Properties

Label 2112.2.a.i.1.1
Level $2112$
Weight $2$
Character 2112.1
Self dual yes
Analytic conductor $16.864$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2112.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} -6.00000 q^{17} +4.00000 q^{19} -2.00000 q^{21} +6.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{33} +10.0000 q^{37} -4.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} -6.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} -4.00000 q^{57} -8.00000 q^{61} +2.00000 q^{63} +4.00000 q^{67} -6.00000 q^{69} +6.00000 q^{71} +2.00000 q^{73} +5.00000 q^{75} +2.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.00000 q^{87} -6.00000 q^{89} +8.00000 q^{91} -8.00000 q^{93} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 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
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −10.0000 −0.755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) −20.0000 −1.10940
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) −16.0000 −0.774294
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 0 0
\(453\) 10.0000 0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 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
\(501\) −12.0000 −0.536120
\(502\) 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
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 10.0000 0.436436
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 24.0000 1.03568
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 28.0000 1.19068
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) −30.0000 −1.25109
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) 28.0000 1.07454
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) 42.0000 1.42970 0.714848 0.699280i \(-0.246496\pi\)
0.714848 + 0.699280i \(0.246496\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −50.0000 −1.64399
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 18.0000 0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 20.0000 0.640513
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −44.0000 −1.39349 −0.696747 0.717317i \(-0.745370\pi\)
−0.696747 + 0.717317i \(0.745370\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 2112.2.a.i.1.1 1
3.2 odd 2 6336.2.a.bj.1.1 1
4.3 odd 2 2112.2.a.v.1.1 1
8.3 odd 2 528.2.a.d.1.1 1
8.5 even 2 66.2.a.a.1.1 1
12.11 even 2 6336.2.a.bf.1.1 1
24.5 odd 2 198.2.a.e.1.1 1
24.11 even 2 1584.2.a.h.1.1 1
40.13 odd 4 1650.2.c.d.199.2 2
40.29 even 2 1650.2.a.m.1.1 1
40.37 odd 4 1650.2.c.d.199.1 2
56.13 odd 2 3234.2.a.d.1.1 1
72.5 odd 6 1782.2.e.f.1189.1 2
72.13 even 6 1782.2.e.s.1189.1 2
72.29 odd 6 1782.2.e.f.595.1 2
72.61 even 6 1782.2.e.s.595.1 2
88.5 even 10 726.2.e.k.487.1 4
88.13 odd 10 726.2.e.b.565.1 4
88.21 odd 2 726.2.a.i.1.1 1
88.29 odd 10 726.2.e.b.511.1 4
88.37 even 10 726.2.e.k.511.1 4
88.43 even 2 5808.2.a.l.1.1 1
88.53 even 10 726.2.e.k.565.1 4
88.61 odd 10 726.2.e.b.487.1 4
88.69 even 10 726.2.e.k.493.1 4
88.85 odd 10 726.2.e.b.493.1 4
120.29 odd 2 4950.2.a.g.1.1 1
120.53 even 4 4950.2.c.r.199.1 2
120.77 even 4 4950.2.c.r.199.2 2
168.125 even 2 9702.2.a.bu.1.1 1
264.197 even 2 2178.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 8.5 even 2
198.2.a.e.1.1 1 24.5 odd 2
528.2.a.d.1.1 1 8.3 odd 2
726.2.a.i.1.1 1 88.21 odd 2
726.2.e.b.487.1 4 88.61 odd 10
726.2.e.b.493.1 4 88.85 odd 10
726.2.e.b.511.1 4 88.29 odd 10
726.2.e.b.565.1 4 88.13 odd 10
726.2.e.k.487.1 4 88.5 even 10
726.2.e.k.493.1 4 88.69 even 10
726.2.e.k.511.1 4 88.37 even 10
726.2.e.k.565.1 4 88.53 even 10
1584.2.a.h.1.1 1 24.11 even 2
1650.2.a.m.1.1 1 40.29 even 2
1650.2.c.d.199.1 2 40.37 odd 4
1650.2.c.d.199.2 2 40.13 odd 4
1782.2.e.f.595.1 2 72.29 odd 6
1782.2.e.f.1189.1 2 72.5 odd 6
1782.2.e.s.595.1 2 72.61 even 6
1782.2.e.s.1189.1 2 72.13 even 6
2112.2.a.i.1.1 1 1.1 even 1 trivial
2112.2.a.v.1.1 1 4.3 odd 2
2178.2.a.b.1.1 1 264.197 even 2
3234.2.a.d.1.1 1 56.13 odd 2
4950.2.a.g.1.1 1 120.29 odd 2
4950.2.c.r.199.1 2 120.53 even 4
4950.2.c.r.199.2 2 120.77 even 4
5808.2.a.l.1.1 1 88.43 even 2
6336.2.a.bf.1.1 1 12.11 even 2
6336.2.a.bj.1.1 1 3.2 odd 2
9702.2.a.bu.1.1 1 168.125 even 2