Properties

Label 1216.2.a.u.1.3
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
Defining polynomial: \(x^{3} - x^{2} - 10 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.08387\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.08387 q^{3} -0.786802 q^{5} +4.29707 q^{7} +6.51027 q^{9} +O(q^{10})\) \(q+3.08387 q^{3} -0.786802 q^{5} +4.29707 q^{7} +6.51027 q^{9} +1.21320 q^{11} -5.08387 q^{13} -2.42640 q^{15} -2.29707 q^{17} +1.00000 q^{19} +13.2516 q^{21} +7.67801 q^{23} -4.38094 q^{25} +10.8252 q^{27} +0.489731 q^{29} +3.74135 q^{33} -3.38094 q^{35} +2.00000 q^{37} -15.6780 q^{39} -4.16774 q^{41} -12.9545 q^{43} -5.12229 q^{45} -5.80734 q^{47} +11.4648 q^{49} -7.08387 q^{51} -1.93667 q^{53} -0.954547 q^{55} +3.08387 q^{57} +11.0839 q^{59} +5.38094 q^{61} +27.9751 q^{63} +4.00000 q^{65} -2.48973 q^{67} +23.6780 q^{69} -11.7413 q^{71} -8.46482 q^{73} -13.5103 q^{75} +5.21320 q^{77} +1.83226 q^{79} +13.8528 q^{81} +7.02054 q^{83} +1.80734 q^{85} +1.51027 q^{87} +13.7413 q^{89} -21.8458 q^{91} -0.786802 q^{95} -3.57360 q^{97} +7.89825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + O(q^{10}) \) \( 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41} - 17 q^{43} + 27 q^{45} - q^{47} + 5 q^{49} - 11 q^{51} - q^{53} + 19 q^{55} - q^{57} + 23 q^{59} - 3 q^{61} + 47 q^{63} + 12 q^{65} - 15 q^{67} + 43 q^{69} - 12 q^{71} + 4 q^{73} - 33 q^{75} + 17 q^{77} + 26 q^{79} + 47 q^{81} + 6 q^{83} - 11 q^{85} - 3 q^{87} + 18 q^{89} - 17 q^{91} - q^{95} - 8 q^{97} + 51 q^{99} + O(q^{100}) \)

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 0
\(3\) 3.08387 1.78047 0.890237 0.455497i \(-0.150538\pi\)
0.890237 + 0.455497i \(0.150538\pi\)
\(4\) 0 0
\(5\) −0.786802 −0.351868 −0.175934 0.984402i \(-0.556295\pi\)
−0.175934 + 0.984402i \(0.556295\pi\)
\(6\) 0 0
\(7\) 4.29707 1.62414 0.812070 0.583560i \(-0.198341\pi\)
0.812070 + 0.583560i \(0.198341\pi\)
\(8\) 0 0
\(9\) 6.51027 2.17009
\(10\) 0 0
\(11\) 1.21320 0.365793 0.182897 0.983132i \(-0.441453\pi\)
0.182897 + 0.983132i \(0.441453\pi\)
\(12\) 0 0
\(13\) −5.08387 −1.41001 −0.705006 0.709201i \(-0.749056\pi\)
−0.705006 + 0.709201i \(0.749056\pi\)
\(14\) 0 0
\(15\) −2.42640 −0.626493
\(16\) 0 0
\(17\) −2.29707 −0.557121 −0.278561 0.960419i \(-0.589857\pi\)
−0.278561 + 0.960419i \(0.589857\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 13.2516 2.89174
\(22\) 0 0
\(23\) 7.67801 1.60098 0.800488 0.599348i \(-0.204574\pi\)
0.800488 + 0.599348i \(0.204574\pi\)
\(24\) 0 0
\(25\) −4.38094 −0.876189
\(26\) 0 0
\(27\) 10.8252 2.08331
\(28\) 0 0
\(29\) 0.489731 0.0909408 0.0454704 0.998966i \(-0.485521\pi\)
0.0454704 + 0.998966i \(0.485521\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.74135 0.651285
\(34\) 0 0
\(35\) −3.38094 −0.571484
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −15.6780 −2.51049
\(40\) 0 0
\(41\) −4.16774 −0.650892 −0.325446 0.945561i \(-0.605514\pi\)
−0.325446 + 0.945561i \(0.605514\pi\)
\(42\) 0 0
\(43\) −12.9545 −1.97555 −0.987775 0.155887i \(-0.950176\pi\)
−0.987775 + 0.155887i \(0.950176\pi\)
\(44\) 0 0
\(45\) −5.12229 −0.763586
\(46\) 0 0
\(47\) −5.80734 −0.847087 −0.423544 0.905876i \(-0.639214\pi\)
−0.423544 + 0.905876i \(0.639214\pi\)
\(48\) 0 0
\(49\) 11.4648 1.63783
\(50\) 0 0
\(51\) −7.08387 −0.991941
\(52\) 0 0
\(53\) −1.93667 −0.266021 −0.133011 0.991115i \(-0.542464\pi\)
−0.133011 + 0.991115i \(0.542464\pi\)
\(54\) 0 0
\(55\) −0.954547 −0.128711
\(56\) 0 0
\(57\) 3.08387 0.408469
\(58\) 0 0
\(59\) 11.0839 1.44300 0.721499 0.692416i \(-0.243454\pi\)
0.721499 + 0.692416i \(0.243454\pi\)
\(60\) 0 0
\(61\) 5.38094 0.688959 0.344480 0.938794i \(-0.388055\pi\)
0.344480 + 0.938794i \(0.388055\pi\)
\(62\) 0 0
\(63\) 27.9751 3.52453
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −2.48973 −0.304169 −0.152085 0.988367i \(-0.548599\pi\)
−0.152085 + 0.988367i \(0.548599\pi\)
\(68\) 0 0
\(69\) 23.6780 2.85050
\(70\) 0 0
\(71\) −11.7413 −1.39344 −0.696721 0.717342i \(-0.745358\pi\)
−0.696721 + 0.717342i \(0.745358\pi\)
\(72\) 0 0
\(73\) −8.46482 −0.990732 −0.495366 0.868684i \(-0.664966\pi\)
−0.495366 + 0.868684i \(0.664966\pi\)
\(74\) 0 0
\(75\) −13.5103 −1.56003
\(76\) 0 0
\(77\) 5.21320 0.594099
\(78\) 0 0
\(79\) 1.83226 0.206145 0.103072 0.994674i \(-0.467133\pi\)
0.103072 + 0.994674i \(0.467133\pi\)
\(80\) 0 0
\(81\) 13.8528 1.53920
\(82\) 0 0
\(83\) 7.02054 0.770604 0.385302 0.922791i \(-0.374097\pi\)
0.385302 + 0.922791i \(0.374097\pi\)
\(84\) 0 0
\(85\) 1.80734 0.196033
\(86\) 0 0
\(87\) 1.51027 0.161918
\(88\) 0 0
\(89\) 13.7413 1.45658 0.728290 0.685269i \(-0.240315\pi\)
0.728290 + 0.685269i \(0.240315\pi\)
\(90\) 0 0
\(91\) −21.8458 −2.29006
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.786802 −0.0807242
\(96\) 0 0
\(97\) −3.57360 −0.362844 −0.181422 0.983405i \(-0.558070\pi\)
−0.181422 + 0.983405i \(0.558070\pi\)
\(98\) 0 0
\(99\) 7.89825 0.793804
\(100\) 0 0
\(101\) 6.33549 0.630405 0.315202 0.949024i \(-0.397928\pi\)
0.315202 + 0.949024i \(0.397928\pi\)
\(102\) 0 0
\(103\) −5.44693 −0.536702 −0.268351 0.963321i \(-0.586479\pi\)
−0.268351 + 0.963321i \(0.586479\pi\)
\(104\) 0 0
\(105\) −10.4264 −1.01751
\(106\) 0 0
\(107\) −3.08387 −0.298129 −0.149065 0.988827i \(-0.547626\pi\)
−0.149065 + 0.988827i \(0.547626\pi\)
\(108\) 0 0
\(109\) −8.10441 −0.776262 −0.388131 0.921604i \(-0.626879\pi\)
−0.388131 + 0.921604i \(0.626879\pi\)
\(110\) 0 0
\(111\) 6.16774 0.585416
\(112\) 0 0
\(113\) 14.3355 1.34857 0.674285 0.738471i \(-0.264452\pi\)
0.674285 + 0.738471i \(0.264452\pi\)
\(114\) 0 0
\(115\) −6.04107 −0.563333
\(116\) 0 0
\(117\) −33.0974 −3.05985
\(118\) 0 0
\(119\) −9.87067 −0.904843
\(120\) 0 0
\(121\) −9.52815 −0.866195
\(122\) 0 0
\(123\) −12.8528 −1.15890
\(124\) 0 0
\(125\) 7.38094 0.660172
\(126\) 0 0
\(127\) −3.74135 −0.331991 −0.165995 0.986127i \(-0.553084\pi\)
−0.165995 + 0.986127i \(0.553084\pi\)
\(128\) 0 0
\(129\) −39.9502 −3.51742
\(130\) 0 0
\(131\) 12.9545 1.13184 0.565922 0.824459i \(-0.308520\pi\)
0.565922 + 0.824459i \(0.308520\pi\)
\(132\) 0 0
\(133\) 4.29707 0.372603
\(134\) 0 0
\(135\) −8.51730 −0.733053
\(136\) 0 0
\(137\) −6.63256 −0.566658 −0.283329 0.959023i \(-0.591439\pi\)
−0.283329 + 0.959023i \(0.591439\pi\)
\(138\) 0 0
\(139\) 4.23374 0.359101 0.179550 0.983749i \(-0.442536\pi\)
0.179550 + 0.983749i \(0.442536\pi\)
\(140\) 0 0
\(141\) −17.9091 −1.50822
\(142\) 0 0
\(143\) −6.16774 −0.515773
\(144\) 0 0
\(145\) −0.385321 −0.0319992
\(146\) 0 0
\(147\) 35.3560 2.91612
\(148\) 0 0
\(149\) 9.63959 0.789706 0.394853 0.918744i \(-0.370795\pi\)
0.394853 + 0.918744i \(0.370795\pi\)
\(150\) 0 0
\(151\) −5.44693 −0.443265 −0.221633 0.975130i \(-0.571139\pi\)
−0.221633 + 0.975130i \(0.571139\pi\)
\(152\) 0 0
\(153\) −14.9545 −1.20900
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −5.97243 −0.473644
\(160\) 0 0
\(161\) 32.9930 2.60021
\(162\) 0 0
\(163\) 16.3355 1.27949 0.639747 0.768585i \(-0.279039\pi\)
0.639747 + 0.768585i \(0.279039\pi\)
\(164\) 0 0
\(165\) −2.94370 −0.229167
\(166\) 0 0
\(167\) −13.5736 −1.05036 −0.525178 0.850992i \(-0.676001\pi\)
−0.525178 + 0.850992i \(0.676001\pi\)
\(168\) 0 0
\(169\) 12.8458 0.988135
\(170\) 0 0
\(171\) 6.51027 0.497853
\(172\) 0 0
\(173\) −18.3355 −1.39402 −0.697011 0.717061i \(-0.745487\pi\)
−0.697011 + 0.717061i \(0.745487\pi\)
\(174\) 0 0
\(175\) −18.8252 −1.42305
\(176\) 0 0
\(177\) 34.1812 2.56922
\(178\) 0 0
\(179\) −19.9502 −1.49115 −0.745573 0.666424i \(-0.767824\pi\)
−0.745573 + 0.666424i \(0.767824\pi\)
\(180\) 0 0
\(181\) 6.85279 0.509364 0.254682 0.967025i \(-0.418029\pi\)
0.254682 + 0.967025i \(0.418029\pi\)
\(182\) 0 0
\(183\) 16.5941 1.22667
\(184\) 0 0
\(185\) −1.57360 −0.115694
\(186\) 0 0
\(187\) −2.78680 −0.203791
\(188\) 0 0
\(189\) 46.5167 3.38359
\(190\) 0 0
\(191\) 19.7798 1.43121 0.715607 0.698503i \(-0.246150\pi\)
0.715607 + 0.698503i \(0.246150\pi\)
\(192\) 0 0
\(193\) −9.61468 −0.692080 −0.346040 0.938220i \(-0.612474\pi\)
−0.346040 + 0.938220i \(0.612474\pi\)
\(194\) 0 0
\(195\) 12.3355 0.883363
\(196\) 0 0
\(197\) −4.16774 −0.296940 −0.148470 0.988917i \(-0.547435\pi\)
−0.148470 + 0.988917i \(0.547435\pi\)
\(198\) 0 0
\(199\) 5.48535 0.388846 0.194423 0.980918i \(-0.437717\pi\)
0.194423 + 0.980918i \(0.437717\pi\)
\(200\) 0 0
\(201\) −7.67801 −0.541565
\(202\) 0 0
\(203\) 2.10441 0.147701
\(204\) 0 0
\(205\) 3.27919 0.229029
\(206\) 0 0
\(207\) 49.9859 3.47426
\(208\) 0 0
\(209\) 1.21320 0.0839187
\(210\) 0 0
\(211\) 1.89559 0.130498 0.0652489 0.997869i \(-0.479216\pi\)
0.0652489 + 0.997869i \(0.479216\pi\)
\(212\) 0 0
\(213\) −36.2088 −2.48099
\(214\) 0 0
\(215\) 10.1927 0.695134
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −26.1044 −1.76397
\(220\) 0 0
\(221\) 11.6780 0.785548
\(222\) 0 0
\(223\) 3.74135 0.250539 0.125270 0.992123i \(-0.460020\pi\)
0.125270 + 0.992123i \(0.460020\pi\)
\(224\) 0 0
\(225\) −28.5211 −1.90141
\(226\) 0 0
\(227\) −19.6780 −1.30608 −0.653038 0.757325i \(-0.726506\pi\)
−0.653038 + 0.757325i \(0.726506\pi\)
\(228\) 0 0
\(229\) −21.7164 −1.43506 −0.717531 0.696526i \(-0.754728\pi\)
−0.717531 + 0.696526i \(0.754728\pi\)
\(230\) 0 0
\(231\) 16.0768 1.05778
\(232\) 0 0
\(233\) −10.5692 −0.692413 −0.346206 0.938158i \(-0.612530\pi\)
−0.346206 + 0.938158i \(0.612530\pi\)
\(234\) 0 0
\(235\) 4.56923 0.298063
\(236\) 0 0
\(237\) 5.65044 0.367036
\(238\) 0 0
\(239\) −16.0384 −1.03744 −0.518720 0.854944i \(-0.673591\pi\)
−0.518720 + 0.854944i \(0.673591\pi\)
\(240\) 0 0
\(241\) −17.0205 −1.09639 −0.548195 0.836351i \(-0.684685\pi\)
−0.548195 + 0.836351i \(0.684685\pi\)
\(242\) 0 0
\(243\) 10.2446 0.657190
\(244\) 0 0
\(245\) −9.02054 −0.576301
\(246\) 0 0
\(247\) −5.08387 −0.323479
\(248\) 0 0
\(249\) 21.6504 1.37204
\(250\) 0 0
\(251\) 19.1223 1.20699 0.603494 0.797367i \(-0.293775\pi\)
0.603494 + 0.797367i \(0.293775\pi\)
\(252\) 0 0
\(253\) 9.31495 0.585626
\(254\) 0 0
\(255\) 5.57360 0.349033
\(256\) 0 0
\(257\) −0.553066 −0.0344993 −0.0172497 0.999851i \(-0.505491\pi\)
−0.0172497 + 0.999851i \(0.505491\pi\)
\(258\) 0 0
\(259\) 8.59414 0.534014
\(260\) 0 0
\(261\) 3.18828 0.197350
\(262\) 0 0
\(263\) 7.04545 0.434441 0.217221 0.976123i \(-0.430301\pi\)
0.217221 + 0.976123i \(0.430301\pi\)
\(264\) 0 0
\(265\) 1.52377 0.0936045
\(266\) 0 0
\(267\) 42.3766 2.59340
\(268\) 0 0
\(269\) −12.7619 −0.778106 −0.389053 0.921215i \(-0.627198\pi\)
−0.389053 + 0.921215i \(0.627198\pi\)
\(270\) 0 0
\(271\) 15.6780 0.952371 0.476186 0.879345i \(-0.342019\pi\)
0.476186 + 0.879345i \(0.342019\pi\)
\(272\) 0 0
\(273\) −67.3695 −4.07739
\(274\) 0 0
\(275\) −5.31495 −0.320504
\(276\) 0 0
\(277\) 5.97508 0.359008 0.179504 0.983757i \(-0.442551\pi\)
0.179504 + 0.983757i \(0.442551\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.2088 1.80211 0.901054 0.433708i \(-0.142795\pi\)
0.901054 + 0.433708i \(0.142795\pi\)
\(282\) 0 0
\(283\) 1.21320 0.0721171 0.0360586 0.999350i \(-0.488520\pi\)
0.0360586 + 0.999350i \(0.488520\pi\)
\(284\) 0 0
\(285\) −2.42640 −0.143727
\(286\) 0 0
\(287\) −17.9091 −1.05714
\(288\) 0 0
\(289\) −11.7235 −0.689616
\(290\) 0 0
\(291\) −11.0205 −0.646035
\(292\) 0 0
\(293\) −0.104410 −0.00609969 −0.00304984 0.999995i \(-0.500971\pi\)
−0.00304984 + 0.999995i \(0.500971\pi\)
\(294\) 0 0
\(295\) −8.72081 −0.507745
\(296\) 0 0
\(297\) 13.1331 0.762062
\(298\) 0 0
\(299\) −39.0340 −2.25740
\(300\) 0 0
\(301\) −55.6666 −3.20857
\(302\) 0 0
\(303\) 19.5378 1.12242
\(304\) 0 0
\(305\) −4.23374 −0.242423
\(306\) 0 0
\(307\) −0.852793 −0.0486715 −0.0243357 0.999704i \(-0.507747\pi\)
−0.0243357 + 0.999704i \(0.507747\pi\)
\(308\) 0 0
\(309\) −16.7976 −0.955585
\(310\) 0 0
\(311\) −6.12933 −0.347562 −0.173781 0.984784i \(-0.555599\pi\)
−0.173781 + 0.984784i \(0.555599\pi\)
\(312\) 0 0
\(313\) −3.17478 −0.179449 −0.0897246 0.995967i \(-0.528599\pi\)
−0.0897246 + 0.995967i \(0.528599\pi\)
\(314\) 0 0
\(315\) −22.0108 −1.24017
\(316\) 0 0
\(317\) 20.6986 1.16255 0.581273 0.813708i \(-0.302555\pi\)
0.581273 + 0.813708i \(0.302555\pi\)
\(318\) 0 0
\(319\) 0.594141 0.0332655
\(320\) 0 0
\(321\) −9.51027 −0.530811
\(322\) 0 0
\(323\) −2.29707 −0.127812
\(324\) 0 0
\(325\) 22.2722 1.23544
\(326\) 0 0
\(327\) −24.9930 −1.38211
\(328\) 0 0
\(329\) −24.9545 −1.37579
\(330\) 0 0
\(331\) −9.84576 −0.541172 −0.270586 0.962696i \(-0.587217\pi\)
−0.270586 + 0.962696i \(0.587217\pi\)
\(332\) 0 0
\(333\) 13.0205 0.713521
\(334\) 0 0
\(335\) 1.95893 0.107028
\(336\) 0 0
\(337\) −18.9296 −1.03116 −0.515581 0.856841i \(-0.672424\pi\)
−0.515581 + 0.856841i \(0.672424\pi\)
\(338\) 0 0
\(339\) 44.2088 2.40109
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.1856 1.03593
\(344\) 0 0
\(345\) −18.6299 −1.00300
\(346\) 0 0
\(347\) 6.78680 0.364335 0.182167 0.983268i \(-0.441689\pi\)
0.182167 + 0.983268i \(0.441689\pi\)
\(348\) 0 0
\(349\) −6.23374 −0.333684 −0.166842 0.985984i \(-0.553357\pi\)
−0.166842 + 0.985984i \(0.553357\pi\)
\(350\) 0 0
\(351\) −55.0340 −2.93750
\(352\) 0 0
\(353\) 31.7191 1.68824 0.844118 0.536157i \(-0.180124\pi\)
0.844118 + 0.536157i \(0.180124\pi\)
\(354\) 0 0
\(355\) 9.23811 0.490308
\(356\) 0 0
\(357\) −30.4399 −1.61105
\(358\) 0 0
\(359\) −23.3944 −1.23471 −0.617356 0.786684i \(-0.711796\pi\)
−0.617356 + 0.786684i \(0.711796\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −29.3836 −1.54224
\(364\) 0 0
\(365\) 6.66013 0.348607
\(366\) 0 0
\(367\) −20.3355 −1.06150 −0.530752 0.847527i \(-0.678090\pi\)
−0.530752 + 0.847527i \(0.678090\pi\)
\(368\) 0 0
\(369\) −27.1331 −1.41249
\(370\) 0 0
\(371\) −8.32199 −0.432056
\(372\) 0 0
\(373\) 36.9021 1.91072 0.955358 0.295450i \(-0.0954697\pi\)
0.955358 + 0.295450i \(0.0954697\pi\)
\(374\) 0 0
\(375\) 22.7619 1.17542
\(376\) 0 0
\(377\) −2.48973 −0.128228
\(378\) 0 0
\(379\) 15.9367 0.818611 0.409306 0.912397i \(-0.365771\pi\)
0.409306 + 0.912397i \(0.365771\pi\)
\(380\) 0 0
\(381\) −11.5378 −0.591101
\(382\) 0 0
\(383\) −19.0205 −0.971904 −0.485952 0.873985i \(-0.661527\pi\)
−0.485952 + 0.873985i \(0.661527\pi\)
\(384\) 0 0
\(385\) −4.10175 −0.209045
\(386\) 0 0
\(387\) −84.3376 −4.28712
\(388\) 0 0
\(389\) −2.61906 −0.132791 −0.0663957 0.997793i \(-0.521150\pi\)
−0.0663957 + 0.997793i \(0.521150\pi\)
\(390\) 0 0
\(391\) −17.6369 −0.891938
\(392\) 0 0
\(393\) 39.9502 2.01522
\(394\) 0 0
\(395\) −1.44162 −0.0725359
\(396\) 0 0
\(397\) 31.1634 1.56404 0.782022 0.623251i \(-0.214188\pi\)
0.782022 + 0.623251i \(0.214188\pi\)
\(398\) 0 0
\(399\) 13.2516 0.663411
\(400\) 0 0
\(401\) 21.7413 1.08571 0.542856 0.839826i \(-0.317343\pi\)
0.542856 + 0.839826i \(0.317343\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −10.8994 −0.541596
\(406\) 0 0
\(407\) 2.42640 0.120272
\(408\) 0 0
\(409\) −23.2651 −1.15039 −0.575193 0.818018i \(-0.695073\pi\)
−0.575193 + 0.818018i \(0.695073\pi\)
\(410\) 0 0
\(411\) −20.4540 −1.00892
\(412\) 0 0
\(413\) 47.6282 2.34363
\(414\) 0 0
\(415\) −5.52377 −0.271151
\(416\) 0 0
\(417\) 13.0563 0.639370
\(418\) 0 0
\(419\) −8.33549 −0.407215 −0.203608 0.979053i \(-0.565267\pi\)
−0.203608 + 0.979053i \(0.565267\pi\)
\(420\) 0 0
\(421\) −0.748383 −0.0364740 −0.0182370 0.999834i \(-0.505805\pi\)
−0.0182370 + 0.999834i \(0.505805\pi\)
\(422\) 0 0
\(423\) −37.8073 −1.83826
\(424\) 0 0
\(425\) 10.0633 0.488143
\(426\) 0 0
\(427\) 23.1223 1.11897
\(428\) 0 0
\(429\) −19.0205 −0.918320
\(430\) 0 0
\(431\) 31.3560 1.51037 0.755183 0.655514i \(-0.227548\pi\)
0.755183 + 0.655514i \(0.227548\pi\)
\(432\) 0 0
\(433\) 9.48270 0.455709 0.227855 0.973695i \(-0.426829\pi\)
0.227855 + 0.973695i \(0.426829\pi\)
\(434\) 0 0
\(435\) −1.18828 −0.0569738
\(436\) 0 0
\(437\) 7.67801 0.367289
\(438\) 0 0
\(439\) −31.9502 −1.52490 −0.762449 0.647048i \(-0.776003\pi\)
−0.762449 + 0.647048i \(0.776003\pi\)
\(440\) 0 0
\(441\) 74.6390 3.55424
\(442\) 0 0
\(443\) −2.91878 −0.138676 −0.0693378 0.997593i \(-0.522089\pi\)
−0.0693378 + 0.997593i \(0.522089\pi\)
\(444\) 0 0
\(445\) −10.8117 −0.512525
\(446\) 0 0
\(447\) 29.7273 1.40605
\(448\) 0 0
\(449\) 0.0909069 0.00429016 0.00214508 0.999998i \(-0.499317\pi\)
0.00214508 + 0.999998i \(0.499317\pi\)
\(450\) 0 0
\(451\) −5.05630 −0.238092
\(452\) 0 0
\(453\) −16.7976 −0.789222
\(454\) 0 0
\(455\) 17.1883 0.805799
\(456\) 0 0
\(457\) 19.1499 0.895793 0.447896 0.894085i \(-0.352173\pi\)
0.447896 + 0.894085i \(0.352173\pi\)
\(458\) 0 0
\(459\) −24.8663 −1.16066
\(460\) 0 0
\(461\) −33.9253 −1.58006 −0.790028 0.613070i \(-0.789934\pi\)
−0.790028 + 0.613070i \(0.789934\pi\)
\(462\) 0 0
\(463\) 27.3809 1.27250 0.636250 0.771483i \(-0.280485\pi\)
0.636250 + 0.771483i \(0.280485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.2132 −0.796532 −0.398266 0.917270i \(-0.630388\pi\)
−0.398266 + 0.917270i \(0.630388\pi\)
\(468\) 0 0
\(469\) −10.6986 −0.494013
\(470\) 0 0
\(471\) −43.1742 −1.98936
\(472\) 0 0
\(473\) −15.7164 −0.722642
\(474\) 0 0
\(475\) −4.38094 −0.201011
\(476\) 0 0
\(477\) −12.6082 −0.577290
\(478\) 0 0
\(479\) 15.4827 0.707422 0.353711 0.935355i \(-0.384920\pi\)
0.353711 + 0.935355i \(0.384920\pi\)
\(480\) 0 0
\(481\) −10.1677 −0.463609
\(482\) 0 0
\(483\) 101.746 4.62961
\(484\) 0 0
\(485\) 2.81172 0.127674
\(486\) 0 0
\(487\) 30.1179 1.36477 0.682386 0.730992i \(-0.260942\pi\)
0.682386 + 0.730992i \(0.260942\pi\)
\(488\) 0 0
\(489\) 50.3766 2.27811
\(490\) 0 0
\(491\) 18.2944 0.825615 0.412808 0.910818i \(-0.364548\pi\)
0.412808 + 0.910818i \(0.364548\pi\)
\(492\) 0 0
\(493\) −1.12495 −0.0506651
\(494\) 0 0
\(495\) −6.21435 −0.279314
\(496\) 0 0
\(497\) −50.4534 −2.26314
\(498\) 0 0
\(499\) −10.5282 −0.471305 −0.235652 0.971837i \(-0.575723\pi\)
−0.235652 + 0.971837i \(0.575723\pi\)
\(500\) 0 0
\(501\) −41.8593 −1.87013
\(502\) 0 0
\(503\) 37.2018 1.65875 0.829373 0.558696i \(-0.188698\pi\)
0.829373 + 0.558696i \(0.188698\pi\)
\(504\) 0 0
\(505\) −4.98477 −0.221820
\(506\) 0 0
\(507\) 39.6147 1.75935
\(508\) 0 0
\(509\) 20.4264 0.905384 0.452692 0.891667i \(-0.350464\pi\)
0.452692 + 0.891667i \(0.350464\pi\)
\(510\) 0 0
\(511\) −36.3739 −1.60909
\(512\) 0 0
\(513\) 10.8252 0.477945
\(514\) 0 0
\(515\) 4.28566 0.188849
\(516\) 0 0
\(517\) −7.04545 −0.309859
\(518\) 0 0
\(519\) −56.5443 −2.48202
\(520\) 0 0
\(521\) −37.8235 −1.65708 −0.828539 0.559932i \(-0.810827\pi\)
−0.828539 + 0.559932i \(0.810827\pi\)
\(522\) 0 0
\(523\) −13.0428 −0.570322 −0.285161 0.958480i \(-0.592047\pi\)
−0.285161 + 0.958480i \(0.592047\pi\)
\(524\) 0 0
\(525\) −58.0546 −2.53371
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 35.9519 1.56313
\(530\) 0 0
\(531\) 72.1590 3.13143
\(532\) 0 0
\(533\) 21.1883 0.917766
\(534\) 0 0
\(535\) 2.42640 0.104902
\(536\) 0 0
\(537\) −61.5238 −2.65495
\(538\) 0 0
\(539\) 13.9091 0.599107
\(540\) 0 0
\(541\) 27.6754 1.18986 0.594928 0.803779i \(-0.297180\pi\)
0.594928 + 0.803779i \(0.297180\pi\)
\(542\) 0 0
\(543\) 21.1331 0.906910
\(544\) 0 0
\(545\) 6.37656 0.273142
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 35.0314 1.49510
\(550\) 0 0
\(551\) 0.489731 0.0208633
\(552\) 0 0
\(553\) 7.87333 0.334808
\(554\) 0 0
\(555\) −4.85279 −0.205990
\(556\) 0 0
\(557\) 36.2695 1.53679 0.768394 0.639977i \(-0.221056\pi\)
0.768394 + 0.639977i \(0.221056\pi\)
\(558\) 0 0
\(559\) 65.8593 2.78555
\(560\) 0 0
\(561\) −8.59414 −0.362845
\(562\) 0 0
\(563\) −13.9091 −0.586198 −0.293099 0.956082i \(-0.594687\pi\)
−0.293099 + 0.956082i \(0.594687\pi\)
\(564\) 0 0
\(565\) −11.2792 −0.474519
\(566\) 0 0
\(567\) 59.5264 2.49987
\(568\) 0 0
\(569\) 29.7413 1.24682 0.623411 0.781894i \(-0.285746\pi\)
0.623411 + 0.781894i \(0.285746\pi\)
\(570\) 0 0
\(571\) 36.6710 1.53463 0.767316 0.641269i \(-0.221592\pi\)
0.767316 + 0.641269i \(0.221592\pi\)
\(572\) 0 0
\(573\) 60.9983 2.54824
\(574\) 0 0
\(575\) −33.6369 −1.40276
\(576\) 0 0
\(577\) 20.4648 0.851961 0.425981 0.904732i \(-0.359929\pi\)
0.425981 + 0.904732i \(0.359929\pi\)
\(578\) 0 0
\(579\) −29.6504 −1.23223
\(580\) 0 0
\(581\) 30.1677 1.25157
\(582\) 0 0
\(583\) −2.34956 −0.0973088
\(584\) 0 0
\(585\) 26.0411 1.07667
\(586\) 0 0
\(587\) −18.2748 −0.754282 −0.377141 0.926156i \(-0.623093\pi\)
−0.377141 + 0.926156i \(0.623093\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.8528 −0.528693
\(592\) 0 0
\(593\) 31.5238 1.29453 0.647263 0.762267i \(-0.275914\pi\)
0.647263 + 0.762267i \(0.275914\pi\)
\(594\) 0 0
\(595\) 7.76626 0.318386
\(596\) 0 0
\(597\) 16.9161 0.692331
\(598\) 0 0
\(599\) −26.3766 −1.07772 −0.538859 0.842396i \(-0.681144\pi\)
−0.538859 + 0.842396i \(0.681144\pi\)
\(600\) 0 0
\(601\) −27.3971 −1.11755 −0.558776 0.829319i \(-0.688729\pi\)
−0.558776 + 0.829319i \(0.688729\pi\)
\(602\) 0 0
\(603\) −16.2088 −0.660074
\(604\) 0 0
\(605\) 7.49677 0.304787
\(606\) 0 0
\(607\) 33.9859 1.37945 0.689723 0.724073i \(-0.257732\pi\)
0.689723 + 0.724073i \(0.257732\pi\)
\(608\) 0 0
\(609\) 6.48973 0.262977
\(610\) 0 0
\(611\) 29.5238 1.19440
\(612\) 0 0
\(613\) −3.08653 −0.124664 −0.0623319 0.998055i \(-0.519854\pi\)
−0.0623319 + 0.998055i \(0.519854\pi\)
\(614\) 0 0
\(615\) 10.1126 0.407779
\(616\) 0 0
\(617\) −26.1018 −1.05082 −0.525409 0.850850i \(-0.676087\pi\)
−0.525409 + 0.850850i \(0.676087\pi\)
\(618\) 0 0
\(619\) −29.0616 −1.16808 −0.584042 0.811723i \(-0.698530\pi\)
−0.584042 + 0.811723i \(0.698530\pi\)
\(620\) 0 0
\(621\) 83.1162 3.33534
\(622\) 0 0
\(623\) 59.0475 2.36569
\(624\) 0 0
\(625\) 16.0974 0.643895
\(626\) 0 0
\(627\) 3.74135 0.149415
\(628\) 0 0
\(629\) −4.59414 −0.183180
\(630\) 0 0
\(631\) 20.1018 0.800238 0.400119 0.916463i \(-0.368969\pi\)
0.400119 + 0.916463i \(0.368969\pi\)
\(632\) 0 0
\(633\) 5.84576 0.232348
\(634\) 0 0
\(635\) 2.94370 0.116817
\(636\) 0 0
\(637\) −58.2857 −2.30936
\(638\) 0 0
\(639\) −76.4393 −3.02389
\(640\) 0 0
\(641\) 24.7619 0.978036 0.489018 0.872274i \(-0.337355\pi\)
0.489018 + 0.872274i \(0.337355\pi\)
\(642\) 0 0
\(643\) 11.8431 0.467046 0.233523 0.972351i \(-0.424975\pi\)
0.233523 + 0.972351i \(0.424975\pi\)
\(644\) 0 0
\(645\) 31.4329 1.23767
\(646\) 0 0
\(647\) 37.4854 1.47370 0.736851 0.676056i \(-0.236312\pi\)
0.736851 + 0.676056i \(0.236312\pi\)
\(648\) 0 0
\(649\) 13.4469 0.527838
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.8279 1.36292 0.681460 0.731855i \(-0.261345\pi\)
0.681460 + 0.731855i \(0.261345\pi\)
\(654\) 0 0
\(655\) −10.1927 −0.398260
\(656\) 0 0
\(657\) −55.1082 −2.14998
\(658\) 0 0
\(659\) −28.2722 −1.10133 −0.550663 0.834727i \(-0.685625\pi\)
−0.550663 + 0.834727i \(0.685625\pi\)
\(660\) 0 0
\(661\) −2.58064 −0.100375 −0.0501876 0.998740i \(-0.515982\pi\)
−0.0501876 + 0.998740i \(0.515982\pi\)
\(662\) 0 0
\(663\) 36.0135 1.39865
\(664\) 0 0
\(665\) −3.38094 −0.131107
\(666\) 0 0
\(667\) 3.76016 0.145594
\(668\) 0 0
\(669\) 11.5378 0.446079
\(670\) 0 0
\(671\) 6.52815 0.252016
\(672\) 0 0
\(673\) 50.5443 1.94834 0.974170 0.225816i \(-0.0725048\pi\)
0.974170 + 0.225816i \(0.0725048\pi\)
\(674\) 0 0
\(675\) −47.4247 −1.82538
\(676\) 0 0
\(677\) −0.825221 −0.0317158 −0.0158579 0.999874i \(-0.505048\pi\)
−0.0158579 + 0.999874i \(0.505048\pi\)
\(678\) 0 0
\(679\) −15.3560 −0.589310
\(680\) 0 0
\(681\) −60.6845 −2.32543
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 5.21851 0.199389
\(686\) 0 0
\(687\) −66.9707 −2.55509
\(688\) 0 0
\(689\) 9.84576 0.375094
\(690\) 0 0
\(691\) −0.0249160 −0.000947850 0 −0.000473925 1.00000i \(-0.500151\pi\)
−0.000473925 1.00000i \(0.500151\pi\)
\(692\) 0 0
\(693\) 33.9393 1.28925
\(694\) 0 0
\(695\) −3.33111 −0.126356
\(696\) 0 0
\(697\) 9.57360 0.362626
\(698\) 0 0
\(699\) −32.5941 −1.23282
\(700\) 0 0
\(701\) −29.3560 −1.10876 −0.554381 0.832263i \(-0.687045\pi\)
−0.554381 + 0.832263i \(0.687045\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 14.0909 0.530694
\(706\) 0 0
\(707\) 27.2240 1.02387
\(708\) 0 0
\(709\) 17.4827 0.656576 0.328288 0.944578i \(-0.393528\pi\)
0.328288 + 0.944578i \(0.393528\pi\)
\(710\) 0 0
\(711\) 11.9285 0.447353
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 4.85279 0.181484
\(716\) 0 0
\(717\) −49.4604 −1.84713
\(718\) 0 0
\(719\) 18.5915 0.693345 0.346673 0.937986i \(-0.387311\pi\)
0.346673 + 0.937986i \(0.387311\pi\)
\(720\) 0 0
\(721\) −23.4059 −0.871680
\(722\) 0 0
\(723\) −52.4892 −1.95209
\(724\) 0 0
\(725\) −2.14548 −0.0796813
\(726\) 0 0
\(727\) −8.50589 −0.315466 −0.157733 0.987482i \(-0.550419\pi\)
−0.157733 + 0.987482i \(0.550419\pi\)
\(728\) 0 0
\(729\) −9.96539 −0.369089
\(730\) 0 0
\(731\) 29.7575 1.10062
\(732\) 0 0
\(733\) −24.5032 −0.905048 −0.452524 0.891752i \(-0.649476\pi\)
−0.452524 + 0.891752i \(0.649476\pi\)
\(734\) 0 0
\(735\) −27.8182 −1.02609
\(736\) 0 0
\(737\) −3.02054 −0.111263
\(738\) 0 0
\(739\) 20.9047 0.768992 0.384496 0.923127i \(-0.374375\pi\)
0.384496 + 0.923127i \(0.374375\pi\)
\(740\) 0 0
\(741\) −15.6780 −0.575946
\(742\) 0 0
\(743\) 21.5238 0.789631 0.394815 0.918761i \(-0.370809\pi\)
0.394815 + 0.918761i \(0.370809\pi\)
\(744\) 0 0
\(745\) −7.58445 −0.277873
\(746\) 0 0
\(747\) 45.7056 1.67228
\(748\) 0 0
\(749\) −13.2516 −0.484204
\(750\) 0 0
\(751\) 2.37656 0.0867221 0.0433610 0.999059i \(-0.486193\pi\)
0.0433610 + 0.999059i \(0.486193\pi\)
\(752\) 0 0
\(753\) 58.9707 2.14901
\(754\) 0 0
\(755\) 4.28566 0.155971
\(756\) 0 0
\(757\) −46.9545 −1.70659 −0.853296 0.521427i \(-0.825400\pi\)
−0.853296 + 0.521427i \(0.825400\pi\)
\(758\) 0 0
\(759\) 28.7261 1.04269
\(760\) 0 0
\(761\) −5.44428 −0.197355 −0.0986775 0.995119i \(-0.531461\pi\)
−0.0986775 + 0.995119i \(0.531461\pi\)
\(762\) 0 0
\(763\) −34.8252 −1.26076
\(764\) 0 0
\(765\) 11.7663 0.425410
\(766\) 0 0
\(767\) −56.3490 −2.03464
\(768\) 0 0
\(769\) 8.92697 0.321915 0.160957 0.986961i \(-0.448542\pi\)
0.160957 + 0.986961i \(0.448542\pi\)
\(770\) 0 0
\(771\) −1.70559 −0.0614252
\(772\) 0 0
\(773\) 17.8047 0.640390 0.320195 0.947352i \(-0.396252\pi\)
0.320195 + 0.947352i \(0.396252\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 26.5032 0.950798
\(778\) 0 0
\(779\) −4.16774 −0.149325
\(780\) 0 0
\(781\) −14.2446 −0.509711
\(782\) 0 0
\(783\) 5.30145 0.189458
\(784\) 0 0
\(785\) 11.0152 0.393150
\(786\) 0 0
\(787\) −13.3836 −0.477074 −0.238537 0.971133i \(-0.576668\pi\)
−0.238537 + 0.971133i \(0.576668\pi\)
\(788\) 0 0
\(789\) 21.7273 0.773512
\(790\) 0 0
\(791\) 61.6006 2.19027
\(792\) 0 0
\(793\) −27.3560 −0.971441
\(794\) 0 0
\(795\) 4.69912 0.166661
\(796\) 0 0
\(797\) −28.5666 −1.01188 −0.505940 0.862569i \(-0.668854\pi\)
−0.505940 + 0.862569i \(0.668854\pi\)
\(798\) 0 0
\(799\) 13.3399 0.471931
\(800\) 0 0
\(801\) 89.4599 3.16091
\(802\) 0 0
\(803\) −10.2695 −0.362403
\(804\) 0 0
\(805\) −25.9589 −0.914932
\(806\) 0 0
\(807\) −39.3560 −1.38540
\(808\) 0 0
\(809\) 45.2625 1.59134 0.795672 0.605728i \(-0.207118\pi\)
0.795672 + 0.605728i \(0.207118\pi\)
\(810\) 0 0
\(811\) 49.4551 1.73660 0.868302 0.496036i \(-0.165211\pi\)
0.868302 + 0.496036i \(0.165211\pi\)
\(812\) 0 0
\(813\) 48.3490 1.69567
\(814\) 0 0
\(815\) −12.8528 −0.450214
\(816\) 0 0
\(817\) −12.9545 −0.453222
\(818\) 0 0
\(819\) −142.222 −4.96963
\(820\) 0 0
\(821\) 41.7933 1.45860 0.729298 0.684197i \(-0.239847\pi\)
0.729298 + 0.684197i \(0.239847\pi\)
\(822\) 0 0
\(823\) −8.68239 −0.302649 −0.151325 0.988484i \(-0.548354\pi\)
−0.151325 + 0.988484i \(0.548354\pi\)
\(824\) 0 0
\(825\) −16.3906 −0.570649
\(826\) 0 0
\(827\) −13.1196 −0.456214 −0.228107 0.973636i \(-0.573254\pi\)
−0.228107 + 0.973636i \(0.573254\pi\)
\(828\) 0 0
\(829\) 8.83053 0.306697 0.153349 0.988172i \(-0.450994\pi\)
0.153349 + 0.988172i \(0.450994\pi\)
\(830\) 0 0
\(831\) 18.4264 0.639205
\(832\) 0 0
\(833\) −26.3355 −0.912471
\(834\) 0 0
\(835\) 10.6797 0.369588
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.80296 0.165817 0.0829083 0.996557i \(-0.473579\pi\)
0.0829083 + 0.996557i \(0.473579\pi\)
\(840\) 0 0
\(841\) −28.7602 −0.991730
\(842\) 0 0
\(843\) 93.1601 3.20861
\(844\) 0 0
\(845\) −10.1071 −0.347694
\(846\) 0 0
\(847\) −40.9431 −1.40682
\(848\) 0 0
\(849\) 3.74135 0.128403
\(850\) 0 0
\(851\) 15.3560 0.526398
\(852\) 0 0
\(853\) 2.51730 0.0861908 0.0430954 0.999071i \(-0.486278\pi\)
0.0430954 + 0.999071i \(0.486278\pi\)
\(854\) 0 0
\(855\) −5.12229 −0.175179
\(856\) 0 0
\(857\) −37.3560 −1.27606 −0.638029 0.770013i \(-0.720250\pi\)
−0.638029 + 0.770013i \(0.720250\pi\)
\(858\) 0 0
\(859\) −40.6461 −1.38683 −0.693413 0.720540i \(-0.743894\pi\)
−0.693413 + 0.720540i \(0.743894\pi\)
\(860\) 0 0
\(861\) −55.2294 −1.88221
\(862\) 0 0
\(863\) −7.40586 −0.252098 −0.126049 0.992024i \(-0.540230\pi\)
−0.126049 + 0.992024i \(0.540230\pi\)
\(864\) 0 0
\(865\) 14.4264 0.490512
\(866\) 0 0
\(867\) −36.1537 −1.22784
\(868\) 0 0
\(869\) 2.22289 0.0754063
\(870\) 0 0
\(871\) 12.6575 0.428882
\(872\) 0 0
\(873\) −23.2651 −0.787405
\(874\) 0 0
\(875\) 31.7164 1.07221
\(876\) 0 0
\(877\) −50.5308 −1.70630 −0.853152 0.521662i \(-0.825312\pi\)
−0.853152 + 0.521662i \(0.825312\pi\)
\(878\) 0 0
\(879\) −0.321987 −0.0108603
\(880\) 0 0
\(881\) 33.4631 1.12740 0.563700 0.825980i \(-0.309377\pi\)
0.563700 + 0.825980i \(0.309377\pi\)
\(882\) 0 0
\(883\) −30.0162 −1.01012 −0.505062 0.863083i \(-0.668530\pi\)
−0.505062 + 0.863083i \(0.668530\pi\)
\(884\) 0 0
\(885\) −26.8939 −0.904027
\(886\) 0 0
\(887\) 19.0704 0.640320 0.320160 0.947363i \(-0.396263\pi\)
0.320160 + 0.947363i \(0.396263\pi\)
\(888\) 0 0
\(889\) −16.0768 −0.539200
\(890\) 0 0
\(891\) 16.8062 0.563028
\(892\) 0 0
\(893\) −5.80734 −0.194335
\(894\) 0 0
\(895\) 15.6968 0.524687
\(896\) 0 0
\(897\) −120.376 −4.01924
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 4.44866 0.148206
\(902\) 0 0
\(903\) −171.669 −5.71278
\(904\) 0 0
\(905\) −5.39179 −0.179229
\(906\) 0 0
\(907\) 48.0135 1.59426 0.797131 0.603806i \(-0.206350\pi\)
0.797131 + 0.603806i \(0.206350\pi\)
\(908\) 0 0
\(909\) 41.2457 1.36803
\(910\) 0 0
\(911\) −46.2446 −1.53215 −0.766076 0.642750i \(-0.777793\pi\)
−0.766076 + 0.642750i \(0.777793\pi\)
\(912\) 0 0
\(913\) 8.51730 0.281882
\(914\) 0 0
\(915\) −13.0563 −0.431628
\(916\) 0 0
\(917\) 55.6666 1.83827
\(918\) 0 0
\(919\) 41.5103 1.36930 0.684649 0.728873i \(-0.259956\pi\)
0.684649 + 0.728873i \(0.259956\pi\)
\(920\) 0 0
\(921\) −2.62990 −0.0866583
\(922\) 0 0
\(923\) 59.6915 1.96477
\(924\) 0 0
\(925\) −8.76189 −0.288089
\(926\) 0 0
\(927\) −35.4610 −1.16469
\(928\) 0 0
\(929\) 55.4551 1.81942 0.909712 0.415240i \(-0.136302\pi\)
0.909712 + 0.415240i \(0.136302\pi\)
\(930\) 0 0
\(931\) 11.4648 0.375744
\(932\) 0 0
\(933\) −18.9021 −0.618826
\(934\) 0 0
\(935\) 2.19266 0.0717077
\(936\) 0 0
\(937\) −6.08825 −0.198894 −0.0994472 0.995043i \(-0.531707\pi\)
−0.0994472 + 0.995043i \(0.531707\pi\)
\(938\) 0 0
\(939\) −9.79061 −0.319505
\(940\) 0 0
\(941\) 11.3836 0.371095 0.185547 0.982635i \(-0.440594\pi\)
0.185547 + 0.982635i \(0.440594\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) −36.5995 −1.19058
\(946\) 0 0
\(947\) 38.3766 1.24707 0.623535 0.781795i \(-0.285696\pi\)
0.623535 + 0.781795i \(0.285696\pi\)
\(948\) 0 0
\(949\) 43.0340 1.39694
\(950\) 0 0
\(951\) 63.8317 2.06988
\(952\) 0 0
\(953\) 30.2857 0.981049 0.490524 0.871427i \(-0.336805\pi\)
0.490524 + 0.871427i \(0.336805\pi\)
\(954\) 0 0
\(955\) −15.5628 −0.503599
\(956\) 0 0
\(957\) 1.83226 0.0592284
\(958\) 0 0
\(959\) −28.5006 −0.920332
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −20.0768 −0.646967
\(964\) 0 0
\(965\) 7.56485 0.243521
\(966\) 0 0
\(967\) −24.6710 −0.793365 −0.396683 0.917956i \(-0.629839\pi\)
−0.396683 + 0.917956i \(0.629839\pi\)
\(968\) 0 0
\(969\) −7.08387 −0.227567
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 18.1927 0.583230
\(974\) 0 0
\(975\) 68.6845 2.19966
\(976\) 0 0
\(977\) −21.4059 −0.684834 −0.342417 0.939548i \(-0.611246\pi\)
−0.342417 + 0.939548i \(0.611246\pi\)
\(978\) 0 0
\(979\) 16.6710 0.532807
\(980\) 0 0
\(981\) −52.7619 −1.68456
\(982\) 0 0
\(983\) 38.1677 1.21736 0.608681 0.793415i \(-0.291699\pi\)
0.608681 + 0.793415i \(0.291699\pi\)
\(984\) 0 0
\(985\) 3.27919 0.104484
\(986\) 0 0
\(987\) −76.9566 −2.44956
\(988\) 0 0
\(989\) −99.4652 −3.16281
\(990\) 0 0
\(991\) 16.4675 0.523106 0.261553 0.965189i \(-0.415765\pi\)
0.261553 + 0.965189i \(0.415765\pi\)
\(992\) 0 0
\(993\) −30.3631 −0.963543
\(994\) 0 0
\(995\) −4.31589 −0.136823
\(996\) 0 0
\(997\) −13.8431 −0.438415 −0.219208 0.975678i \(-0.570347\pi\)
−0.219208 + 0.975678i \(0.570347\pi\)
\(998\) 0 0
\(999\) 21.6504 0.684990
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 1216.2.a.u.1.3 3
4.3 odd 2 1216.2.a.v.1.1 3
8.3 odd 2 304.2.a.g.1.3 3
8.5 even 2 152.2.a.c.1.1 3
24.5 odd 2 1368.2.a.n.1.2 3
24.11 even 2 2736.2.a.bd.1.2 3
40.13 odd 4 3800.2.d.j.3649.2 6
40.19 odd 2 7600.2.a.bv.1.1 3
40.29 even 2 3800.2.a.r.1.3 3
40.37 odd 4 3800.2.d.j.3649.5 6
56.13 odd 2 7448.2.a.bf.1.3 3
152.37 odd 2 2888.2.a.o.1.3 3
152.75 even 2 5776.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.1 3 8.5 even 2
304.2.a.g.1.3 3 8.3 odd 2
1216.2.a.u.1.3 3 1.1 even 1 trivial
1216.2.a.v.1.1 3 4.3 odd 2
1368.2.a.n.1.2 3 24.5 odd 2
2736.2.a.bd.1.2 3 24.11 even 2
2888.2.a.o.1.3 3 152.37 odd 2
3800.2.a.r.1.3 3 40.29 even 2
3800.2.d.j.3649.2 6 40.13 odd 4
3800.2.d.j.3649.5 6 40.37 odd 4
5776.2.a.bp.1.1 3 152.75 even 2
7448.2.a.bf.1.3 3 56.13 odd 2
7600.2.a.bv.1.1 3 40.19 odd 2