Properties

Label 4368.2.a.e.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +9.00000 q^{29} -4.00000 q^{31} +5.00000 q^{33} +1.00000 q^{35} -11.0000 q^{37} +1.00000 q^{39} +5.00000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} -2.00000 q^{53} +5.00000 q^{55} -1.00000 q^{57} -4.00000 q^{59} -15.0000 q^{61} -1.00000 q^{63} +1.00000 q^{65} +2.00000 q^{67} +3.00000 q^{69} +12.0000 q^{71} +11.0000 q^{73} +4.00000 q^{75} +5.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +14.0000 q^{83} +3.00000 q^{85} -9.00000 q^{87} +6.00000 q^{89} +1.00000 q^{91} +4.00000 q^{93} -1.00000 q^{95} -14.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 15.0000 1.47799 0.738997 0.673709i \(-0.235300\pi\)
0.738997 + 0.673709i \(0.235300\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) −13.0000 −1.13582 −0.567908 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 23.0000 1.87171 0.935857 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) 11.0000 0.808736
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −5.00000 −0.340997
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) 0 0
\(303\) −4.00000 −0.229794
\(304\) 0 0
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −15.0000 −0.853320
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) −45.0000 −2.51952
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 9.00000 0.497701
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) −11.0000 −0.602796
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 29.0000 1.48183 0.740915 0.671598i \(-0.234392\pi\)
0.740915 + 0.671598i \(0.234392\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) 0 0
\(387\) 5.00000 0.254164
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 13.0000 0.655763
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 55.0000 2.72625
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) −19.0000 −0.937201
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −14.0000 −0.687233
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.00000 −0.341972 −0.170986 0.985273i \(-0.554695\pi\)
−0.170986 + 0.985273i \(0.554695\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 15.0000 0.725901
\(428\) 0 0
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) 0 0
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −13.0000 −0.620456 −0.310228 0.950662i \(-0.600405\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 4.00000 0.189194
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −23.0000 −1.08063
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −35.0000 −1.63011 −0.815056 0.579382i \(-0.803294\pi\)
−0.815056 + 0.579382i \(0.803294\pi\)
\(462\) 0 0
\(463\) −9.00000 −0.418265 −0.209133 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) −25.0000 −1.14950
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 31.0000 1.41643 0.708213 0.705999i \(-0.249502\pi\)
0.708213 + 0.705999i \(0.249502\pi\)
\(480\) 0 0
\(481\) 11.0000 0.501557
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 26.0000 1.17336 0.586682 0.809818i \(-0.300434\pi\)
0.586682 + 0.809818i \(0.300434\pi\)
\(492\) 0 0
\(493\) −27.0000 −1.21602
\(494\) 0 0
\(495\) 5.00000 0.224733
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 13.0000 0.576215 0.288107 0.957598i \(-0.406974\pi\)
0.288107 + 0.957598i \(0.406974\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −15.0000 −0.660979
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.0000 0.432338
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) −15.0000 −0.640184
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) −11.0000 −0.466924
\(556\) 0 0
\(557\) −44.0000 −1.86434 −0.932170 0.362021i \(-0.882087\pi\)
−0.932170 + 0.362021i \(0.882087\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) 11.0000 0.450200
\(598\) 0 0
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 43.0000 1.74532 0.872658 0.488332i \(-0.162394\pi\)
0.872658 + 0.488332i \(0.162394\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −9.00000 −0.363507 −0.181753 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.00000 0.362326 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 5.00000 0.199681
\(628\) 0 0
\(629\) 33.0000 1.31580
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) 0 0
\(633\) 9.00000 0.357718
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 5.00000 0.196875
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 31.0000 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) 0 0
\(655\) 13.0000 0.507952
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −27.0000 −1.04544
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 75.0000 2.89534
\(672\) 0 0
\(673\) −33.0000 −1.27206 −0.636028 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) −19.0000 −0.725953
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −11.0000 −0.414873
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 0 0
\(723\) −18.0000 −0.669427
\(724\) 0 0
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 35.0000 1.29808 0.649039 0.760755i \(-0.275171\pi\)
0.649039 + 0.760755i \(0.275171\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.0000 −0.554795
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) 0 0
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −7.00000 −0.255094
\(754\) 0 0
\(755\) −23.0000 −0.837056
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 0 0
\(763\) 9.00000 0.325822
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) −3.00000 −0.107903 −0.0539513 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) −11.0000 −0.394623
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 3.00000 0.107075
\(786\) 0 0
\(787\) −39.0000 −1.39020 −0.695100 0.718913i \(-0.744640\pi\)
−0.695100 + 0.718913i \(0.744640\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 15.0000 0.532666
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −55.0000 −1.94091
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 0 0
\(815\) 10.0000 0.350285
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) 19.0000 0.660695 0.330347 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) −21.0000 −0.726735
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 28.0000 0.966667 0.483334 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 26.0000 0.895488
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 33.0000 1.13123
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38.0000 −1.29354 −0.646768 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 50.0000 1.69613
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 0 0
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) −51.0000 −1.71629 −0.858143 0.513410i \(-0.828382\pi\)
−0.858143 + 0.513410i \(0.828382\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) 5.00000 0.166390
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 19.0000 0.629498 0.314749 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) 0 0
\(913\) −70.0000 −2.31666
\(914\) 0 0
\(915\) −15.0000 −0.495885
\(916\) 0 0
\(917\) 13.0000 0.429298
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 44.0000 1.44671
\(926\) 0 0
\(927\) 15.0000 0.492665
\(928\) 0 0
\(929\) −8.00000 −0.262471 −0.131236 0.991351i \(-0.541894\pi\)
−0.131236 + 0.991351i \(0.541894\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −10.0000 −0.327385
\(934\) 0 0
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 39.0000 1.26733 0.633665 0.773608i \(-0.281550\pi\)
0.633665 + 0.773608i \(0.281550\pi\)
\(948\) 0 0
\(949\) −11.0000 −0.357075
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 16.0000 0.518291 0.259145 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) 0 0
\(957\) 45.0000 1.45464
\(958\) 0 0
\(959\) −19.0000 −0.613542
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −10.0000 −0.322245
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 5.00000 0.160789 0.0803946 0.996763i \(-0.474382\pi\)
0.0803946 + 0.996763i \(0.474382\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −51.0000 −1.63163 −0.815817 0.578310i \(-0.803713\pi\)
−0.815817 + 0.578310i \(0.803713\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) −9.00000 −0.287348
\(982\) 0 0
\(983\) −25.0000 −0.797376 −0.398688 0.917087i \(-0.630534\pi\)
−0.398688 + 0.917087i \(0.630534\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) 11.0000 0.348723
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) 11.0000 0.348025
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 4368.2.a.e.1.1 1
4.3 odd 2 546.2.a.f.1.1 1
12.11 even 2 1638.2.a.h.1.1 1
28.27 even 2 3822.2.a.w.1.1 1
52.51 odd 2 7098.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.f.1.1 1 4.3 odd 2
1638.2.a.h.1.1 1 12.11 even 2
3822.2.a.w.1.1 1 28.27 even 2
4368.2.a.e.1.1 1 1.1 even 1 trivial
7098.2.a.n.1.1 1 52.51 odd 2