Properties

Label 5571.2.a.g.1.23
Level 5571
Weight 2
Character 5571.1
Self dual yes
Analytic conductor 44.485
Analytic rank 1
Dimension 30
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5571 = 3^{2} \cdot 619 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5571.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.4846589661\)
Analytic rank: \(1\)
Dimension: \(30\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 619)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.23
Character \(\chi\) = 5571.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +O(q^{10})\) \(q+1.37056 q^{2} -0.121559 q^{4} +0.225133 q^{5} +2.47282 q^{7} -2.90773 q^{8} +0.308559 q^{10} +4.06256 q^{11} +2.26536 q^{13} +3.38915 q^{14} -3.74211 q^{16} -7.19086 q^{17} -7.47015 q^{19} -0.0273670 q^{20} +5.56800 q^{22} +0.178059 q^{23} -4.94931 q^{25} +3.10482 q^{26} -0.300594 q^{28} -10.5609 q^{29} +1.27650 q^{31} +0.686670 q^{32} -9.85552 q^{34} +0.556714 q^{35} +2.08508 q^{37} -10.2383 q^{38} -0.654627 q^{40} -4.23367 q^{41} +8.52011 q^{43} -0.493842 q^{44} +0.244041 q^{46} -9.96677 q^{47} -0.885174 q^{49} -6.78334 q^{50} -0.275375 q^{52} -0.619984 q^{53} +0.914619 q^{55} -7.19028 q^{56} -14.4744 q^{58} -11.8540 q^{59} +6.30300 q^{61} +1.74952 q^{62} +8.42533 q^{64} +0.510008 q^{65} +12.5388 q^{67} +0.874114 q^{68} +0.763011 q^{70} +9.50401 q^{71} +5.14906 q^{73} +2.85773 q^{74} +0.908065 q^{76} +10.0460 q^{77} -16.3932 q^{79} -0.842473 q^{80} -5.80251 q^{82} -4.79273 q^{83} -1.61890 q^{85} +11.6773 q^{86} -11.8128 q^{88} +1.64933 q^{89} +5.60182 q^{91} -0.0216447 q^{92} -13.6601 q^{94} -1.68178 q^{95} +3.34400 q^{97} -1.21319 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30q - 9q^{2} + 33q^{4} - 21q^{5} + 2q^{7} - 27q^{8} + O(q^{10}) \) \( 30q - 9q^{2} + 33q^{4} - 21q^{5} + 2q^{7} - 27q^{8} + 5q^{10} - 23q^{11} + 9q^{13} - 7q^{14} + 35q^{16} - 4q^{17} - q^{19} - 29q^{20} - 4q^{23} + 35q^{25} - q^{26} - 13q^{28} - 90q^{29} + 2q^{31} - 43q^{32} - 9q^{34} - 9q^{35} + 19q^{37} - 5q^{38} - 12q^{40} - 59q^{41} - 4q^{43} - 52q^{44} - q^{46} - 4q^{47} + 30q^{49} - 31q^{50} - 12q^{52} - 34q^{53} - 17q^{55} - 2q^{56} + 6q^{58} - 13q^{59} + 16q^{61} - 28q^{62} + 37q^{64} - 31q^{65} - 11q^{67} + 52q^{68} - 40q^{70} - 42q^{71} - 4q^{73} - 16q^{74} - 42q^{76} - 29q^{77} + 3q^{79} - 21q^{80} - 43q^{82} + 11q^{83} + 19q^{85} + 11q^{86} - 47q^{88} - 58q^{89} - 39q^{91} + 7q^{92} - 46q^{94} - 23q^{95} - 9q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37056 0.969134 0.484567 0.874754i \(-0.338977\pi\)
0.484567 + 0.874754i \(0.338977\pi\)
\(3\) 0 0
\(4\) −0.121559 −0.0607796
\(5\) 0.225133 0.100683 0.0503414 0.998732i \(-0.483969\pi\)
0.0503414 + 0.998732i \(0.483969\pi\)
\(6\) 0 0
\(7\) 2.47282 0.934637 0.467319 0.884089i \(-0.345220\pi\)
0.467319 + 0.884089i \(0.345220\pi\)
\(8\) −2.90773 −1.02804
\(9\) 0 0
\(10\) 0.308559 0.0975750
\(11\) 4.06256 1.22491 0.612455 0.790506i \(-0.290182\pi\)
0.612455 + 0.790506i \(0.290182\pi\)
\(12\) 0 0
\(13\) 2.26536 0.628298 0.314149 0.949374i \(-0.398281\pi\)
0.314149 + 0.949374i \(0.398281\pi\)
\(14\) 3.38915 0.905788
\(15\) 0 0
\(16\) −3.74211 −0.935526
\(17\) −7.19086 −1.74404 −0.872019 0.489471i \(-0.837190\pi\)
−0.872019 + 0.489471i \(0.837190\pi\)
\(18\) 0 0
\(19\) −7.47015 −1.71377 −0.856885 0.515507i \(-0.827603\pi\)
−0.856885 + 0.515507i \(0.827603\pi\)
\(20\) −0.0273670 −0.00611945
\(21\) 0 0
\(22\) 5.56800 1.18710
\(23\) 0.178059 0.0371279 0.0185639 0.999828i \(-0.494091\pi\)
0.0185639 + 0.999828i \(0.494091\pi\)
\(24\) 0 0
\(25\) −4.94931 −0.989863
\(26\) 3.10482 0.608905
\(27\) 0 0
\(28\) −0.300594 −0.0568068
\(29\) −10.5609 −1.96111 −0.980557 0.196232i \(-0.937129\pi\)
−0.980557 + 0.196232i \(0.937129\pi\)
\(30\) 0 0
\(31\) 1.27650 0.229266 0.114633 0.993408i \(-0.463431\pi\)
0.114633 + 0.993408i \(0.463431\pi\)
\(32\) 0.686670 0.121387
\(33\) 0 0
\(34\) −9.85552 −1.69021
\(35\) 0.556714 0.0941018
\(36\) 0 0
\(37\) 2.08508 0.342784 0.171392 0.985203i \(-0.445173\pi\)
0.171392 + 0.985203i \(0.445173\pi\)
\(38\) −10.2383 −1.66087
\(39\) 0 0
\(40\) −0.654627 −0.103506
\(41\) −4.23367 −0.661188 −0.330594 0.943773i \(-0.607249\pi\)
−0.330594 + 0.943773i \(0.607249\pi\)
\(42\) 0 0
\(43\) 8.52011 1.29930 0.649652 0.760232i \(-0.274915\pi\)
0.649652 + 0.760232i \(0.274915\pi\)
\(44\) −0.493842 −0.0744495
\(45\) 0 0
\(46\) 0.244041 0.0359819
\(47\) −9.96677 −1.45380 −0.726901 0.686742i \(-0.759040\pi\)
−0.726901 + 0.686742i \(0.759040\pi\)
\(48\) 0 0
\(49\) −0.885174 −0.126453
\(50\) −6.78334 −0.959310
\(51\) 0 0
\(52\) −0.275375 −0.0381877
\(53\) −0.619984 −0.0851614 −0.0425807 0.999093i \(-0.513558\pi\)
−0.0425807 + 0.999093i \(0.513558\pi\)
\(54\) 0 0
\(55\) 0.914619 0.123327
\(56\) −7.19028 −0.960842
\(57\) 0 0
\(58\) −14.4744 −1.90058
\(59\) −11.8540 −1.54326 −0.771628 0.636074i \(-0.780557\pi\)
−0.771628 + 0.636074i \(0.780557\pi\)
\(60\) 0 0
\(61\) 6.30300 0.807017 0.403508 0.914976i \(-0.367791\pi\)
0.403508 + 0.914976i \(0.367791\pi\)
\(62\) 1.74952 0.222190
\(63\) 0 0
\(64\) 8.42533 1.05317
\(65\) 0.510008 0.0632587
\(66\) 0 0
\(67\) 12.5388 1.53186 0.765928 0.642927i \(-0.222280\pi\)
0.765928 + 0.642927i \(0.222280\pi\)
\(68\) 0.874114 0.106002
\(69\) 0 0
\(70\) 0.763011 0.0911972
\(71\) 9.50401 1.12792 0.563959 0.825803i \(-0.309277\pi\)
0.563959 + 0.825803i \(0.309277\pi\)
\(72\) 0 0
\(73\) 5.14906 0.602652 0.301326 0.953521i \(-0.402571\pi\)
0.301326 + 0.953521i \(0.402571\pi\)
\(74\) 2.85773 0.332204
\(75\) 0 0
\(76\) 0.908065 0.104162
\(77\) 10.0460 1.14485
\(78\) 0 0
\(79\) −16.3932 −1.84438 −0.922190 0.386737i \(-0.873602\pi\)
−0.922190 + 0.386737i \(0.873602\pi\)
\(80\) −0.842473 −0.0941913
\(81\) 0 0
\(82\) −5.80251 −0.640780
\(83\) −4.79273 −0.526071 −0.263035 0.964786i \(-0.584724\pi\)
−0.263035 + 0.964786i \(0.584724\pi\)
\(84\) 0 0
\(85\) −1.61890 −0.175595
\(86\) 11.6773 1.25920
\(87\) 0 0
\(88\) −11.8128 −1.25925
\(89\) 1.64933 0.174829 0.0874143 0.996172i \(-0.472140\pi\)
0.0874143 + 0.996172i \(0.472140\pi\)
\(90\) 0 0
\(91\) 5.60182 0.587231
\(92\) −0.0216447 −0.00225662
\(93\) 0 0
\(94\) −13.6601 −1.40893
\(95\) −1.68178 −0.172547
\(96\) 0 0
\(97\) 3.34400 0.339531 0.169766 0.985484i \(-0.445699\pi\)
0.169766 + 0.985484i \(0.445699\pi\)
\(98\) −1.21319 −0.122550
\(99\) 0 0
\(100\) 0.601634 0.0601634
\(101\) 12.6806 1.26176 0.630882 0.775878i \(-0.282693\pi\)
0.630882 + 0.775878i \(0.282693\pi\)
\(102\) 0 0
\(103\) −4.47808 −0.441238 −0.220619 0.975360i \(-0.570808\pi\)
−0.220619 + 0.975360i \(0.570808\pi\)
\(104\) −6.58705 −0.645914
\(105\) 0 0
\(106\) −0.849727 −0.0825328
\(107\) −15.0754 −1.45740 −0.728699 0.684834i \(-0.759875\pi\)
−0.728699 + 0.684834i \(0.759875\pi\)
\(108\) 0 0
\(109\) 3.66599 0.351138 0.175569 0.984467i \(-0.443823\pi\)
0.175569 + 0.984467i \(0.443823\pi\)
\(110\) 1.25354 0.119521
\(111\) 0 0
\(112\) −9.25354 −0.874378
\(113\) −6.54316 −0.615528 −0.307764 0.951463i \(-0.599581\pi\)
−0.307764 + 0.951463i \(0.599581\pi\)
\(114\) 0 0
\(115\) 0.0400870 0.00373814
\(116\) 1.28378 0.119196
\(117\) 0 0
\(118\) −16.2466 −1.49562
\(119\) −17.7817 −1.63004
\(120\) 0 0
\(121\) 5.50443 0.500403
\(122\) 8.63866 0.782107
\(123\) 0 0
\(124\) −0.155170 −0.0139347
\(125\) −2.23992 −0.200345
\(126\) 0 0
\(127\) −15.0494 −1.33542 −0.667711 0.744420i \(-0.732726\pi\)
−0.667711 + 0.744420i \(0.732726\pi\)
\(128\) 10.1741 0.899272
\(129\) 0 0
\(130\) 0.698998 0.0613062
\(131\) −9.57374 −0.836462 −0.418231 0.908341i \(-0.637350\pi\)
−0.418231 + 0.908341i \(0.637350\pi\)
\(132\) 0 0
\(133\) −18.4723 −1.60175
\(134\) 17.1852 1.48457
\(135\) 0 0
\(136\) 20.9091 1.79294
\(137\) −2.15272 −0.183919 −0.0919595 0.995763i \(-0.529313\pi\)
−0.0919595 + 0.995763i \(0.529313\pi\)
\(138\) 0 0
\(139\) 5.62106 0.476772 0.238386 0.971171i \(-0.423382\pi\)
0.238386 + 0.971171i \(0.423382\pi\)
\(140\) −0.0676736 −0.00571947
\(141\) 0 0
\(142\) 13.0258 1.09310
\(143\) 9.20317 0.769608
\(144\) 0 0
\(145\) −2.37762 −0.197450
\(146\) 7.05711 0.584050
\(147\) 0 0
\(148\) −0.253460 −0.0208343
\(149\) −15.1382 −1.24017 −0.620086 0.784534i \(-0.712902\pi\)
−0.620086 + 0.784534i \(0.712902\pi\)
\(150\) 0 0
\(151\) −19.7193 −1.60474 −0.802369 0.596828i \(-0.796427\pi\)
−0.802369 + 0.596828i \(0.796427\pi\)
\(152\) 21.7212 1.76182
\(153\) 0 0
\(154\) 13.7686 1.10951
\(155\) 0.287383 0.0230831
\(156\) 0 0
\(157\) 15.7111 1.25389 0.626943 0.779065i \(-0.284306\pi\)
0.626943 + 0.779065i \(0.284306\pi\)
\(158\) −22.4679 −1.78745
\(159\) 0 0
\(160\) 0.154592 0.0122216
\(161\) 0.440308 0.0347011
\(162\) 0 0
\(163\) −11.8717 −0.929864 −0.464932 0.885346i \(-0.653921\pi\)
−0.464932 + 0.885346i \(0.653921\pi\)
\(164\) 0.514641 0.0401867
\(165\) 0 0
\(166\) −6.56874 −0.509833
\(167\) 5.37936 0.416267 0.208134 0.978100i \(-0.433261\pi\)
0.208134 + 0.978100i \(0.433261\pi\)
\(168\) 0 0
\(169\) −7.86814 −0.605242
\(170\) −2.21881 −0.170175
\(171\) 0 0
\(172\) −1.03570 −0.0789711
\(173\) 18.6393 1.41712 0.708560 0.705651i \(-0.249345\pi\)
0.708560 + 0.705651i \(0.249345\pi\)
\(174\) 0 0
\(175\) −12.2388 −0.925163
\(176\) −15.2025 −1.14593
\(177\) 0 0
\(178\) 2.26051 0.169432
\(179\) 22.6264 1.69117 0.845587 0.533837i \(-0.179250\pi\)
0.845587 + 0.533837i \(0.179250\pi\)
\(180\) 0 0
\(181\) 12.1002 0.899398 0.449699 0.893180i \(-0.351531\pi\)
0.449699 + 0.893180i \(0.351531\pi\)
\(182\) 7.67765 0.569105
\(183\) 0 0
\(184\) −0.517747 −0.0381688
\(185\) 0.469420 0.0345124
\(186\) 0 0
\(187\) −29.2133 −2.13629
\(188\) 1.21155 0.0883615
\(189\) 0 0
\(190\) −2.30498 −0.167221
\(191\) 0.166765 0.0120667 0.00603336 0.999982i \(-0.498080\pi\)
0.00603336 + 0.999982i \(0.498080\pi\)
\(192\) 0 0
\(193\) −16.5260 −1.18956 −0.594782 0.803887i \(-0.702762\pi\)
−0.594782 + 0.803887i \(0.702762\pi\)
\(194\) 4.58316 0.329051
\(195\) 0 0
\(196\) 0.107601 0.00768578
\(197\) −9.82274 −0.699841 −0.349921 0.936779i \(-0.613791\pi\)
−0.349921 + 0.936779i \(0.613791\pi\)
\(198\) 0 0
\(199\) 11.5691 0.820113 0.410056 0.912060i \(-0.365509\pi\)
0.410056 + 0.912060i \(0.365509\pi\)
\(200\) 14.3913 1.01762
\(201\) 0 0
\(202\) 17.3795 1.22282
\(203\) −26.1152 −1.83293
\(204\) 0 0
\(205\) −0.953140 −0.0665702
\(206\) −6.13749 −0.427619
\(207\) 0 0
\(208\) −8.47722 −0.587789
\(209\) −30.3480 −2.09921
\(210\) 0 0
\(211\) −19.2316 −1.32396 −0.661978 0.749523i \(-0.730283\pi\)
−0.661978 + 0.749523i \(0.730283\pi\)
\(212\) 0.0753647 0.00517607
\(213\) 0 0
\(214\) −20.6618 −1.41241
\(215\) 1.91816 0.130817
\(216\) 0 0
\(217\) 3.15655 0.214281
\(218\) 5.02447 0.340300
\(219\) 0 0
\(220\) −0.111180 −0.00749577
\(221\) −16.2899 −1.09578
\(222\) 0 0
\(223\) −2.26463 −0.151651 −0.0758255 0.997121i \(-0.524159\pi\)
−0.0758255 + 0.997121i \(0.524159\pi\)
\(224\) 1.69801 0.113453
\(225\) 0 0
\(226\) −8.96780 −0.596529
\(227\) −12.6616 −0.840378 −0.420189 0.907437i \(-0.638036\pi\)
−0.420189 + 0.907437i \(0.638036\pi\)
\(228\) 0 0
\(229\) 27.7524 1.83393 0.916967 0.398964i \(-0.130630\pi\)
0.916967 + 0.398964i \(0.130630\pi\)
\(230\) 0.0549418 0.00362275
\(231\) 0 0
\(232\) 30.7083 2.01610
\(233\) 10.9164 0.715157 0.357579 0.933883i \(-0.383602\pi\)
0.357579 + 0.933883i \(0.383602\pi\)
\(234\) 0 0
\(235\) −2.24385 −0.146373
\(236\) 1.44096 0.0937984
\(237\) 0 0
\(238\) −24.3709 −1.57973
\(239\) −6.41476 −0.414936 −0.207468 0.978242i \(-0.566522\pi\)
−0.207468 + 0.978242i \(0.566522\pi\)
\(240\) 0 0
\(241\) 9.85126 0.634576 0.317288 0.948329i \(-0.397228\pi\)
0.317288 + 0.948329i \(0.397228\pi\)
\(242\) 7.54417 0.484957
\(243\) 0 0
\(244\) −0.766188 −0.0490501
\(245\) −0.199282 −0.0127317
\(246\) 0 0
\(247\) −16.9226 −1.07676
\(248\) −3.71172 −0.235694
\(249\) 0 0
\(250\) −3.06995 −0.194161
\(251\) −5.56947 −0.351542 −0.175771 0.984431i \(-0.556242\pi\)
−0.175771 + 0.984431i \(0.556242\pi\)
\(252\) 0 0
\(253\) 0.723376 0.0454783
\(254\) −20.6262 −1.29420
\(255\) 0 0
\(256\) −2.90642 −0.181651
\(257\) −24.0328 −1.49913 −0.749564 0.661932i \(-0.769737\pi\)
−0.749564 + 0.661932i \(0.769737\pi\)
\(258\) 0 0
\(259\) 5.15601 0.320379
\(260\) −0.0619962 −0.00384484
\(261\) 0 0
\(262\) −13.1214 −0.810643
\(263\) −14.9598 −0.922463 −0.461232 0.887280i \(-0.652592\pi\)
−0.461232 + 0.887280i \(0.652592\pi\)
\(264\) 0 0
\(265\) −0.139579 −0.00857428
\(266\) −25.3175 −1.55231
\(267\) 0 0
\(268\) −1.52420 −0.0931055
\(269\) −13.4385 −0.819357 −0.409679 0.912230i \(-0.634359\pi\)
−0.409679 + 0.912230i \(0.634359\pi\)
\(270\) 0 0
\(271\) 5.72605 0.347833 0.173916 0.984760i \(-0.444358\pi\)
0.173916 + 0.984760i \(0.444358\pi\)
\(272\) 26.9089 1.63159
\(273\) 0 0
\(274\) −2.95043 −0.178242
\(275\) −20.1069 −1.21249
\(276\) 0 0
\(277\) 12.0981 0.726905 0.363452 0.931613i \(-0.381598\pi\)
0.363452 + 0.931613i \(0.381598\pi\)
\(278\) 7.70401 0.462056
\(279\) 0 0
\(280\) −1.61877 −0.0967402
\(281\) 6.21778 0.370921 0.185461 0.982652i \(-0.440622\pi\)
0.185461 + 0.982652i \(0.440622\pi\)
\(282\) 0 0
\(283\) 12.9505 0.769826 0.384913 0.922953i \(-0.374231\pi\)
0.384913 + 0.922953i \(0.374231\pi\)
\(284\) −1.15530 −0.0685544
\(285\) 0 0
\(286\) 12.6135 0.745853
\(287\) −10.4691 −0.617971
\(288\) 0 0
\(289\) 34.7084 2.04167
\(290\) −3.25867 −0.191356
\(291\) 0 0
\(292\) −0.625915 −0.0366289
\(293\) −14.3490 −0.838279 −0.419139 0.907922i \(-0.637668\pi\)
−0.419139 + 0.907922i \(0.637668\pi\)
\(294\) 0 0
\(295\) −2.66873 −0.155379
\(296\) −6.06283 −0.352395
\(297\) 0 0
\(298\) −20.7479 −1.20189
\(299\) 0.403368 0.0233274
\(300\) 0 0
\(301\) 21.0687 1.21438
\(302\) −27.0266 −1.55521
\(303\) 0 0
\(304\) 27.9541 1.60328
\(305\) 1.41902 0.0812526
\(306\) 0 0
\(307\) −28.7422 −1.64040 −0.820202 0.572075i \(-0.806139\pi\)
−0.820202 + 0.572075i \(0.806139\pi\)
\(308\) −1.22118 −0.0695832
\(309\) 0 0
\(310\) 0.393876 0.0223707
\(311\) 29.9677 1.69931 0.849655 0.527339i \(-0.176810\pi\)
0.849655 + 0.527339i \(0.176810\pi\)
\(312\) 0 0
\(313\) −13.6402 −0.770990 −0.385495 0.922710i \(-0.625969\pi\)
−0.385495 + 0.922710i \(0.625969\pi\)
\(314\) 21.5331 1.21518
\(315\) 0 0
\(316\) 1.99274 0.112101
\(317\) −12.9433 −0.726967 −0.363484 0.931601i \(-0.618413\pi\)
−0.363484 + 0.931601i \(0.618413\pi\)
\(318\) 0 0
\(319\) −42.9044 −2.40219
\(320\) 1.89682 0.106036
\(321\) 0 0
\(322\) 0.603469 0.0336300
\(323\) 53.7168 2.98888
\(324\) 0 0
\(325\) −11.2120 −0.621929
\(326\) −16.2709 −0.901163
\(327\) 0 0
\(328\) 12.3104 0.679726
\(329\) −24.6460 −1.35878
\(330\) 0 0
\(331\) 8.48346 0.466293 0.233147 0.972442i \(-0.425098\pi\)
0.233147 + 0.972442i \(0.425098\pi\)
\(332\) 0.582600 0.0319744
\(333\) 0 0
\(334\) 7.37275 0.403419
\(335\) 2.82290 0.154231
\(336\) 0 0
\(337\) −28.4431 −1.54939 −0.774697 0.632332i \(-0.782098\pi\)
−0.774697 + 0.632332i \(0.782098\pi\)
\(338\) −10.7838 −0.586560
\(339\) 0 0
\(340\) 0.196792 0.0106726
\(341\) 5.18586 0.280830
\(342\) 0 0
\(343\) −19.4986 −1.05283
\(344\) −24.7742 −1.33573
\(345\) 0 0
\(346\) 25.5463 1.37338
\(347\) −6.37057 −0.341990 −0.170995 0.985272i \(-0.554698\pi\)
−0.170995 + 0.985272i \(0.554698\pi\)
\(348\) 0 0
\(349\) −12.7281 −0.681317 −0.340659 0.940187i \(-0.610650\pi\)
−0.340659 + 0.940187i \(0.610650\pi\)
\(350\) −16.7740 −0.896607
\(351\) 0 0
\(352\) 2.78964 0.148688
\(353\) −1.48049 −0.0787984 −0.0393992 0.999224i \(-0.512544\pi\)
−0.0393992 + 0.999224i \(0.512544\pi\)
\(354\) 0 0
\(355\) 2.13967 0.113562
\(356\) −0.200491 −0.0106260
\(357\) 0 0
\(358\) 31.0109 1.63897
\(359\) 27.5753 1.45537 0.727685 0.685911i \(-0.240596\pi\)
0.727685 + 0.685911i \(0.240596\pi\)
\(360\) 0 0
\(361\) 36.8032 1.93701
\(362\) 16.5840 0.871637
\(363\) 0 0
\(364\) −0.680953 −0.0356916
\(365\) 1.15923 0.0606766
\(366\) 0 0
\(367\) 6.08957 0.317873 0.158936 0.987289i \(-0.449193\pi\)
0.158936 + 0.987289i \(0.449193\pi\)
\(368\) −0.666316 −0.0347341
\(369\) 0 0
\(370\) 0.643369 0.0334472
\(371\) −1.53311 −0.0795950
\(372\) 0 0
\(373\) 22.4949 1.16474 0.582371 0.812923i \(-0.302125\pi\)
0.582371 + 0.812923i \(0.302125\pi\)
\(374\) −40.0387 −2.07035
\(375\) 0 0
\(376\) 28.9807 1.49456
\(377\) −23.9243 −1.23216
\(378\) 0 0
\(379\) −14.1157 −0.725075 −0.362538 0.931969i \(-0.618090\pi\)
−0.362538 + 0.931969i \(0.618090\pi\)
\(380\) 0.204436 0.0104873
\(381\) 0 0
\(382\) 0.228562 0.0116943
\(383\) −24.3617 −1.24482 −0.622412 0.782690i \(-0.713847\pi\)
−0.622412 + 0.782690i \(0.713847\pi\)
\(384\) 0 0
\(385\) 2.26169 0.115266
\(386\) −22.6498 −1.15285
\(387\) 0 0
\(388\) −0.406493 −0.0206366
\(389\) 16.7319 0.848341 0.424170 0.905582i \(-0.360566\pi\)
0.424170 + 0.905582i \(0.360566\pi\)
\(390\) 0 0
\(391\) −1.28040 −0.0647525
\(392\) 2.57385 0.129999
\(393\) 0 0
\(394\) −13.4627 −0.678240
\(395\) −3.69066 −0.185697
\(396\) 0 0
\(397\) −21.7689 −1.09255 −0.546274 0.837607i \(-0.683954\pi\)
−0.546274 + 0.837607i \(0.683954\pi\)
\(398\) 15.8562 0.794799
\(399\) 0 0
\(400\) 18.5209 0.926043
\(401\) −20.2269 −1.01008 −0.505041 0.863095i \(-0.668523\pi\)
−0.505041 + 0.863095i \(0.668523\pi\)
\(402\) 0 0
\(403\) 2.89173 0.144047
\(404\) −1.54144 −0.0766895
\(405\) 0 0
\(406\) −35.7926 −1.77636
\(407\) 8.47075 0.419880
\(408\) 0 0
\(409\) 28.5461 1.41151 0.705757 0.708454i \(-0.250607\pi\)
0.705757 + 0.708454i \(0.250607\pi\)
\(410\) −1.30634 −0.0645155
\(411\) 0 0
\(412\) 0.544352 0.0268183
\(413\) −29.3127 −1.44238
\(414\) 0 0
\(415\) −1.07900 −0.0529662
\(416\) 1.55555 0.0762673
\(417\) 0 0
\(418\) −41.5938 −2.03442
\(419\) 36.3207 1.77438 0.887192 0.461400i \(-0.152653\pi\)
0.887192 + 0.461400i \(0.152653\pi\)
\(420\) 0 0
\(421\) −20.8986 −1.01853 −0.509267 0.860609i \(-0.670083\pi\)
−0.509267 + 0.860609i \(0.670083\pi\)
\(422\) −26.3581 −1.28309
\(423\) 0 0
\(424\) 1.80275 0.0875491
\(425\) 35.5898 1.72636
\(426\) 0 0
\(427\) 15.5862 0.754268
\(428\) 1.83256 0.0885800
\(429\) 0 0
\(430\) 2.62896 0.126780
\(431\) −5.65795 −0.272534 −0.136267 0.990672i \(-0.543510\pi\)
−0.136267 + 0.990672i \(0.543510\pi\)
\(432\) 0 0
\(433\) −0.797593 −0.0383299 −0.0191649 0.999816i \(-0.506101\pi\)
−0.0191649 + 0.999816i \(0.506101\pi\)
\(434\) 4.32625 0.207667
\(435\) 0 0
\(436\) −0.445635 −0.0213420
\(437\) −1.33013 −0.0636286
\(438\) 0 0
\(439\) −18.2826 −0.872581 −0.436290 0.899806i \(-0.643708\pi\)
−0.436290 + 0.899806i \(0.643708\pi\)
\(440\) −2.65946 −0.126785
\(441\) 0 0
\(442\) −22.3263 −1.06195
\(443\) 18.4038 0.874391 0.437195 0.899367i \(-0.355972\pi\)
0.437195 + 0.899367i \(0.355972\pi\)
\(444\) 0 0
\(445\) 0.371319 0.0176022
\(446\) −3.10382 −0.146970
\(447\) 0 0
\(448\) 20.8343 0.984329
\(449\) −2.95767 −0.139581 −0.0697905 0.997562i \(-0.522233\pi\)
−0.0697905 + 0.997562i \(0.522233\pi\)
\(450\) 0 0
\(451\) −17.1996 −0.809896
\(452\) 0.795381 0.0374116
\(453\) 0 0
\(454\) −17.3535 −0.814439
\(455\) 1.26116 0.0591240
\(456\) 0 0
\(457\) 18.1383 0.848472 0.424236 0.905552i \(-0.360543\pi\)
0.424236 + 0.905552i \(0.360543\pi\)
\(458\) 38.0365 1.77733
\(459\) 0 0
\(460\) −0.00487295 −0.000227202 0
\(461\) 16.5572 0.771145 0.385573 0.922677i \(-0.374004\pi\)
0.385573 + 0.922677i \(0.374004\pi\)
\(462\) 0 0
\(463\) 4.01151 0.186430 0.0932152 0.995646i \(-0.470286\pi\)
0.0932152 + 0.995646i \(0.470286\pi\)
\(464\) 39.5201 1.83467
\(465\) 0 0
\(466\) 14.9616 0.693083
\(467\) 4.61263 0.213447 0.106724 0.994289i \(-0.465964\pi\)
0.106724 + 0.994289i \(0.465964\pi\)
\(468\) 0 0
\(469\) 31.0061 1.43173
\(470\) −3.07534 −0.141855
\(471\) 0 0
\(472\) 34.4681 1.58652
\(473\) 34.6135 1.59153
\(474\) 0 0
\(475\) 36.9721 1.69640
\(476\) 2.16153 0.0990733
\(477\) 0 0
\(478\) −8.79182 −0.402129
\(479\) 37.2886 1.70376 0.851879 0.523739i \(-0.175463\pi\)
0.851879 + 0.523739i \(0.175463\pi\)
\(480\) 0 0
\(481\) 4.72345 0.215371
\(482\) 13.5018 0.614989
\(483\) 0 0
\(484\) −0.669114 −0.0304143
\(485\) 0.752845 0.0341849
\(486\) 0 0
\(487\) 25.0176 1.13366 0.566828 0.823836i \(-0.308170\pi\)
0.566828 + 0.823836i \(0.308170\pi\)
\(488\) −18.3274 −0.829644
\(489\) 0 0
\(490\) −0.273129 −0.0123387
\(491\) 23.7021 1.06966 0.534830 0.844960i \(-0.320376\pi\)
0.534830 + 0.844960i \(0.320376\pi\)
\(492\) 0 0
\(493\) 75.9421 3.42026
\(494\) −23.1935 −1.04352
\(495\) 0 0
\(496\) −4.77680 −0.214485
\(497\) 23.5017 1.05419
\(498\) 0 0
\(499\) 27.1748 1.21651 0.608255 0.793742i \(-0.291870\pi\)
0.608255 + 0.793742i \(0.291870\pi\)
\(500\) 0.272283 0.0121769
\(501\) 0 0
\(502\) −7.63331 −0.340691
\(503\) −12.0746 −0.538381 −0.269191 0.963087i \(-0.586756\pi\)
−0.269191 + 0.963087i \(0.586756\pi\)
\(504\) 0 0
\(505\) 2.85482 0.127038
\(506\) 0.991432 0.0440745
\(507\) 0 0
\(508\) 1.82940 0.0811664
\(509\) 43.7176 1.93775 0.968873 0.247557i \(-0.0796279\pi\)
0.968873 + 0.247557i \(0.0796279\pi\)
\(510\) 0 0
\(511\) 12.7327 0.563261
\(512\) −24.3316 −1.07532
\(513\) 0 0
\(514\) −32.9385 −1.45286
\(515\) −1.00817 −0.0444251
\(516\) 0 0
\(517\) −40.4906 −1.78078
\(518\) 7.06663 0.310490
\(519\) 0 0
\(520\) −1.48297 −0.0650323
\(521\) 35.3638 1.54932 0.774659 0.632379i \(-0.217922\pi\)
0.774659 + 0.632379i \(0.217922\pi\)
\(522\) 0 0
\(523\) 19.8938 0.869897 0.434949 0.900455i \(-0.356767\pi\)
0.434949 + 0.900455i \(0.356767\pi\)
\(524\) 1.16378 0.0508398
\(525\) 0 0
\(526\) −20.5034 −0.893990
\(527\) −9.17913 −0.399849
\(528\) 0 0
\(529\) −22.9683 −0.998622
\(530\) −0.191302 −0.00830962
\(531\) 0 0
\(532\) 2.24548 0.0973539
\(533\) −9.59079 −0.415423
\(534\) 0 0
\(535\) −3.39399 −0.146735
\(536\) −36.4594 −1.57480
\(537\) 0 0
\(538\) −18.4182 −0.794067
\(539\) −3.59608 −0.154894
\(540\) 0 0
\(541\) −15.3119 −0.658311 −0.329156 0.944276i \(-0.606764\pi\)
−0.329156 + 0.944276i \(0.606764\pi\)
\(542\) 7.84790 0.337096
\(543\) 0 0
\(544\) −4.93774 −0.211704
\(545\) 0.825337 0.0353535
\(546\) 0 0
\(547\) 42.3037 1.80878 0.904388 0.426711i \(-0.140328\pi\)
0.904388 + 0.426711i \(0.140328\pi\)
\(548\) 0.261682 0.0111785
\(549\) 0 0
\(550\) −27.5578 −1.17507
\(551\) 78.8917 3.36090
\(552\) 0 0
\(553\) −40.5374 −1.72383
\(554\) 16.5812 0.704468
\(555\) 0 0
\(556\) −0.683291 −0.0289780
\(557\) −7.44351 −0.315392 −0.157696 0.987488i \(-0.550407\pi\)
−0.157696 + 0.987488i \(0.550407\pi\)
\(558\) 0 0
\(559\) 19.3011 0.816350
\(560\) −2.08328 −0.0880347
\(561\) 0 0
\(562\) 8.52185 0.359472
\(563\) 30.6551 1.29196 0.645978 0.763356i \(-0.276450\pi\)
0.645978 + 0.763356i \(0.276450\pi\)
\(564\) 0 0
\(565\) −1.47308 −0.0619731
\(566\) 17.7494 0.746064
\(567\) 0 0
\(568\) −27.6351 −1.15954
\(569\) −45.0445 −1.88836 −0.944182 0.329425i \(-0.893145\pi\)
−0.944182 + 0.329425i \(0.893145\pi\)
\(570\) 0 0
\(571\) −20.5533 −0.860127 −0.430063 0.902799i \(-0.641509\pi\)
−0.430063 + 0.902799i \(0.641509\pi\)
\(572\) −1.11873 −0.0467764
\(573\) 0 0
\(574\) −14.3485 −0.598897
\(575\) −0.881270 −0.0367515
\(576\) 0 0
\(577\) 2.43980 0.101570 0.0507851 0.998710i \(-0.483828\pi\)
0.0507851 + 0.998710i \(0.483828\pi\)
\(578\) 47.5700 1.97865
\(579\) 0 0
\(580\) 0.289021 0.0120009
\(581\) −11.8516 −0.491685
\(582\) 0 0
\(583\) −2.51873 −0.104315
\(584\) −14.9721 −0.619549
\(585\) 0 0
\(586\) −19.6662 −0.812404
\(587\) −28.0624 −1.15826 −0.579129 0.815236i \(-0.696607\pi\)
−0.579129 + 0.815236i \(0.696607\pi\)
\(588\) 0 0
\(589\) −9.53565 −0.392910
\(590\) −3.65765 −0.150583
\(591\) 0 0
\(592\) −7.80257 −0.320684
\(593\) 20.1600 0.827873 0.413937 0.910306i \(-0.364153\pi\)
0.413937 + 0.910306i \(0.364153\pi\)
\(594\) 0 0
\(595\) −4.00325 −0.164117
\(596\) 1.84019 0.0753771
\(597\) 0 0
\(598\) 0.552841 0.0226073
\(599\) −23.5060 −0.960427 −0.480214 0.877152i \(-0.659441\pi\)
−0.480214 + 0.877152i \(0.659441\pi\)
\(600\) 0 0
\(601\) −20.9839 −0.855952 −0.427976 0.903790i \(-0.640773\pi\)
−0.427976 + 0.903790i \(0.640773\pi\)
\(602\) 28.8759 1.17689
\(603\) 0 0
\(604\) 2.39707 0.0975353
\(605\) 1.23923 0.0503819
\(606\) 0 0
\(607\) 32.2511 1.30903 0.654516 0.756048i \(-0.272872\pi\)
0.654516 + 0.756048i \(0.272872\pi\)
\(608\) −5.12953 −0.208030
\(609\) 0 0
\(610\) 1.94485 0.0787447
\(611\) −22.5783 −0.913421
\(612\) 0 0
\(613\) −37.2120 −1.50298 −0.751489 0.659746i \(-0.770664\pi\)
−0.751489 + 0.659746i \(0.770664\pi\)
\(614\) −39.3930 −1.58977
\(615\) 0 0
\(616\) −29.2110 −1.17694
\(617\) −38.3479 −1.54383 −0.771915 0.635726i \(-0.780701\pi\)
−0.771915 + 0.635726i \(0.780701\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934
\(620\) −0.0349340 −0.00140298
\(621\) 0 0
\(622\) 41.0726 1.64686
\(623\) 4.07849 0.163401
\(624\) 0 0
\(625\) 24.2423 0.969692
\(626\) −18.6948 −0.747193
\(627\) 0 0
\(628\) −1.90983 −0.0762106
\(629\) −14.9935 −0.597829
\(630\) 0 0
\(631\) 38.9829 1.55188 0.775942 0.630804i \(-0.217275\pi\)
0.775942 + 0.630804i \(0.217275\pi\)
\(632\) 47.6670 1.89609
\(633\) 0 0
\(634\) −17.7396 −0.704529
\(635\) −3.38813 −0.134454
\(636\) 0 0
\(637\) −2.00524 −0.0794504
\(638\) −58.8032 −2.32804
\(639\) 0 0
\(640\) 2.29053 0.0905412
\(641\) 0.319675 0.0126264 0.00631321 0.999980i \(-0.497990\pi\)
0.00631321 + 0.999980i \(0.497990\pi\)
\(642\) 0 0
\(643\) 0.159017 0.00627101 0.00313550 0.999995i \(-0.499002\pi\)
0.00313550 + 0.999995i \(0.499002\pi\)
\(644\) −0.0535234 −0.00210912
\(645\) 0 0
\(646\) 73.6222 2.89663
\(647\) −24.6688 −0.969829 −0.484915 0.874562i \(-0.661149\pi\)
−0.484915 + 0.874562i \(0.661149\pi\)
\(648\) 0 0
\(649\) −48.1575 −1.89035
\(650\) −15.3667 −0.602732
\(651\) 0 0
\(652\) 1.44311 0.0565167
\(653\) 28.3461 1.10927 0.554635 0.832094i \(-0.312858\pi\)
0.554635 + 0.832094i \(0.312858\pi\)
\(654\) 0 0
\(655\) −2.15537 −0.0842172
\(656\) 15.8428 0.618559
\(657\) 0 0
\(658\) −33.7789 −1.31684
\(659\) −39.5061 −1.53894 −0.769470 0.638683i \(-0.779479\pi\)
−0.769470 + 0.638683i \(0.779479\pi\)
\(660\) 0 0
\(661\) −15.7041 −0.610818 −0.305409 0.952221i \(-0.598793\pi\)
−0.305409 + 0.952221i \(0.598793\pi\)
\(662\) 11.6271 0.451900
\(663\) 0 0
\(664\) 13.9360 0.540820
\(665\) −4.15874 −0.161269
\(666\) 0 0
\(667\) −1.88047 −0.0728120
\(668\) −0.653910 −0.0253006
\(669\) 0 0
\(670\) 3.86896 0.149471
\(671\) 25.6064 0.988523
\(672\) 0 0
\(673\) −23.0243 −0.887522 −0.443761 0.896145i \(-0.646356\pi\)
−0.443761 + 0.896145i \(0.646356\pi\)
\(674\) −38.9830 −1.50157
\(675\) 0 0
\(676\) 0.956445 0.0367863
\(677\) −7.63096 −0.293282 −0.146641 0.989190i \(-0.546846\pi\)
−0.146641 + 0.989190i \(0.546846\pi\)
\(678\) 0 0
\(679\) 8.26909 0.317339
\(680\) 4.70733 0.180518
\(681\) 0 0
\(682\) 7.10755 0.272162
\(683\) −0.644724 −0.0246697 −0.0123348 0.999924i \(-0.503926\pi\)
−0.0123348 + 0.999924i \(0.503926\pi\)
\(684\) 0 0
\(685\) −0.484648 −0.0185175
\(686\) −26.7240 −1.02033
\(687\) 0 0
\(688\) −31.8831 −1.21553
\(689\) −1.40449 −0.0535067
\(690\) 0 0
\(691\) −29.5024 −1.12232 −0.561162 0.827706i \(-0.689646\pi\)
−0.561162 + 0.827706i \(0.689646\pi\)
\(692\) −2.26578 −0.0861319
\(693\) 0 0
\(694\) −8.73126 −0.331434
\(695\) 1.26549 0.0480027
\(696\) 0 0
\(697\) 30.4437 1.15314
\(698\) −17.4446 −0.660288
\(699\) 0 0
\(700\) 1.48773 0.0562310
\(701\) 30.4332 1.14944 0.574722 0.818348i \(-0.305110\pi\)
0.574722 + 0.818348i \(0.305110\pi\)
\(702\) 0 0
\(703\) −15.5758 −0.587453
\(704\) 34.2285 1.29003
\(705\) 0 0
\(706\) −2.02910 −0.0763662
\(707\) 31.3568 1.17929
\(708\) 0 0
\(709\) 10.4754 0.393413 0.196707 0.980462i \(-0.436975\pi\)
0.196707 + 0.980462i \(0.436975\pi\)
\(710\) 2.93255 0.110057
\(711\) 0 0
\(712\) −4.79580 −0.179730
\(713\) 0.227292 0.00851217
\(714\) 0 0
\(715\) 2.07194 0.0774862
\(716\) −2.75044 −0.102789
\(717\) 0 0
\(718\) 37.7937 1.41045
\(719\) −0.815327 −0.0304066 −0.0152033 0.999884i \(-0.504840\pi\)
−0.0152033 + 0.999884i \(0.504840\pi\)
\(720\) 0 0
\(721\) −11.0735 −0.412398
\(722\) 50.4410 1.87722
\(723\) 0 0
\(724\) −1.47089 −0.0546650
\(725\) 52.2694 1.94123
\(726\) 0 0
\(727\) 5.58478 0.207128 0.103564 0.994623i \(-0.466975\pi\)
0.103564 + 0.994623i \(0.466975\pi\)
\(728\) −16.2886 −0.603695
\(729\) 0 0
\(730\) 1.58879 0.0588038
\(731\) −61.2669 −2.26604
\(732\) 0 0
\(733\) 27.7414 1.02465 0.512327 0.858791i \(-0.328784\pi\)
0.512327 + 0.858791i \(0.328784\pi\)
\(734\) 8.34613 0.308061
\(735\) 0 0
\(736\) 0.122268 0.00450685
\(737\) 50.9396 1.87638
\(738\) 0 0
\(739\) −10.9980 −0.404569 −0.202285 0.979327i \(-0.564837\pi\)
−0.202285 + 0.979327i \(0.564837\pi\)
\(740\) −0.0570623 −0.00209765
\(741\) 0 0
\(742\) −2.10122 −0.0771382
\(743\) −27.5706 −1.01147 −0.505733 0.862690i \(-0.668778\pi\)
−0.505733 + 0.862690i \(0.668778\pi\)
\(744\) 0 0
\(745\) −3.40812 −0.124864
\(746\) 30.8307 1.12879
\(747\) 0 0
\(748\) 3.55115 0.129843
\(749\) −37.2788 −1.36214
\(750\) 0 0
\(751\) 2.39041 0.0872274 0.0436137 0.999048i \(-0.486113\pi\)
0.0436137 + 0.999048i \(0.486113\pi\)
\(752\) 37.2967 1.36007
\(753\) 0 0
\(754\) −32.7897 −1.19413
\(755\) −4.43948 −0.161569
\(756\) 0 0
\(757\) 22.8502 0.830505 0.415253 0.909706i \(-0.363693\pi\)
0.415253 + 0.909706i \(0.363693\pi\)
\(758\) −19.3465 −0.702695
\(759\) 0 0
\(760\) 4.89016 0.177385
\(761\) −35.2537 −1.27795 −0.638974 0.769229i \(-0.720641\pi\)
−0.638974 + 0.769229i \(0.720641\pi\)
\(762\) 0 0
\(763\) 9.06532 0.328187
\(764\) −0.0202719 −0.000733410 0
\(765\) 0 0
\(766\) −33.3892 −1.20640
\(767\) −26.8535 −0.969625
\(768\) 0 0
\(769\) 14.0283 0.505875 0.252937 0.967483i \(-0.418603\pi\)
0.252937 + 0.967483i \(0.418603\pi\)
\(770\) 3.09978 0.111708
\(771\) 0 0
\(772\) 2.00888 0.0723012
\(773\) −2.05297 −0.0738401 −0.0369200 0.999318i \(-0.511755\pi\)
−0.0369200 + 0.999318i \(0.511755\pi\)
\(774\) 0 0
\(775\) −6.31780 −0.226942
\(776\) −9.72344 −0.349051
\(777\) 0 0
\(778\) 22.9321 0.822156
\(779\) 31.6262 1.13312
\(780\) 0 0
\(781\) 38.6107 1.38160
\(782\) −1.75486 −0.0627538
\(783\) 0 0
\(784\) 3.31241 0.118300
\(785\) 3.53710 0.126245
\(786\) 0 0
\(787\) −7.63128 −0.272026 −0.136013 0.990707i \(-0.543429\pi\)
−0.136013 + 0.990707i \(0.543429\pi\)
\(788\) 1.19404 0.0425361
\(789\) 0 0
\(790\) −5.05828 −0.179965
\(791\) −16.1800 −0.575296
\(792\) 0 0
\(793\) 14.2786 0.507047
\(794\) −29.8356 −1.05882
\(795\) 0 0
\(796\) −1.40633 −0.0498461
\(797\) −23.8432 −0.844570 −0.422285 0.906463i \(-0.638772\pi\)
−0.422285 + 0.906463i \(0.638772\pi\)
\(798\) 0 0
\(799\) 71.6696 2.53549
\(800\) −3.39854 −0.120157
\(801\) 0 0
\(802\) −27.7222 −0.978905
\(803\) 20.9184 0.738194
\(804\) 0 0
\(805\) 0.0991279 0.00349380
\(806\) 3.96330 0.139601
\(807\) 0 0
\(808\) −36.8717 −1.29714
\(809\) −19.9428 −0.701152 −0.350576 0.936534i \(-0.614014\pi\)
−0.350576 + 0.936534i \(0.614014\pi\)
\(810\) 0 0
\(811\) −24.3626 −0.855488 −0.427744 0.903900i \(-0.640692\pi\)
−0.427744 + 0.903900i \(0.640692\pi\)
\(812\) 3.17455 0.111405
\(813\) 0 0
\(814\) 11.6097 0.406920
\(815\) −2.67272 −0.0936212
\(816\) 0 0
\(817\) −63.6465 −2.22671
\(818\) 39.1242 1.36795
\(819\) 0 0
\(820\) 0.115863 0.00404611
\(821\) 15.6976 0.547850 0.273925 0.961751i \(-0.411678\pi\)
0.273925 + 0.961751i \(0.411678\pi\)
\(822\) 0 0
\(823\) −7.38747 −0.257511 −0.128755 0.991676i \(-0.541098\pi\)
−0.128755 + 0.991676i \(0.541098\pi\)
\(824\) 13.0210 0.453609
\(825\) 0 0
\(826\) −40.1749 −1.39786
\(827\) −11.5262 −0.400806 −0.200403 0.979714i \(-0.564225\pi\)
−0.200403 + 0.979714i \(0.564225\pi\)
\(828\) 0 0
\(829\) −17.7336 −0.615914 −0.307957 0.951400i \(-0.599645\pi\)
−0.307957 + 0.951400i \(0.599645\pi\)
\(830\) −1.47884 −0.0513314
\(831\) 0 0
\(832\) 19.0864 0.661702
\(833\) 6.36516 0.220540
\(834\) 0 0
\(835\) 1.21107 0.0419109
\(836\) 3.68907 0.127589
\(837\) 0 0
\(838\) 49.7798 1.71962
\(839\) 28.6447 0.988924 0.494462 0.869199i \(-0.335365\pi\)
0.494462 + 0.869199i \(0.335365\pi\)
\(840\) 0 0
\(841\) 82.5332 2.84597
\(842\) −28.6428 −0.987095
\(843\) 0 0
\(844\) 2.33777 0.0804695
\(845\) −1.77138 −0.0609374
\(846\) 0 0
\(847\) 13.6115 0.467695
\(848\) 2.32005 0.0796707
\(849\) 0 0
\(850\) 48.7781 1.67307
\(851\) 0.371266 0.0127269
\(852\) 0 0
\(853\) 41.6536 1.42619 0.713097 0.701066i \(-0.247292\pi\)
0.713097 + 0.701066i \(0.247292\pi\)
\(854\) 21.3618 0.730987
\(855\) 0 0
\(856\) 43.8353 1.49826
\(857\) 9.05282 0.309238 0.154619 0.987974i \(-0.450585\pi\)
0.154619 + 0.987974i \(0.450585\pi\)
\(858\) 0 0
\(859\) 9.47579 0.323310 0.161655 0.986847i \(-0.448317\pi\)
0.161655 + 0.986847i \(0.448317\pi\)
\(860\) −0.233170 −0.00795103
\(861\) 0 0
\(862\) −7.75457 −0.264122
\(863\) 34.6314 1.17887 0.589434 0.807817i \(-0.299351\pi\)
0.589434 + 0.807817i \(0.299351\pi\)
\(864\) 0 0
\(865\) 4.19633 0.142679
\(866\) −1.09315 −0.0371468
\(867\) 0 0
\(868\) −0.383708 −0.0130239
\(869\) −66.5985 −2.25920
\(870\) 0 0
\(871\) 28.4048 0.962461
\(872\) −10.6597 −0.360983
\(873\) 0 0
\(874\) −1.82302 −0.0616647
\(875\) −5.53892 −0.187250
\(876\) 0 0
\(877\) 15.5441 0.524886 0.262443 0.964947i \(-0.415472\pi\)
0.262443 + 0.964947i \(0.415472\pi\)
\(878\) −25.0574 −0.845648
\(879\) 0 0
\(880\) −3.42260 −0.115376
\(881\) −52.4905 −1.76845 −0.884224 0.467063i \(-0.845312\pi\)
−0.884224 + 0.467063i \(0.845312\pi\)
\(882\) 0 0
\(883\) −32.8989 −1.10713 −0.553567 0.832804i \(-0.686734\pi\)
−0.553567 + 0.832804i \(0.686734\pi\)
\(884\) 1.98018 0.0666008
\(885\) 0 0
\(886\) 25.2236 0.847402
\(887\) 40.9458 1.37483 0.687413 0.726267i \(-0.258746\pi\)
0.687413 + 0.726267i \(0.258746\pi\)
\(888\) 0 0
\(889\) −37.2145 −1.24814
\(890\) 0.508916 0.0170589
\(891\) 0 0
\(892\) 0.275287 0.00921729
\(893\) 74.4533 2.49148
\(894\) 0 0
\(895\) 5.09395 0.170272
\(896\) 25.1587 0.840493
\(897\) 0 0
\(898\) −4.05367 −0.135273
\(899\) −13.4810 −0.449617
\(900\) 0 0
\(901\) 4.45822 0.148525
\(902\) −23.5731 −0.784897
\(903\) 0 0
\(904\) 19.0257 0.632786
\(905\) 2.72415 0.0905539
\(906\) 0 0
\(907\) −22.3371 −0.741692 −0.370846 0.928694i \(-0.620932\pi\)
−0.370846 + 0.928694i \(0.620932\pi\)
\(908\) 1.53913 0.0510778
\(909\) 0 0
\(910\) 1.72849 0.0572990
\(911\) −15.9284 −0.527732 −0.263866 0.964559i \(-0.584998\pi\)
−0.263866 + 0.964559i \(0.584998\pi\)
\(912\) 0 0
\(913\) −19.4708 −0.644389
\(914\) 24.8596 0.822283
\(915\) 0 0
\(916\) −3.37356 −0.111466
\(917\) −23.6741 −0.781788
\(918\) 0 0
\(919\) −40.6296 −1.34025 −0.670124 0.742249i \(-0.733759\pi\)
−0.670124 + 0.742249i \(0.733759\pi\)
\(920\) −0.116562 −0.00384294
\(921\) 0 0
\(922\) 22.6927 0.747343
\(923\) 21.5300 0.708669
\(924\) 0 0
\(925\) −10.3197 −0.339309
\(926\) 5.49802 0.180676
\(927\) 0 0
\(928\) −7.25187 −0.238054
\(929\) 38.1089 1.25031 0.625155 0.780500i \(-0.285036\pi\)
0.625155 + 0.780500i \(0.285036\pi\)
\(930\) 0 0
\(931\) 6.61238 0.216712
\(932\) −1.32699 −0.0434670
\(933\) 0 0
\(934\) 6.32190 0.206859
\(935\) −6.57689 −0.215087
\(936\) 0 0
\(937\) 20.9882 0.685653 0.342827 0.939399i \(-0.388616\pi\)
0.342827 + 0.939399i \(0.388616\pi\)
\(938\) 42.4958 1.38754
\(939\) 0 0
\(940\) 0.272761 0.00889648
\(941\) −0.271913 −0.00886412 −0.00443206 0.999990i \(-0.501411\pi\)
−0.00443206 + 0.999990i \(0.501411\pi\)
\(942\) 0 0
\(943\) −0.753843 −0.0245485
\(944\) 44.3588 1.44376
\(945\) 0 0
\(946\) 47.4399 1.54241
\(947\) −18.9489 −0.615756 −0.307878 0.951426i \(-0.599619\pi\)
−0.307878 + 0.951426i \(0.599619\pi\)
\(948\) 0 0
\(949\) 11.6645 0.378645
\(950\) 50.6726 1.64404
\(951\) 0 0
\(952\) 51.7043 1.67575
\(953\) 2.67357 0.0866055 0.0433027 0.999062i \(-0.486212\pi\)
0.0433027 + 0.999062i \(0.486212\pi\)
\(954\) 0 0
\(955\) 0.0375445 0.00121491
\(956\) 0.779772 0.0252196
\(957\) 0 0
\(958\) 51.1063 1.65117
\(959\) −5.32327 −0.171897
\(960\) 0 0
\(961\) −29.3705 −0.947437
\(962\) 6.47378 0.208723
\(963\) 0 0
\(964\) −1.19751 −0.0385692
\(965\) −3.72054 −0.119769
\(966\) 0 0
\(967\) 20.1196 0.647004 0.323502 0.946227i \(-0.395140\pi\)
0.323502 + 0.946227i \(0.395140\pi\)
\(968\) −16.0054 −0.514433
\(969\) 0 0
\(970\) 1.03182 0.0331298
\(971\) −15.9577 −0.512106 −0.256053 0.966663i \(-0.582422\pi\)
−0.256053 + 0.966663i \(0.582422\pi\)
\(972\) 0 0
\(973\) 13.8998 0.445609
\(974\) 34.2882 1.09866
\(975\) 0 0
\(976\) −23.5865 −0.754985
\(977\) −42.1759 −1.34933 −0.674663 0.738126i \(-0.735711\pi\)
−0.674663 + 0.738126i \(0.735711\pi\)
\(978\) 0 0
\(979\) 6.70051 0.214149
\(980\) 0.0242246 0.000773826 0
\(981\) 0 0
\(982\) 32.4852 1.03664
\(983\) 31.8772 1.01673 0.508363 0.861143i \(-0.330251\pi\)
0.508363 + 0.861143i \(0.330251\pi\)
\(984\) 0 0
\(985\) −2.21143 −0.0704619
\(986\) 104.083 3.31469
\(987\) 0 0
\(988\) 2.05709 0.0654449
\(989\) 1.51708 0.0482404
\(990\) 0 0
\(991\) 28.4135 0.902584 0.451292 0.892376i \(-0.350963\pi\)
0.451292 + 0.892376i \(0.350963\pi\)
\(992\) 0.876534 0.0278300
\(993\) 0 0
\(994\) 32.2105 1.02166
\(995\) 2.60459 0.0825712
\(996\) 0 0
\(997\) −38.9058 −1.23216 −0.616079 0.787684i \(-0.711280\pi\)
−0.616079 + 0.787684i \(0.711280\pi\)
\(998\) 37.2447 1.17896
\(999\) 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 5571.2.a.g.1.23 30
3.2 odd 2 619.2.a.b.1.8 30
12.11 even 2 9904.2.a.n.1.18 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
619.2.a.b.1.8 30 3.2 odd 2
5571.2.a.g.1.23 30 1.1 even 1 trivial
9904.2.a.n.1.18 30 12.11 even 2