Properties

Label 8624.2.a.bs.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{15} -5.65685 q^{17} -6.00000 q^{23} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +1.41421 q^{31} -1.41421 q^{33} -10.0000 q^{37} -11.3137 q^{41} +8.00000 q^{43} -4.24264 q^{45} -4.24264 q^{47} +8.00000 q^{51} +8.00000 q^{53} +4.24264 q^{55} -1.41421 q^{59} -2.82843 q^{61} -2.00000 q^{67} +8.48528 q^{69} +2.00000 q^{71} +8.48528 q^{73} -18.3848 q^{75} -16.0000 q^{79} -5.00000 q^{81} +16.9706 q^{83} -24.0000 q^{85} -2.82843 q^{87} -7.07107 q^{89} -2.00000 q^{93} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 4.24264 1.89737 0.948683 0.316228i \(-0.102416\pi\)
0.948683 + 0.316228i \(0.102416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −6.00000 −1.54919
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 13.0000 2.60000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) −1.41421 −0.246183
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.3137 −1.76690 −0.883452 0.468521i \(-0.844787\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −4.24264 −0.632456
\(46\) 0 0
\(47\) −4.24264 −0.618853 −0.309426 0.950923i \(-0.600137\pi\)
−0.309426 + 0.950923i \(0.600137\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 4.24264 0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.41421 −0.184115 −0.0920575 0.995754i \(-0.529344\pi\)
−0.0920575 + 0.995754i \(0.529344\pi\)
\(60\) 0 0
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 0 0
\(75\) −18.3848 −2.12289
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 16.9706 1.86276 0.931381 0.364047i \(-0.118605\pi\)
0.931381 + 0.364047i \(0.118605\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 0 0
\(87\) −2.82843 −0.303239
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 5.65685 0.562878 0.281439 0.959579i \(-0.409188\pi\)
0.281439 + 0.959579i \(0.409188\pi\)
\(102\) 0 0
\(103\) −18.3848 −1.81151 −0.905753 0.423806i \(-0.860694\pi\)
−0.905753 + 0.423806i \(0.860694\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 14.1421 1.34231
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −25.4558 −2.37377
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) 33.9411 3.03579
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) 19.7990 1.72985 0.864923 0.501905i \(-0.167367\pi\)
0.864923 + 0.501905i \(0.167367\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 24.0000 2.06559
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −11.3137 −0.959616 −0.479808 0.877373i \(-0.659294\pi\)
−0.479808 + 0.877373i \(0.659294\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.48528 0.704664
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 5.65685 0.457330
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 4.24264 0.338600 0.169300 0.985565i \(-0.445849\pi\)
0.169300 + 0.985565i \(0.445849\pi\)
\(158\) 0 0
\(159\) −11.3137 −0.897235
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.3137 −0.860165 −0.430083 0.902790i \(-0.641516\pi\)
−0.430083 + 0.902790i \(0.641516\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −7.07107 −0.525588 −0.262794 0.964852i \(-0.584644\pi\)
−0.262794 + 0.964852i \(0.584644\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) −42.4264 −3.11925
\(186\) 0 0
\(187\) −5.65685 −0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 1.41421 0.100251 0.0501255 0.998743i \(-0.484038\pi\)
0.0501255 + 0.998743i \(0.484038\pi\)
\(200\) 0 0
\(201\) 2.82843 0.199502
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −48.0000 −3.35247
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −2.82843 −0.193801
\(214\) 0 0
\(215\) 33.9411 2.31477
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 21.2132 1.42054 0.710271 0.703929i \(-0.248573\pi\)
0.710271 + 0.703929i \(0.248573\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.866667
\(226\) 0 0
\(227\) 14.1421 0.938647 0.469323 0.883026i \(-0.344498\pi\)
0.469323 + 0.883026i \(0.344498\pi\)
\(228\) 0 0
\(229\) 9.89949 0.654177 0.327089 0.944994i \(-0.393932\pi\)
0.327089 + 0.944994i \(0.393932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) 22.6274 1.46981
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) −18.3848 −1.16044 −0.580218 0.814461i \(-0.697033\pi\)
−0.580218 + 0.814461i \(0.697033\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 33.9411 2.12548
\(256\) 0 0
\(257\) 1.41421 0.0882162 0.0441081 0.999027i \(-0.485955\pi\)
0.0441081 + 0.999027i \(0.485955\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 33.9411 2.08499
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 0 0
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.0000 0.783929
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −1.41421 −0.0846668
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 19.7990 1.17693 0.588464 0.808523i \(-0.299733\pi\)
0.588464 + 0.808523i \(0.299733\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 25.4558 1.45284 0.726421 0.687250i \(-0.241182\pi\)
0.726421 + 0.687250i \(0.241182\pi\)
\(308\) 0 0
\(309\) 26.0000 1.47909
\(310\) 0 0
\(311\) −18.3848 −1.04251 −0.521253 0.853402i \(-0.674535\pi\)
−0.521253 + 0.853402i \(0.674535\pi\)
\(312\) 0 0
\(313\) −12.7279 −0.719425 −0.359712 0.933063i \(-0.617125\pi\)
−0.359712 + 0.933063i \(0.617125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 22.6274 1.26294
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.82843 0.156412
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) −8.48528 −0.463600
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 2.82843 0.153619
\(340\) 0 0
\(341\) 1.41421 0.0765840
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 36.0000 1.93817
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 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
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 0 0
\(355\) 8.48528 0.450352
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −1.41421 −0.0742270
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) 21.2132 1.10732 0.553660 0.832743i \(-0.313231\pi\)
0.553660 + 0.832743i \(0.313231\pi\)
\(368\) 0 0
\(369\) 11.3137 0.588968
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −48.0000 −2.47871
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) 22.6274 1.15924
\(382\) 0 0
\(383\) −15.5563 −0.794892 −0.397446 0.917625i \(-0.630103\pi\)
−0.397446 + 0.917625i \(0.630103\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 33.9411 1.71648
\(392\) 0 0
\(393\) −28.0000 −1.41241
\(394\) 0 0
\(395\) −67.8823 −3.41553
\(396\) 0 0
\(397\) 12.7279 0.638796 0.319398 0.947621i \(-0.396519\pi\)
0.319398 + 0.947621i \(0.396519\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −21.2132 −1.05409
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −2.82843 −0.139857 −0.0699284 0.997552i \(-0.522277\pi\)
−0.0699284 + 0.997552i \(0.522277\pi\)
\(410\) 0 0
\(411\) 25.4558 1.25564
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 72.0000 3.53434
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 4.24264 0.206284
\(424\) 0 0
\(425\) −73.5391 −3.56717
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −12.7279 −0.611665 −0.305832 0.952085i \(-0.598935\pi\)
−0.305832 + 0.952085i \(0.598935\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 25.4558 1.21494 0.607471 0.794342i \(-0.292184\pi\)
0.607471 + 0.794342i \(0.292184\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −30.0000 −1.42214
\(446\) 0 0
\(447\) 14.1421 0.668900
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −11.3137 −0.532742
\(452\) 0 0
\(453\) 5.65685 0.265782
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) −32.0000 −1.49363
\(460\) 0 0
\(461\) 2.82843 0.131733 0.0658665 0.997828i \(-0.479019\pi\)
0.0658665 + 0.997828i \(0.479019\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 0 0
\(465\) −8.48528 −0.393496
\(466\) 0 0
\(467\) 4.24264 0.196326 0.0981630 0.995170i \(-0.468703\pi\)
0.0981630 + 0.995170i \(0.468703\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.00000 −0.366295
\(478\) 0 0
\(479\) −2.82843 −0.129234 −0.0646171 0.997910i \(-0.520583\pi\)
−0.0646171 + 0.997910i \(0.520583\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −42.0000 −1.90712
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −14.1421 −0.639529
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −11.3137 −0.509544
\(494\) 0 0
\(495\) −4.24264 −0.190693
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −31.1127 −1.38725 −0.693623 0.720338i \(-0.743987\pi\)
−0.693623 + 0.720338i \(0.743987\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 18.3848 0.816497
\(508\) 0 0
\(509\) 12.7279 0.564155 0.282078 0.959392i \(-0.408976\pi\)
0.282078 + 0.959392i \(0.408976\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −78.0000 −3.43709
\(516\) 0 0
\(517\) −4.24264 −0.186591
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 1.41421 0.0619578 0.0309789 0.999520i \(-0.490138\pi\)
0.0309789 + 0.999520i \(0.490138\pi\)
\(522\) 0 0
\(523\) −8.48528 −0.371035 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 1.41421 0.0613716
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −67.8823 −2.93481
\(536\) 0 0
\(537\) 16.9706 0.732334
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −8.48528 −0.363470
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) 2.82843 0.120714
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 60.0000 2.54686
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) −22.6274 −0.953632 −0.476816 0.879003i \(-0.658209\pi\)
−0.476816 + 0.879003i \(0.658209\pi\)
\(564\) 0 0
\(565\) −8.48528 −0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 22.6274 0.945274
\(574\) 0 0
\(575\) −78.0000 −3.25282
\(576\) 0 0
\(577\) −7.07107 −0.294372 −0.147186 0.989109i \(-0.547022\pi\)
−0.147186 + 0.989109i \(0.547022\pi\)
\(578\) 0 0
\(579\) 8.48528 0.352636
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.8701 1.10905 0.554523 0.832168i \(-0.312901\pi\)
0.554523 + 0.832168i \(0.312901\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −31.1127 −1.27981
\(592\) 0 0
\(593\) 42.4264 1.74224 0.871122 0.491067i \(-0.163393\pi\)
0.871122 + 0.491067i \(0.163393\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.00000 −0.0818546
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) 5.65685 0.230748 0.115374 0.993322i \(-0.463193\pi\)
0.115374 + 0.993322i \(0.463193\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 4.24264 0.172488
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 67.8823 2.73728
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) −9.89949 −0.397894 −0.198947 0.980010i \(-0.563752\pi\)
−0.198947 + 0.980010i \(0.563752\pi\)
\(620\) 0 0
\(621\) −33.9411 −1.36201
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 79.0000 3.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 56.5685 2.25554
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) 0 0
\(633\) 11.3137 0.449680
\(634\) 0 0
\(635\) −67.8823 −2.69382
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 38.1838 1.50582 0.752910 0.658123i \(-0.228649\pi\)
0.752910 + 0.658123i \(0.228649\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) 15.5563 0.611583 0.305792 0.952098i \(-0.401079\pi\)
0.305792 + 0.952098i \(0.401079\pi\)
\(648\) 0 0
\(649\) −1.41421 −0.0555127
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 0 0
\(655\) 84.0000 3.28215
\(656\) 0 0
\(657\) −8.48528 −0.331042
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 12.7279 0.495059 0.247529 0.968880i \(-0.420381\pi\)
0.247529 + 0.968880i \(0.420381\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) −30.0000 −1.15987
\(670\) 0 0
\(671\) −2.82843 −0.109190
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 73.5391 2.83052
\(676\) 0 0
\(677\) −39.5980 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) −76.3675 −2.91785
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −15.5563 −0.591791 −0.295896 0.955220i \(-0.595618\pi\)
−0.295896 + 0.955220i \(0.595618\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −48.0000 −1.82074
\(696\) 0 0
\(697\) 64.0000 2.42417
\(698\) 0 0
\(699\) −19.7990 −0.748867
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 25.4558 0.958723
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) −8.48528 −0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.9706 −0.633777
\(718\) 0 0
\(719\) −18.3848 −0.685636 −0.342818 0.939402i \(-0.611381\pi\)
−0.342818 + 0.939402i \(0.611381\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 26.0000 0.965616
\(726\) 0 0
\(727\) −12.7279 −0.472052 −0.236026 0.971747i \(-0.575845\pi\)
−0.236026 + 0.971747i \(0.575845\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −45.2548 −1.67381
\(732\) 0 0
\(733\) 14.1421 0.522352 0.261176 0.965291i \(-0.415890\pi\)
0.261176 + 0.965291i \(0.415890\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.0000 1.61420 0.807102 0.590412i \(-0.201035\pi\)
0.807102 + 0.590412i \(0.201035\pi\)
\(744\) 0 0
\(745\) −42.4264 −1.55438
\(746\) 0 0
\(747\) −16.9706 −0.620920
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) 26.0000 0.947493
\(754\) 0 0
\(755\) −16.9706 −0.617622
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) 8.48528 0.307996
\(760\) 0 0
\(761\) −42.4264 −1.53796 −0.768978 0.639275i \(-0.779234\pi\)
−0.768978 + 0.639275i \(0.779234\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 24.0000 0.867722
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 19.7990 0.713970 0.356985 0.934110i \(-0.383805\pi\)
0.356985 + 0.934110i \(0.383805\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) 1.41421 0.0508657 0.0254329 0.999677i \(-0.491904\pi\)
0.0254329 + 0.999677i \(0.491904\pi\)
\(774\) 0 0
\(775\) 18.3848 0.660401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 0 0
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 5.65685 0.201645 0.100823 0.994904i \(-0.467853\pi\)
0.100823 + 0.994904i \(0.467853\pi\)
\(788\) 0 0
\(789\) −11.3137 −0.402779
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −48.0000 −1.70238
\(796\) 0 0
\(797\) 7.07107 0.250470 0.125235 0.992127i \(-0.460032\pi\)
0.125235 + 0.992127i \(0.460032\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 7.07107 0.249844
\(802\) 0 0
\(803\) 8.48528 0.299439
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −26.0000 −0.915243
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) −36.7696 −1.29115 −0.645577 0.763695i \(-0.723383\pi\)
−0.645577 + 0.763695i \(0.723383\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 42.4264 1.48613
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −18.3848 −0.640076
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) −4.24264 −0.147353 −0.0736765 0.997282i \(-0.523473\pi\)
−0.0736765 + 0.997282i \(0.523473\pi\)
\(830\) 0 0
\(831\) 2.82843 0.0981170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) −1.41421 −0.0488241 −0.0244120 0.999702i \(-0.507771\pi\)
−0.0244120 + 0.999702i \(0.507771\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 19.7990 0.681913
\(844\) 0 0
\(845\) −55.1543 −1.89737
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 28.2843 0.968435 0.484218 0.874948i \(-0.339104\pi\)
0.484218 + 0.874948i \(0.339104\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.4558 −0.869555 −0.434778 0.900538i \(-0.643173\pi\)
−0.434778 + 0.900538i \(0.643173\pi\)
\(858\) 0 0
\(859\) −12.7279 −0.434271 −0.217136 0.976141i \(-0.569671\pi\)
−0.217136 + 0.976141i \(0.569671\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9.89949 0.335047
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −29.6985 −1.00057 −0.500284 0.865862i \(-0.666771\pi\)
−0.500284 + 0.865862i \(0.666771\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 8.48528 0.285230
\(886\) 0 0
\(887\) −36.7696 −1.23460 −0.617300 0.786728i \(-0.711774\pi\)
−0.617300 + 0.786728i \(0.711774\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −50.9117 −1.70179
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.82843 0.0943333
\(900\) 0 0
\(901\) −45.2548 −1.50766
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 0 0
\(909\) −5.65685 −0.187626
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 16.9706 0.561644
\(914\) 0 0
\(915\) 16.9706 0.561029
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −130.000 −4.27437
\(926\) 0 0
\(927\) 18.3848 0.603835
\(928\) 0 0
\(929\) −32.5269 −1.06717 −0.533587 0.845745i \(-0.679156\pi\)
−0.533587 + 0.845745i \(0.679156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26.0000 0.851202
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 25.4558 0.831606 0.415803 0.909455i \(-0.363501\pi\)
0.415803 + 0.909455i \(0.363501\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 31.1127 1.01424 0.507122 0.861874i \(-0.330709\pi\)
0.507122 + 0.861874i \(0.330709\pi\)
\(942\) 0 0
\(943\) 67.8823 2.21055
\(944\) 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
\(949\) 0 0
\(950\) 0 0
\(951\) −42.4264 −1.37577
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) −67.8823 −2.19662
\(956\) 0 0
\(957\) −2.82843 −0.0914301
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) −25.4558 −0.819453
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −52.3259 −1.67922 −0.839609 0.543191i \(-0.817216\pi\)
−0.839609 + 0.543191i \(0.817216\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.0000 1.66363 0.831814 0.555055i \(-0.187303\pi\)
0.831814 + 0.555055i \(0.187303\pi\)
\(978\) 0 0
\(979\) −7.07107 −0.225992
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −1.41421 −0.0451064 −0.0225532 0.999746i \(-0.507180\pi\)
−0.0225532 + 0.999746i \(0.507180\pi\)
\(984\) 0 0
\(985\) 93.3381 2.97400
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 46.0000 1.46124 0.730619 0.682785i \(-0.239232\pi\)
0.730619 + 0.682785i \(0.239232\pi\)
\(992\) 0 0
\(993\) 28.2843 0.897574
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) 16.9706 0.537463 0.268732 0.963215i \(-0.413396\pi\)
0.268732 + 0.963215i \(0.413396\pi\)
\(998\) 0 0
\(999\) −56.5685 −1.78975
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 8624.2.a.bs.1.1 2
4.3 odd 2 1078.2.a.u.1.2 yes 2
7.6 odd 2 inner 8624.2.a.bs.1.2 2
12.11 even 2 9702.2.a.cp.1.1 2
28.3 even 6 1078.2.e.p.177.2 4
28.11 odd 6 1078.2.e.p.177.1 4
28.19 even 6 1078.2.e.p.67.2 4
28.23 odd 6 1078.2.e.p.67.1 4
28.27 even 2 1078.2.a.u.1.1 2
84.83 odd 2 9702.2.a.cp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 2 28.27 even 2
1078.2.a.u.1.2 yes 2 4.3 odd 2
1078.2.e.p.67.1 4 28.23 odd 6
1078.2.e.p.67.2 4 28.19 even 6
1078.2.e.p.177.1 4 28.11 odd 6
1078.2.e.p.177.2 4 28.3 even 6
8624.2.a.bs.1.1 2 1.1 even 1 trivial
8624.2.a.bs.1.2 2 7.6 odd 2 inner
9702.2.a.cp.1.1 2 12.11 even 2
9702.2.a.cp.1.2 2 84.83 odd 2