Properties

Label 4608.2.a.l.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{5} +4.24264 q^{7} +O(q^{10})\) \(q+1.41421 q^{5} +4.24264 q^{7} +6.00000 q^{11} -5.65685 q^{13} +6.00000 q^{17} -4.00000 q^{19} +2.82843 q^{23} -3.00000 q^{25} +1.41421 q^{29} +1.41421 q^{31} +6.00000 q^{35} +8.48528 q^{37} +2.00000 q^{41} +2.82843 q^{47} +11.0000 q^{49} -9.89949 q^{53} +8.48528 q^{55} +4.00000 q^{59} +8.48528 q^{61} -8.00000 q^{65} -8.00000 q^{67} -2.82843 q^{71} +8.00000 q^{73} +25.4558 q^{77} -12.7279 q^{79} +2.00000 q^{83} +8.48528 q^{85} -2.00000 q^{89} -24.0000 q^{91} -5.65685 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{11} + 12 q^{17} - 8 q^{19} - 6 q^{25} + 12 q^{35} + 4 q^{41} + 22 q^{49} + 8 q^{59} - 16 q^{65} - 16 q^{67} + 16 q^{73} + 4 q^{83} - 4 q^{89} - 48 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.89949 −1.35980 −0.679900 0.733305i \(-0.737977\pi\)
−0.679900 + 0.733305i \(0.737977\pi\)
\(54\) 0 0
\(55\) 8.48528 1.14416
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 8.48528 1.08643 0.543214 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 25.4558 2.90096
\(78\) 0 0
\(79\) −12.7279 −1.43200 −0.716002 0.698099i \(-0.754030\pi\)
−0.716002 + 0.698099i \(0.754030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 8.48528 0.920358
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) 0 0
\(103\) −7.07107 −0.696733 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −16.9706 −1.62549 −0.812743 0.582623i \(-0.802026\pi\)
−0.812743 + 0.582623i \(0.802026\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 25.4558 2.33353
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −12.7279 −1.12942 −0.564710 0.825289i \(-0.691012\pi\)
−0.564710 + 0.825289i \(0.691012\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −16.9706 −1.47153
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −33.9411 −2.83830
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.24264 −0.347571 −0.173785 0.984784i \(-0.555600\pi\)
−0.173785 + 0.984784i \(0.555600\pi\)
\(150\) 0 0
\(151\) 12.7279 1.03578 0.517892 0.855446i \(-0.326717\pi\)
0.517892 + 0.855446i \(0.326717\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −2.82843 −0.225733 −0.112867 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.6274 −1.75096 −0.875481 0.483252i \(-0.839455\pi\)
−0.875481 + 0.483252i \(0.839455\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.07107 −0.537603 −0.268802 0.963196i \(-0.586628\pi\)
−0.268802 + 0.963196i \(0.586628\pi\)
\(174\) 0 0
\(175\) −12.7279 −0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 16.9706 1.26141 0.630706 0.776022i \(-0.282765\pi\)
0.630706 + 0.776022i \(0.282765\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.24264 −0.302276 −0.151138 0.988513i \(-0.548294\pi\)
−0.151138 + 0.988513i \(0.548294\pi\)
\(198\) 0 0
\(199\) 4.24264 0.300753 0.150376 0.988629i \(-0.451951\pi\)
0.150376 + 0.988629i \(0.451951\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 2.82843 0.197546
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −33.9411 −2.28313
\(222\) 0 0
\(223\) −15.5563 −1.04173 −0.520865 0.853639i \(-0.674391\pi\)
−0.520865 + 0.853639i \(0.674391\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.5563 0.993859
\(246\) 0 0
\(247\) 22.6274 1.43975
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 16.9706 1.06693
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 36.0000 2.23693
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.9706 1.04645 0.523225 0.852195i \(-0.324729\pi\)
0.523225 + 0.852195i \(0.324729\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.7279 −0.776035 −0.388018 0.921652i \(-0.626840\pi\)
−0.388018 + 0.921652i \(0.626840\pi\)
\(270\) 0 0
\(271\) −4.24264 −0.257722 −0.128861 0.991663i \(-0.541132\pi\)
−0.128861 + 0.991663i \(0.541132\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.0000 −1.08544
\(276\) 0 0
\(277\) −16.9706 −1.01966 −0.509831 0.860274i \(-0.670292\pi\)
−0.509831 + 0.860274i \(0.670292\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.0416 1.40453 0.702264 0.711917i \(-0.252173\pi\)
0.702264 + 0.711917i \(0.252173\pi\)
\(294\) 0 0
\(295\) 5.65685 0.329355
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.6274 1.28308 0.641542 0.767088i \(-0.278295\pi\)
0.641542 + 0.767088i \(0.278295\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.2132 1.19145 0.595726 0.803188i \(-0.296864\pi\)
0.595726 + 0.803188i \(0.296864\pi\)
\(318\) 0 0
\(319\) 8.48528 0.475085
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 16.9706 0.941357
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528 0.459504
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 14.1421 0.757011 0.378506 0.925599i \(-0.376438\pi\)
0.378506 + 0.925599i \(0.376438\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −31.1127 −1.64207 −0.821033 0.570881i \(-0.806602\pi\)
−0.821033 + 0.570881i \(0.806602\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) 12.7279 0.664392 0.332196 0.943210i \(-0.392210\pi\)
0.332196 + 0.943210i \(0.392210\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −42.0000 −2.18053
\(372\) 0 0
\(373\) −19.7990 −1.02515 −0.512576 0.858642i \(-0.671309\pi\)
−0.512576 + 0.858642i \(0.671309\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) 36.0000 1.83473
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.41421 −0.0717035 −0.0358517 0.999357i \(-0.511414\pi\)
−0.0358517 + 0.999357i \(0.511414\pi\)
\(390\) 0 0
\(391\) 16.9706 0.858238
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −18.0000 −0.905678
\(396\) 0 0
\(397\) 31.1127 1.56150 0.780751 0.624843i \(-0.214837\pi\)
0.780751 + 0.624843i \(0.214837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 50.9117 2.52360
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.9706 0.835067
\(414\) 0 0
\(415\) 2.82843 0.138842
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.0000 1.07477 0.537385 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(420\) 0 0
\(421\) 11.3137 0.551396 0.275698 0.961244i \(-0.411091\pi\)
0.275698 + 0.961244i \(0.411091\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.0000 −0.873128
\(426\) 0 0
\(427\) 36.0000 1.74216
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.7990 −0.953684 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) 9.89949 0.472477 0.236239 0.971695i \(-0.424085\pi\)
0.236239 + 0.971695i \(0.424085\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −22.0000 −1.04525 −0.522626 0.852562i \(-0.675047\pi\)
−0.522626 + 0.852562i \(0.675047\pi\)
\(444\) 0 0
\(445\) −2.82843 −0.134080
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −33.9411 −1.59118
\(456\) 0 0
\(457\) 42.0000 1.96468 0.982339 0.187112i \(-0.0599128\pi\)
0.982339 + 0.187112i \(0.0599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −35.3553 −1.64666 −0.823331 0.567561i \(-0.807887\pi\)
−0.823331 + 0.567561i \(0.807887\pi\)
\(462\) 0 0
\(463\) −29.6985 −1.38021 −0.690103 0.723711i \(-0.742435\pi\)
−0.690103 + 0.723711i \(0.742435\pi\)
\(464\) 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
\(469\) −33.9411 −1.56726
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.48528 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.82843 0.128432
\(486\) 0 0
\(487\) −35.3553 −1.60210 −0.801052 0.598595i \(-0.795726\pi\)
−0.801052 + 0.598595i \(0.795726\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.1421 0.630567 0.315283 0.948998i \(-0.397900\pi\)
0.315283 + 0.948998i \(0.397900\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.89949 −0.438787 −0.219394 0.975636i \(-0.570408\pi\)
−0.219394 + 0.975636i \(0.570408\pi\)
\(510\) 0 0
\(511\) 33.9411 1.50147
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) 16.9706 0.746364
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.48528 0.369625
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −11.3137 −0.490051
\(534\) 0 0
\(535\) −5.65685 −0.244567
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 66.0000 2.84282
\(540\) 0 0
\(541\) −22.6274 −0.972829 −0.486414 0.873728i \(-0.661695\pi\)
−0.486414 + 0.873728i \(0.661695\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.65685 −0.240990
\(552\) 0 0
\(553\) −54.0000 −2.29631
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.07107 0.299611 0.149805 0.988716i \(-0.452135\pi\)
0.149805 + 0.988716i \(0.452135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 46.0000 1.93867 0.969334 0.245745i \(-0.0790327\pi\)
0.969334 + 0.245745i \(0.0790327\pi\)
\(564\) 0 0
\(565\) 14.1421 0.594964
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.48528 0.352029
\(582\) 0 0
\(583\) −59.3970 −2.45997
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) −5.65685 −0.233087
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 36.0000 1.47586
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −31.1127 −1.27123 −0.635615 0.772006i \(-0.719253\pi\)
−0.635615 + 0.772006i \(0.719253\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 35.3553 1.43740
\(606\) 0 0
\(607\) 9.89949 0.401808 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −2.82843 −0.114239 −0.0571195 0.998367i \(-0.518192\pi\)
−0.0571195 + 0.998367i \(0.518192\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.48528 −0.339956
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.9117 2.02998
\(630\) 0 0
\(631\) −1.41421 −0.0562990 −0.0281495 0.999604i \(-0.508961\pi\)
−0.0281495 + 0.999604i \(0.508961\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −18.0000 −0.714308
\(636\) 0 0
\(637\) −62.2254 −2.46546
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −46.0000 −1.81689 −0.908445 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −29.6985 −1.16219 −0.581096 0.813835i \(-0.697376\pi\)
−0.581096 + 0.813835i \(0.697376\pi\)
\(654\) 0 0
\(655\) −16.9706 −0.663095
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 50.9117 1.96542
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −43.8406 −1.68493 −0.842466 0.538750i \(-0.818897\pi\)
−0.842466 + 0.538750i \(0.818897\pi\)
\(678\) 0 0
\(679\) 8.48528 0.325635
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 25.4558 0.972618
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 56.0000 2.13343
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.1838 1.44218 0.721090 0.692841i \(-0.243641\pi\)
0.721090 + 0.692841i \(0.243641\pi\)
\(702\) 0 0
\(703\) −33.9411 −1.28011
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 16.9706 0.637343 0.318671 0.947865i \(-0.396763\pi\)
0.318671 + 0.947865i \(0.396763\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) −48.0000 −1.79510
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.1127 1.16031 0.580154 0.814507i \(-0.302992\pi\)
0.580154 + 0.814507i \(0.302992\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.24264 −0.157568
\(726\) 0 0
\(727\) −26.8701 −0.996555 −0.498278 0.867018i \(-0.666034\pi\)
−0.498278 + 0.867018i \(0.666034\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −28.2843 −1.04470 −0.522352 0.852730i \(-0.674945\pi\)
−0.522352 + 0.852730i \(0.674945\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.2843 −1.03765 −0.518825 0.854881i \(-0.673630\pi\)
−0.518825 + 0.854881i \(0.673630\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.9706 −0.620091
\(750\) 0 0
\(751\) −7.07107 −0.258027 −0.129013 0.991643i \(-0.541181\pi\)
−0.129013 + 0.991643i \(0.541181\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.0000 0.655087
\(756\) 0 0
\(757\) 45.2548 1.64481 0.822407 0.568899i \(-0.192630\pi\)
0.822407 + 0.568899i \(0.192630\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) 0 0
\(763\) −72.0000 −2.60658
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.3848 −0.661254 −0.330627 0.943761i \(-0.607260\pi\)
−0.330627 + 0.943761i \(0.607260\pi\)
\(774\) 0 0
\(775\) −4.24264 −0.152400
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −16.9706 −0.607254
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0416 0.851598 0.425799 0.904818i \(-0.359993\pi\)
0.425799 + 0.904818i \(0.359993\pi\)
\(798\) 0 0
\(799\) 16.9706 0.600375
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 48.0000 1.69388
\(804\) 0 0
\(805\) 16.9706 0.598134
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.3137 −0.396302
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.41421 −0.0493564 −0.0246782 0.999695i \(-0.507856\pi\)
−0.0246782 + 0.999695i \(0.507856\pi\)
\(822\) 0 0
\(823\) −41.0122 −1.42960 −0.714798 0.699331i \(-0.753481\pi\)
−0.714798 + 0.699331i \(0.753481\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 28.2843 0.982353 0.491177 0.871060i \(-0.336567\pi\)
0.491177 + 0.871060i \(0.336567\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −19.7990 −0.683537 −0.341769 0.939784i \(-0.611026\pi\)
−0.341769 + 0.939784i \(0.611026\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 26.8701 0.924358
\(846\) 0 0
\(847\) 106.066 3.64447
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 42.4264 1.45265 0.726326 0.687350i \(-0.241226\pi\)
0.726326 + 0.687350i \(0.241226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.3137 −0.385123 −0.192562 0.981285i \(-0.561680\pi\)
−0.192562 + 0.981285i \(0.561680\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −76.3675 −2.59059
\(870\) 0 0
\(871\) 45.2548 1.53340
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 36.7696 1.24162 0.620810 0.783961i \(-0.286804\pi\)
0.620810 + 0.783961i \(0.286804\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) −24.0000 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −39.5980 −1.32957 −0.664785 0.747035i \(-0.731477\pi\)
−0.664785 + 0.747035i \(0.731477\pi\)
\(888\) 0 0
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.3137 −0.378599
\(894\) 0 0
\(895\) 28.2843 0.945439
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −59.3970 −1.97880
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.2843 −0.937100 −0.468550 0.883437i \(-0.655223\pi\)
−0.468550 + 0.883437i \(0.655223\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −50.9117 −1.68125
\(918\) 0 0
\(919\) 26.8701 0.886361 0.443181 0.896432i \(-0.353850\pi\)
0.443181 + 0.896432i \(0.353850\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −25.4558 −0.836983
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −44.0000 −1.44204
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 50.9117 1.66499
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.41421 −0.0461020 −0.0230510 0.999734i \(-0.507338\pi\)
−0.0230510 + 0.999734i \(0.507338\pi\)
\(942\) 0 0
\(943\) 5.65685 0.184213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) −45.2548 −1.46903
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 76.3675 2.46604
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16.9706 0.546302
\(966\) 0 0
\(967\) −21.2132 −0.682171 −0.341085 0.940032i \(-0.610795\pi\)
−0.341085 + 0.940032i \(0.610795\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 14.0000 0.449281 0.224641 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −28.2843 −0.902128 −0.451064 0.892492i \(-0.648955\pi\)
−0.451064 + 0.892492i \(0.648955\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 4.24264 0.134772 0.0673860 0.997727i \(-0.478534\pi\)
0.0673860 + 0.997727i \(0.478534\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) 31.1127 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(998\) 0 0
\(999\) 0 0
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 4608.2.a.l.1.2 2
3.2 odd 2 1536.2.a.i.1.1 yes 2
4.3 odd 2 4608.2.a.g.1.2 2
8.3 odd 2 inner 4608.2.a.l.1.1 2
8.5 even 2 4608.2.a.g.1.1 2
12.11 even 2 1536.2.a.d.1.1 2
16.3 odd 4 4608.2.d.n.2305.2 4
16.5 even 4 4608.2.d.n.2305.3 4
16.11 odd 4 4608.2.d.n.2305.4 4
16.13 even 4 4608.2.d.n.2305.1 4
24.5 odd 2 1536.2.a.d.1.2 yes 2
24.11 even 2 1536.2.a.i.1.2 yes 2
48.5 odd 4 1536.2.d.c.769.1 4
48.11 even 4 1536.2.d.c.769.3 4
48.29 odd 4 1536.2.d.c.769.4 4
48.35 even 4 1536.2.d.c.769.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.d.1.1 2 12.11 even 2
1536.2.a.d.1.2 yes 2 24.5 odd 2
1536.2.a.i.1.1 yes 2 3.2 odd 2
1536.2.a.i.1.2 yes 2 24.11 even 2
1536.2.d.c.769.1 4 48.5 odd 4
1536.2.d.c.769.2 4 48.35 even 4
1536.2.d.c.769.3 4 48.11 even 4
1536.2.d.c.769.4 4 48.29 odd 4
4608.2.a.g.1.1 2 8.5 even 2
4608.2.a.g.1.2 2 4.3 odd 2
4608.2.a.l.1.1 2 8.3 odd 2 inner
4608.2.a.l.1.2 2 1.1 even 1 trivial
4608.2.d.n.2305.1 4 16.13 even 4
4608.2.d.n.2305.2 4 16.3 odd 4
4608.2.d.n.2305.3 4 16.5 even 4
4608.2.d.n.2305.4 4 16.11 odd 4