Properties

Label 9216.2.a.w.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.84776\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.41421 q^{5} -1.53073 q^{7} +O(q^{10})\) \(q-3.41421 q^{5} -1.53073 q^{7} -4.77791 q^{11} +0.585786 q^{13} +2.82843 q^{17} +0.448342 q^{19} +5.86030 q^{23} +6.65685 q^{25} -4.58579 q^{29} +7.39104 q^{31} +5.22625 q^{35} -5.07107 q^{37} +4.00000 q^{41} +2.61313 q^{43} -7.39104 q^{47} -4.65685 q^{49} -7.41421 q^{53} +16.3128 q^{55} +2.61313 q^{59} +13.0711 q^{61} -2.00000 q^{65} +10.0042 q^{67} -11.9832 q^{71} +10.4853 q^{73} +7.31371 q^{77} +6.12293 q^{79} +3.50981 q^{83} -9.65685 q^{85} +0.828427 q^{89} -0.896683 q^{91} -1.53073 q^{95} +10.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + O(q^{10}) \) \( 4q - 8q^{5} + 8q^{13} + 4q^{25} - 24q^{29} + 8q^{37} + 16q^{41} + 4q^{49} - 24q^{53} + 24q^{61} - 8q^{65} + 8q^{73} - 16q^{77} - 16q^{85} - 8q^{89} + 32q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) −1.53073 −0.578563 −0.289281 0.957244i \(-0.593416\pi\)
−0.289281 + 0.957244i \(0.593416\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.77791 −1.44059 −0.720297 0.693666i \(-0.755995\pi\)
−0.720297 + 0.693666i \(0.755995\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 0.448342 0.102857 0.0514283 0.998677i \(-0.483623\pi\)
0.0514283 + 0.998677i \(0.483623\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.86030 1.22196 0.610979 0.791647i \(-0.290776\pi\)
0.610979 + 0.791647i \(0.290776\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.58579 −0.851559 −0.425780 0.904827i \(-0.640000\pi\)
−0.425780 + 0.904827i \(0.640000\pi\)
\(30\) 0 0
\(31\) 7.39104 1.32747 0.663735 0.747968i \(-0.268970\pi\)
0.663735 + 0.747968i \(0.268970\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.22625 0.883398
\(36\) 0 0
\(37\) −5.07107 −0.833678 −0.416839 0.908980i \(-0.636862\pi\)
−0.416839 + 0.908980i \(0.636862\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 2.61313 0.398498 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.39104 −1.07809 −0.539047 0.842276i \(-0.681215\pi\)
−0.539047 + 0.842276i \(0.681215\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.41421 −1.01842 −0.509210 0.860642i \(-0.670062\pi\)
−0.509210 + 0.860642i \(0.670062\pi\)
\(54\) 0 0
\(55\) 16.3128 2.19962
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.61313 0.340200 0.170100 0.985427i \(-0.445591\pi\)
0.170100 + 0.985427i \(0.445591\pi\)
\(60\) 0 0
\(61\) 13.0711 1.67358 0.836789 0.547525i \(-0.184430\pi\)
0.836789 + 0.547525i \(0.184430\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 10.0042 1.22220 0.611101 0.791552i \(-0.290727\pi\)
0.611101 + 0.791552i \(0.290727\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.9832 −1.42215 −0.711074 0.703117i \(-0.751791\pi\)
−0.711074 + 0.703117i \(0.751791\pi\)
\(72\) 0 0
\(73\) 10.4853 1.22721 0.613605 0.789613i \(-0.289719\pi\)
0.613605 + 0.789613i \(0.289719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.31371 0.833474
\(78\) 0 0
\(79\) 6.12293 0.688884 0.344442 0.938808i \(-0.388068\pi\)
0.344442 + 0.938808i \(0.388068\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.50981 0.385252 0.192626 0.981272i \(-0.438300\pi\)
0.192626 + 0.981272i \(0.438300\pi\)
\(84\) 0 0
\(85\) −9.65685 −1.04743
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.828427 0.0878131 0.0439065 0.999036i \(-0.486020\pi\)
0.0439065 + 0.999036i \(0.486020\pi\)
\(90\) 0 0
\(91\) −0.896683 −0.0939979
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.53073 −0.157050
\(96\) 0 0
\(97\) 10.8284 1.09946 0.549730 0.835342i \(-0.314731\pi\)
0.549730 + 0.835342i \(0.314731\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.92893 −0.291440 −0.145720 0.989326i \(-0.546550\pi\)
−0.145720 + 0.989326i \(0.546550\pi\)
\(102\) 0 0
\(103\) −2.79884 −0.275777 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0042 0.967139 0.483569 0.875306i \(-0.339340\pi\)
0.483569 + 0.875306i \(0.339340\pi\)
\(108\) 0 0
\(109\) 1.07107 0.102590 0.0512948 0.998684i \(-0.483665\pi\)
0.0512948 + 0.998684i \(0.483665\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.65685 −0.344008 −0.172004 0.985096i \(-0.555024\pi\)
−0.172004 + 0.985096i \(0.555024\pi\)
\(114\) 0 0
\(115\) −20.0083 −1.86579
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.32957 −0.396891
\(120\) 0 0
\(121\) 11.8284 1.07531
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −13.5140 −1.19917 −0.599586 0.800311i \(-0.704668\pi\)
−0.599586 + 0.800311i \(0.704668\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.10748 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(132\) 0 0
\(133\) −0.686292 −0.0595090
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.9706 1.79164 0.895818 0.444421i \(-0.146591\pi\)
0.895818 + 0.444421i \(0.146591\pi\)
\(138\) 0 0
\(139\) 19.1886 1.62755 0.813776 0.581178i \(-0.197408\pi\)
0.813776 + 0.581178i \(0.197408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.79884 −0.234050
\(144\) 0 0
\(145\) 15.6569 1.30023
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.2426 −0.839110 −0.419555 0.907730i \(-0.637814\pi\)
−0.419555 + 0.907730i \(0.637814\pi\)
\(150\) 0 0
\(151\) 8.92177 0.726043 0.363022 0.931781i \(-0.381745\pi\)
0.363022 + 0.931781i \(0.381745\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −25.2346 −2.02689
\(156\) 0 0
\(157\) −6.24264 −0.498217 −0.249108 0.968476i \(-0.580138\pi\)
−0.249108 + 0.968476i \(0.580138\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.97056 −0.706979
\(162\) 0 0
\(163\) −13.0656 −1.02338 −0.511690 0.859170i \(-0.670980\pi\)
−0.511690 + 0.859170i \(0.670980\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.53073 0.118452 0.0592259 0.998245i \(-0.481137\pi\)
0.0592259 + 0.998245i \(0.481137\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −23.2132 −1.76487 −0.882434 0.470437i \(-0.844096\pi\)
−0.882434 + 0.470437i \(0.844096\pi\)
\(174\) 0 0
\(175\) −10.1899 −0.770282
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.71644 0.128293 0.0641465 0.997940i \(-0.479568\pi\)
0.0641465 + 0.997940i \(0.479568\pi\)
\(180\) 0 0
\(181\) −5.75736 −0.427941 −0.213971 0.976840i \(-0.568640\pi\)
−0.213971 + 0.976840i \(0.568640\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.3137 1.27293
\(186\) 0 0
\(187\) −13.5140 −0.988239
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.5140 −0.977837 −0.488918 0.872330i \(-0.662608\pi\)
−0.488918 + 0.872330i \(0.662608\pi\)
\(192\) 0 0
\(193\) −16.4853 −1.18664 −0.593318 0.804968i \(-0.702182\pi\)
−0.593318 + 0.804968i \(0.702182\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.7279 −1.04932 −0.524660 0.851312i \(-0.675808\pi\)
−0.524660 + 0.851312i \(0.675808\pi\)
\(198\) 0 0
\(199\) −22.4357 −1.59043 −0.795214 0.606329i \(-0.792641\pi\)
−0.795214 + 0.606329i \(0.792641\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.01962 0.492681
\(204\) 0 0
\(205\) −13.6569 −0.953836
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.14214 −0.148175
\(210\) 0 0
\(211\) −26.9510 −1.85538 −0.927692 0.373346i \(-0.878211\pi\)
−0.927692 + 0.373346i \(0.878211\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.92177 −0.608460
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.65685 0.111452
\(222\) 0 0
\(223\) 7.39104 0.494940 0.247470 0.968896i \(-0.420401\pi\)
0.247470 + 0.968896i \(0.420401\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.61313 0.173439 0.0867196 0.996233i \(-0.472362\pi\)
0.0867196 + 0.996233i \(0.472362\pi\)
\(228\) 0 0
\(229\) −15.8995 −1.05067 −0.525334 0.850896i \(-0.676060\pi\)
−0.525334 + 0.850896i \(0.676060\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.1716 −0.993923 −0.496961 0.867773i \(-0.665551\pi\)
−0.496961 + 0.867773i \(0.665551\pi\)
\(234\) 0 0
\(235\) 25.2346 1.64612
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.5641 1.91235 0.956173 0.292803i \(-0.0945880\pi\)
0.956173 + 0.292803i \(0.0945880\pi\)
\(240\) 0 0
\(241\) −18.8284 −1.21285 −0.606423 0.795142i \(-0.707396\pi\)
−0.606423 + 0.795142i \(0.707396\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.8995 1.01578
\(246\) 0 0
\(247\) 0.262632 0.0167109
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.9623 −0.881293 −0.440647 0.897681i \(-0.645251\pi\)
−0.440647 + 0.897681i \(0.645251\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.9706 −1.18335 −0.591676 0.806176i \(-0.701533\pi\)
−0.591676 + 0.806176i \(0.701533\pi\)
\(258\) 0 0
\(259\) 7.76245 0.482335
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.6424 −1.27286 −0.636432 0.771333i \(-0.719590\pi\)
−0.636432 + 0.771333i \(0.719590\pi\)
\(264\) 0 0
\(265\) 25.3137 1.55501
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.55635 0.338777 0.169388 0.985549i \(-0.445821\pi\)
0.169388 + 0.985549i \(0.445821\pi\)
\(270\) 0 0
\(271\) 28.2960 1.71886 0.859431 0.511252i \(-0.170818\pi\)
0.859431 + 0.511252i \(0.170818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −31.8059 −1.91797
\(276\) 0 0
\(277\) −0.585786 −0.0351965 −0.0175982 0.999845i \(-0.505602\pi\)
−0.0175982 + 0.999845i \(0.505602\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.7990 1.30042 0.650209 0.759755i \(-0.274681\pi\)
0.650209 + 0.759755i \(0.274681\pi\)
\(282\) 0 0
\(283\) −9.10748 −0.541383 −0.270692 0.962666i \(-0.587252\pi\)
−0.270692 + 0.962666i \(0.587252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.12293 −0.361425
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.41421 −0.199460 −0.0997302 0.995015i \(-0.531798\pi\)
−0.0997302 + 0.995015i \(0.531798\pi\)
\(294\) 0 0
\(295\) −8.92177 −0.519446
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.43289 0.198529
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −44.6274 −2.55536
\(306\) 0 0
\(307\) 25.6829 1.46580 0.732901 0.680336i \(-0.238166\pi\)
0.732901 + 0.680336i \(0.238166\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.12840 0.404215 0.202107 0.979363i \(-0.435221\pi\)
0.202107 + 0.979363i \(0.435221\pi\)
\(312\) 0 0
\(313\) −8.68629 −0.490978 −0.245489 0.969399i \(-0.578949\pi\)
−0.245489 + 0.969399i \(0.578949\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.5858 −0.706888 −0.353444 0.935456i \(-0.614990\pi\)
−0.353444 + 0.935456i \(0.614990\pi\)
\(318\) 0 0
\(319\) 21.9105 1.22675
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.26810 0.0705590
\(324\) 0 0
\(325\) 3.89949 0.216305
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.3137 0.623745
\(330\) 0 0
\(331\) −5.67459 −0.311904 −0.155952 0.987765i \(-0.549844\pi\)
−0.155952 + 0.987765i \(0.549844\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −34.1563 −1.86616
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −35.3137 −1.91234
\(342\) 0 0
\(343\) 17.8435 0.963461
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.7248 1.16625 0.583123 0.812384i \(-0.301830\pi\)
0.583123 + 0.812384i \(0.301830\pi\)
\(348\) 0 0
\(349\) −22.7279 −1.21660 −0.608299 0.793708i \(-0.708148\pi\)
−0.608299 + 0.793708i \(0.708148\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.9706 0.583904 0.291952 0.956433i \(-0.405695\pi\)
0.291952 + 0.956433i \(0.405695\pi\)
\(354\) 0 0
\(355\) 40.9133 2.17145
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.92177 0.470873 0.235437 0.971890i \(-0.424348\pi\)
0.235437 + 0.971890i \(0.424348\pi\)
\(360\) 0 0
\(361\) −18.7990 −0.989421
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −35.7990 −1.87380
\(366\) 0 0
\(367\) −22.1731 −1.15743 −0.578713 0.815531i \(-0.696445\pi\)
−0.578713 + 0.815531i \(0.696445\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3492 0.589220
\(372\) 0 0
\(373\) 18.7279 0.969695 0.484848 0.874599i \(-0.338875\pi\)
0.484848 + 0.874599i \(0.338875\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.68629 −0.138351
\(378\) 0 0
\(379\) −4.40649 −0.226346 −0.113173 0.993575i \(-0.536101\pi\)
−0.113173 + 0.993575i \(0.536101\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.65914 0.442461 0.221231 0.975222i \(-0.428993\pi\)
0.221231 + 0.975222i \(0.428993\pi\)
\(384\) 0 0
\(385\) −24.9706 −1.27262
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.3848 0.830741 0.415371 0.909652i \(-0.363652\pi\)
0.415371 + 0.909652i \(0.363652\pi\)
\(390\) 0 0
\(391\) 16.5754 0.838256
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −20.9050 −1.05185
\(396\) 0 0
\(397\) 9.07107 0.455264 0.227632 0.973747i \(-0.426902\pi\)
0.227632 + 0.973747i \(0.426902\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.4853 −1.22274 −0.611368 0.791346i \(-0.709381\pi\)
−0.611368 + 0.791346i \(0.709381\pi\)
\(402\) 0 0
\(403\) 4.32957 0.215671
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.2291 1.20099
\(408\) 0 0
\(409\) −8.68629 −0.429509 −0.214755 0.976668i \(-0.568895\pi\)
−0.214755 + 0.976668i \(0.568895\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −11.9832 −0.588234
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.3337 −0.700249 −0.350124 0.936703i \(-0.613861\pi\)
−0.350124 + 0.936703i \(0.613861\pi\)
\(420\) 0 0
\(421\) 11.4142 0.556295 0.278147 0.960538i \(-0.410280\pi\)
0.278147 + 0.960538i \(0.410280\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.8284 0.913313
\(426\) 0 0
\(427\) −20.0083 −0.968271
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −25.4558 −1.22333 −0.611665 0.791117i \(-0.709500\pi\)
−0.611665 + 0.791117i \(0.709500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.62742 0.125686
\(438\) 0 0
\(439\) 18.8490 0.899614 0.449807 0.893126i \(-0.351493\pi\)
0.449807 + 0.893126i \(0.351493\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.0042 −0.475312 −0.237656 0.971349i \(-0.576379\pi\)
−0.237656 + 0.971349i \(0.576379\pi\)
\(444\) 0 0
\(445\) −2.82843 −0.134080
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.1421 −1.04495 −0.522476 0.852654i \(-0.674992\pi\)
−0.522476 + 0.852654i \(0.674992\pi\)
\(450\) 0 0
\(451\) −19.1116 −0.899932
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.06147 0.143524
\(456\) 0 0
\(457\) 18.6274 0.871354 0.435677 0.900103i \(-0.356509\pi\)
0.435677 + 0.900103i \(0.356509\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −29.7574 −1.38594 −0.692969 0.720967i \(-0.743698\pi\)
−0.692969 + 0.720967i \(0.743698\pi\)
\(462\) 0 0
\(463\) 2.53620 0.117867 0.0589337 0.998262i \(-0.481230\pi\)
0.0589337 + 0.998262i \(0.481230\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.9205 0.829260 0.414630 0.909990i \(-0.363911\pi\)
0.414630 + 0.909990i \(0.363911\pi\)
\(468\) 0 0
\(469\) −15.3137 −0.707121
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.4853 −0.574074
\(474\) 0 0
\(475\) 2.98454 0.136940
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 40.5419 1.85241 0.926204 0.377024i \(-0.123052\pi\)
0.926204 + 0.377024i \(0.123052\pi\)
\(480\) 0 0
\(481\) −2.97056 −0.135446
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −36.9706 −1.67875
\(486\) 0 0
\(487\) −37.2178 −1.68650 −0.843250 0.537521i \(-0.819361\pi\)
−0.843250 + 0.537521i \(0.819361\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.3952 −0.785034 −0.392517 0.919745i \(-0.628396\pi\)
−0.392517 + 0.919745i \(0.628396\pi\)
\(492\) 0 0
\(493\) −12.9706 −0.584165
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.3431 0.822803
\(498\) 0 0
\(499\) −16.4985 −0.738575 −0.369287 0.929315i \(-0.620398\pi\)
−0.369287 + 0.929315i \(0.620398\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.1062 0.807314 0.403657 0.914910i \(-0.367739\pi\)
0.403657 + 0.914910i \(0.367739\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.7574 −0.787081 −0.393541 0.919307i \(-0.628750\pi\)
−0.393541 + 0.919307i \(0.628750\pi\)
\(510\) 0 0
\(511\) −16.0502 −0.710018
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.55582 0.421080
\(516\) 0 0
\(517\) 35.3137 1.55310
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −17.6569 −0.773561 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(522\) 0 0
\(523\) −39.5683 −1.73020 −0.865101 0.501598i \(-0.832746\pi\)
−0.865101 + 0.501598i \(0.832746\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.9050 0.910636
\(528\) 0 0
\(529\) 11.3431 0.493180
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.34315 0.101493
\(534\) 0 0
\(535\) −34.1563 −1.47671
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.2500 0.958377
\(540\) 0 0
\(541\) 1.75736 0.0755548 0.0377774 0.999286i \(-0.487972\pi\)
0.0377774 + 0.999286i \(0.487972\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.65685 −0.156642
\(546\) 0 0
\(547\) 1.34502 0.0575091 0.0287545 0.999587i \(-0.490846\pi\)
0.0287545 + 0.999587i \(0.490846\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.05600 −0.0875885
\(552\) 0 0
\(553\) −9.37258 −0.398563
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.58579 −0.363791 −0.181896 0.983318i \(-0.558223\pi\)
−0.181896 + 0.983318i \(0.558223\pi\)
\(558\) 0 0
\(559\) 1.53073 0.0647431
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.3533 −0.899936 −0.449968 0.893045i \(-0.648565\pi\)
−0.449968 + 0.893045i \(0.648565\pi\)
\(564\) 0 0
\(565\) 12.4853 0.525260
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.3431 0.601296 0.300648 0.953735i \(-0.402797\pi\)
0.300648 + 0.953735i \(0.402797\pi\)
\(570\) 0 0
\(571\) 22.2500 0.931135 0.465567 0.885012i \(-0.345850\pi\)
0.465567 + 0.885012i \(0.345850\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 39.0112 1.62688
\(576\) 0 0
\(577\) −5.31371 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.37258 −0.222892
\(582\) 0 0
\(583\) 35.4244 1.46713
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.1158 1.20174 0.600869 0.799348i \(-0.294821\pi\)
0.600869 + 0.799348i \(0.294821\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.3137 0.546728 0.273364 0.961911i \(-0.411864\pi\)
0.273364 + 0.961911i \(0.411864\pi\)
\(594\) 0 0
\(595\) 14.7821 0.606006
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.7875 0.645061 0.322531 0.946559i \(-0.395466\pi\)
0.322531 + 0.946559i \(0.395466\pi\)
\(600\) 0 0
\(601\) 15.1716 0.618861 0.309431 0.950922i \(-0.399862\pi\)
0.309431 + 0.950922i \(0.399862\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −40.3848 −1.64187
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.32957 −0.175156
\(612\) 0 0
\(613\) −36.5858 −1.47769 −0.738843 0.673878i \(-0.764627\pi\)
−0.738843 + 0.673878i \(0.764627\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.5147 −0.866150 −0.433075 0.901358i \(-0.642571\pi\)
−0.433075 + 0.901358i \(0.642571\pi\)
\(618\) 0 0
\(619\) 4.77791 0.192040 0.0960202 0.995379i \(-0.469389\pi\)
0.0960202 + 0.995379i \(0.469389\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.26810 −0.0508054
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.3431 −0.571899
\(630\) 0 0
\(631\) 2.79884 0.111420 0.0557099 0.998447i \(-0.482258\pi\)
0.0557099 + 0.998447i \(0.482258\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 46.1396 1.83099
\(636\) 0 0
\(637\) −2.72792 −0.108084
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.85786 0.389362 0.194681 0.980867i \(-0.437633\pi\)
0.194681 + 0.980867i \(0.437633\pi\)
\(642\) 0 0
\(643\) 1.34502 0.0530426 0.0265213 0.999648i \(-0.491557\pi\)
0.0265213 + 0.999648i \(0.491557\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.262632 0.0103251 0.00516257 0.999987i \(-0.498357\pi\)
0.00516257 + 0.999987i \(0.498357\pi\)
\(648\) 0 0
\(649\) −12.4853 −0.490090
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.5563 1.46969 0.734847 0.678233i \(-0.237254\pi\)
0.734847 + 0.678233i \(0.237254\pi\)
\(654\) 0 0
\(655\) −31.0949 −1.21498
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 33.0740 1.28838 0.644189 0.764866i \(-0.277195\pi\)
0.644189 + 0.764866i \(0.277195\pi\)
\(660\) 0 0
\(661\) 14.9289 0.580668 0.290334 0.956925i \(-0.406234\pi\)
0.290334 + 0.956925i \(0.406234\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.34315 0.0908633
\(666\) 0 0
\(667\) −26.8741 −1.04057
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −62.4524 −2.41095
\(672\) 0 0
\(673\) −21.1716 −0.816104 −0.408052 0.912959i \(-0.633792\pi\)
−0.408052 + 0.912959i \(0.633792\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.5563 −0.674745 −0.337373 0.941371i \(-0.609538\pi\)
−0.337373 + 0.941371i \(0.609538\pi\)
\(678\) 0 0
\(679\) −16.5754 −0.636107
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.0852 0.768541 0.384270 0.923221i \(-0.374453\pi\)
0.384270 + 0.923221i \(0.374453\pi\)
\(684\) 0 0
\(685\) −71.5980 −2.73562
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.34315 −0.165461
\(690\) 0 0
\(691\) 19.9314 0.758226 0.379113 0.925350i \(-0.376229\pi\)
0.379113 + 0.925350i \(0.376229\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −65.5139 −2.48508
\(696\) 0 0
\(697\) 11.3137 0.428537
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.8701 1.09041 0.545204 0.838304i \(-0.316452\pi\)
0.545204 + 0.838304i \(0.316452\pi\)
\(702\) 0 0
\(703\) −2.27357 −0.0857493
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.48342 0.168616
\(708\) 0 0
\(709\) −35.6985 −1.34068 −0.670342 0.742052i \(-0.733853\pi\)
−0.670342 + 0.742052i \(0.733853\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.3137 1.62211
\(714\) 0 0
\(715\) 9.55582 0.357367
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.2459 −0.456694 −0.228347 0.973580i \(-0.573332\pi\)
−0.228347 + 0.973580i \(0.573332\pi\)
\(720\) 0 0
\(721\) 4.28427 0.159555
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −30.5269 −1.13374
\(726\) 0 0
\(727\) −42.8155 −1.58794 −0.793969 0.607958i \(-0.791989\pi\)
−0.793969 + 0.607958i \(0.791989\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.39104 0.273367
\(732\) 0 0
\(733\) 46.5269 1.71851 0.859255 0.511547i \(-0.170927\pi\)
0.859255 + 0.511547i \(0.170927\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −47.7990 −1.76070
\(738\) 0 0
\(739\) −31.4344 −1.15633 −0.578167 0.815918i \(-0.696232\pi\)
−0.578167 + 0.815918i \(0.696232\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.5587 1.04772 0.523858 0.851806i \(-0.324492\pi\)
0.523858 + 0.851806i \(0.324492\pi\)
\(744\) 0 0
\(745\) 34.9706 1.28122
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.3137 −0.559551
\(750\) 0 0
\(751\) −6.12293 −0.223429 −0.111715 0.993740i \(-0.535634\pi\)
−0.111715 + 0.993740i \(0.535634\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −30.4608 −1.10858
\(756\) 0 0
\(757\) −33.0711 −1.20199 −0.600994 0.799253i \(-0.705229\pi\)
−0.600994 + 0.799253i \(0.705229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.6274 0.385244 0.192622 0.981273i \(-0.438301\pi\)
0.192622 + 0.981273i \(0.438301\pi\)
\(762\) 0 0
\(763\) −1.63952 −0.0593546
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.53073 0.0552716
\(768\) 0 0
\(769\) 10.8284 0.390483 0.195242 0.980755i \(-0.437451\pi\)
0.195242 + 0.980755i \(0.437451\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 19.6985 0.708505 0.354253 0.935150i \(-0.384735\pi\)
0.354253 + 0.935150i \(0.384735\pi\)
\(774\) 0 0
\(775\) 49.2011 1.76735
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.79337 0.0642540
\(780\) 0 0
\(781\) 57.2548 2.04874
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 21.3137 0.760719
\(786\) 0 0
\(787\) 48.7527 1.73785 0.868923 0.494947i \(-0.164813\pi\)
0.868923 + 0.494947i \(0.164813\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.59767 0.199030
\(792\) 0 0
\(793\) 7.65685 0.271903
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.6985 −0.697756 −0.348878 0.937168i \(-0.613437\pi\)
−0.348878 + 0.937168i \(0.613437\pi\)
\(798\) 0 0
\(799\) −20.9050 −0.739566
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −50.0977 −1.76791
\(804\) 0 0
\(805\) 30.6274 1.07947
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 47.3137 1.66346 0.831731 0.555179i \(-0.187350\pi\)
0.831731 + 0.555179i \(0.187350\pi\)
\(810\) 0 0
\(811\) −20.4567 −0.718331 −0.359165 0.933274i \(-0.616939\pi\)
−0.359165 + 0.933274i \(0.616939\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 44.6088 1.56258
\(816\) 0 0
\(817\) 1.17157 0.0409881
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.5269 0.925796 0.462898 0.886412i \(-0.346810\pi\)
0.462898 + 0.886412i \(0.346810\pi\)
\(822\) 0 0
\(823\) −42.8155 −1.49245 −0.746227 0.665692i \(-0.768137\pi\)
−0.746227 + 0.665692i \(0.768137\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.1158 −1.01246 −0.506228 0.862400i \(-0.668961\pi\)
−0.506228 + 0.862400i \(0.668961\pi\)
\(828\) 0 0
\(829\) 24.3848 0.846918 0.423459 0.905915i \(-0.360816\pi\)
0.423459 + 0.905915i \(0.360816\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −13.1716 −0.456368
\(834\) 0 0
\(835\) −5.22625 −0.180862
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.7540 1.37246 0.686231 0.727384i \(-0.259264\pi\)
0.686231 + 0.727384i \(0.259264\pi\)
\(840\) 0 0
\(841\) −7.97056 −0.274847
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 43.2132 1.48658
\(846\) 0 0
\(847\) −18.1062 −0.622135
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −29.7180 −1.01872
\(852\) 0 0
\(853\) −51.8995 −1.77700 −0.888502 0.458872i \(-0.848254\pi\)
−0.888502 + 0.458872i \(0.848254\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.2843 1.10281 0.551405 0.834238i \(-0.314092\pi\)
0.551405 + 0.834238i \(0.314092\pi\)
\(858\) 0 0
\(859\) 28.7444 0.980746 0.490373 0.871513i \(-0.336861\pi\)
0.490373 + 0.871513i \(0.336861\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.0781 −1.46640 −0.733198 0.680015i \(-0.761973\pi\)
−0.733198 + 0.680015i \(0.761973\pi\)
\(864\) 0 0
\(865\) 79.2548 2.69475
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −29.2548 −0.992402
\(870\) 0 0
\(871\) 5.86030 0.198569
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.65914 0.292732
\(876\) 0 0
\(877\) 18.4437 0.622798 0.311399 0.950279i \(-0.399202\pi\)
0.311399 + 0.950279i \(0.399202\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27.9411 −0.941360 −0.470680 0.882304i \(-0.655991\pi\)
−0.470680 + 0.882304i \(0.655991\pi\)
\(882\) 0 0
\(883\) 2.98454 0.100438 0.0502190 0.998738i \(-0.484008\pi\)
0.0502190 + 0.998738i \(0.484008\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.1899 −0.342142 −0.171071 0.985259i \(-0.554723\pi\)
−0.171071 + 0.985259i \(0.554723\pi\)
\(888\) 0 0
\(889\) 20.6863 0.693796
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.31371 −0.110889
\(894\) 0 0
\(895\) −5.86030 −0.195888
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −33.8937 −1.13042
\(900\) 0 0
\(901\) −20.9706 −0.698631
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.6569 0.653416
\(906\) 0 0
\(907\) 9.47890 0.314742 0.157371 0.987540i \(-0.449698\pi\)
0.157371 + 0.987540i \(0.449698\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0502 −0.531766 −0.265883 0.964005i \(-0.585663\pi\)
−0.265883 + 0.964005i \(0.585663\pi\)
\(912\) 0 0
\(913\) −16.7696 −0.554991
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.9411 −0.460377
\(918\) 0 0
\(919\) −35.4244 −1.16854 −0.584272 0.811558i \(-0.698620\pi\)
−0.584272 + 0.811558i \(0.698620\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.01962 −0.231054
\(924\) 0 0
\(925\) −33.7574 −1.10994
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 37.1716 1.21956 0.609780 0.792571i \(-0.291258\pi\)
0.609780 + 0.792571i \(0.291258\pi\)
\(930\) 0 0
\(931\) −2.08786 −0.0684269
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 46.1396 1.50893
\(936\) 0 0
\(937\) −7.45584 −0.243572 −0.121786 0.992556i \(-0.538862\pi\)
−0.121786 + 0.992556i \(0.538862\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.0711 −0.556501 −0.278250 0.960509i \(-0.589755\pi\)
−0.278250 + 0.960509i \(0.589755\pi\)
\(942\) 0 0
\(943\) 23.4412 0.763351
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.0321 1.20338 0.601691 0.798729i \(-0.294494\pi\)
0.601691 + 0.798729i \(0.294494\pi\)
\(948\) 0 0
\(949\) 6.14214 0.199382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.9706 −1.45674 −0.728370 0.685184i \(-0.759722\pi\)
−0.728370 + 0.685184i \(0.759722\pi\)
\(954\) 0 0
\(955\) 46.1396 1.49304
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −32.1003 −1.03657
\(960\) 0 0
\(961\) 23.6274 0.762175
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 56.2843 1.81185
\(966\) 0 0
\(967\) 4.59220 0.147675 0.0738376 0.997270i \(-0.476475\pi\)
0.0738376 + 0.997270i \(0.476475\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.58221 0.275416 0.137708 0.990473i \(-0.456026\pi\)
0.137708 + 0.990473i \(0.456026\pi\)
\(972\) 0 0
\(973\) −29.3726 −0.941642
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.1127 −0.483498 −0.241749 0.970339i \(-0.577721\pi\)
−0.241749 + 0.970339i \(0.577721\pi\)
\(978\) 0 0
\(979\) −3.95815 −0.126503
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.05600 0.0655762 0.0327881 0.999462i \(-0.489561\pi\)
0.0327881 + 0.999462i \(0.489561\pi\)
\(984\) 0 0
\(985\) 50.2843 1.60219
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 15.3137 0.486948
\(990\) 0 0
\(991\) −23.4412 −0.744635 −0.372317 0.928106i \(-0.621437\pi\)
−0.372317 + 0.928106i \(0.621437\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 76.6004 2.42840
\(996\) 0 0
\(997\) −8.38478 −0.265549 −0.132774 0.991146i \(-0.542389\pi\)
−0.132774 + 0.991146i \(0.542389\pi\)
\(998\) 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 9216.2.a.w.1.1 4
3.2 odd 2 1024.2.a.i.1.1 4
4.3 odd 2 inner 9216.2.a.w.1.2 4
8.3 odd 2 9216.2.a.bp.1.4 4
8.5 even 2 9216.2.a.bp.1.3 4
12.11 even 2 1024.2.a.i.1.4 4
24.5 odd 2 1024.2.a.h.1.4 4
24.11 even 2 1024.2.a.h.1.1 4
32.3 odd 8 4608.2.k.bi.1153.3 8
32.5 even 8 4608.2.k.bd.3457.1 8
32.11 odd 8 4608.2.k.bi.3457.4 8
32.13 even 8 4608.2.k.bd.1153.2 8
32.19 odd 8 4608.2.k.bd.1153.1 8
32.21 even 8 4608.2.k.bi.3457.3 8
32.27 odd 8 4608.2.k.bd.3457.2 8
32.29 even 8 4608.2.k.bi.1153.4 8
48.5 odd 4 1024.2.b.g.513.8 8
48.11 even 4 1024.2.b.g.513.2 8
48.29 odd 4 1024.2.b.g.513.1 8
48.35 even 4 1024.2.b.g.513.7 8
96.5 odd 8 512.2.e.j.385.4 yes 8
96.11 even 8 512.2.e.i.385.4 yes 8
96.29 odd 8 512.2.e.i.129.1 8
96.35 even 8 512.2.e.i.129.4 yes 8
96.53 odd 8 512.2.e.i.385.1 yes 8
96.59 even 8 512.2.e.j.385.1 yes 8
96.77 odd 8 512.2.e.j.129.4 yes 8
96.83 even 8 512.2.e.j.129.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.i.129.1 8 96.29 odd 8
512.2.e.i.129.4 yes 8 96.35 even 8
512.2.e.i.385.1 yes 8 96.53 odd 8
512.2.e.i.385.4 yes 8 96.11 even 8
512.2.e.j.129.1 yes 8 96.83 even 8
512.2.e.j.129.4 yes 8 96.77 odd 8
512.2.e.j.385.1 yes 8 96.59 even 8
512.2.e.j.385.4 yes 8 96.5 odd 8
1024.2.a.h.1.1 4 24.11 even 2
1024.2.a.h.1.4 4 24.5 odd 2
1024.2.a.i.1.1 4 3.2 odd 2
1024.2.a.i.1.4 4 12.11 even 2
1024.2.b.g.513.1 8 48.29 odd 4
1024.2.b.g.513.2 8 48.11 even 4
1024.2.b.g.513.7 8 48.35 even 4
1024.2.b.g.513.8 8 48.5 odd 4
4608.2.k.bd.1153.1 8 32.19 odd 8
4608.2.k.bd.1153.2 8 32.13 even 8
4608.2.k.bd.3457.1 8 32.5 even 8
4608.2.k.bd.3457.2 8 32.27 odd 8
4608.2.k.bi.1153.3 8 32.3 odd 8
4608.2.k.bi.1153.4 8 32.29 even 8
4608.2.k.bi.3457.3 8 32.21 even 8
4608.2.k.bi.3457.4 8 32.11 odd 8
9216.2.a.w.1.1 4 1.1 even 1 trivial
9216.2.a.w.1.2 4 4.3 odd 2 inner
9216.2.a.bp.1.3 4 8.5 even 2
9216.2.a.bp.1.4 4 8.3 odd 2