Properties

Label 1216.2.a.w.1.1
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69353\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +0.644529 q^{11} +3.94860 q^{13} +6.77410 q^{15} +5.56802 q^{17} -1.00000 q^{19} +0.463560 q^{21} -4.46356 q^{23} +1.99353 q^{25} -1.43845 q^{27} +3.94860 q^{29} -5.48504 q^{31} -1.65100 q^{33} +0.478577 q^{35} -7.48504 q^{37} -10.1146 q^{39} +8.41216 q^{41} +5.76763 q^{43} -9.41863 q^{45} -11.2527 q^{47} -6.96725 q^{49} -14.2628 q^{51} -7.58667 q^{53} -1.70448 q^{55} +2.56155 q^{57} -11.8506 q^{59} +2.47858 q^{61} -0.644529 q^{63} -10.4422 q^{65} +8.97372 q^{67} +11.4336 q^{69} -7.63806 q^{71} +10.6911 q^{73} -5.10654 q^{75} -0.116639 q^{77} +10.6081 q^{79} -7.00000 q^{81} -5.65100 q^{83} -14.7248 q^{85} -10.1146 q^{87} -17.1733 q^{89} -0.714573 q^{91} +14.0502 q^{93} +2.64453 q^{95} +3.28906 q^{97} +2.29552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - q^{5} - q^{7} + 6q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - q^{5} - q^{7} + 6q^{9} - 7q^{11} - 10q^{13} - 8q^{15} + 5q^{17} - 4q^{19} - 8q^{21} - 8q^{23} + 17q^{25} - 14q^{27} - 10q^{29} - 6q^{31} + 12q^{33} - 5q^{35} - 14q^{37} - 12q^{39} - 2q^{41} - 3q^{43} + 7q^{45} - 3q^{47} + 7q^{49} + 6q^{51} - 4q^{53} - 35q^{55} + 2q^{57} - 20q^{59} + 3q^{61} + 7q^{63} - 12q^{65} - 8q^{67} + 4q^{69} - 30q^{71} + 9q^{73} - 34q^{75} + 7q^{77} + 10q^{79} - 28q^{81} - 4q^{83} - 19q^{85} - 12q^{87} - 16q^{89} - 10q^{91} + 20q^{93} + q^{95} - 6q^{97} - 19q^{99} + O(q^{100}) \)

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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) −2.64453 −1.18267 −0.591335 0.806426i \(-0.701399\pi\)
−0.591335 + 0.806426i \(0.701399\pi\)
\(6\) 0 0
\(7\) −0.180969 −0.0683997 −0.0341998 0.999415i \(-0.510888\pi\)
−0.0341998 + 0.999415i \(0.510888\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 0.644529 0.194333 0.0971664 0.995268i \(-0.469022\pi\)
0.0971664 + 0.995268i \(0.469022\pi\)
\(12\) 0 0
\(13\) 3.94860 1.09515 0.547573 0.836758i \(-0.315552\pi\)
0.547573 + 0.836758i \(0.315552\pi\)
\(14\) 0 0
\(15\) 6.77410 1.74907
\(16\) 0 0
\(17\) 5.56802 1.35044 0.675221 0.737615i \(-0.264048\pi\)
0.675221 + 0.737615i \(0.264048\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.463560 0.101157
\(22\) 0 0
\(23\) −4.46356 −0.930717 −0.465358 0.885122i \(-0.654075\pi\)
−0.465358 + 0.885122i \(0.654075\pi\)
\(24\) 0 0
\(25\) 1.99353 0.398707
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) 3.94860 0.733237 0.366619 0.930371i \(-0.380515\pi\)
0.366619 + 0.930371i \(0.380515\pi\)
\(30\) 0 0
\(31\) −5.48504 −0.985143 −0.492571 0.870272i \(-0.663943\pi\)
−0.492571 + 0.870272i \(0.663943\pi\)
\(32\) 0 0
\(33\) −1.65100 −0.287401
\(34\) 0 0
\(35\) 0.478577 0.0808942
\(36\) 0 0
\(37\) −7.48504 −1.23053 −0.615267 0.788319i \(-0.710952\pi\)
−0.615267 + 0.788319i \(0.710952\pi\)
\(38\) 0 0
\(39\) −10.1146 −1.61963
\(40\) 0 0
\(41\) 8.41216 1.31376 0.656880 0.753995i \(-0.271876\pi\)
0.656880 + 0.753995i \(0.271876\pi\)
\(42\) 0 0
\(43\) 5.76763 0.879556 0.439778 0.898107i \(-0.355057\pi\)
0.439778 + 0.898107i \(0.355057\pi\)
\(44\) 0 0
\(45\) −9.41863 −1.40405
\(46\) 0 0
\(47\) −11.2527 −1.64137 −0.820686 0.571380i \(-0.806408\pi\)
−0.820686 + 0.571380i \(0.806408\pi\)
\(48\) 0 0
\(49\) −6.96725 −0.995321
\(50\) 0 0
\(51\) −14.2628 −1.99719
\(52\) 0 0
\(53\) −7.58667 −1.04211 −0.521054 0.853523i \(-0.674461\pi\)
−0.521054 + 0.853523i \(0.674461\pi\)
\(54\) 0 0
\(55\) −1.70448 −0.229831
\(56\) 0 0
\(57\) 2.56155 0.339286
\(58\) 0 0
\(59\) −11.8506 −1.54282 −0.771409 0.636340i \(-0.780448\pi\)
−0.771409 + 0.636340i \(0.780448\pi\)
\(60\) 0 0
\(61\) 2.47858 0.317349 0.158675 0.987331i \(-0.449278\pi\)
0.158675 + 0.987331i \(0.449278\pi\)
\(62\) 0 0
\(63\) −0.644529 −0.0812030
\(64\) 0 0
\(65\) −10.4422 −1.29519
\(66\) 0 0
\(67\) 8.97372 1.09631 0.548157 0.836375i \(-0.315330\pi\)
0.548157 + 0.836375i \(0.315330\pi\)
\(68\) 0 0
\(69\) 11.4336 1.37645
\(70\) 0 0
\(71\) −7.63806 −0.906471 −0.453236 0.891391i \(-0.649730\pi\)
−0.453236 + 0.891391i \(0.649730\pi\)
\(72\) 0 0
\(73\) 10.6911 1.25130 0.625651 0.780103i \(-0.284834\pi\)
0.625651 + 0.780103i \(0.284834\pi\)
\(74\) 0 0
\(75\) −5.10654 −0.589653
\(76\) 0 0
\(77\) −0.116639 −0.0132923
\(78\) 0 0
\(79\) 10.6081 1.19351 0.596755 0.802424i \(-0.296456\pi\)
0.596755 + 0.802424i \(0.296456\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −5.65100 −0.620277 −0.310139 0.950691i \(-0.600375\pi\)
−0.310139 + 0.950691i \(0.600375\pi\)
\(84\) 0 0
\(85\) −14.7248 −1.59713
\(86\) 0 0
\(87\) −10.1146 −1.08439
\(88\) 0 0
\(89\) −17.1733 −1.82037 −0.910185 0.414202i \(-0.864061\pi\)
−0.910185 + 0.414202i \(0.864061\pi\)
\(90\) 0 0
\(91\) −0.714573 −0.0749076
\(92\) 0 0
\(93\) 14.0502 1.45694
\(94\) 0 0
\(95\) 2.64453 0.271323
\(96\) 0 0
\(97\) 3.28906 0.333953 0.166977 0.985961i \(-0.446600\pi\)
0.166977 + 0.985961i \(0.446600\pi\)
\(98\) 0 0
\(99\) 2.29552 0.230709
\(100\) 0 0
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 0 0
\(103\) 14.4122 1.42007 0.710036 0.704165i \(-0.248678\pi\)
0.710036 + 0.704165i \(0.248678\pi\)
\(104\) 0 0
\(105\) −1.22590 −0.119636
\(106\) 0 0
\(107\) −9.69759 −0.937501 −0.468751 0.883330i \(-0.655296\pi\)
−0.468751 + 0.883330i \(0.655296\pi\)
\(108\) 0 0
\(109\) −19.4839 −1.86622 −0.933108 0.359596i \(-0.882915\pi\)
−0.933108 + 0.359596i \(0.882915\pi\)
\(110\) 0 0
\(111\) 19.1733 1.81985
\(112\) 0 0
\(113\) −7.28906 −0.685697 −0.342848 0.939391i \(-0.611392\pi\)
−0.342848 + 0.939391i \(0.611392\pi\)
\(114\) 0 0
\(115\) 11.8040 1.10073
\(116\) 0 0
\(117\) 14.0632 1.30014
\(118\) 0 0
\(119\) −1.00764 −0.0923699
\(120\) 0 0
\(121\) −10.5846 −0.962235
\(122\) 0 0
\(123\) −21.5482 −1.94294
\(124\) 0 0
\(125\) 7.95069 0.711131
\(126\) 0 0
\(127\) −0.595216 −0.0528169 −0.0264084 0.999651i \(-0.508407\pi\)
−0.0264084 + 0.999651i \(0.508407\pi\)
\(128\) 0 0
\(129\) −14.7741 −1.30079
\(130\) 0 0
\(131\) 15.7806 1.37875 0.689377 0.724403i \(-0.257884\pi\)
0.689377 + 0.724403i \(0.257884\pi\)
\(132\) 0 0
\(133\) 0.180969 0.0156920
\(134\) 0 0
\(135\) 3.80402 0.327398
\(136\) 0 0
\(137\) −15.9802 −1.36528 −0.682640 0.730755i \(-0.739168\pi\)
−0.682640 + 0.730755i \(0.739168\pi\)
\(138\) 0 0
\(139\) −4.47858 −0.379868 −0.189934 0.981797i \(-0.560827\pi\)
−0.189934 + 0.981797i \(0.560827\pi\)
\(140\) 0 0
\(141\) 28.8243 2.42745
\(142\) 0 0
\(143\) 2.54499 0.212823
\(144\) 0 0
\(145\) −10.4422 −0.867177
\(146\) 0 0
\(147\) 17.8470 1.47199
\(148\) 0 0
\(149\) −10.3758 −0.850017 −0.425009 0.905189i \(-0.639729\pi\)
−0.425009 + 0.905189i \(0.639729\pi\)
\(150\) 0 0
\(151\) −16.6584 −1.35564 −0.677820 0.735228i \(-0.737075\pi\)
−0.677820 + 0.735228i \(0.737075\pi\)
\(152\) 0 0
\(153\) 19.8308 1.60322
\(154\) 0 0
\(155\) 14.5054 1.16510
\(156\) 0 0
\(157\) 11.0203 0.879517 0.439758 0.898116i \(-0.355064\pi\)
0.439758 + 0.898116i \(0.355064\pi\)
\(158\) 0 0
\(159\) 19.4336 1.54119
\(160\) 0 0
\(161\) 0.807764 0.0636607
\(162\) 0 0
\(163\) 7.80402 0.611258 0.305629 0.952151i \(-0.401133\pi\)
0.305629 + 0.952151i \(0.401133\pi\)
\(164\) 0 0
\(165\) 4.36610 0.339901
\(166\) 0 0
\(167\) −10.2462 −0.792876 −0.396438 0.918062i \(-0.629754\pi\)
−0.396438 + 0.918062i \(0.629754\pi\)
\(168\) 0 0
\(169\) 2.59147 0.199344
\(170\) 0 0
\(171\) −3.56155 −0.272359
\(172\) 0 0
\(173\) 1.80402 0.137157 0.0685784 0.997646i \(-0.478154\pi\)
0.0685784 + 0.997646i \(0.478154\pi\)
\(174\) 0 0
\(175\) −0.360767 −0.0272714
\(176\) 0 0
\(177\) 30.3560 2.28169
\(178\) 0 0
\(179\) −1.75379 −0.131084 −0.0655422 0.997850i \(-0.520878\pi\)
−0.0655422 + 0.997850i \(0.520878\pi\)
\(180\) 0 0
\(181\) −6.52789 −0.485214 −0.242607 0.970125i \(-0.578003\pi\)
−0.242607 + 0.970125i \(0.578003\pi\)
\(182\) 0 0
\(183\) −6.34900 −0.469332
\(184\) 0 0
\(185\) 19.7944 1.45531
\(186\) 0 0
\(187\) 3.58875 0.262435
\(188\) 0 0
\(189\) 0.260314 0.0189350
\(190\) 0 0
\(191\) −7.45709 −0.539576 −0.269788 0.962920i \(-0.586954\pi\)
−0.269788 + 0.962920i \(0.586954\pi\)
\(192\) 0 0
\(193\) 19.0203 1.36911 0.684556 0.728960i \(-0.259996\pi\)
0.684556 + 0.728960i \(0.259996\pi\)
\(194\) 0 0
\(195\) 26.7482 1.91548
\(196\) 0 0
\(197\) −0.153020 −0.0109022 −0.00545112 0.999985i \(-0.501735\pi\)
−0.00545112 + 0.999985i \(0.501735\pi\)
\(198\) 0 0
\(199\) 21.3672 1.51468 0.757341 0.653019i \(-0.226498\pi\)
0.757341 + 0.653019i \(0.226498\pi\)
\(200\) 0 0
\(201\) −22.9866 −1.62135
\(202\) 0 0
\(203\) −0.714573 −0.0501532
\(204\) 0 0
\(205\) −22.2462 −1.55374
\(206\) 0 0
\(207\) −15.8972 −1.10493
\(208\) 0 0
\(209\) −0.644529 −0.0445830
\(210\) 0 0
\(211\) −6.82070 −0.469556 −0.234778 0.972049i \(-0.575436\pi\)
−0.234778 + 0.972049i \(0.575436\pi\)
\(212\) 0 0
\(213\) 19.5653 1.34059
\(214\) 0 0
\(215\) −15.2527 −1.04022
\(216\) 0 0
\(217\) 0.992620 0.0673835
\(218\) 0 0
\(219\) −27.3859 −1.85057
\(220\) 0 0
\(221\) 21.9859 1.47893
\(222\) 0 0
\(223\) −8.42510 −0.564186 −0.282093 0.959387i \(-0.591029\pi\)
−0.282093 + 0.959387i \(0.591029\pi\)
\(224\) 0 0
\(225\) 7.10008 0.473338
\(226\) 0 0
\(227\) −23.3859 −1.55218 −0.776088 0.630625i \(-0.782799\pi\)
−0.776088 + 0.630625i \(0.782799\pi\)
\(228\) 0 0
\(229\) 3.37567 0.223070 0.111535 0.993760i \(-0.464423\pi\)
0.111535 + 0.993760i \(0.464423\pi\)
\(230\) 0 0
\(231\) 0.298778 0.0196582
\(232\) 0 0
\(233\) −15.8308 −1.03711 −0.518555 0.855044i \(-0.673530\pi\)
−0.518555 + 0.855044i \(0.673530\pi\)
\(234\) 0 0
\(235\) 29.7580 1.94120
\(236\) 0 0
\(237\) −27.1733 −1.76510
\(238\) 0 0
\(239\) 2.39715 0.155059 0.0775293 0.996990i \(-0.475297\pi\)
0.0775293 + 0.996990i \(0.475297\pi\)
\(240\) 0 0
\(241\) 5.44208 0.350555 0.175278 0.984519i \(-0.443918\pi\)
0.175278 + 0.984519i \(0.443918\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) 18.4251 1.17714
\(246\) 0 0
\(247\) −3.94860 −0.251244
\(248\) 0 0
\(249\) 14.4753 0.917336
\(250\) 0 0
\(251\) −12.8405 −0.810486 −0.405243 0.914209i \(-0.632813\pi\)
−0.405243 + 0.914209i \(0.632813\pi\)
\(252\) 0 0
\(253\) −2.87689 −0.180869
\(254\) 0 0
\(255\) 37.7183 2.36201
\(256\) 0 0
\(257\) −19.1863 −1.19681 −0.598403 0.801195i \(-0.704198\pi\)
−0.598403 + 0.801195i \(0.704198\pi\)
\(258\) 0 0
\(259\) 1.35456 0.0841681
\(260\) 0 0
\(261\) 14.0632 0.870488
\(262\) 0 0
\(263\) 20.2098 1.24619 0.623096 0.782146i \(-0.285875\pi\)
0.623096 + 0.782146i \(0.285875\pi\)
\(264\) 0 0
\(265\) 20.0632 1.23247
\(266\) 0 0
\(267\) 43.9904 2.69217
\(268\) 0 0
\(269\) −16.4422 −1.00250 −0.501249 0.865303i \(-0.667126\pi\)
−0.501249 + 0.865303i \(0.667126\pi\)
\(270\) 0 0
\(271\) 8.02886 0.487719 0.243859 0.969811i \(-0.421586\pi\)
0.243859 + 0.969811i \(0.421586\pi\)
\(272\) 0 0
\(273\) 1.83042 0.110782
\(274\) 0 0
\(275\) 1.28489 0.0774818
\(276\) 0 0
\(277\) −9.68738 −0.582058 −0.291029 0.956714i \(-0.593998\pi\)
−0.291029 + 0.956714i \(0.593998\pi\)
\(278\) 0 0
\(279\) −19.5353 −1.16955
\(280\) 0 0
\(281\) −8.15302 −0.486368 −0.243184 0.969980i \(-0.578192\pi\)
−0.243184 + 0.969980i \(0.578192\pi\)
\(282\) 0 0
\(283\) −27.9742 −1.66289 −0.831447 0.555604i \(-0.812487\pi\)
−0.831447 + 0.555604i \(0.812487\pi\)
\(284\) 0 0
\(285\) −6.77410 −0.401263
\(286\) 0 0
\(287\) −1.52234 −0.0898607
\(288\) 0 0
\(289\) 14.0028 0.823696
\(290\) 0 0
\(291\) −8.42510 −0.493888
\(292\) 0 0
\(293\) 12.5438 0.732818 0.366409 0.930454i \(-0.380587\pi\)
0.366409 + 0.930454i \(0.380587\pi\)
\(294\) 0 0
\(295\) 31.3393 1.82464
\(296\) 0 0
\(297\) −0.927121 −0.0537970
\(298\) 0 0
\(299\) −17.6248 −1.01927
\(300\) 0 0
\(301\) −1.04376 −0.0601614
\(302\) 0 0
\(303\) 0.630683 0.0362318
\(304\) 0 0
\(305\) −6.55467 −0.375319
\(306\) 0 0
\(307\) 12.9572 0.739504 0.369752 0.929131i \(-0.379443\pi\)
0.369752 + 0.929131i \(0.379443\pi\)
\(308\) 0 0
\(309\) −36.9175 −2.10016
\(310\) 0 0
\(311\) 31.8467 1.80586 0.902931 0.429786i \(-0.141411\pi\)
0.902931 + 0.429786i \(0.141411\pi\)
\(312\) 0 0
\(313\) 16.2628 0.919226 0.459613 0.888119i \(-0.347988\pi\)
0.459613 + 0.888119i \(0.347988\pi\)
\(314\) 0 0
\(315\) 1.70448 0.0960363
\(316\) 0 0
\(317\) 6.88866 0.386905 0.193453 0.981110i \(-0.438031\pi\)
0.193453 + 0.981110i \(0.438031\pi\)
\(318\) 0 0
\(319\) 2.54499 0.142492
\(320\) 0 0
\(321\) 24.8409 1.38648
\(322\) 0 0
\(323\) −5.56802 −0.309813
\(324\) 0 0
\(325\) 7.87167 0.436642
\(326\) 0 0
\(327\) 49.9090 2.75997
\(328\) 0 0
\(329\) 2.03638 0.112269
\(330\) 0 0
\(331\) −25.6976 −1.41247 −0.706234 0.707979i \(-0.749607\pi\)
−0.706234 + 0.707979i \(0.749607\pi\)
\(332\) 0 0
\(333\) −26.6584 −1.46087
\(334\) 0 0
\(335\) −23.7313 −1.29658
\(336\) 0 0
\(337\) 7.78382 0.424012 0.212006 0.977268i \(-0.432000\pi\)
0.212006 + 0.977268i \(0.432000\pi\)
\(338\) 0 0
\(339\) 18.6713 1.01409
\(340\) 0 0
\(341\) −3.53527 −0.191446
\(342\) 0 0
\(343\) 2.52763 0.136479
\(344\) 0 0
\(345\) −30.2366 −1.62788
\(346\) 0 0
\(347\) 32.3758 1.73802 0.869012 0.494792i \(-0.164756\pi\)
0.869012 + 0.494792i \(0.164756\pi\)
\(348\) 0 0
\(349\) −25.0567 −1.34125 −0.670627 0.741795i \(-0.733975\pi\)
−0.670627 + 0.741795i \(0.733975\pi\)
\(350\) 0 0
\(351\) −5.67986 −0.303168
\(352\) 0 0
\(353\) −2.70731 −0.144096 −0.0720478 0.997401i \(-0.522953\pi\)
−0.0720478 + 0.997401i \(0.522953\pi\)
\(354\) 0 0
\(355\) 20.1991 1.07206
\(356\) 0 0
\(357\) 2.58111 0.136607
\(358\) 0 0
\(359\) −27.6862 −1.46122 −0.730611 0.682794i \(-0.760765\pi\)
−0.730611 + 0.682794i \(0.760765\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 27.1130 1.42306
\(364\) 0 0
\(365\) −28.2730 −1.47988
\(366\) 0 0
\(367\) 15.3020 0.798757 0.399379 0.916786i \(-0.369226\pi\)
0.399379 + 0.916786i \(0.369226\pi\)
\(368\) 0 0
\(369\) 29.9604 1.55967
\(370\) 0 0
\(371\) 1.37295 0.0712799
\(372\) 0 0
\(373\) −27.2506 −1.41098 −0.705491 0.708719i \(-0.749274\pi\)
−0.705491 + 0.708719i \(0.749274\pi\)
\(374\) 0 0
\(375\) −20.3661 −1.05170
\(376\) 0 0
\(377\) 15.5915 0.803001
\(378\) 0 0
\(379\) −30.0968 −1.54597 −0.772985 0.634424i \(-0.781237\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(380\) 0 0
\(381\) 1.52468 0.0781116
\(382\) 0 0
\(383\) 37.8146 1.93224 0.966118 0.258100i \(-0.0830966\pi\)
0.966118 + 0.258100i \(0.0830966\pi\)
\(384\) 0 0
\(385\) 0.308457 0.0157204
\(386\) 0 0
\(387\) 20.5417 1.04420
\(388\) 0 0
\(389\) −2.44128 −0.123778 −0.0618890 0.998083i \(-0.519712\pi\)
−0.0618890 + 0.998083i \(0.519712\pi\)
\(390\) 0 0
\(391\) −24.8532 −1.25688
\(392\) 0 0
\(393\) −40.4228 −2.03906
\(394\) 0 0
\(395\) −28.0536 −1.41153
\(396\) 0 0
\(397\) −16.4260 −0.824398 −0.412199 0.911094i \(-0.635239\pi\)
−0.412199 + 0.911094i \(0.635239\pi\)
\(398\) 0 0
\(399\) −0.463560 −0.0232071
\(400\) 0 0
\(401\) −30.0932 −1.50278 −0.751391 0.659857i \(-0.770617\pi\)
−0.751391 + 0.659857i \(0.770617\pi\)
\(402\) 0 0
\(403\) −21.6583 −1.07887
\(404\) 0 0
\(405\) 18.5117 0.919854
\(406\) 0 0
\(407\) −4.82433 −0.239133
\(408\) 0 0
\(409\) −39.4454 −1.95045 −0.975225 0.221216i \(-0.928998\pi\)
−0.975225 + 0.221216i \(0.928998\pi\)
\(410\) 0 0
\(411\) 40.9341 2.01913
\(412\) 0 0
\(413\) 2.14459 0.105528
\(414\) 0 0
\(415\) 14.9442 0.733583
\(416\) 0 0
\(417\) 11.4721 0.561792
\(418\) 0 0
\(419\) −13.3522 −0.652298 −0.326149 0.945318i \(-0.605751\pi\)
−0.326149 + 0.945318i \(0.605751\pi\)
\(420\) 0 0
\(421\) −21.8587 −1.06533 −0.532665 0.846326i \(-0.678809\pi\)
−0.532665 + 0.846326i \(0.678809\pi\)
\(422\) 0 0
\(423\) −40.0770 −1.94861
\(424\) 0 0
\(425\) 11.1000 0.538431
\(426\) 0 0
\(427\) −0.448544 −0.0217066
\(428\) 0 0
\(429\) −6.51912 −0.314746
\(430\) 0 0
\(431\) −14.3490 −0.691167 −0.345584 0.938388i \(-0.612319\pi\)
−0.345584 + 0.938388i \(0.612319\pi\)
\(432\) 0 0
\(433\) 19.2536 0.925269 0.462634 0.886549i \(-0.346904\pi\)
0.462634 + 0.886549i \(0.346904\pi\)
\(434\) 0 0
\(435\) 26.7482 1.28248
\(436\) 0 0
\(437\) 4.46356 0.213521
\(438\) 0 0
\(439\) −19.3020 −0.921234 −0.460617 0.887599i \(-0.652372\pi\)
−0.460617 + 0.887599i \(0.652372\pi\)
\(440\) 0 0
\(441\) −24.8142 −1.18163
\(442\) 0 0
\(443\) −4.61866 −0.219439 −0.109720 0.993963i \(-0.534995\pi\)
−0.109720 + 0.993963i \(0.534995\pi\)
\(444\) 0 0
\(445\) 45.4154 2.15290
\(446\) 0 0
\(447\) 26.5781 1.25710
\(448\) 0 0
\(449\) 8.54082 0.403066 0.201533 0.979482i \(-0.435408\pi\)
0.201533 + 0.979482i \(0.435408\pi\)
\(450\) 0 0
\(451\) 5.42188 0.255307
\(452\) 0 0
\(453\) 42.6713 2.00487
\(454\) 0 0
\(455\) 1.88971 0.0885909
\(456\) 0 0
\(457\) 25.7340 1.20378 0.601892 0.798577i \(-0.294414\pi\)
0.601892 + 0.798577i \(0.294414\pi\)
\(458\) 0 0
\(459\) −8.00930 −0.373842
\(460\) 0 0
\(461\) 12.3629 0.575795 0.287898 0.957661i \(-0.407044\pi\)
0.287898 + 0.957661i \(0.407044\pi\)
\(462\) 0 0
\(463\) −21.1069 −0.980922 −0.490461 0.871463i \(-0.663172\pi\)
−0.490461 + 0.871463i \(0.663172\pi\)
\(464\) 0 0
\(465\) −37.1562 −1.72308
\(466\) 0 0
\(467\) −25.2729 −1.16949 −0.584745 0.811218i \(-0.698805\pi\)
−0.584745 + 0.811218i \(0.698805\pi\)
\(468\) 0 0
\(469\) −1.62396 −0.0749875
\(470\) 0 0
\(471\) −28.2291 −1.30073
\(472\) 0 0
\(473\) 3.71741 0.170927
\(474\) 0 0
\(475\) −1.99353 −0.0914696
\(476\) 0 0
\(477\) −27.0203 −1.23718
\(478\) 0 0
\(479\) 3.27613 0.149690 0.0748450 0.997195i \(-0.476154\pi\)
0.0748450 + 0.997195i \(0.476154\pi\)
\(480\) 0 0
\(481\) −29.5555 −1.34761
\(482\) 0 0
\(483\) −2.06913 −0.0941487
\(484\) 0 0
\(485\) −8.69801 −0.394956
\(486\) 0 0
\(487\) −14.0202 −0.635316 −0.317658 0.948205i \(-0.602896\pi\)
−0.317658 + 0.948205i \(0.602896\pi\)
\(488\) 0 0
\(489\) −19.9904 −0.903997
\(490\) 0 0
\(491\) 21.6487 0.976990 0.488495 0.872567i \(-0.337546\pi\)
0.488495 + 0.872567i \(0.337546\pi\)
\(492\) 0 0
\(493\) 21.9859 0.990195
\(494\) 0 0
\(495\) −6.07058 −0.272852
\(496\) 0 0
\(497\) 1.38225 0.0620023
\(498\) 0 0
\(499\) 34.4187 1.54079 0.770397 0.637564i \(-0.220058\pi\)
0.770397 + 0.637564i \(0.220058\pi\)
\(500\) 0 0
\(501\) 26.2462 1.17259
\(502\) 0 0
\(503\) −14.3178 −0.638399 −0.319200 0.947687i \(-0.603414\pi\)
−0.319200 + 0.947687i \(0.603414\pi\)
\(504\) 0 0
\(505\) 0.651113 0.0289741
\(506\) 0 0
\(507\) −6.63818 −0.294812
\(508\) 0 0
\(509\) −17.3393 −0.768550 −0.384275 0.923219i \(-0.625549\pi\)
−0.384275 + 0.923219i \(0.625549\pi\)
\(510\) 0 0
\(511\) −1.93476 −0.0855886
\(512\) 0 0
\(513\) 1.43845 0.0635090
\(514\) 0 0
\(515\) −38.1134 −1.67948
\(516\) 0 0
\(517\) −7.25268 −0.318972
\(518\) 0 0
\(519\) −4.62108 −0.202843
\(520\) 0 0
\(521\) 30.5856 1.33998 0.669990 0.742370i \(-0.266298\pi\)
0.669990 + 0.742370i \(0.266298\pi\)
\(522\) 0 0
\(523\) −22.7275 −0.993804 −0.496902 0.867807i \(-0.665529\pi\)
−0.496902 + 0.867807i \(0.665529\pi\)
\(524\) 0 0
\(525\) 0.924124 0.0403321
\(526\) 0 0
\(527\) −30.5408 −1.33038
\(528\) 0 0
\(529\) −3.07663 −0.133766
\(530\) 0 0
\(531\) −42.2066 −1.83161
\(532\) 0 0
\(533\) 33.2163 1.43876
\(534\) 0 0
\(535\) 25.6456 1.10875
\(536\) 0 0
\(537\) 4.49242 0.193862
\(538\) 0 0
\(539\) −4.49060 −0.193424
\(540\) 0 0
\(541\) 31.9913 1.37541 0.687707 0.725988i \(-0.258617\pi\)
0.687707 + 0.725988i \(0.258617\pi\)
\(542\) 0 0
\(543\) 16.7215 0.717590
\(544\) 0 0
\(545\) 51.5257 2.20712
\(546\) 0 0
\(547\) 16.1005 0.688406 0.344203 0.938895i \(-0.388149\pi\)
0.344203 + 0.938895i \(0.388149\pi\)
\(548\) 0 0
\(549\) 8.82758 0.376752
\(550\) 0 0
\(551\) −3.94860 −0.168216
\(552\) 0 0
\(553\) −1.91974 −0.0816357
\(554\) 0 0
\(555\) −50.7044 −2.15228
\(556\) 0 0
\(557\) 14.3126 0.606445 0.303223 0.952920i \(-0.401937\pi\)
0.303223 + 0.952920i \(0.401937\pi\)
\(558\) 0 0
\(559\) 22.7741 0.963242
\(560\) 0 0
\(561\) −9.19277 −0.388119
\(562\) 0 0
\(563\) −13.9143 −0.586418 −0.293209 0.956048i \(-0.594723\pi\)
−0.293209 + 0.956048i \(0.594723\pi\)
\(564\) 0 0
\(565\) 19.2761 0.810953
\(566\) 0 0
\(567\) 1.26678 0.0531998
\(568\) 0 0
\(569\) −7.71415 −0.323394 −0.161697 0.986840i \(-0.551697\pi\)
−0.161697 + 0.986840i \(0.551697\pi\)
\(570\) 0 0
\(571\) 9.42188 0.394294 0.197147 0.980374i \(-0.436832\pi\)
0.197147 + 0.980374i \(0.436832\pi\)
\(572\) 0 0
\(573\) 19.1017 0.797987
\(574\) 0 0
\(575\) −8.89826 −0.371083
\(576\) 0 0
\(577\) −10.3851 −0.432337 −0.216168 0.976356i \(-0.569356\pi\)
−0.216168 + 0.976356i \(0.569356\pi\)
\(578\) 0 0
\(579\) −48.7215 −2.02480
\(580\) 0 0
\(581\) 1.02265 0.0424268
\(582\) 0 0
\(583\) −4.88983 −0.202516
\(584\) 0 0
\(585\) −37.1904 −1.53764
\(586\) 0 0
\(587\) 38.8866 1.60502 0.802510 0.596638i \(-0.203497\pi\)
0.802510 + 0.596638i \(0.203497\pi\)
\(588\) 0 0
\(589\) 5.48504 0.226007
\(590\) 0 0
\(591\) 0.391969 0.0161235
\(592\) 0 0
\(593\) 19.1304 0.785590 0.392795 0.919626i \(-0.371508\pi\)
0.392795 + 0.919626i \(0.371508\pi\)
\(594\) 0 0
\(595\) 2.66472 0.109243
\(596\) 0 0
\(597\) −54.7333 −2.24008
\(598\) 0 0
\(599\) −27.5353 −1.12506 −0.562530 0.826777i \(-0.690172\pi\)
−0.562530 + 0.826777i \(0.690172\pi\)
\(600\) 0 0
\(601\) −5.73669 −0.234004 −0.117002 0.993132i \(-0.537328\pi\)
−0.117002 + 0.993132i \(0.537328\pi\)
\(602\) 0 0
\(603\) 31.9604 1.30153
\(604\) 0 0
\(605\) 27.9912 1.13801
\(606\) 0 0
\(607\) 25.3191 1.02767 0.513835 0.857889i \(-0.328224\pi\)
0.513835 + 0.857889i \(0.328224\pi\)
\(608\) 0 0
\(609\) 1.83042 0.0741722
\(610\) 0 0
\(611\) −44.4324 −1.79754
\(612\) 0 0
\(613\) −1.55872 −0.0629560 −0.0314780 0.999504i \(-0.510021\pi\)
−0.0314780 + 0.999504i \(0.510021\pi\)
\(614\) 0 0
\(615\) 56.9848 2.29785
\(616\) 0 0
\(617\) −11.6317 −0.468275 −0.234138 0.972203i \(-0.575227\pi\)
−0.234138 + 0.972203i \(0.575227\pi\)
\(618\) 0 0
\(619\) −15.6754 −0.630046 −0.315023 0.949084i \(-0.602012\pi\)
−0.315023 + 0.949084i \(0.602012\pi\)
\(620\) 0 0
\(621\) 6.42060 0.257650
\(622\) 0 0
\(623\) 3.10783 0.124513
\(624\) 0 0
\(625\) −30.9935 −1.23974
\(626\) 0 0
\(627\) 1.65100 0.0659344
\(628\) 0 0
\(629\) −41.6769 −1.66177
\(630\) 0 0
\(631\) −9.63171 −0.383432 −0.191716 0.981450i \(-0.561405\pi\)
−0.191716 + 0.981450i \(0.561405\pi\)
\(632\) 0 0
\(633\) 17.4716 0.694433
\(634\) 0 0
\(635\) 1.57407 0.0624649
\(636\) 0 0
\(637\) −27.5109 −1.09002
\(638\) 0 0
\(639\) −27.2034 −1.07615
\(640\) 0 0
\(641\) −16.5150 −0.652302 −0.326151 0.945318i \(-0.605752\pi\)
−0.326151 + 0.945318i \(0.605752\pi\)
\(642\) 0 0
\(643\) 18.3887 0.725180 0.362590 0.931949i \(-0.381893\pi\)
0.362590 + 0.931949i \(0.381893\pi\)
\(644\) 0 0
\(645\) 39.0705 1.53840
\(646\) 0 0
\(647\) 17.3414 0.681760 0.340880 0.940107i \(-0.389275\pi\)
0.340880 + 0.940107i \(0.389275\pi\)
\(648\) 0 0
\(649\) −7.63806 −0.299820
\(650\) 0 0
\(651\) −2.54265 −0.0996543
\(652\) 0 0
\(653\) 3.46886 0.135747 0.0678734 0.997694i \(-0.478379\pi\)
0.0678734 + 0.997694i \(0.478379\pi\)
\(654\) 0 0
\(655\) −41.7322 −1.63061
\(656\) 0 0
\(657\) 38.0770 1.48553
\(658\) 0 0
\(659\) −36.6677 −1.42837 −0.714185 0.699957i \(-0.753202\pi\)
−0.714185 + 0.699957i \(0.753202\pi\)
\(660\) 0 0
\(661\) 48.6960 1.89405 0.947027 0.321153i \(-0.104070\pi\)
0.947027 + 0.321153i \(0.104070\pi\)
\(662\) 0 0
\(663\) −56.3180 −2.18721
\(664\) 0 0
\(665\) −0.478577 −0.0185584
\(666\) 0 0
\(667\) −17.6248 −0.682436
\(668\) 0 0
\(669\) 21.5813 0.834382
\(670\) 0 0
\(671\) 1.59751 0.0616714
\(672\) 0 0
\(673\) 24.2535 0.934903 0.467451 0.884019i \(-0.345172\pi\)
0.467451 + 0.884019i \(0.345172\pi\)
\(674\) 0 0
\(675\) −2.86759 −0.110374
\(676\) 0 0
\(677\) −16.7471 −0.643642 −0.321821 0.946801i \(-0.604295\pi\)
−0.321821 + 0.946801i \(0.604295\pi\)
\(678\) 0 0
\(679\) −0.595216 −0.0228423
\(680\) 0 0
\(681\) 59.9042 2.29553
\(682\) 0 0
\(683\) −17.0299 −0.651632 −0.325816 0.945433i \(-0.605639\pi\)
−0.325816 + 0.945433i \(0.605639\pi\)
\(684\) 0 0
\(685\) 42.2601 1.61467
\(686\) 0 0
\(687\) −8.64695 −0.329902
\(688\) 0 0
\(689\) −29.9567 −1.14126
\(690\) 0 0
\(691\) −38.8811 −1.47911 −0.739554 0.673097i \(-0.764964\pi\)
−0.739554 + 0.673097i \(0.764964\pi\)
\(692\) 0 0
\(693\) −0.415418 −0.0157804
\(694\) 0 0
\(695\) 11.8437 0.449258
\(696\) 0 0
\(697\) 46.8391 1.77416
\(698\) 0 0
\(699\) 40.5514 1.53380
\(700\) 0 0
\(701\) 8.17888 0.308912 0.154456 0.988000i \(-0.450637\pi\)
0.154456 + 0.988000i \(0.450637\pi\)
\(702\) 0 0
\(703\) 7.48504 0.282304
\(704\) 0 0
\(705\) −76.2268 −2.87087
\(706\) 0 0
\(707\) 0.0445565 0.00167572
\(708\) 0 0
\(709\) −23.3522 −0.877011 −0.438505 0.898729i \(-0.644492\pi\)
−0.438505 + 0.898729i \(0.644492\pi\)
\(710\) 0 0
\(711\) 37.7815 1.41692
\(712\) 0 0
\(713\) 24.4828 0.916889
\(714\) 0 0
\(715\) −6.73030 −0.251699
\(716\) 0 0
\(717\) −6.14042 −0.229318
\(718\) 0 0
\(719\) −24.1810 −0.901798 −0.450899 0.892575i \(-0.648897\pi\)
−0.450899 + 0.892575i \(0.648897\pi\)
\(720\) 0 0
\(721\) −2.60815 −0.0971325
\(722\) 0 0
\(723\) −13.9402 −0.518441
\(724\) 0 0
\(725\) 7.87167 0.292347
\(726\) 0 0
\(727\) 10.3503 0.383870 0.191935 0.981408i \(-0.438524\pi\)
0.191935 + 0.981408i \(0.438524\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 32.1143 1.18779
\(732\) 0 0
\(733\) 1.81128 0.0669011 0.0334505 0.999440i \(-0.489350\pi\)
0.0334505 + 0.999440i \(0.489350\pi\)
\(734\) 0 0
\(735\) −47.1969 −1.74088
\(736\) 0 0
\(737\) 5.78382 0.213050
\(738\) 0 0
\(739\) 17.0342 0.626612 0.313306 0.949652i \(-0.398564\pi\)
0.313306 + 0.949652i \(0.398564\pi\)
\(740\) 0 0
\(741\) 10.1146 0.371567
\(742\) 0 0
\(743\) 0.992620 0.0364157 0.0182079 0.999834i \(-0.494204\pi\)
0.0182079 + 0.999834i \(0.494204\pi\)
\(744\) 0 0
\(745\) 27.4391 1.00529
\(746\) 0 0
\(747\) −20.1263 −0.736383
\(748\) 0 0
\(749\) 1.75496 0.0641248
\(750\) 0 0
\(751\) 40.1263 1.46423 0.732115 0.681181i \(-0.238533\pi\)
0.732115 + 0.681181i \(0.238533\pi\)
\(752\) 0 0
\(753\) 32.8917 1.19864
\(754\) 0 0
\(755\) 44.0536 1.60327
\(756\) 0 0
\(757\) 41.8810 1.52219 0.761096 0.648640i \(-0.224662\pi\)
0.761096 + 0.648640i \(0.224662\pi\)
\(758\) 0 0
\(759\) 7.36932 0.267489
\(760\) 0 0
\(761\) 23.8142 0.863265 0.431633 0.902050i \(-0.357938\pi\)
0.431633 + 0.902050i \(0.357938\pi\)
\(762\) 0 0
\(763\) 3.52597 0.127649
\(764\) 0 0
\(765\) −52.4431 −1.89608
\(766\) 0 0
\(767\) −46.7934 −1.68961
\(768\) 0 0
\(769\) −47.5828 −1.71588 −0.857939 0.513751i \(-0.828256\pi\)
−0.857939 + 0.513751i \(0.828256\pi\)
\(770\) 0 0
\(771\) 49.1466 1.76997
\(772\) 0 0
\(773\) −33.4240 −1.20218 −0.601090 0.799182i \(-0.705267\pi\)
−0.601090 + 0.799182i \(0.705267\pi\)
\(774\) 0 0
\(775\) −10.9346 −0.392783
\(776\) 0 0
\(777\) −3.46977 −0.124477
\(778\) 0 0
\(779\) −8.41216 −0.301397
\(780\) 0 0
\(781\) −4.92295 −0.176157
\(782\) 0 0
\(783\) −5.67986 −0.202982
\(784\) 0 0
\(785\) −29.1435 −1.04018
\(786\) 0 0
\(787\) −0.315342 −0.0112407 −0.00562036 0.999984i \(-0.501789\pi\)
−0.00562036 + 0.999984i \(0.501789\pi\)
\(788\) 0 0
\(789\) −51.7685 −1.84301
\(790\) 0 0
\(791\) 1.31909 0.0469015
\(792\) 0 0
\(793\) 9.78692 0.347544
\(794\) 0 0
\(795\) −51.3928 −1.82272
\(796\) 0 0
\(797\) −11.0942 −0.392978 −0.196489 0.980506i \(-0.562954\pi\)
−0.196489 + 0.980506i \(0.562954\pi\)
\(798\) 0 0
\(799\) −62.6551 −2.21658
\(800\) 0 0
\(801\) −61.1637 −2.16111
\(802\) 0 0
\(803\) 6.89074 0.243169
\(804\) 0 0
\(805\) −2.13616 −0.0752896
\(806\) 0 0
\(807\) 42.1176 1.48261
\(808\) 0 0
\(809\) 17.4021 0.611824 0.305912 0.952060i \(-0.401039\pi\)
0.305912 + 0.952060i \(0.401039\pi\)
\(810\) 0 0
\(811\) 44.2703 1.55454 0.777270 0.629167i \(-0.216604\pi\)
0.777270 + 0.629167i \(0.216604\pi\)
\(812\) 0 0
\(813\) −20.5664 −0.721294
\(814\) 0 0
\(815\) −20.6379 −0.722916
\(816\) 0 0
\(817\) −5.76763 −0.201784
\(818\) 0 0
\(819\) −2.54499 −0.0889291
\(820\) 0 0
\(821\) −5.12402 −0.178830 −0.0894148 0.995994i \(-0.528500\pi\)
−0.0894148 + 0.995994i \(0.528500\pi\)
\(822\) 0 0
\(823\) 6.29435 0.219407 0.109704 0.993964i \(-0.465010\pi\)
0.109704 + 0.993964i \(0.465010\pi\)
\(824\) 0 0
\(825\) −3.29131 −0.114589
\(826\) 0 0
\(827\) 28.9996 1.00841 0.504207 0.863583i \(-0.331785\pi\)
0.504207 + 0.863583i \(0.331785\pi\)
\(828\) 0 0
\(829\) 11.0215 0.382792 0.191396 0.981513i \(-0.438699\pi\)
0.191396 + 0.981513i \(0.438699\pi\)
\(830\) 0 0
\(831\) 24.8147 0.860813
\(832\) 0 0
\(833\) −38.7938 −1.34412
\(834\) 0 0
\(835\) 27.0964 0.937710
\(836\) 0 0
\(837\) 7.88994 0.272716
\(838\) 0 0
\(839\) 14.8727 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(840\) 0 0
\(841\) −13.4085 −0.462363
\(842\) 0 0
\(843\) 20.8844 0.719297
\(844\) 0 0
\(845\) −6.85321 −0.235758
\(846\) 0 0
\(847\) 1.91548 0.0658166
\(848\) 0 0
\(849\) 71.6574 2.45928
\(850\) 0 0
\(851\) 33.4099 1.14528
\(852\) 0 0
\(853\) −19.8801 −0.680682 −0.340341 0.940302i \(-0.610542\pi\)
−0.340341 + 0.940302i \(0.610542\pi\)
\(854\) 0 0
\(855\) 9.41863 0.322110
\(856\) 0 0
\(857\) −23.9474 −0.818029 −0.409014 0.912528i \(-0.634127\pi\)
−0.409014 + 0.912528i \(0.634127\pi\)
\(858\) 0 0
\(859\) 44.5394 1.51966 0.759832 0.650119i \(-0.225281\pi\)
0.759832 + 0.650119i \(0.225281\pi\)
\(860\) 0 0
\(861\) 3.89955 0.132896
\(862\) 0 0
\(863\) −7.47937 −0.254601 −0.127300 0.991864i \(-0.540631\pi\)
−0.127300 + 0.991864i \(0.540631\pi\)
\(864\) 0 0
\(865\) −4.77077 −0.162211
\(866\) 0 0
\(867\) −35.8690 −1.21818
\(868\) 0 0
\(869\) 6.83726 0.231938
\(870\) 0 0
\(871\) 35.4336 1.20062
\(872\) 0 0
\(873\) 11.7142 0.396464
\(874\) 0 0
\(875\) −1.43882 −0.0486411
\(876\) 0 0
\(877\) −43.0919 −1.45511 −0.727555 0.686049i \(-0.759343\pi\)
−0.727555 + 0.686049i \(0.759343\pi\)
\(878\) 0 0
\(879\) −32.1317 −1.08377
\(880\) 0 0
\(881\) 12.4283 0.418722 0.209361 0.977838i \(-0.432862\pi\)
0.209361 + 0.977838i \(0.432862\pi\)
\(882\) 0 0
\(883\) −34.0244 −1.14501 −0.572507 0.819900i \(-0.694029\pi\)
−0.572507 + 0.819900i \(0.694029\pi\)
\(884\) 0 0
\(885\) −80.2772 −2.69849
\(886\) 0 0
\(887\) 45.2609 1.51971 0.759855 0.650092i \(-0.225270\pi\)
0.759855 + 0.650092i \(0.225270\pi\)
\(888\) 0 0
\(889\) 0.107715 0.00361266
\(890\) 0 0
\(891\) −4.51170 −0.151148
\(892\) 0 0
\(893\) 11.2527 0.376556
\(894\) 0 0
\(895\) 4.63795 0.155029
\(896\) 0 0
\(897\) 45.1469 1.50741
\(898\) 0 0
\(899\) −21.6583 −0.722343
\(900\) 0 0
\(901\) −42.2427 −1.40731
\(902\) 0 0
\(903\) 2.67365 0.0889734
\(904\) 0 0
\(905\) 17.2632 0.573848
\(906\) 0 0
\(907\) −24.7681 −0.822412 −0.411206 0.911542i \(-0.634892\pi\)
−0.411206 + 0.911542i \(0.634892\pi\)
\(908\) 0 0
\(909\) −0.876894 −0.0290848
\(910\) 0 0
\(911\) 12.8875 0.426981 0.213491 0.976945i \(-0.431517\pi\)
0.213491 + 0.976945i \(0.431517\pi\)
\(912\) 0 0
\(913\) −3.64223 −0.120540
\(914\) 0 0
\(915\) 16.7901 0.555064
\(916\) 0 0
\(917\) −2.85579 −0.0943064
\(918\) 0 0
\(919\) −33.0846 −1.09136 −0.545681 0.837993i \(-0.683729\pi\)
−0.545681 + 0.837993i \(0.683729\pi\)
\(920\) 0 0
\(921\) −33.1904 −1.09366
\(922\) 0 0
\(923\) −30.1597 −0.992718
\(924\) 0 0
\(925\) −14.9217 −0.490622
\(926\) 0 0
\(927\) 51.3297 1.68589
\(928\) 0 0
\(929\) 48.0572 1.57671 0.788353 0.615224i \(-0.210934\pi\)
0.788353 + 0.615224i \(0.210934\pi\)
\(930\) 0 0
\(931\) 6.96725 0.228342
\(932\) 0 0
\(933\) −81.5771 −2.67071
\(934\) 0 0
\(935\) −9.49055 −0.310374
\(936\) 0 0
\(937\) −4.71132 −0.153912 −0.0769560 0.997034i \(-0.524520\pi\)
−0.0769560 + 0.997034i \(0.524520\pi\)
\(938\) 0 0
\(939\) −41.6580 −1.35946
\(940\) 0 0
\(941\) 55.0727 1.79532 0.897659 0.440690i \(-0.145266\pi\)
0.897659 + 0.440690i \(0.145266\pi\)
\(942\) 0 0
\(943\) −37.5482 −1.22274
\(944\) 0 0
\(945\) −0.688407 −0.0223939
\(946\) 0 0
\(947\) −23.5984 −0.766846 −0.383423 0.923573i \(-0.625255\pi\)
−0.383423 + 0.923573i \(0.625255\pi\)
\(948\) 0 0
\(949\) 42.2150 1.37036
\(950\) 0 0
\(951\) −17.6457 −0.572200
\(952\) 0 0
\(953\) 35.4123 1.14712 0.573558 0.819165i \(-0.305563\pi\)
0.573558 + 0.819165i \(0.305563\pi\)
\(954\) 0 0
\(955\) 19.7205 0.638140
\(956\) 0 0
\(957\) −6.51912 −0.210733
\(958\) 0 0
\(959\) 2.89191 0.0933847
\(960\) 0 0
\(961\) −0.914306 −0.0294938
\(962\) 0 0
\(963\) −34.5385 −1.11299
\(964\) 0 0
\(965\) −50.2998 −1.61921
\(966\) 0 0
\(967\) 58.6786 1.88698 0.943488 0.331407i \(-0.107523\pi\)
0.943488 + 0.331407i \(0.107523\pi\)
\(968\) 0 0
\(969\) 14.2628 0.458186
\(970\) 0 0
\(971\) 20.7839 0.666988 0.333494 0.942752i \(-0.391772\pi\)
0.333494 + 0.942752i \(0.391772\pi\)
\(972\) 0 0
\(973\) 0.810482 0.0259828
\(974\) 0 0
\(975\) −20.1637 −0.645756
\(976\) 0 0
\(977\) 11.7111 0.374670 0.187335 0.982296i \(-0.440015\pi\)
0.187335 + 0.982296i \(0.440015\pi\)
\(978\) 0 0
\(979\) −11.0687 −0.353758
\(980\) 0 0
\(981\) −69.3928 −2.21554
\(982\) 0 0
\(983\) −55.7921 −1.77949 −0.889745 0.456457i \(-0.849118\pi\)
−0.889745 + 0.456457i \(0.849118\pi\)
\(984\) 0 0
\(985\) 0.404666 0.0128937
\(986\) 0 0
\(987\) −5.21630 −0.166037
\(988\) 0 0
\(989\) −25.7442 −0.818617
\(990\) 0 0
\(991\) −52.1167 −1.65554 −0.827771 0.561066i \(-0.810391\pi\)
−0.827771 + 0.561066i \(0.810391\pi\)
\(992\) 0 0
\(993\) 65.8257 2.08892
\(994\) 0 0
\(995\) −56.5063 −1.79137
\(996\) 0 0
\(997\) 35.9871 1.13972 0.569862 0.821740i \(-0.306997\pi\)
0.569862 + 0.821740i \(0.306997\pi\)
\(998\) 0 0
\(999\) 10.7668 0.340648
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.w.1.1 4
4.3 odd 2 1216.2.a.x.1.3 4
8.3 odd 2 608.2.a.i.1.2 4
8.5 even 2 608.2.a.j.1.4 yes 4
24.5 odd 2 5472.2.a.bs.1.2 4
24.11 even 2 5472.2.a.bt.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.2 4 8.3 odd 2
608.2.a.j.1.4 yes 4 8.5 even 2
1216.2.a.w.1.1 4 1.1 even 1 trivial
1216.2.a.x.1.3 4 4.3 odd 2
5472.2.a.bs.1.2 4 24.5 odd 2
5472.2.a.bt.1.2 4 24.11 even 2