\)
0.226217 + 0.974077i \(0.427364\pi\)
\(564\) 0 0
\(565\) −5.54942 −0.233466
\(566\) 6.87776 0.289094
\(567\) 0 0
\(568\) −27.3503 −1.14759
\(569\) −19.1468 −0.802677 −0.401338 0.915930i \(-0.631455\pi\)
−0.401338 + 0.915930i \(0.631455\pi\)
\(570\) 0 0
\(571\) −22.6449 −0.947661 −0.473831 0.880616i \(-0.657129\pi\)
−0.473831 + 0.880616i \(0.657129\pi\)
\(572\) 4.92736 0.206023
\(573\) 0 0
\(574\) 8.33581 0.347930
\(575\) 2.26995 0.0946637
\(576\) 0 0
\(577\) −24.6727 −1.02714 −0.513570 0.858048i \(-0.671677\pi\)
−0.513570 + 0.858048i \(0.671677\pi\)
\(578\) −6.43487 −0.267655
\(579\) 0 0
\(580\) −6.52663 −0.271003
\(581\) −18.2563 −0.757398
\(582\) 0 0
\(583\) −2.36885 −0.0981078
\(584\) 10.8765 0.450071
\(585\) 0 0
\(586\) −7.44456 −0.307532
\(587\) 14.3041 0.590395 0.295198 0.955436i \(-0.404615\pi\)
0.295198 + 0.955436i \(0.404615\pi\)
\(588\) 0 0
\(589\) 3.73562 0.153923
\(590\) 3.71694 0.153024
\(591\) 0 0
\(592\) −15.9800 −0.656775
\(593\) −31.7751 −1.30485 −0.652423 0.757855i \(-0.726248\pi\)
−0.652423 + 0.757855i \(0.726248\pi\)
\(594\) 0 0
\(595\) −14.5301 −0.595675
\(596\) 37.2581 1.52615
\(597\) 0 0
\(598\) 9.18432 0.375575
\(599\) 33.7567 1.37926 0.689631 0.724161i \(-0.257773\pi\)
0.689631 + 0.724161i \(0.257773\pi\)
\(600\) 0 0
\(601\) −34.6759 −1.41446 −0.707229 0.706985i \(-0.750055\pi\)
−0.707229 + 0.706985i \(0.750055\pi\)
\(602\) −11.1773 −0.455554
\(603\) 0 0
\(604\) −14.6437 −0.595845
\(605\) 10.8153 0.439706
\(606\) 0 0
\(607\) −12.8661 −0.522217 −0.261109 0.965309i \(-0.584088\pi\)
−0.261109 + 0.965309i \(0.584088\pi\)
\(608\) 8.98862 0.364537
\(609\) 0 0
\(610\) 5.26649 0.213234
\(611\) 57.7336 2.33565
\(612\) 0 0
\(613\) 16.6348 0.671872 0.335936 0.941885i \(-0.390947\pi\)
0.335936 + 0.941885i \(0.390947\pi\)
\(614\) 2.68526 0.108368
\(615\) 0 0
\(616\) −2.52484 −0.101729
\(617\) −33.5170 −1.34934 −0.674671 0.738118i \(-0.735715\pi\)
−0.674671 + 0.738118i \(0.735715\pi\)
\(618\) 0 0
\(619\) 34.1671 1.37329 0.686646 0.726992i \(-0.259082\pi\)
0.686646 + 0.726992i \(0.259082\pi\)
\(620\) −3.78025 −0.151819
\(621\) 0 0
\(622\) −9.13352 −0.366221
\(623\) 2.74591 0.110013
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.0799 −0.522779
\(627\) 0 0
\(628\) 7.25885 0.289660
\(629\) 40.9700 1.63358
\(630\) 0 0
\(631\) −2.76256 −0.109976 −0.0549878 0.998487i \(-0.517512\pi\)
−0.0549878 + 0.998487i \(0.517512\pi\)
\(632\) −2.85376 −0.113516
\(633\) 0 0
\(634\) 11.3455 0.450588
\(635\) 16.6399 0.660335
\(636\) 0 0
\(637\) −3.73527 −0.147997
\(638\) 0.989652 0.0391807
\(639\) 0 0
\(640\) −11.5107 −0.455001
\(641\) −46.4551 −1.83487 −0.917434 0.397889i \(-0.869743\pi\)
−0.917434 + 0.397889i \(0.869743\pi\)
\(642\) 0 0
\(643\) −45.9246 −1.81109 −0.905545 0.424249i \(-0.860538\pi\)
−0.905545 + 0.424249i \(0.860538\pi\)
\(644\) 10.3333 0.407187
\(645\) 0 0
\(646\) −5.07099 −0.199516
\(647\) −14.4744 −0.569048 −0.284524 0.958669i \(-0.591836\pi\)
−0.284524 + 0.958669i \(0.591836\pi\)
\(648\) 0 0
\(649\) 2.73046 0.107180
\(650\) 4.04604 0.158699
\(651\) 0 0
\(652\) 28.7009 1.12401
\(653\) −29.4446 −1.15226 −0.576129 0.817359i \(-0.695437\pi\)
−0.576129 + 0.817359i \(0.695437\pi\)
\(654\) 0 0
\(655\) 0.840072 0.0328243
\(656\) 10.7106 0.418181
\(657\) 0 0
\(658\) −13.4079 −0.522695
\(659\) 25.5581 0.995601 0.497800 0.867292i \(-0.334141\pi\)
0.497800 + 0.867292i \(0.334141\pi\)
\(660\) 0 0
\(661\) −22.7900 −0.886428 −0.443214 0.896416i \(-0.646162\pi\)
−0.443214 + 0.896416i \(0.646162\pi\)
\(662\) −14.7690 −0.574013
\(663\) 0 0
\(664\) 14.2260 0.552076
\(665\) −4.49843 −0.174442
\(666\) 0 0
\(667\) −8.93661 −0.346027
\(668\) 8.11373 0.313930
\(669\) 0 0
\(670\) −4.90920 −0.189659
\(671\) 3.86877 0.149352
\(672\) 0 0
\(673\) 26.0465 1.00402 0.502009 0.864862i \(-0.332594\pi\)
0.502009 + 0.864862i \(0.332594\pi\)
\(674\) −2.37451 −0.0914627
\(675\) 0 0
\(676\) −57.7567 −2.22141
\(677\) −8.03655 −0.308870 −0.154435 0.988003i \(-0.549356\pi\)
−0.154435 + 0.988003i \(0.549356\pi\)
\(678\) 0 0
\(679\) −2.27381 −0.0872606
\(680\) 11.3224 0.434195
\(681\) 0 0
\(682\) 0.573211 0.0219494
\(683\) 8.08860 0.309502 0.154751 0.987954i \(-0.450543\pi\)
0.154751 + 0.987954i \(0.450543\pi\)
\(684\) 0 0
\(685\) −15.0870 −0.576445
\(686\) −10.3766 −0.396180
\(687\) 0 0
\(688\) −14.3617 −0.547535
\(689\) 38.1278 1.45255
\(690\) 0 0
\(691\) 27.2840 1.03793 0.518966 0.854795i \(-0.326317\pi\)
0.518966 + 0.854795i \(0.326317\pi\)
\(692\) −3.46350 −0.131662
\(693\) 0 0
\(694\) 14.7432 0.559645
\(695\) 4.93675 0.187262
\(696\) 0 0
\(697\) −27.4602 −1.04013
\(698\) 6.93202 0.262381
\(699\) 0 0
\(700\) 4.55219 0.172056
\(701\) 9.03883 0.341392 0.170696 0.985324i \(-0.445398\pi\)
0.170696 + 0.985324i \(0.445398\pi\)
\(702\) 0 0
\(703\) 12.6841 0.478390
\(704\) −0.394571 −0.0148710
\(705\) 0 0
\(706\) 8.17621 0.307716
\(707\) −3.94145 −0.148233
\(708\) 0 0
\(709\) −46.5488 −1.74818 −0.874088 0.485767i \(-0.838540\pi\)
−0.874088 + 0.485767i \(0.838540\pi\)
\(710\) −7.47725 −0.280616
\(711\) 0 0
\(712\) −2.13973 −0.0801896
\(713\) −5.17613 −0.193848
\(714\) 0 0
\(715\) 2.97222 0.111155
\(716\) −28.9542 −1.08207
\(717\) 0 0
\(718\) 7.39416 0.275948
\(719\) 49.3092 1.83892 0.919462 0.393180i \(-0.128625\pi\)
0.919462 + 0.393180i \(0.128625\pi\)
\(720\) 0 0
\(721\) 16.5573 0.616625
\(722\) 9.54458 0.355213
\(723\) 0 0
\(724\) −19.9724 −0.742269
\(725\) −3.93691 −0.146213
\(726\) 0 0
\(727\) −39.4930 −1.46471 −0.732357 0.680921i \(-0.761580\pi\)
−0.732357 + 0.680921i \(0.761580\pi\)
\(728\) 40.6385 1.50616
\(729\) 0 0
\(730\) 2.97349 0.110054
\(731\) 36.8209 1.36187
\(732\) 0 0
\(733\) −4.32917 −0.159902 −0.0799508 0.996799i \(-0.525476\pi\)
−0.0799508 + 0.996799i \(0.525476\pi\)
\(734\) 10.8300 0.399741
\(735\) 0 0
\(736\) −12.4548 −0.459089
\(737\) −3.60630 −0.132840
\(738\) 0 0
\(739\) 2.52150 0.0927549 0.0463775 0.998924i \(-0.485232\pi\)
0.0463775 + 0.998924i \(0.485232\pi\)
\(740\) −12.8357 −0.471848
\(741\) 0 0
\(742\) −8.85470 −0.325066
\(743\) −43.3061 −1.58875 −0.794373 0.607431i \(-0.792200\pi\)
−0.794373 + 0.607431i \(0.792200\pi\)
\(744\) 0 0
\(745\) 22.4744 0.823398
\(746\) −17.0296 −0.623497
\(747\) 0 0
\(748\) 3.76967 0.137833
\(749\) −30.3991 −1.11076
\(750\) 0 0
\(751\) 43.4241 1.58457 0.792284 0.610153i \(-0.208892\pi\)
0.792284 + 0.610153i \(0.208892\pi\)
\(752\) −17.2278 −0.628232
\(753\) 0 0
\(754\) −15.9289 −0.580096
\(755\) −8.83321 −0.321474
\(756\) 0 0
\(757\) 46.8153 1.70153 0.850765 0.525547i \(-0.176139\pi\)
0.850765 + 0.525547i \(0.176139\pi\)
\(758\) −9.74421 −0.353926
\(759\) 0 0
\(760\) 3.50536 0.127153
\(761\) 48.9144 1.77315 0.886573 0.462588i \(-0.153079\pi\)
0.886573 + 0.462588i \(0.153079\pi\)
\(762\) 0 0
\(763\) −24.6916 −0.893896
\(764\) −13.1127 −0.474400
\(765\) 0 0
\(766\) 3.80104 0.137337
\(767\) −43.9480 −1.58687
\(768\) 0 0
\(769\) 12.2118 0.440370 0.220185 0.975458i \(-0.429334\pi\)
0.220185 + 0.975458i \(0.429334\pi\)
\(770\) −0.690261 −0.0248753
\(771\) 0 0
\(772\) −3.97298 −0.142991
\(773\) 23.1151 0.831391 0.415695 0.909504i \(-0.363538\pi\)
0.415695 + 0.909504i \(0.363538\pi\)
\(774\) 0 0
\(775\) −2.28028 −0.0819100
\(776\) 1.77184 0.0636053
\(777\) 0 0
\(778\) 15.7586 0.564973
\(779\) −8.50154 −0.304599
\(780\) 0 0
\(781\) −5.49279 −0.196548
\(782\) 7.02645 0.251265
\(783\) 0 0
\(784\) 1.11461 0.0398074
\(785\) 4.37859 0.156279
\(786\) 0 0
\(787\) −24.5229 −0.874147 −0.437073 0.899426i \(-0.643985\pi\)
−0.437073 + 0.899426i \(0.643985\pi\)
\(788\) −28.0757 −1.00016
\(789\) 0 0
\(790\) −0.780183 −0.0277577
\(791\) −15.2382 −0.541809
\(792\) 0 0
\(793\) −62.2696 −2.21126
\(794\) −15.2607 −0.541582
\(795\) 0 0
\(796\) 41.4833 1.47034
\(797\) 46.4290 1.64460 0.822299 0.569055i \(-0.192691\pi\)
0.822299 + 0.569055i \(0.192691\pi\)
\(798\) 0 0
\(799\) 44.1690 1.56259
\(800\) −5.48679 −0.193987
\(801\) 0 0
\(802\) 22.0939 0.780163
\(803\) 2.18433 0.0770833
\(804\) 0 0
\(805\) 6.23310 0.219688
\(806\) −9.22610 −0.324975
\(807\) 0 0
\(808\) 3.07133 0.108049
\(809\) 9.67473 0.340145 0.170073 0.985432i \(-0.445600\pi\)
0.170073 + 0.985432i \(0.445600\pi\)
\(810\) 0 0
\(811\) 27.8332 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(812\) −17.9216 −0.628923
\(813\) 0 0
\(814\) 1.94631 0.0682181
\(815\) 17.3126 0.606434
\(816\) 0 0
\(817\) 11.3996 0.398820
\(818\) 19.2326 0.672451
\(819\) 0 0
\(820\) 8.60312 0.300434
\(821\) 37.2035 1.29841 0.649206 0.760612i \(-0.275101\pi\)
0.649206 + 0.760612i \(0.275101\pi\)
\(822\) 0 0
\(823\) −25.9987 −0.906257 −0.453129 0.891445i \(-0.649692\pi\)
−0.453129 + 0.891445i \(0.649692\pi\)
\(824\) −12.9021 −0.449465
\(825\) 0 0
\(826\) 10.2064 0.355126
\(827\) −10.2724 −0.357207 −0.178603 0.983921i \(-0.557158\pi\)
−0.178603 + 0.983921i \(0.557158\pi\)
\(828\) 0 0
\(829\) 13.8050 0.479466 0.239733 0.970839i \(-0.422940\pi\)
0.239733 + 0.970839i \(0.422940\pi\)
\(830\) 3.88922 0.134997
\(831\) 0 0
\(832\) 6.35081 0.220175
\(833\) −2.85766 −0.0990120
\(834\) 0 0
\(835\) 4.89427 0.169373
\(836\) 1.16707 0.0403639
\(837\) 0 0
\(838\) −10.2375 −0.353647
\(839\) −36.6994 −1.26700 −0.633502 0.773741i \(-0.718383\pi\)
−0.633502 + 0.773741i \(0.718383\pi\)
\(840\) 0 0
\(841\) −13.5007 −0.465542
\(842\) −1.62969 −0.0561627
\(843\) 0 0
\(844\) −14.7575 −0.507975
\(845\) −34.8393 −1.19851
\(846\) 0 0
\(847\) 29.6980 1.02043
\(848\) −11.3774 −0.390700
\(849\) 0 0
\(850\) 3.09541 0.106172
\(851\) −17.5753 −0.602473
\(852\) 0 0
\(853\) −42.5986 −1.45855 −0.729275 0.684221i \(-0.760142\pi\)
−0.729275 + 0.684221i \(0.760142\pi\)
\(854\) 14.4613 0.494857
\(855\) 0 0
\(856\) 23.6882 0.809646
\(857\) −53.8456 −1.83933 −0.919665 0.392704i \(-0.871540\pi\)
−0.919665 + 0.392704i \(0.871540\pi\)
\(858\) 0 0
\(859\) −3.35768 −0.114563 −0.0572813 0.998358i \(-0.518243\pi\)
−0.0572813 + 0.998358i \(0.518243\pi\)
\(860\) −11.5358 −0.393367
\(861\) 0 0
\(862\) 8.41959 0.286772
\(863\) −58.0162 −1.97489 −0.987447 0.157949i \(-0.949512\pi\)
−0.987447 + 0.157949i \(0.949512\pi\)
\(864\) 0 0
\(865\) −2.08921 −0.0710353
\(866\) −4.04570 −0.137478
\(867\) 0 0
\(868\) −10.3803 −0.352329
\(869\) −0.573123 −0.0194419
\(870\) 0 0
\(871\) 58.0450 1.96678
\(872\) 19.2407 0.651572
\(873\) 0 0
\(874\) 2.17535 0.0735824
\(875\) 2.74591 0.0928288
\(876\) 0 0
\(877\) −33.8297 −1.14235 −0.571173 0.820830i \(-0.693512\pi\)
−0.571173 + 0.820830i \(0.693512\pi\)
\(878\) −19.4881 −0.657692
\(879\) 0 0
\(880\) −0.886914 −0.0298979
\(881\) −17.7123 −0.596744 −0.298372 0.954450i \(-0.596444\pi\)
−0.298372 + 0.954450i \(0.596444\pi\)
\(882\) 0 0
\(883\) −46.8318 −1.57602 −0.788008 0.615665i \(-0.788888\pi\)
−0.788008 + 0.615665i \(0.788888\pi\)
\(884\) −60.6745 −2.04070
\(885\) 0 0
\(886\) −10.3521 −0.347786
\(887\) 19.8302 0.665832 0.332916 0.942957i \(-0.391967\pi\)
0.332916 + 0.942957i \(0.391967\pi\)
\(888\) 0 0
\(889\) 45.6918 1.53245
\(890\) −0.584976 −0.0196084
\(891\) 0 0
\(892\) −46.6169 −1.56085
\(893\) 13.6745 0.457599
\(894\) 0 0
\(895\) −17.4654 −0.583803
\(896\) −31.6074 −1.05593
\(897\) 0 0
\(898\) 24.5824 0.820326
\(899\) 8.97726 0.299408
\(900\) 0 0
\(901\) 29.1696 0.971779
\(902\) −1.30452 −0.0434357
\(903\) 0 0
\(904\) 11.8742 0.394931
\(905\) −12.0475 −0.400473
\(906\) 0 0
\(907\) −4.41800 −0.146697 −0.0733487 0.997306i \(-0.523369\pi\)
−0.0733487 + 0.997306i \(0.523369\pi\)
\(908\) −26.5595 −0.881407
\(909\) 0 0
\(910\) 11.1101 0.368295
\(911\) −43.9432 −1.45590 −0.727951 0.685629i \(-0.759527\pi\)
−0.727951 + 0.685629i \(0.759527\pi\)
\(912\) 0 0
\(913\) 2.85702 0.0945537
\(914\) 9.89526 0.327306
\(915\) 0 0
\(916\) −11.1328 −0.367840
\(917\) 2.30676 0.0761761
\(918\) 0 0
\(919\) 54.2143 1.78836 0.894182 0.447704i \(-0.147758\pi\)
0.894182 + 0.447704i \(0.147758\pi\)
\(920\) −4.85708 −0.160133
\(921\) 0 0
\(922\) −5.71766 −0.188301
\(923\) 88.4090 2.91002
\(924\) 0 0
\(925\) −7.74257 −0.254574
\(926\) −23.6811 −0.778210
\(927\) 0 0
\(928\) 21.6010 0.709089
\(929\) 0.0310742 0.00101951 0.000509756 1.00000i \(-0.499838\pi\)
0.000509756 1.00000i \(0.499838\pi\)
\(930\) 0 0
\(931\) −0.884716 −0.0289954
\(932\) −15.2574 −0.499772
\(933\) 0 0
\(934\) −12.7365 −0.416750
\(935\) 2.27389 0.0743642
\(936\) 0 0
\(937\) −18.6118 −0.608020 −0.304010 0.952669i \(-0.598326\pi\)
−0.304010 + 0.952669i \(0.598326\pi\)
\(938\) −13.4802 −0.440145
\(939\) 0 0
\(940\) −13.8379 −0.451342
\(941\) −4.95121 −0.161405 −0.0807025 0.996738i \(-0.525716\pi\)
−0.0807025 + 0.996738i \(0.525716\pi\)
\(942\) 0 0
\(943\) 11.7799 0.383605
\(944\) 13.1141 0.426829
\(945\) 0 0
\(946\) 1.74920 0.0568715
\(947\) −33.9363 −1.10278 −0.551391 0.834247i \(-0.685903\pi\)
−0.551391 + 0.834247i \(0.685903\pi\)
\(948\) 0 0
\(949\) −35.1578 −1.14127
\(950\) 0.958324 0.0310921
\(951\) 0 0
\(952\) 31.0904 1.00764
\(953\) 5.44095 0.176250 0.0881249 0.996109i \(-0.471913\pi\)
0.0881249 + 0.996109i \(0.471913\pi\)
\(954\) 0 0
\(955\) −7.90966 −0.255951
\(956\) −1.67770 −0.0542607
\(957\) 0 0
\(958\) −8.42223 −0.272110
\(959\) −41.4276 −1.33777
\(960\) 0 0
\(961\) −25.8003 −0.832269
\(962\) −31.3267 −1.01001
\(963\) 0 0
\(964\) −10.1855 −0.328054
\(965\) −2.39653 −0.0771470
\(966\) 0 0
\(967\) 0.538092 0.0173039 0.00865193 0.999963i \(-0.497246\pi\)
0.00865193 + 0.999963i \(0.497246\pi\)
\(968\) −23.1419 −0.743807
\(969\) 0 0
\(970\) 0.484400 0.0155531
\(971\) 6.65013 0.213413 0.106706 0.994291i \(-0.465969\pi\)
0.106706 + 0.994291i \(0.465969\pi\)
\(972\) 0 0
\(973\) 13.5559 0.434582
\(974\) 17.4129 0.557945
\(975\) 0 0
\(976\) 18.5813 0.594773
\(977\) 19.8824 0.636094 0.318047 0.948075i \(-0.396973\pi\)
0.318047 + 0.948075i \(0.396973\pi\)
\(978\) 0 0
\(979\) −0.429723 −0.0137340
\(980\) 0.895288 0.0285989
\(981\) 0 0
\(982\) −18.8433 −0.601314
\(983\) 13.2974 0.424121 0.212060 0.977257i \(-0.431983\pi\)
0.212060 + 0.977257i \(0.431983\pi\)
\(984\) 0 0
\(985\) −16.9355 −0.539610
\(986\) −12.1864 −0.388093
\(987\) 0 0
\(988\) −18.7845 −0.597615
\(989\) −15.7954 −0.502265
\(990\) 0 0
\(991\) −24.2737 −0.771081 −0.385540 0.922691i \(-0.625985\pi\)
−0.385540 + 0.922691i \(0.625985\pi\)
\(992\) 12.5114 0.397238
\(993\) 0 0
\(994\) −20.5319 −0.651232
\(995\) 25.0231 0.793284
\(996\) 0 0
\(997\) −4.40285 −0.139440 −0.0697199 0.997567i \(-0.522211\pi\)
−0.0697199 + 0.997567i \(0.522211\pi\)
\(998\) −12.1096 −0.383323
\(999\) 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 4005.2.a.o.1.3 7
3.2 odd 2 445.2.a.f.1.5 7
12.11 even 2 7120.2.a.bj.1.2 7
15.14 odd 2 2225.2.a.k.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.5 7 3.2 odd 2
2225.2.a.k.1.3 7 15.14 odd 2
4005.2.a.o.1.3 7 1.1 even 1 trivial
7120.2.a.bj.1.2 7 12.11 even 2