Properties

Label 7728.2.a.f.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -6.00000 q^{13} -7.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +10.0000 q^{31} +1.00000 q^{33} +6.00000 q^{37} +6.00000 q^{39} -5.00000 q^{41} -4.00000 q^{43} +1.00000 q^{47} +1.00000 q^{49} -3.00000 q^{53} +7.00000 q^{57} +3.00000 q^{59} +9.00000 q^{61} +1.00000 q^{63} +4.00000 q^{67} +1.00000 q^{69} +12.0000 q^{71} -4.00000 q^{73} +5.00000 q^{75} -1.00000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +10.0000 q^{87} +18.0000 q^{89} -6.00000 q^{91} -10.0000 q^{93} -6.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 9.00000 1.15233 0.576166 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 0 0
\(103\) 19.0000 1.87213 0.936063 0.351833i \(-0.114441\pi\)
0.936063 + 0.351833i \(0.114441\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0000 1.45241 0.726204 0.687479i \(-0.241283\pi\)
0.726204 + 0.687479i \(0.241283\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.00000 0.386912 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −9.00000 −0.665299
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 42.0000 2.67240
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.00000 −0.295141
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 1.00000 0.0574485
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −19.0000 −1.08087
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 30.0000 1.66410
\(326\) 0 0
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) 1.00000 0.0551318
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) 0 0
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60.0000 3.09016
\(378\) 0 0
\(379\) 38.0000 1.95193 0.975964 0.217930i \(-0.0699304\pi\)
0.975964 + 0.217930i \(0.0699304\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.00000 −0.0504433
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) −60.0000 −2.98881
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) −17.0000 −0.838548
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 1.00000 0.0486217
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.00000 0.435541
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 11.0000 0.529851 0.264926 0.964269i \(-0.414653\pi\)
0.264926 + 0.964269i \(0.414653\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.00000 0.334855
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 0 0
\(453\) −5.00000 −0.234920
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 35.0000 1.60591
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 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
\(501\) −5.00000 −0.223384
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) −1.00000 −0.0443242 −0.0221621 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) 7.00000 0.309058
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.00000 −0.0439799
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 9.00000 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 9.00000 0.384111
\(550\) 0 0
\(551\) 70.0000 2.98210
\(552\) 0 0
\(553\) 2.00000 0.0850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 5.00000 0.208514
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 3.00000 0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.0000 0.536567 0.268284 0.963340i \(-0.413544\pi\)
0.268284 + 0.963340i \(0.413544\pi\)
\(588\) 0 0
\(589\) −70.0000 −2.88430
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.00000 −0.204636
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −7.00000 −0.279553
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 0 0
\(633\) −25.0000 −0.993661
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) 3.00000 0.118308 0.0591542 0.998249i \(-0.481160\pi\)
0.0591542 + 0.998249i \(0.481160\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −9.00000 −0.347441
\(672\) 0 0
\(673\) 33.0000 1.27206 0.636028 0.771666i \(-0.280576\pi\)
0.636028 + 0.771666i \(0.280576\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 28.0000 1.07613 0.538064 0.842904i \(-0.319156\pi\)
0.538064 + 0.842904i \(0.319156\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000 0.267067
\(688\) 0 0
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 16.0000 0.605176
\(700\) 0 0
\(701\) −3.00000 −0.113308 −0.0566542 0.998394i \(-0.518043\pi\)
−0.0566542 + 0.998394i \(0.518043\pi\)
\(702\) 0 0
\(703\) −42.0000 −1.58406
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.00000 −0.0376089
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.00000 −0.149383
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 19.0000 0.707597
\(722\) 0 0
\(723\) −7.00000 −0.260333
\(724\) 0 0
\(725\) 50.0000 1.85695
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −42.0000 −1.54291
\(742\) 0 0
\(743\) −49.0000 −1.79764 −0.898818 0.438322i \(-0.855573\pi\)
−0.898818 + 0.438322i \(0.855573\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 10.0000 0.364420
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) 52.0000 1.87031 0.935155 0.354239i \(-0.115260\pi\)
0.935155 + 0.354239i \(0.115260\pi\)
\(774\) 0 0
\(775\) −50.0000 −1.79605
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −5.00000 −0.178231 −0.0891154 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 0 0
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −54.0000 −1.91760
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 28.0000 0.979596
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) −21.0000 −0.732014 −0.366007 0.930612i \(-0.619275\pi\)
−0.366007 + 0.930612i \(0.619275\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −11.0000 −0.381586
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 22.0000 0.757720
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 5.00000 0.170400
\(862\) 0 0
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −29.0000 −0.979260 −0.489630 0.871930i \(-0.662868\pi\)
−0.489630 + 0.871930i \(0.662868\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −7.00000 −0.234246
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −100.000 −3.33519
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 0 0
\(909\) −1.00000 −0.0331679
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 2.00000 0.0661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.00000 0.0330229
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −30.0000 −0.986394
\(926\) 0 0
\(927\) 19.0000 0.624042
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 0 0
\(933\) −15.0000 −0.491078
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.00000 0.0980057 0.0490029 0.998799i \(-0.484396\pi\)
0.0490029 + 0.998799i \(0.484396\pi\)
\(938\) 0 0
\(939\) 23.0000 0.750577
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 5.00000 0.162822
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 41.0000 1.32812 0.664060 0.747679i \(-0.268832\pi\)
0.664060 + 0.747679i \(0.268832\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −10.0000 −0.323254
\(958\) 0 0
\(959\) 17.0000 0.548959
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) −30.0000 −0.960769
\(976\) 0 0
\(977\) −35.0000 −1.11975 −0.559875 0.828577i \(-0.689151\pi\)
−0.559875 + 0.828577i \(0.689151\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.00000 −0.0318304
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 45.0000 1.42947 0.714736 0.699394i \(-0.246547\pi\)
0.714736 + 0.699394i \(0.246547\pi\)
\(992\) 0 0
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 0 0
\(999\) −6.00000 −0.189832
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 7728.2.a.f.1.1 1
4.3 odd 2 3864.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.c.1.1 1 4.3 odd 2
7728.2.a.f.1.1 1 1.1 even 1 trivial