Properties

Label 2898.2.a.bb.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.70156 q^{10} +4.00000 q^{11} +5.70156 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -5.40312 q^{19} -3.70156 q^{20} +4.00000 q^{22} -1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} +1.00000 q^{28} -0.298438 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -3.70156 q^{35} +4.29844 q^{37} -5.40312 q^{38} -3.70156 q^{40} +0.298438 q^{41} -1.70156 q^{43} +4.00000 q^{44} -1.00000 q^{46} +11.1047 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} +9.40312 q^{53} -14.8062 q^{55} +1.00000 q^{56} -0.298438 q^{58} +7.40312 q^{59} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -21.1047 q^{65} +14.8062 q^{67} -4.00000 q^{68} -3.70156 q^{70} +7.40312 q^{71} +1.40312 q^{73} +4.29844 q^{74} -5.40312 q^{76} +4.00000 q^{77} +8.00000 q^{79} -3.70156 q^{80} +0.298438 q^{82} -13.4031 q^{83} +14.8062 q^{85} -1.70156 q^{86} +4.00000 q^{88} +11.4031 q^{89} +5.70156 q^{91} -1.00000 q^{92} +11.1047 q^{94} +20.0000 q^{95} +6.29844 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} + 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} + 2q^{7} + 2q^{8} - q^{10} + 8q^{11} + 5q^{13} + 2q^{14} + 2q^{16} - 8q^{17} + 2q^{19} - q^{20} + 8q^{22} - 2q^{23} + 11q^{25} + 5q^{26} + 2q^{28} - 7q^{29} - 4q^{31} + 2q^{32} - 8q^{34} - q^{35} + 15q^{37} + 2q^{38} - q^{40} + 7q^{41} + 3q^{43} + 8q^{44} - 2q^{46} + 3q^{47} + 2q^{49} + 11q^{50} + 5q^{52} + 6q^{53} - 4q^{55} + 2q^{56} - 7q^{58} + 2q^{59} - 4q^{61} - 4q^{62} + 2q^{64} - 23q^{65} + 4q^{67} - 8q^{68} - q^{70} + 2q^{71} - 10q^{73} + 15q^{74} + 2q^{76} + 8q^{77} + 16q^{79} - q^{80} + 7q^{82} - 14q^{83} + 4q^{85} + 3q^{86} + 8q^{88} + 10q^{89} + 5q^{91} - 2q^{92} + 3q^{94} + 40q^{95} + 19q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.70156 −1.65539 −0.827694 0.561179i \(-0.810348\pi\)
−0.827694 + 0.561179i \(0.810348\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.70156 −1.17054
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 5.70156 1.58133 0.790664 0.612250i \(-0.209735\pi\)
0.790664 + 0.612250i \(0.209735\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −5.40312 −1.23956 −0.619781 0.784775i \(-0.712779\pi\)
−0.619781 + 0.784775i \(0.712779\pi\)
\(20\) −3.70156 −0.827694
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.70156 1.74031
\(26\) 5.70156 1.11817
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −0.298438 −0.0554185 −0.0277093 0.999616i \(-0.508821\pi\)
−0.0277093 + 0.999616i \(0.508821\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −3.70156 −0.625678
\(36\) 0 0
\(37\) 4.29844 0.706659 0.353329 0.935499i \(-0.385049\pi\)
0.353329 + 0.935499i \(0.385049\pi\)
\(38\) −5.40312 −0.876502
\(39\) 0 0
\(40\) −3.70156 −0.585268
\(41\) 0.298438 0.0466082 0.0233041 0.999728i \(-0.492581\pi\)
0.0233041 + 0.999728i \(0.492581\pi\)
\(42\) 0 0
\(43\) −1.70156 −0.259486 −0.129743 0.991548i \(-0.541415\pi\)
−0.129743 + 0.991548i \(0.541415\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.1047 1.61978 0.809892 0.586578i \(-0.199525\pi\)
0.809892 + 0.586578i \(0.199525\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.70156 1.23059
\(51\) 0 0
\(52\) 5.70156 0.790664
\(53\) 9.40312 1.29162 0.645809 0.763499i \(-0.276520\pi\)
0.645809 + 0.763499i \(0.276520\pi\)
\(54\) 0 0
\(55\) −14.8062 −1.99647
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −0.298438 −0.0391868
\(59\) 7.40312 0.963805 0.481902 0.876225i \(-0.339946\pi\)
0.481902 + 0.876225i \(0.339946\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −21.1047 −2.61771
\(66\) 0 0
\(67\) 14.8062 1.80887 0.904436 0.426610i \(-0.140292\pi\)
0.904436 + 0.426610i \(0.140292\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) −3.70156 −0.442421
\(71\) 7.40312 0.878589 0.439295 0.898343i \(-0.355228\pi\)
0.439295 + 0.898343i \(0.355228\pi\)
\(72\) 0 0
\(73\) 1.40312 0.164223 0.0821116 0.996623i \(-0.473834\pi\)
0.0821116 + 0.996623i \(0.473834\pi\)
\(74\) 4.29844 0.499683
\(75\) 0 0
\(76\) −5.40312 −0.619781
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −3.70156 −0.413847
\(81\) 0 0
\(82\) 0.298438 0.0329570
\(83\) −13.4031 −1.47118 −0.735592 0.677425i \(-0.763096\pi\)
−0.735592 + 0.677425i \(0.763096\pi\)
\(84\) 0 0
\(85\) 14.8062 1.60596
\(86\) −1.70156 −0.183484
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 11.4031 1.20873 0.604364 0.796708i \(-0.293427\pi\)
0.604364 + 0.796708i \(0.293427\pi\)
\(90\) 0 0
\(91\) 5.70156 0.597686
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 11.1047 1.14536
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 6.29844 0.639509 0.319755 0.947500i \(-0.396399\pi\)
0.319755 + 0.947500i \(0.396399\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 8.70156 0.870156
\(101\) 3.40312 0.338624 0.169312 0.985563i \(-0.445846\pi\)
0.169312 + 0.985563i \(0.445846\pi\)
\(102\) 0 0
\(103\) 2.29844 0.226472 0.113236 0.993568i \(-0.463878\pi\)
0.113236 + 0.993568i \(0.463878\pi\)
\(104\) 5.70156 0.559084
\(105\) 0 0
\(106\) 9.40312 0.913312
\(107\) −18.8062 −1.81807 −0.909034 0.416721i \(-0.863179\pi\)
−0.909034 + 0.416721i \(0.863179\pi\)
\(108\) 0 0
\(109\) −7.70156 −0.737676 −0.368838 0.929494i \(-0.620244\pi\)
−0.368838 + 0.929494i \(0.620244\pi\)
\(110\) −14.8062 −1.41172
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −7.10469 −0.668353 −0.334176 0.942511i \(-0.608458\pi\)
−0.334176 + 0.942511i \(0.608458\pi\)
\(114\) 0 0
\(115\) 3.70156 0.345172
\(116\) −0.298438 −0.0277093
\(117\) 0 0
\(118\) 7.40312 0.681513
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −13.7016 −1.22550
\(126\) 0 0
\(127\) −5.70156 −0.505932 −0.252966 0.967475i \(-0.581406\pi\)
−0.252966 + 0.967475i \(0.581406\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −21.1047 −1.85100
\(131\) 7.40312 0.646814 0.323407 0.946260i \(-0.395172\pi\)
0.323407 + 0.946260i \(0.395172\pi\)
\(132\) 0 0
\(133\) −5.40312 −0.468510
\(134\) 14.8062 1.27907
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 19.7016 1.68322 0.841609 0.540087i \(-0.181609\pi\)
0.841609 + 0.540087i \(0.181609\pi\)
\(138\) 0 0
\(139\) 17.7016 1.50143 0.750713 0.660628i \(-0.229710\pi\)
0.750713 + 0.660628i \(0.229710\pi\)
\(140\) −3.70156 −0.312839
\(141\) 0 0
\(142\) 7.40312 0.621256
\(143\) 22.8062 1.90715
\(144\) 0 0
\(145\) 1.10469 0.0917392
\(146\) 1.40312 0.116123
\(147\) 0 0
\(148\) 4.29844 0.353329
\(149\) 20.2094 1.65562 0.827808 0.561011i \(-0.189588\pi\)
0.827808 + 0.561011i \(0.189588\pi\)
\(150\) 0 0
\(151\) 23.9109 1.94584 0.972922 0.231133i \(-0.0742433\pi\)
0.972922 + 0.231133i \(0.0742433\pi\)
\(152\) −5.40312 −0.438251
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 7.40312 0.594633
\(156\) 0 0
\(157\) −1.40312 −0.111982 −0.0559908 0.998431i \(-0.517832\pi\)
−0.0559908 + 0.998431i \(0.517832\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −3.70156 −0.292634
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −10.8062 −0.846411 −0.423205 0.906034i \(-0.639095\pi\)
−0.423205 + 0.906034i \(0.639095\pi\)
\(164\) 0.298438 0.0233041
\(165\) 0 0
\(166\) −13.4031 −1.04028
\(167\) 13.4031 1.03716 0.518582 0.855028i \(-0.326460\pi\)
0.518582 + 0.855028i \(0.326460\pi\)
\(168\) 0 0
\(169\) 19.5078 1.50060
\(170\) 14.8062 1.13559
\(171\) 0 0
\(172\) −1.70156 −0.129743
\(173\) −19.4031 −1.47519 −0.737596 0.675242i \(-0.764039\pi\)
−0.737596 + 0.675242i \(0.764039\pi\)
\(174\) 0 0
\(175\) 8.70156 0.657776
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 11.4031 0.854700
\(179\) 25.1047 1.87641 0.938206 0.346077i \(-0.112486\pi\)
0.938206 + 0.346077i \(0.112486\pi\)
\(180\) 0 0
\(181\) −16.8062 −1.24920 −0.624599 0.780945i \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) 5.70156 0.422628
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −15.9109 −1.16980
\(186\) 0 0
\(187\) −16.0000 −1.17004
\(188\) 11.1047 0.809892
\(189\) 0 0
\(190\) 20.0000 1.45095
\(191\) −22.8062 −1.65020 −0.825101 0.564985i \(-0.808882\pi\)
−0.825101 + 0.564985i \(0.808882\pi\)
\(192\) 0 0
\(193\) −23.1047 −1.66311 −0.831556 0.555441i \(-0.812549\pi\)
−0.831556 + 0.555441i \(0.812549\pi\)
\(194\) 6.29844 0.452201
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −22.5078 −1.60362 −0.801808 0.597582i \(-0.796128\pi\)
−0.801808 + 0.597582i \(0.796128\pi\)
\(198\) 0 0
\(199\) −2.29844 −0.162932 −0.0814660 0.996676i \(-0.525960\pi\)
−0.0814660 + 0.996676i \(0.525960\pi\)
\(200\) 8.70156 0.615293
\(201\) 0 0
\(202\) 3.40312 0.239443
\(203\) −0.298438 −0.0209462
\(204\) 0 0
\(205\) −1.10469 −0.0771546
\(206\) 2.29844 0.160140
\(207\) 0 0
\(208\) 5.70156 0.395332
\(209\) −21.6125 −1.49497
\(210\) 0 0
\(211\) 10.8062 0.743933 0.371966 0.928246i \(-0.378684\pi\)
0.371966 + 0.928246i \(0.378684\pi\)
\(212\) 9.40312 0.645809
\(213\) 0 0
\(214\) −18.8062 −1.28557
\(215\) 6.29844 0.429550
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −7.70156 −0.521616
\(219\) 0 0
\(220\) −14.8062 −0.998237
\(221\) −22.8062 −1.53411
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −7.10469 −0.472597
\(227\) 23.1047 1.53351 0.766756 0.641939i \(-0.221870\pi\)
0.766756 + 0.641939i \(0.221870\pi\)
\(228\) 0 0
\(229\) −14.5969 −0.964589 −0.482294 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(230\) 3.70156 0.244074
\(231\) 0 0
\(232\) −0.298438 −0.0195934
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −41.1047 −2.68137
\(236\) 7.40312 0.481902
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −10.8062 −0.698998 −0.349499 0.936937i \(-0.613648\pi\)
−0.349499 + 0.936937i \(0.613648\pi\)
\(240\) 0 0
\(241\) −15.9109 −1.02491 −0.512457 0.858713i \(-0.671264\pi\)
−0.512457 + 0.858713i \(0.671264\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −3.70156 −0.236484
\(246\) 0 0
\(247\) −30.8062 −1.96015
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −13.7016 −0.866563
\(251\) −19.7016 −1.24355 −0.621776 0.783195i \(-0.713588\pi\)
−0.621776 + 0.783195i \(0.713588\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −5.70156 −0.357748
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 4.29844 0.267092
\(260\) −21.1047 −1.30886
\(261\) 0 0
\(262\) 7.40312 0.457367
\(263\) −9.70156 −0.598224 −0.299112 0.954218i \(-0.596690\pi\)
−0.299112 + 0.954218i \(0.596690\pi\)
\(264\) 0 0
\(265\) −34.8062 −2.13813
\(266\) −5.40312 −0.331287
\(267\) 0 0
\(268\) 14.8062 0.904436
\(269\) 10.2094 0.622476 0.311238 0.950332i \(-0.399256\pi\)
0.311238 + 0.950332i \(0.399256\pi\)
\(270\) 0 0
\(271\) 1.40312 0.0852337 0.0426169 0.999091i \(-0.486431\pi\)
0.0426169 + 0.999091i \(0.486431\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 19.7016 1.19021
\(275\) 34.8062 2.09890
\(276\) 0 0
\(277\) −21.4031 −1.28599 −0.642995 0.765871i \(-0.722308\pi\)
−0.642995 + 0.765871i \(0.722308\pi\)
\(278\) 17.7016 1.06167
\(279\) 0 0
\(280\) −3.70156 −0.221211
\(281\) −14.5078 −0.865463 −0.432732 0.901523i \(-0.642450\pi\)
−0.432732 + 0.901523i \(0.642450\pi\)
\(282\) 0 0
\(283\) 4.80625 0.285702 0.142851 0.989744i \(-0.454373\pi\)
0.142851 + 0.989744i \(0.454373\pi\)
\(284\) 7.40312 0.439295
\(285\) 0 0
\(286\) 22.8062 1.34856
\(287\) 0.298438 0.0176162
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 1.10469 0.0648694
\(291\) 0 0
\(292\) 1.40312 0.0821116
\(293\) 20.8062 1.21551 0.607757 0.794123i \(-0.292069\pi\)
0.607757 + 0.794123i \(0.292069\pi\)
\(294\) 0 0
\(295\) −27.4031 −1.59547
\(296\) 4.29844 0.249842
\(297\) 0 0
\(298\) 20.2094 1.17070
\(299\) −5.70156 −0.329730
\(300\) 0 0
\(301\) −1.70156 −0.0980764
\(302\) 23.9109 1.37592
\(303\) 0 0
\(304\) −5.40312 −0.309890
\(305\) 7.40312 0.423902
\(306\) 0 0
\(307\) 11.9109 0.679793 0.339896 0.940463i \(-0.389608\pi\)
0.339896 + 0.940463i \(0.389608\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 7.40312 0.420469
\(311\) −13.4031 −0.760021 −0.380011 0.924982i \(-0.624080\pi\)
−0.380011 + 0.924982i \(0.624080\pi\)
\(312\) 0 0
\(313\) −6.20937 −0.350974 −0.175487 0.984482i \(-0.556150\pi\)
−0.175487 + 0.984482i \(0.556150\pi\)
\(314\) −1.40312 −0.0791829
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 8.29844 0.466087 0.233043 0.972466i \(-0.425132\pi\)
0.233043 + 0.972466i \(0.425132\pi\)
\(318\) 0 0
\(319\) −1.19375 −0.0668373
\(320\) −3.70156 −0.206924
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 21.6125 1.20255
\(324\) 0 0
\(325\) 49.6125 2.75201
\(326\) −10.8062 −0.598503
\(327\) 0 0
\(328\) 0.298438 0.0164785
\(329\) 11.1047 0.612221
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −13.4031 −0.735592
\(333\) 0 0
\(334\) 13.4031 0.733386
\(335\) −54.8062 −2.99439
\(336\) 0 0
\(337\) −4.20937 −0.229299 −0.114650 0.993406i \(-0.536575\pi\)
−0.114650 + 0.993406i \(0.536575\pi\)
\(338\) 19.5078 1.06109
\(339\) 0 0
\(340\) 14.8062 0.802982
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −1.70156 −0.0917421
\(345\) 0 0
\(346\) −19.4031 −1.04312
\(347\) 10.2984 0.552849 0.276425 0.961036i \(-0.410850\pi\)
0.276425 + 0.961036i \(0.410850\pi\)
\(348\) 0 0
\(349\) 12.5969 0.674295 0.337148 0.941452i \(-0.390538\pi\)
0.337148 + 0.941452i \(0.390538\pi\)
\(350\) 8.70156 0.465118
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −4.29844 −0.228783 −0.114391 0.993436i \(-0.536492\pi\)
−0.114391 + 0.993436i \(0.536492\pi\)
\(354\) 0 0
\(355\) −27.4031 −1.45441
\(356\) 11.4031 0.604364
\(357\) 0 0
\(358\) 25.1047 1.32682
\(359\) 2.89531 0.152809 0.0764044 0.997077i \(-0.475656\pi\)
0.0764044 + 0.997077i \(0.475656\pi\)
\(360\) 0 0
\(361\) 10.1938 0.536513
\(362\) −16.8062 −0.883317
\(363\) 0 0
\(364\) 5.70156 0.298843
\(365\) −5.19375 −0.271853
\(366\) 0 0
\(367\) −17.1047 −0.892857 −0.446429 0.894819i \(-0.647304\pi\)
−0.446429 + 0.894819i \(0.647304\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −15.9109 −0.827170
\(371\) 9.40312 0.488186
\(372\) 0 0
\(373\) −0.806248 −0.0417460 −0.0208730 0.999782i \(-0.506645\pi\)
−0.0208730 + 0.999782i \(0.506645\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 11.1047 0.572680
\(377\) −1.70156 −0.0876349
\(378\) 0 0
\(379\) 13.7016 0.703802 0.351901 0.936037i \(-0.385535\pi\)
0.351901 + 0.936037i \(0.385535\pi\)
\(380\) 20.0000 1.02598
\(381\) 0 0
\(382\) −22.8062 −1.16687
\(383\) −29.6125 −1.51313 −0.756564 0.653920i \(-0.773123\pi\)
−0.756564 + 0.653920i \(0.773123\pi\)
\(384\) 0 0
\(385\) −14.8062 −0.754596
\(386\) −23.1047 −1.17600
\(387\) 0 0
\(388\) 6.29844 0.319755
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −22.5078 −1.13393
\(395\) −29.6125 −1.48997
\(396\) 0 0
\(397\) 8.59688 0.431465 0.215732 0.976453i \(-0.430786\pi\)
0.215732 + 0.976453i \(0.430786\pi\)
\(398\) −2.29844 −0.115210
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −11.4031 −0.568030
\(404\) 3.40312 0.169312
\(405\) 0 0
\(406\) −0.298438 −0.0148112
\(407\) 17.1938 0.852263
\(408\) 0 0
\(409\) 5.40312 0.267167 0.133584 0.991038i \(-0.457352\pi\)
0.133584 + 0.991038i \(0.457352\pi\)
\(410\) −1.10469 −0.0545566
\(411\) 0 0
\(412\) 2.29844 0.113236
\(413\) 7.40312 0.364284
\(414\) 0 0
\(415\) 49.6125 2.43538
\(416\) 5.70156 0.279542
\(417\) 0 0
\(418\) −21.6125 −1.05710
\(419\) 0.209373 0.0102285 0.00511426 0.999987i \(-0.498372\pi\)
0.00511426 + 0.999987i \(0.498372\pi\)
\(420\) 0 0
\(421\) −15.7016 −0.765247 −0.382624 0.923904i \(-0.624979\pi\)
−0.382624 + 0.923904i \(0.624979\pi\)
\(422\) 10.8062 0.526040
\(423\) 0 0
\(424\) 9.40312 0.456656
\(425\) −34.8062 −1.68835
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −18.8062 −0.909034
\(429\) 0 0
\(430\) 6.29844 0.303738
\(431\) 3.91093 0.188383 0.0941916 0.995554i \(-0.469973\pi\)
0.0941916 + 0.995554i \(0.469973\pi\)
\(432\) 0 0
\(433\) 31.3141 1.50486 0.752429 0.658674i \(-0.228882\pi\)
0.752429 + 0.658674i \(0.228882\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −7.70156 −0.368838
\(437\) 5.40312 0.258466
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) −14.8062 −0.705860
\(441\) 0 0
\(442\) −22.8062 −1.08478
\(443\) 13.7016 0.650981 0.325490 0.945545i \(-0.394471\pi\)
0.325490 + 0.945545i \(0.394471\pi\)
\(444\) 0 0
\(445\) −42.2094 −2.00092
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 38.4187 1.81309 0.906546 0.422106i \(-0.138709\pi\)
0.906546 + 0.422106i \(0.138709\pi\)
\(450\) 0 0
\(451\) 1.19375 0.0562116
\(452\) −7.10469 −0.334176
\(453\) 0 0
\(454\) 23.1047 1.08436
\(455\) −21.1047 −0.989403
\(456\) 0 0
\(457\) 28.2094 1.31958 0.659789 0.751451i \(-0.270645\pi\)
0.659789 + 0.751451i \(0.270645\pi\)
\(458\) −14.5969 −0.682067
\(459\) 0 0
\(460\) 3.70156 0.172586
\(461\) 2.20937 0.102901 0.0514504 0.998676i \(-0.483616\pi\)
0.0514504 + 0.998676i \(0.483616\pi\)
\(462\) 0 0
\(463\) −22.2984 −1.03630 −0.518148 0.855291i \(-0.673378\pi\)
−0.518148 + 0.855291i \(0.673378\pi\)
\(464\) −0.298438 −0.0138546
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 3.70156 0.171288 0.0856439 0.996326i \(-0.472705\pi\)
0.0856439 + 0.996326i \(0.472705\pi\)
\(468\) 0 0
\(469\) 14.8062 0.683689
\(470\) −41.1047 −1.89602
\(471\) 0 0
\(472\) 7.40312 0.340756
\(473\) −6.80625 −0.312952
\(474\) 0 0
\(475\) −47.0156 −2.15722
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) −10.8062 −0.494266
\(479\) −16.5969 −0.758331 −0.379165 0.925329i \(-0.623789\pi\)
−0.379165 + 0.925329i \(0.623789\pi\)
\(480\) 0 0
\(481\) 24.5078 1.11746
\(482\) −15.9109 −0.724723
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −23.3141 −1.05864
\(486\) 0 0
\(487\) −6.29844 −0.285409 −0.142705 0.989765i \(-0.545580\pi\)
−0.142705 + 0.989765i \(0.545580\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −3.70156 −0.167220
\(491\) 9.61250 0.433806 0.216903 0.976193i \(-0.430404\pi\)
0.216903 + 0.976193i \(0.430404\pi\)
\(492\) 0 0
\(493\) 1.19375 0.0537639
\(494\) −30.8062 −1.38604
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 7.40312 0.332076
\(498\) 0 0
\(499\) −23.4031 −1.04767 −0.523834 0.851820i \(-0.675499\pi\)
−0.523834 + 0.851820i \(0.675499\pi\)
\(500\) −13.7016 −0.612752
\(501\) 0 0
\(502\) −19.7016 −0.879324
\(503\) −4.59688 −0.204965 −0.102482 0.994735i \(-0.532678\pi\)
−0.102482 + 0.994735i \(0.532678\pi\)
\(504\) 0 0
\(505\) −12.5969 −0.560554
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −5.70156 −0.252966
\(509\) −37.6125 −1.66714 −0.833572 0.552410i \(-0.813708\pi\)
−0.833572 + 0.552410i \(0.813708\pi\)
\(510\) 0 0
\(511\) 1.40312 0.0620706
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −8.50781 −0.374899
\(516\) 0 0
\(517\) 44.4187 1.95353
\(518\) 4.29844 0.188863
\(519\) 0 0
\(520\) −21.1047 −0.925502
\(521\) 20.5969 0.902366 0.451183 0.892432i \(-0.351002\pi\)
0.451183 + 0.892432i \(0.351002\pi\)
\(522\) 0 0
\(523\) 20.8062 0.909794 0.454897 0.890544i \(-0.349676\pi\)
0.454897 + 0.890544i \(0.349676\pi\)
\(524\) 7.40312 0.323407
\(525\) 0 0
\(526\) −9.70156 −0.423008
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −34.8062 −1.51189
\(531\) 0 0
\(532\) −5.40312 −0.234255
\(533\) 1.70156 0.0737028
\(534\) 0 0
\(535\) 69.6125 3.00961
\(536\) 14.8062 0.639533
\(537\) 0 0
\(538\) 10.2094 0.440157
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 33.4031 1.43611 0.718056 0.695985i \(-0.245032\pi\)
0.718056 + 0.695985i \(0.245032\pi\)
\(542\) 1.40312 0.0602693
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 28.5078 1.22114
\(546\) 0 0
\(547\) 29.0156 1.24062 0.620309 0.784357i \(-0.287007\pi\)
0.620309 + 0.784357i \(0.287007\pi\)
\(548\) 19.7016 0.841609
\(549\) 0 0
\(550\) 34.8062 1.48414
\(551\) 1.61250 0.0686947
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −21.4031 −0.909332
\(555\) 0 0
\(556\) 17.7016 0.750713
\(557\) 37.4031 1.58482 0.792411 0.609988i \(-0.208826\pi\)
0.792411 + 0.609988i \(0.208826\pi\)
\(558\) 0 0
\(559\) −9.70156 −0.410332
\(560\) −3.70156 −0.156420
\(561\) 0 0
\(562\) −14.5078 −0.611975
\(563\) 3.10469 0.130847 0.0654235 0.997858i \(-0.479160\pi\)
0.0654235 + 0.997858i \(0.479160\pi\)
\(564\) 0 0
\(565\) 26.2984 1.10638
\(566\) 4.80625 0.202022
\(567\) 0 0
\(568\) 7.40312 0.310628
\(569\) −31.7016 −1.32900 −0.664499 0.747289i \(-0.731355\pi\)
−0.664499 + 0.747289i \(0.731355\pi\)
\(570\) 0 0
\(571\) −14.8062 −0.619622 −0.309811 0.950798i \(-0.600266\pi\)
−0.309811 + 0.950798i \(0.600266\pi\)
\(572\) 22.8062 0.953577
\(573\) 0 0
\(574\) 0.298438 0.0124566
\(575\) −8.70156 −0.362880
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 1.10469 0.0458696
\(581\) −13.4031 −0.556055
\(582\) 0 0
\(583\) 37.6125 1.55775
\(584\) 1.40312 0.0580617
\(585\) 0 0
\(586\) 20.8062 0.859498
\(587\) −14.8062 −0.611119 −0.305560 0.952173i \(-0.598844\pi\)
−0.305560 + 0.952173i \(0.598844\pi\)
\(588\) 0 0
\(589\) 10.8062 0.445264
\(590\) −27.4031 −1.12817
\(591\) 0 0
\(592\) 4.29844 0.176665
\(593\) −43.7016 −1.79461 −0.897304 0.441413i \(-0.854477\pi\)
−0.897304 + 0.441413i \(0.854477\pi\)
\(594\) 0 0
\(595\) 14.8062 0.606997
\(596\) 20.2094 0.827808
\(597\) 0 0
\(598\) −5.70156 −0.233154
\(599\) −13.6125 −0.556192 −0.278096 0.960553i \(-0.589703\pi\)
−0.278096 + 0.960553i \(0.589703\pi\)
\(600\) 0 0
\(601\) −13.4031 −0.546725 −0.273362 0.961911i \(-0.588136\pi\)
−0.273362 + 0.961911i \(0.588136\pi\)
\(602\) −1.70156 −0.0693505
\(603\) 0 0
\(604\) 23.9109 0.972922
\(605\) −18.5078 −0.752450
\(606\) 0 0
\(607\) −39.6125 −1.60782 −0.803911 0.594750i \(-0.797251\pi\)
−0.803911 + 0.594750i \(0.797251\pi\)
\(608\) −5.40312 −0.219126
\(609\) 0 0
\(610\) 7.40312 0.299744
\(611\) 63.3141 2.56141
\(612\) 0 0
\(613\) 3.70156 0.149505 0.0747523 0.997202i \(-0.476183\pi\)
0.0747523 + 0.997202i \(0.476183\pi\)
\(614\) 11.9109 0.480686
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 19.1938 0.772711 0.386356 0.922350i \(-0.373734\pi\)
0.386356 + 0.922350i \(0.373734\pi\)
\(618\) 0 0
\(619\) −7.19375 −0.289141 −0.144571 0.989494i \(-0.546180\pi\)
−0.144571 + 0.989494i \(0.546180\pi\)
\(620\) 7.40312 0.297317
\(621\) 0 0
\(622\) −13.4031 −0.537416
\(623\) 11.4031 0.456857
\(624\) 0 0
\(625\) 7.20937 0.288375
\(626\) −6.20937 −0.248176
\(627\) 0 0
\(628\) −1.40312 −0.0559908
\(629\) −17.1938 −0.685560
\(630\) 0 0
\(631\) −29.6125 −1.17885 −0.589427 0.807821i \(-0.700647\pi\)
−0.589427 + 0.807821i \(0.700647\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 8.29844 0.329573
\(635\) 21.1047 0.837514
\(636\) 0 0
\(637\) 5.70156 0.225904
\(638\) −1.19375 −0.0472611
\(639\) 0 0
\(640\) −3.70156 −0.146317
\(641\) 20.7172 0.818280 0.409140 0.912472i \(-0.365829\pi\)
0.409140 + 0.912472i \(0.365829\pi\)
\(642\) 0 0
\(643\) 0.806248 0.0317953 0.0158977 0.999874i \(-0.494939\pi\)
0.0158977 + 0.999874i \(0.494939\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 21.6125 0.850332
\(647\) 6.59688 0.259350 0.129675 0.991557i \(-0.458607\pi\)
0.129675 + 0.991557i \(0.458607\pi\)
\(648\) 0 0
\(649\) 29.6125 1.16239
\(650\) 49.6125 1.94596
\(651\) 0 0
\(652\) −10.8062 −0.423205
\(653\) −7.10469 −0.278028 −0.139014 0.990290i \(-0.544393\pi\)
−0.139014 + 0.990290i \(0.544393\pi\)
\(654\) 0 0
\(655\) −27.4031 −1.07073
\(656\) 0.298438 0.0116520
\(657\) 0 0
\(658\) 11.1047 0.432906
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 42.4187 1.64990 0.824949 0.565207i \(-0.191204\pi\)
0.824949 + 0.565207i \(0.191204\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −13.4031 −0.520142
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) 0.298438 0.0115556
\(668\) 13.4031 0.518582
\(669\) 0 0
\(670\) −54.8062 −2.11735
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 7.70156 0.296873 0.148437 0.988922i \(-0.452576\pi\)
0.148437 + 0.988922i \(0.452576\pi\)
\(674\) −4.20937 −0.162139
\(675\) 0 0
\(676\) 19.5078 0.750300
\(677\) 50.4187 1.93775 0.968875 0.247551i \(-0.0796257\pi\)
0.968875 + 0.247551i \(0.0796257\pi\)
\(678\) 0 0
\(679\) 6.29844 0.241712
\(680\) 14.8062 0.567794
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −33.6125 −1.28615 −0.643073 0.765805i \(-0.722341\pi\)
−0.643073 + 0.765805i \(0.722341\pi\)
\(684\) 0 0
\(685\) −72.9266 −2.78638
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −1.70156 −0.0648714
\(689\) 53.6125 2.04247
\(690\) 0 0
\(691\) −0.507811 −0.0193180 −0.00965901 0.999953i \(-0.503075\pi\)
−0.00965901 + 0.999953i \(0.503075\pi\)
\(692\) −19.4031 −0.737596
\(693\) 0 0
\(694\) 10.2984 0.390923
\(695\) −65.5234 −2.48545
\(696\) 0 0
\(697\) −1.19375 −0.0452166
\(698\) 12.5969 0.476799
\(699\) 0 0
\(700\) 8.70156 0.328888
\(701\) −47.0156 −1.77576 −0.887878 0.460079i \(-0.847821\pi\)
−0.887878 + 0.460079i \(0.847821\pi\)
\(702\) 0 0
\(703\) −23.2250 −0.875947
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −4.29844 −0.161774
\(707\) 3.40312 0.127988
\(708\) 0 0
\(709\) 23.1938 0.871060 0.435530 0.900174i \(-0.356561\pi\)
0.435530 + 0.900174i \(0.356561\pi\)
\(710\) −27.4031 −1.02842
\(711\) 0 0
\(712\) 11.4031 0.427350
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −84.4187 −3.15708
\(716\) 25.1047 0.938206
\(717\) 0 0
\(718\) 2.89531 0.108052
\(719\) 5.91093 0.220441 0.110220 0.993907i \(-0.464844\pi\)
0.110220 + 0.993907i \(0.464844\pi\)
\(720\) 0 0
\(721\) 2.29844 0.0855983
\(722\) 10.1938 0.379372
\(723\) 0 0
\(724\) −16.8062 −0.624599
\(725\) −2.59688 −0.0964455
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 5.70156 0.211314
\(729\) 0 0
\(730\) −5.19375 −0.192229
\(731\) 6.80625 0.251738
\(732\) 0 0
\(733\) 52.2094 1.92840 0.964199 0.265181i \(-0.0854318\pi\)
0.964199 + 0.265181i \(0.0854318\pi\)
\(734\) −17.1047 −0.631345
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 59.2250 2.18158
\(738\) 0 0
\(739\) −29.0156 −1.06736 −0.533678 0.845687i \(-0.679191\pi\)
−0.533678 + 0.845687i \(0.679191\pi\)
\(740\) −15.9109 −0.584898
\(741\) 0 0
\(742\) 9.40312 0.345200
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −74.8062 −2.74069
\(746\) −0.806248 −0.0295189
\(747\) 0 0
\(748\) −16.0000 −0.585018
\(749\) −18.8062 −0.687165
\(750\) 0 0
\(751\) 29.6125 1.08058 0.540288 0.841480i \(-0.318315\pi\)
0.540288 + 0.841480i \(0.318315\pi\)
\(752\) 11.1047 0.404946
\(753\) 0 0
\(754\) −1.70156 −0.0619672
\(755\) −88.5078 −3.22113
\(756\) 0 0
\(757\) −26.4187 −0.960206 −0.480103 0.877212i \(-0.659401\pi\)
−0.480103 + 0.877212i \(0.659401\pi\)
\(758\) 13.7016 0.497663
\(759\) 0 0
\(760\) 20.0000 0.725476
\(761\) 35.6125 1.29095 0.645476 0.763781i \(-0.276659\pi\)
0.645476 + 0.763781i \(0.276659\pi\)
\(762\) 0 0
\(763\) −7.70156 −0.278815
\(764\) −22.8062 −0.825101
\(765\) 0 0
\(766\) −29.6125 −1.06994
\(767\) 42.2094 1.52409
\(768\) 0 0
\(769\) 15.9109 0.573763 0.286881 0.957966i \(-0.407381\pi\)
0.286881 + 0.957966i \(0.407381\pi\)
\(770\) −14.8062 −0.533580
\(771\) 0 0
\(772\) −23.1047 −0.831556
\(773\) 7.10469 0.255538 0.127769 0.991804i \(-0.459218\pi\)
0.127769 + 0.991804i \(0.459218\pi\)
\(774\) 0 0
\(775\) −17.4031 −0.625139
\(776\) 6.29844 0.226101
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −1.61250 −0.0577737
\(780\) 0 0
\(781\) 29.6125 1.05962
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 5.19375 0.185373
\(786\) 0 0
\(787\) −18.5969 −0.662907 −0.331454 0.943472i \(-0.607539\pi\)
−0.331454 + 0.943472i \(0.607539\pi\)
\(788\) −22.5078 −0.801808
\(789\) 0 0
\(790\) −29.6125 −1.05357
\(791\) −7.10469 −0.252614
\(792\) 0 0
\(793\) −11.4031 −0.404937
\(794\) 8.59688 0.305092
\(795\) 0 0
\(796\) −2.29844 −0.0814660
\(797\) −15.7016 −0.556178 −0.278089 0.960555i \(-0.589701\pi\)
−0.278089 + 0.960555i \(0.589701\pi\)
\(798\) 0 0
\(799\) −44.4187 −1.57142
\(800\) 8.70156 0.307647
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) 5.61250 0.198061
\(804\) 0 0
\(805\) 3.70156 0.130463
\(806\) −11.4031 −0.401658
\(807\) 0 0
\(808\) 3.40312 0.119721
\(809\) −31.0156 −1.09045 −0.545226 0.838289i \(-0.683556\pi\)
−0.545226 + 0.838289i \(0.683556\pi\)
\(810\) 0 0
\(811\) 52.5078 1.84380 0.921899 0.387430i \(-0.126637\pi\)
0.921899 + 0.387430i \(0.126637\pi\)
\(812\) −0.298438 −0.0104731
\(813\) 0 0
\(814\) 17.1938 0.602641
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 9.19375 0.321649
\(818\) 5.40312 0.188916
\(819\) 0 0
\(820\) −1.10469 −0.0385773
\(821\) −27.6125 −0.963683 −0.481841 0.876258i \(-0.660032\pi\)
−0.481841 + 0.876258i \(0.660032\pi\)
\(822\) 0 0
\(823\) −13.7016 −0.477606 −0.238803 0.971068i \(-0.576755\pi\)
−0.238803 + 0.971068i \(0.576755\pi\)
\(824\) 2.29844 0.0800699
\(825\) 0 0
\(826\) 7.40312 0.257588
\(827\) 47.4031 1.64837 0.824184 0.566322i \(-0.191634\pi\)
0.824184 + 0.566322i \(0.191634\pi\)
\(828\) 0 0
\(829\) 55.4031 1.92423 0.962115 0.272644i \(-0.0878981\pi\)
0.962115 + 0.272644i \(0.0878981\pi\)
\(830\) 49.6125 1.72207
\(831\) 0 0
\(832\) 5.70156 0.197666
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) −49.6125 −1.71691
\(836\) −21.6125 −0.747484
\(837\) 0 0
\(838\) 0.209373 0.00723266
\(839\) −42.2094 −1.45723 −0.728615 0.684924i \(-0.759835\pi\)
−0.728615 + 0.684924i \(0.759835\pi\)
\(840\) 0 0
\(841\) −28.9109 −0.996929
\(842\) −15.7016 −0.541112
\(843\) 0 0
\(844\) 10.8062 0.371966
\(845\) −72.2094 −2.48408
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 9.40312 0.322905
\(849\) 0 0
\(850\) −34.8062 −1.19384
\(851\) −4.29844 −0.147349
\(852\) 0 0
\(853\) 9.10469 0.311739 0.155869 0.987778i \(-0.450182\pi\)
0.155869 + 0.987778i \(0.450182\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −18.8062 −0.642784
\(857\) 0.298438 0.0101944 0.00509722 0.999987i \(-0.498377\pi\)
0.00509722 + 0.999987i \(0.498377\pi\)
\(858\) 0 0
\(859\) −15.4922 −0.528587 −0.264293 0.964442i \(-0.585139\pi\)
−0.264293 + 0.964442i \(0.585139\pi\)
\(860\) 6.29844 0.214775
\(861\) 0 0
\(862\) 3.91093 0.133207
\(863\) 6.38750 0.217433 0.108717 0.994073i \(-0.465326\pi\)
0.108717 + 0.994073i \(0.465326\pi\)
\(864\) 0 0
\(865\) 71.8219 2.44202
\(866\) 31.3141 1.06410
\(867\) 0 0
\(868\) −2.00000 −0.0678844
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 84.4187 2.86042
\(872\) −7.70156 −0.260808
\(873\) 0 0
\(874\) 5.40312 0.182763
\(875\) −13.7016 −0.463197
\(876\) 0 0
\(877\) −17.4031 −0.587662 −0.293831 0.955857i \(-0.594930\pi\)
−0.293831 + 0.955857i \(0.594930\pi\)
\(878\) −22.0000 −0.742464
\(879\) 0 0
\(880\) −14.8062 −0.499119
\(881\) −6.20937 −0.209199 −0.104600 0.994514i \(-0.533356\pi\)
−0.104600 + 0.994514i \(0.533356\pi\)
\(882\) 0 0
\(883\) −24.5969 −0.827751 −0.413875 0.910334i \(-0.635825\pi\)
−0.413875 + 0.910334i \(0.635825\pi\)
\(884\) −22.8062 −0.767057
\(885\) 0 0
\(886\) 13.7016 0.460313
\(887\) 28.2094 0.947178 0.473589 0.880746i \(-0.342958\pi\)
0.473589 + 0.880746i \(0.342958\pi\)
\(888\) 0 0
\(889\) −5.70156 −0.191224
\(890\) −42.2094 −1.41486
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) −92.9266 −3.10619
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 38.4187 1.28205
\(899\) 0.596876 0.0199069
\(900\) 0 0
\(901\) −37.6125 −1.25305
\(902\) 1.19375 0.0397476
\(903\) 0 0
\(904\) −7.10469 −0.236298
\(905\) 62.2094 2.06791
\(906\) 0 0
\(907\) 20.5078 0.680951 0.340475 0.940253i \(-0.389412\pi\)
0.340475 + 0.940253i \(0.389412\pi\)
\(908\) 23.1047 0.766756
\(909\) 0 0
\(910\) −21.1047 −0.699614
\(911\) −53.1047 −1.75944 −0.879718 0.475495i \(-0.842269\pi\)
−0.879718 + 0.475495i \(0.842269\pi\)
\(912\) 0 0
\(913\) −53.6125 −1.77431
\(914\) 28.2094 0.933083
\(915\) 0 0
\(916\) −14.5969 −0.482294
\(917\) 7.40312 0.244473
\(918\) 0 0
\(919\) −52.4187 −1.72913 −0.864567 0.502517i \(-0.832407\pi\)
−0.864567 + 0.502517i \(0.832407\pi\)
\(920\) 3.70156 0.122037
\(921\) 0 0
\(922\) 2.20937 0.0727618
\(923\) 42.2094 1.38934
\(924\) 0 0
\(925\) 37.4031 1.22981
\(926\) −22.2984 −0.732772
\(927\) 0 0
\(928\) −0.298438 −0.00979670
\(929\) 45.9109 1.50629 0.753144 0.657855i \(-0.228536\pi\)
0.753144 + 0.657855i \(0.228536\pi\)
\(930\) 0 0
\(931\) −5.40312 −0.177080
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) 3.70156 0.121119
\(935\) 59.2250 1.93686
\(936\) 0 0
\(937\) 25.1047 0.820134 0.410067 0.912055i \(-0.365505\pi\)
0.410067 + 0.912055i \(0.365505\pi\)
\(938\) 14.8062 0.483441
\(939\) 0 0
\(940\) −41.1047 −1.34069
\(941\) −25.3141 −0.825215 −0.412607 0.910909i \(-0.635382\pi\)
−0.412607 + 0.910909i \(0.635382\pi\)
\(942\) 0 0
\(943\) −0.298438 −0.00971847
\(944\) 7.40312 0.240951
\(945\) 0 0
\(946\) −6.80625 −0.221290
\(947\) −15.9109 −0.517036 −0.258518 0.966006i \(-0.583234\pi\)
−0.258518 + 0.966006i \(0.583234\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) −47.0156 −1.52539
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) −61.2250 −1.98327 −0.991636 0.129066i \(-0.958802\pi\)
−0.991636 + 0.129066i \(0.958802\pi\)
\(954\) 0 0
\(955\) 84.4187 2.73173
\(956\) −10.8062 −0.349499
\(957\) 0 0
\(958\) −16.5969 −0.536221
\(959\) 19.7016 0.636197
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 24.5078 0.790164
\(963\) 0 0
\(964\) −15.9109 −0.512457
\(965\) 85.5234 2.75310
\(966\) 0 0
\(967\) −14.8062 −0.476137 −0.238068 0.971248i \(-0.576514\pi\)
−0.238068 + 0.971248i \(0.576514\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −23.3141 −0.748569
\(971\) −49.8219 −1.59886 −0.799430 0.600759i \(-0.794865\pi\)
−0.799430 + 0.600759i \(0.794865\pi\)
\(972\) 0 0
\(973\) 17.7016 0.567486
\(974\) −6.29844 −0.201815
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −16.7172 −0.534830 −0.267415 0.963581i \(-0.586169\pi\)
−0.267415 + 0.963581i \(0.586169\pi\)
\(978\) 0 0
\(979\) 45.6125 1.45778
\(980\) −3.70156 −0.118242
\(981\) 0 0
\(982\) 9.61250 0.306747
\(983\) −42.2094 −1.34627 −0.673135 0.739520i \(-0.735053\pi\)
−0.673135 + 0.739520i \(0.735053\pi\)
\(984\) 0 0
\(985\) 83.3141 2.65461
\(986\) 1.19375 0.0380168
\(987\) 0 0
\(988\) −30.8062 −0.980077
\(989\) 1.70156 0.0541065
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 7.40312 0.234813
\(995\) 8.50781 0.269716
\(996\) 0 0
\(997\) −23.4031 −0.741184 −0.370592 0.928796i \(-0.620845\pi\)
−0.370592 + 0.928796i \(0.620845\pi\)
\(998\) −23.4031 −0.740813
\(999\) 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 2898.2.a.bb.1.1 2
3.2 odd 2 966.2.a.n.1.2 2
12.11 even 2 7728.2.a.bc.1.2 2
21.20 even 2 6762.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.2 2 3.2 odd 2
2898.2.a.bb.1.1 2 1.1 even 1 trivial
6762.2.a.bo.1.1 2 21.20 even 2
7728.2.a.bc.1.2 2 12.11 even 2