Properties

Label 1024.2.b.g.513.2
Level $1024$
Weight $2$
Character 1024.513
Analytic conductor $8.177$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 513.2
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1024.513
Dual form 1024.2.b.g.513.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} +O(q^{10})\) \(q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} -4.77791i q^{11} -0.585786i q^{13} +8.92177 q^{15} -2.82843 q^{17} +0.448342i q^{19} +4.00000i q^{21} -5.86030 q^{23} -6.65685 q^{25} +2.16478i q^{27} -4.58579i q^{29} -7.39104 q^{31} -12.4853 q^{33} -5.22625i q^{35} -5.07107i q^{37} -1.53073 q^{39} +4.00000 q^{41} -2.61313i q^{43} -13.0711i q^{45} -7.39104 q^{47} -4.65685 q^{49} +7.39104i q^{51} +7.41421i q^{53} +16.3128 q^{55} +1.17157 q^{57} +2.61313i q^{59} -13.0711i q^{61} +5.86030 q^{63} +2.00000 q^{65} +10.0042i q^{67} +15.3137i q^{69} +11.9832 q^{71} -10.4853 q^{73} +17.3952i q^{75} +7.31371i q^{77} -6.12293 q^{79} -5.82843 q^{81} -3.50981i q^{83} -9.65685i q^{85} -11.9832 q^{87} +0.828427 q^{89} +0.896683i q^{91} +19.3137i q^{93} -1.53073 q^{95} +10.8284 q^{97} +18.2919i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 8q^{25} - 32q^{33} + 32q^{41} + 8q^{49} + 32q^{57} + 16q^{65} - 16q^{73} - 24q^{81} - 16q^{89} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 2.61313i − 1.50869i −0.656479 0.754344i \(-0.727955\pi\)
0.656479 0.754344i \(-0.272045\pi\)
\(4\) 0 0
\(5\) 3.41421i 1.52688i 0.645877 + 0.763441i \(0.276492\pi\)
−0.645877 + 0.763441i \(0.723508\pi\)
\(6\) 0 0
\(7\) −1.53073 −0.578563 −0.289281 0.957244i \(-0.593416\pi\)
−0.289281 + 0.957244i \(0.593416\pi\)
\(8\) 0 0
\(9\) −3.82843 −1.27614
\(10\) 0 0
\(11\) − 4.77791i − 1.44059i −0.693666 0.720297i \(-0.744005\pi\)
0.693666 0.720297i \(-0.255995\pi\)
\(12\) 0 0
\(13\) − 0.585786i − 0.162468i −0.996695 0.0812340i \(-0.974114\pi\)
0.996695 0.0812340i \(-0.0258861\pi\)
\(14\) 0 0
\(15\) 8.92177 2.30359
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 0.448342i 0.102857i 0.998677 + 0.0514283i \(0.0163774\pi\)
−0.998677 + 0.0514283i \(0.983623\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 0 0
\(23\) −5.86030 −1.22196 −0.610979 0.791647i \(-0.709224\pi\)
−0.610979 + 0.791647i \(0.709224\pi\)
\(24\) 0 0
\(25\) −6.65685 −1.33137
\(26\) 0 0
\(27\) 2.16478i 0.416613i
\(28\) 0 0
\(29\) − 4.58579i − 0.851559i −0.904827 0.425780i \(-0.860000\pi\)
0.904827 0.425780i \(-0.140000\pi\)
\(30\) 0 0
\(31\) −7.39104 −1.32747 −0.663735 0.747968i \(-0.731030\pi\)
−0.663735 + 0.747968i \(0.731030\pi\)
\(32\) 0 0
\(33\) −12.4853 −2.17341
\(34\) 0 0
\(35\) − 5.22625i − 0.883398i
\(36\) 0 0
\(37\) − 5.07107i − 0.833678i −0.908980 0.416839i \(-0.863138\pi\)
0.908980 0.416839i \(-0.136862\pi\)
\(38\) 0 0
\(39\) −1.53073 −0.245114
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) − 2.61313i − 0.398498i −0.979949 0.199249i \(-0.936150\pi\)
0.979949 0.199249i \(-0.0638503\pi\)
\(44\) 0 0
\(45\) − 13.0711i − 1.94852i
\(46\) 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
\(51\) 7.39104i 1.03495i
\(52\) 0 0
\(53\) 7.41421i 1.01842i 0.860642 + 0.509210i \(0.170062\pi\)
−0.860642 + 0.509210i \(0.829938\pi\)
\(54\) 0 0
\(55\) 16.3128 2.19962
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 0 0
\(59\) 2.61313i 0.340200i 0.985427 + 0.170100i \(0.0544091\pi\)
−0.985427 + 0.170100i \(0.945591\pi\)
\(60\) 0 0
\(61\) − 13.0711i − 1.67358i −0.547525 0.836789i \(-0.684430\pi\)
0.547525 0.836789i \(-0.315570\pi\)
\(62\) 0 0
\(63\) 5.86030 0.738329
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 10.0042i 1.22220i 0.791552 + 0.611101i \(0.209273\pi\)
−0.791552 + 0.611101i \(0.790727\pi\)
\(68\) 0 0
\(69\) 15.3137i 1.84355i
\(70\) 0 0
\(71\) 11.9832 1.42215 0.711074 0.703117i \(-0.248209\pi\)
0.711074 + 0.703117i \(0.248209\pi\)
\(72\) 0 0
\(73\) −10.4853 −1.22721 −0.613605 0.789613i \(-0.710281\pi\)
−0.613605 + 0.789613i \(0.710281\pi\)
\(74\) 0 0
\(75\) 17.3952i 2.00862i
\(76\) 0 0
\(77\) 7.31371i 0.833474i
\(78\) 0 0
\(79\) −6.12293 −0.688884 −0.344442 0.938808i \(-0.611932\pi\)
−0.344442 + 0.938808i \(0.611932\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 0 0
\(83\) − 3.50981i − 0.385252i −0.981272 0.192626i \(-0.938300\pi\)
0.981272 0.192626i \(-0.0617003\pi\)
\(84\) 0 0
\(85\) − 9.65685i − 1.04743i
\(86\) 0 0
\(87\) −11.9832 −1.28474
\(88\) 0 0
\(89\) 0.828427 0.0878131 0.0439065 0.999036i \(-0.486020\pi\)
0.0439065 + 0.999036i \(0.486020\pi\)
\(90\) 0 0
\(91\) 0.896683i 0.0939979i
\(92\) 0 0
\(93\) 19.3137i 2.00274i
\(94\) 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
\(99\) 18.2919i 1.83840i
\(100\) 0 0
\(101\) 2.92893i 0.291440i 0.989326 + 0.145720i \(0.0465498\pi\)
−0.989326 + 0.145720i \(0.953450\pi\)
\(102\) 0 0
\(103\) −2.79884 −0.275777 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(104\) 0 0
\(105\) −13.6569 −1.33277
\(106\) 0 0
\(107\) 10.0042i 0.967139i 0.875306 + 0.483569i \(0.160660\pi\)
−0.875306 + 0.483569i \(0.839340\pi\)
\(108\) 0 0
\(109\) − 1.07107i − 0.102590i −0.998684 0.0512948i \(-0.983665\pi\)
0.998684 0.0512948i \(-0.0163348\pi\)
\(110\) 0 0
\(111\) −13.2513 −1.25776
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) − 20.0083i − 1.86579i
\(116\) 0 0
\(117\) 2.24264i 0.207332i
\(118\) 0 0
\(119\) 4.32957 0.396891
\(120\) 0 0
\(121\) −11.8284 −1.07531
\(122\) 0 0
\(123\) − 10.4525i − 0.942471i
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 13.5140 1.19917 0.599586 0.800311i \(-0.295332\pi\)
0.599586 + 0.800311i \(0.295332\pi\)
\(128\) 0 0
\(129\) −6.82843 −0.601209
\(130\) 0 0
\(131\) − 9.10748i − 0.795724i −0.917445 0.397862i \(-0.869752\pi\)
0.917445 0.397862i \(-0.130248\pi\)
\(132\) 0 0
\(133\) − 0.686292i − 0.0595090i
\(134\) 0 0
\(135\) −7.39104 −0.636119
\(136\) 0 0
\(137\) 20.9706 1.79164 0.895818 0.444421i \(-0.146591\pi\)
0.895818 + 0.444421i \(0.146591\pi\)
\(138\) 0 0
\(139\) − 19.1886i − 1.62755i −0.581178 0.813776i \(-0.697408\pi\)
0.581178 0.813776i \(-0.302592\pi\)
\(140\) 0 0
\(141\) 19.3137i 1.62651i
\(142\) 0 0
\(143\) −2.79884 −0.234050
\(144\) 0 0
\(145\) 15.6569 1.30023
\(146\) 0 0
\(147\) 12.1689i 1.00368i
\(148\) 0 0
\(149\) 10.2426i 0.839110i 0.907730 + 0.419555i \(0.137814\pi\)
−0.907730 + 0.419555i \(0.862186\pi\)
\(150\) 0 0
\(151\) 8.92177 0.726043 0.363022 0.931781i \(-0.381745\pi\)
0.363022 + 0.931781i \(0.381745\pi\)
\(152\) 0 0
\(153\) 10.8284 0.875426
\(154\) 0 0
\(155\) − 25.2346i − 2.02689i
\(156\) 0 0
\(157\) 6.24264i 0.498217i 0.968476 + 0.249108i \(0.0801376\pi\)
−0.968476 + 0.249108i \(0.919862\pi\)
\(158\) 0 0
\(159\) 19.3743 1.53648
\(160\) 0 0
\(161\) 8.97056 0.706979
\(162\) 0 0
\(163\) − 13.0656i − 1.02338i −0.859170 0.511690i \(-0.829020\pi\)
0.859170 0.511690i \(-0.170980\pi\)
\(164\) 0 0
\(165\) − 42.6274i − 3.31854i
\(166\) 0 0
\(167\) −1.53073 −0.118452 −0.0592259 0.998245i \(-0.518863\pi\)
−0.0592259 + 0.998245i \(0.518863\pi\)
\(168\) 0 0
\(169\) 12.6569 0.973604
\(170\) 0 0
\(171\) − 1.71644i − 0.131260i
\(172\) 0 0
\(173\) − 23.2132i − 1.76487i −0.470437 0.882434i \(-0.655904\pi\)
0.470437 0.882434i \(-0.344096\pi\)
\(174\) 0 0
\(175\) 10.1899 0.770282
\(176\) 0 0
\(177\) 6.82843 0.513256
\(178\) 0 0
\(179\) − 1.71644i − 0.128293i −0.997940 0.0641465i \(-0.979568\pi\)
0.997940 0.0641465i \(-0.0204325\pi\)
\(180\) 0 0
\(181\) − 5.75736i − 0.427941i −0.976840 0.213971i \(-0.931360\pi\)
0.976840 0.213971i \(-0.0686397\pi\)
\(182\) 0 0
\(183\) −34.1563 −2.52491
\(184\) 0 0
\(185\) 17.3137 1.27293
\(186\) 0 0
\(187\) 13.5140i 0.988239i
\(188\) 0 0
\(189\) − 3.31371i − 0.241037i
\(190\) 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
\(195\) − 5.22625i − 0.374260i
\(196\) 0 0
\(197\) 14.7279i 1.04932i 0.851312 + 0.524660i \(0.175808\pi\)
−0.851312 + 0.524660i \(0.824192\pi\)
\(198\) 0 0
\(199\) −22.4357 −1.59043 −0.795214 0.606329i \(-0.792641\pi\)
−0.795214 + 0.606329i \(0.792641\pi\)
\(200\) 0 0
\(201\) 26.1421 1.84392
\(202\) 0 0
\(203\) 7.01962i 0.492681i
\(204\) 0 0
\(205\) 13.6569i 0.953836i
\(206\) 0 0
\(207\) 22.4357 1.55939
\(208\) 0 0
\(209\) 2.14214 0.148175
\(210\) 0 0
\(211\) − 26.9510i − 1.85538i −0.373346 0.927692i \(-0.621789\pi\)
0.373346 0.927692i \(-0.378211\pi\)
\(212\) 0 0
\(213\) − 31.3137i − 2.14558i
\(214\) 0 0
\(215\) 8.92177 0.608460
\(216\) 0 0
\(217\) 11.3137 0.768025
\(218\) 0 0
\(219\) 27.3994i 1.85148i
\(220\) 0 0
\(221\) 1.65685i 0.111452i
\(222\) 0 0
\(223\) −7.39104 −0.494940 −0.247470 0.968896i \(-0.579599\pi\)
−0.247470 + 0.968896i \(0.579599\pi\)
\(224\) 0 0
\(225\) 25.4853 1.69902
\(226\) 0 0
\(227\) − 2.61313i − 0.173439i −0.996233 0.0867196i \(-0.972362\pi\)
0.996233 0.0867196i \(-0.0276384\pi\)
\(228\) 0 0
\(229\) − 15.8995i − 1.05067i −0.850896 0.525334i \(-0.823940\pi\)
0.850896 0.525334i \(-0.176060\pi\)
\(230\) 0 0
\(231\) 19.1116 1.25745
\(232\) 0 0
\(233\) −15.1716 −0.993923 −0.496961 0.867773i \(-0.665551\pi\)
−0.496961 + 0.867773i \(0.665551\pi\)
\(234\) 0 0
\(235\) − 25.2346i − 1.64612i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 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
\(243\) 21.7248i 1.39364i
\(244\) 0 0
\(245\) − 15.8995i − 1.01578i
\(246\) 0 0
\(247\) 0.262632 0.0167109
\(248\) 0 0
\(249\) −9.17157 −0.581225
\(250\) 0 0
\(251\) − 13.9623i − 0.881293i −0.897681 0.440647i \(-0.854749\pi\)
0.897681 0.440647i \(-0.145251\pi\)
\(252\) 0 0
\(253\) 28.0000i 1.76034i
\(254\) 0 0
\(255\) −25.2346 −1.58025
\(256\) 0 0
\(257\) 18.9706 1.18335 0.591676 0.806176i \(-0.298467\pi\)
0.591676 + 0.806176i \(0.298467\pi\)
\(258\) 0 0
\(259\) 7.76245i 0.482335i
\(260\) 0 0
\(261\) 17.5563i 1.08671i
\(262\) 0 0
\(263\) 20.6424 1.27286 0.636432 0.771333i \(-0.280410\pi\)
0.636432 + 0.771333i \(0.280410\pi\)
\(264\) 0 0
\(265\) −25.3137 −1.55501
\(266\) 0 0
\(267\) − 2.16478i − 0.132483i
\(268\) 0 0
\(269\) 5.55635i 0.338777i 0.985549 + 0.169388i \(0.0541792\pi\)
−0.985549 + 0.169388i \(0.945821\pi\)
\(270\) 0 0
\(271\) −28.2960 −1.71886 −0.859431 0.511252i \(-0.829182\pi\)
−0.859431 + 0.511252i \(0.829182\pi\)
\(272\) 0 0
\(273\) 2.34315 0.141814
\(274\) 0 0
\(275\) 31.8059i 1.91797i
\(276\) 0 0
\(277\) − 0.585786i − 0.0351965i −0.999845 0.0175982i \(-0.994398\pi\)
0.999845 0.0175982i \(-0.00560199\pi\)
\(278\) 0 0
\(279\) 28.2960 1.69404
\(280\) 0 0
\(281\) 21.7990 1.30042 0.650209 0.759755i \(-0.274681\pi\)
0.650209 + 0.759755i \(0.274681\pi\)
\(282\) 0 0
\(283\) 9.10748i 0.541383i 0.962666 + 0.270692i \(0.0872524\pi\)
−0.962666 + 0.270692i \(0.912748\pi\)
\(284\) 0 0
\(285\) 4.00000i 0.236940i
\(286\) 0 0
\(287\) −6.12293 −0.361425
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) − 28.2960i − 1.65874i
\(292\) 0 0
\(293\) 3.41421i 0.199460i 0.995015 + 0.0997302i \(0.0317980\pi\)
−0.995015 + 0.0997302i \(0.968202\pi\)
\(294\) 0 0
\(295\) −8.92177 −0.519446
\(296\) 0 0
\(297\) 10.3431 0.600170
\(298\) 0 0
\(299\) 3.43289i 0.198529i
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) 7.65367 0.439692
\(304\) 0 0
\(305\) 44.6274 2.55536
\(306\) 0 0
\(307\) 25.6829i 1.46580i 0.680336 + 0.732901i \(0.261834\pi\)
−0.680336 + 0.732901i \(0.738166\pi\)
\(308\) 0 0
\(309\) 7.31371i 0.416062i
\(310\) 0 0
\(311\) −7.12840 −0.404215 −0.202107 0.979363i \(-0.564779\pi\)
−0.202107 + 0.979363i \(0.564779\pi\)
\(312\) 0 0
\(313\) 8.68629 0.490978 0.245489 0.969399i \(-0.421051\pi\)
0.245489 + 0.969399i \(0.421051\pi\)
\(314\) 0 0
\(315\) 20.0083i 1.12734i
\(316\) 0 0
\(317\) − 12.5858i − 0.706888i −0.935456 0.353444i \(-0.885010\pi\)
0.935456 0.353444i \(-0.114990\pi\)
\(318\) 0 0
\(319\) −21.9105 −1.22675
\(320\) 0 0
\(321\) 26.1421 1.45911
\(322\) 0 0
\(323\) − 1.26810i − 0.0705590i
\(324\) 0 0
\(325\) 3.89949i 0.216305i
\(326\) 0 0
\(327\) −2.79884 −0.154776
\(328\) 0 0
\(329\) 11.3137 0.623745
\(330\) 0 0
\(331\) 5.67459i 0.311904i 0.987765 + 0.155952i \(0.0498445\pi\)
−0.987765 + 0.155952i \(0.950156\pi\)
\(332\) 0 0
\(333\) 19.4142i 1.06389i
\(334\) 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
\(339\) − 9.55582i − 0.519001i
\(340\) 0 0
\(341\) 35.3137i 1.91234i
\(342\) 0 0
\(343\) 17.8435 0.963461
\(344\) 0 0
\(345\) −52.2843 −2.81489
\(346\) 0 0
\(347\) 21.7248i 1.16625i 0.812384 + 0.583123i \(0.198170\pi\)
−0.812384 + 0.583123i \(0.801830\pi\)
\(348\) 0 0
\(349\) 22.7279i 1.21660i 0.793708 + 0.608299i \(0.208148\pi\)
−0.793708 + 0.608299i \(0.791852\pi\)
\(350\) 0 0
\(351\) 1.26810 0.0676862
\(352\) 0 0
\(353\) −10.9706 −0.583904 −0.291952 0.956433i \(-0.594305\pi\)
−0.291952 + 0.956433i \(0.594305\pi\)
\(354\) 0 0
\(355\) 40.9133i 2.17145i
\(356\) 0 0
\(357\) − 11.3137i − 0.598785i
\(358\) 0 0
\(359\) −8.92177 −0.470873 −0.235437 0.971890i \(-0.575652\pi\)
−0.235437 + 0.971890i \(0.575652\pi\)
\(360\) 0 0
\(361\) 18.7990 0.989421
\(362\) 0 0
\(363\) 30.9092i 1.62231i
\(364\) 0 0
\(365\) − 35.7990i − 1.87380i
\(366\) 0 0
\(367\) 22.1731 1.15743 0.578713 0.815531i \(-0.303555\pi\)
0.578713 + 0.815531i \(0.303555\pi\)
\(368\) 0 0
\(369\) −15.3137 −0.797200
\(370\) 0 0
\(371\) − 11.3492i − 0.589220i
\(372\) 0 0
\(373\) 18.7279i 0.969695i 0.874599 + 0.484848i \(0.161125\pi\)
−0.874599 + 0.484848i \(0.838875\pi\)
\(374\) 0 0
\(375\) −14.7821 −0.763343
\(376\) 0 0
\(377\) −2.68629 −0.138351
\(378\) 0 0
\(379\) 4.40649i 0.226346i 0.993575 + 0.113173i \(0.0361015\pi\)
−0.993575 + 0.113173i \(0.963899\pi\)
\(380\) 0 0
\(381\) − 35.3137i − 1.80918i
\(382\) 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
\(387\) 10.0042i 0.508540i
\(388\) 0 0
\(389\) − 16.3848i − 0.830741i −0.909652 0.415371i \(-0.863652\pi\)
0.909652 0.415371i \(-0.136348\pi\)
\(390\) 0 0
\(391\) 16.5754 0.838256
\(392\) 0 0
\(393\) −23.7990 −1.20050
\(394\) 0 0
\(395\) − 20.9050i − 1.05185i
\(396\) 0 0
\(397\) − 9.07107i − 0.455264i −0.973747 0.227632i \(-0.926902\pi\)
0.973747 0.227632i \(-0.0730983\pi\)
\(398\) 0 0
\(399\) −1.79337 −0.0897806
\(400\) 0 0
\(401\) 24.4853 1.22274 0.611368 0.791346i \(-0.290619\pi\)
0.611368 + 0.791346i \(0.290619\pi\)
\(402\) 0 0
\(403\) 4.32957i 0.215671i
\(404\) 0 0
\(405\) − 19.8995i − 0.988814i
\(406\) 0 0
\(407\) −24.2291 −1.20099
\(408\) 0 0
\(409\) 8.68629 0.429509 0.214755 0.976668i \(-0.431105\pi\)
0.214755 + 0.976668i \(0.431105\pi\)
\(410\) 0 0
\(411\) − 54.7987i − 2.70302i
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 11.9832 0.588234
\(416\) 0 0
\(417\) −50.1421 −2.45547
\(418\) 0 0
\(419\) 14.3337i 0.700249i 0.936703 + 0.350124i \(0.113861\pi\)
−0.936703 + 0.350124i \(0.886139\pi\)
\(420\) 0 0
\(421\) 11.4142i 0.556295i 0.960538 + 0.278147i \(0.0897204\pi\)
−0.960538 + 0.278147i \(0.910280\pi\)
\(422\) 0 0
\(423\) 28.2960 1.37580
\(424\) 0 0
\(425\) 18.8284 0.913313
\(426\) 0 0
\(427\) 20.0083i 0.968271i
\(428\) 0 0
\(429\) 7.31371i 0.353109i
\(430\) 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
\(435\) − 40.9133i − 1.96164i
\(436\) 0 0
\(437\) − 2.62742i − 0.125686i
\(438\) 0 0
\(439\) 18.8490 0.899614 0.449807 0.893126i \(-0.351493\pi\)
0.449807 + 0.893126i \(0.351493\pi\)
\(440\) 0 0
\(441\) 17.8284 0.848973
\(442\) 0 0
\(443\) − 10.0042i − 0.475312i −0.971349 0.237656i \(-0.923621\pi\)
0.971349 0.237656i \(-0.0763791\pi\)
\(444\) 0 0
\(445\) 2.82843i 0.134080i
\(446\) 0 0
\(447\) 26.7653 1.26596
\(448\) 0 0
\(449\) 22.1421 1.04495 0.522476 0.852654i \(-0.325008\pi\)
0.522476 + 0.852654i \(0.325008\pi\)
\(450\) 0 0
\(451\) − 19.1116i − 0.899932i
\(452\) 0 0
\(453\) − 23.3137i − 1.09537i
\(454\) 0 0
\(455\) −3.06147 −0.143524
\(456\) 0 0
\(457\) −18.6274 −0.871354 −0.435677 0.900103i \(-0.643491\pi\)
−0.435677 + 0.900103i \(0.643491\pi\)
\(458\) 0 0
\(459\) − 6.12293i − 0.285794i
\(460\) 0 0
\(461\) − 29.7574i − 1.38594i −0.720967 0.692969i \(-0.756302\pi\)
0.720967 0.692969i \(-0.243698\pi\)
\(462\) 0 0
\(463\) −2.53620 −0.117867 −0.0589337 0.998262i \(-0.518770\pi\)
−0.0589337 + 0.998262i \(0.518770\pi\)
\(464\) 0 0
\(465\) −65.9411 −3.05795
\(466\) 0 0
\(467\) − 17.9205i − 0.829260i −0.909990 0.414630i \(-0.863911\pi\)
0.909990 0.414630i \(-0.136089\pi\)
\(468\) 0 0
\(469\) − 15.3137i − 0.707121i
\(470\) 0 0
\(471\) 16.3128 0.751654
\(472\) 0 0
\(473\) −12.4853 −0.574074
\(474\) 0 0
\(475\) − 2.98454i − 0.136940i
\(476\) 0 0
\(477\) − 28.3848i − 1.29965i
\(478\) 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
\(483\) − 23.4412i − 1.06661i
\(484\) 0 0
\(485\) 36.9706i 1.67875i
\(486\) 0 0
\(487\) −37.2178 −1.68650 −0.843250 0.537521i \(-0.819361\pi\)
−0.843250 + 0.537521i \(0.819361\pi\)
\(488\) 0 0
\(489\) −34.1421 −1.54396
\(490\) 0 0
\(491\) − 17.3952i − 0.785034i −0.919745 0.392517i \(-0.871604\pi\)
0.919745 0.392517i \(-0.128396\pi\)
\(492\) 0 0
\(493\) 12.9706i 0.584165i
\(494\) 0 0
\(495\) −62.4524 −2.80703
\(496\) 0 0
\(497\) −18.3431 −0.822803
\(498\) 0 0
\(499\) − 16.4985i − 0.738575i −0.929315 0.369287i \(-0.879602\pi\)
0.929315 0.369287i \(-0.120398\pi\)
\(500\) 0 0
\(501\) 4.00000i 0.178707i
\(502\) 0 0
\(503\) −18.1062 −0.807314 −0.403657 0.914910i \(-0.632261\pi\)
−0.403657 + 0.914910i \(0.632261\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) − 33.0740i − 1.46887i
\(508\) 0 0
\(509\) − 17.7574i − 0.787081i −0.919307 0.393541i \(-0.871250\pi\)
0.919307 0.393541i \(-0.128750\pi\)
\(510\) 0 0
\(511\) 16.0502 0.710018
\(512\) 0 0
\(513\) −0.970563 −0.0428514
\(514\) 0 0
\(515\) − 9.55582i − 0.421080i
\(516\) 0 0
\(517\) 35.3137i 1.55310i
\(518\) 0 0
\(519\) −60.6590 −2.66264
\(520\) 0 0
\(521\) −17.6569 −0.773561 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(522\) 0 0
\(523\) 39.5683i 1.73020i 0.501598 + 0.865101i \(0.332746\pi\)
−0.501598 + 0.865101i \(0.667254\pi\)
\(524\) 0 0
\(525\) − 26.6274i − 1.16212i
\(526\) 0 0
\(527\) 20.9050 0.910636
\(528\) 0 0
\(529\) 11.3431 0.493180
\(530\) 0 0
\(531\) − 10.0042i − 0.434144i
\(532\) 0 0
\(533\) − 2.34315i − 0.101493i
\(534\) 0 0
\(535\) −34.1563 −1.47671
\(536\) 0 0
\(537\) −4.48528 −0.193554
\(538\) 0 0
\(539\) 22.2500i 0.958377i
\(540\) 0 0
\(541\) − 1.75736i − 0.0755548i −0.999286 0.0377774i \(-0.987972\pi\)
0.999286 0.0377774i \(-0.0120278\pi\)
\(542\) 0 0
\(543\) −15.0447 −0.645630
\(544\) 0 0
\(545\) 3.65685 0.156642
\(546\) 0 0
\(547\) 1.34502i 0.0575091i 0.999587 + 0.0287545i \(0.00915412\pi\)
−0.999587 + 0.0287545i \(0.990846\pi\)
\(548\) 0 0
\(549\) 50.0416i 2.13572i
\(550\) 0 0
\(551\) 2.05600 0.0875885
\(552\) 0 0
\(553\) 9.37258 0.398563
\(554\) 0 0
\(555\) − 45.2429i − 1.92045i
\(556\) 0 0
\(557\) − 8.58579i − 0.363791i −0.983318 0.181896i \(-0.941777\pi\)
0.983318 0.181896i \(-0.0582233\pi\)
\(558\) 0 0
\(559\) −1.53073 −0.0647431
\(560\) 0 0
\(561\) 35.3137 1.49095
\(562\) 0 0
\(563\) 21.3533i 0.899936i 0.893045 + 0.449968i \(0.148565\pi\)
−0.893045 + 0.449968i \(0.851435\pi\)
\(564\) 0 0
\(565\) 12.4853i 0.525260i
\(566\) 0 0
\(567\) 8.92177 0.374679
\(568\) 0 0
\(569\) 14.3431 0.601296 0.300648 0.953735i \(-0.402797\pi\)
0.300648 + 0.953735i \(0.402797\pi\)
\(570\) 0 0
\(571\) − 22.2500i − 0.931135i −0.885012 0.465567i \(-0.845850\pi\)
0.885012 0.465567i \(-0.154150\pi\)
\(572\) 0 0
\(573\) 35.3137i 1.47525i
\(574\) 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
\(579\) 43.0781i 1.79027i
\(580\) 0 0
\(581\) 5.37258i 0.222892i
\(582\) 0 0
\(583\) 35.4244 1.46713
\(584\) 0 0
\(585\) −7.65685 −0.316572
\(586\) 0 0
\(587\) 29.1158i 1.20174i 0.799348 + 0.600869i \(0.205179\pi\)
−0.799348 + 0.600869i \(0.794821\pi\)
\(588\) 0 0
\(589\) − 3.31371i − 0.136539i
\(590\) 0 0
\(591\) 38.4859 1.58310
\(592\) 0 0
\(593\) −13.3137 −0.546728 −0.273364 0.961911i \(-0.588136\pi\)
−0.273364 + 0.961911i \(0.588136\pi\)
\(594\) 0 0
\(595\) 14.7821i 0.606006i
\(596\) 0 0
\(597\) 58.6274i 2.39946i
\(598\) 0 0
\(599\) −15.7875 −0.645061 −0.322531 0.946559i \(-0.604534\pi\)
−0.322531 + 0.946559i \(0.604534\pi\)
\(600\) 0 0
\(601\) −15.1716 −0.618861 −0.309431 0.950922i \(-0.600138\pi\)
−0.309431 + 0.950922i \(0.600138\pi\)
\(602\) 0 0
\(603\) − 38.3002i − 1.55970i
\(604\) 0 0
\(605\) − 40.3848i − 1.64187i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 18.3431 0.743302
\(610\) 0 0
\(611\) 4.32957i 0.175156i
\(612\) 0 0
\(613\) − 36.5858i − 1.47769i −0.673878 0.738843i \(-0.735373\pi\)
0.673878 0.738843i \(-0.264627\pi\)
\(614\) 0 0
\(615\) 35.6871 1.43904
\(616\) 0 0
\(617\) −21.5147 −0.866150 −0.433075 0.901358i \(-0.642571\pi\)
−0.433075 + 0.901358i \(0.642571\pi\)
\(618\) 0 0
\(619\) − 4.77791i − 0.192040i −0.995379 0.0960202i \(-0.969389\pi\)
0.995379 0.0960202i \(-0.0306113\pi\)
\(620\) 0 0
\(621\) − 12.6863i − 0.509083i
\(622\) 0 0
\(623\) −1.26810 −0.0508054
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) − 5.59767i − 0.223549i
\(628\) 0 0
\(629\) 14.3431i 0.571899i
\(630\) 0 0
\(631\) 2.79884 0.111420 0.0557099 0.998447i \(-0.482258\pi\)
0.0557099 + 0.998447i \(0.482258\pi\)
\(632\) 0 0
\(633\) −70.4264 −2.79920
\(634\) 0 0
\(635\) 46.1396i 1.83099i
\(636\) 0 0
\(637\) 2.72792i 0.108084i
\(638\) 0 0
\(639\) −45.8770 −1.81486
\(640\) 0 0
\(641\) −9.85786 −0.389362 −0.194681 0.980867i \(-0.562367\pi\)
−0.194681 + 0.980867i \(0.562367\pi\)
\(642\) 0 0
\(643\) 1.34502i 0.0530426i 0.999648 + 0.0265213i \(0.00844298\pi\)
−0.999648 + 0.0265213i \(0.991557\pi\)
\(644\) 0 0
\(645\) − 23.3137i − 0.917976i
\(646\) 0 0
\(647\) −0.262632 −0.0103251 −0.00516257 0.999987i \(-0.501643\pi\)
−0.00516257 + 0.999987i \(0.501643\pi\)
\(648\) 0 0
\(649\) 12.4853 0.490090
\(650\) 0 0
\(651\) − 29.5641i − 1.15871i
\(652\) 0 0
\(653\) 37.5563i 1.46969i 0.678233 + 0.734847i \(0.262746\pi\)
−0.678233 + 0.734847i \(0.737254\pi\)
\(654\) 0 0
\(655\) 31.0949 1.21498
\(656\) 0 0
\(657\) 40.1421 1.56609
\(658\) 0 0
\(659\) − 33.0740i − 1.28838i −0.764866 0.644189i \(-0.777195\pi\)
0.764866 0.644189i \(-0.222805\pi\)
\(660\) 0 0
\(661\) 14.9289i 0.580668i 0.956925 + 0.290334i \(0.0937664\pi\)
−0.956925 + 0.290334i \(0.906234\pi\)
\(662\) 0 0
\(663\) 4.32957 0.168147
\(664\) 0 0
\(665\) 2.34315 0.0908633
\(666\) 0 0
\(667\) 26.8741i 1.04057i
\(668\) 0 0
\(669\) 19.3137i 0.746711i
\(670\) 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
\(675\) − 14.4107i − 0.554666i
\(676\) 0 0
\(677\) 17.5563i 0.674745i 0.941371 + 0.337373i \(0.109538\pi\)
−0.941371 + 0.337373i \(0.890462\pi\)
\(678\) 0 0
\(679\) −16.5754 −0.636107
\(680\) 0 0
\(681\) −6.82843 −0.261666
\(682\) 0 0
\(683\) 20.0852i 0.768541i 0.923221 + 0.384270i \(0.125547\pi\)
−0.923221 + 0.384270i \(0.874453\pi\)
\(684\) 0 0
\(685\) 71.5980i 2.73562i
\(686\) 0 0
\(687\) −41.5474 −1.58513
\(688\) 0 0
\(689\) 4.34315 0.165461
\(690\) 0 0
\(691\) 19.9314i 0.758226i 0.925350 + 0.379113i \(0.123771\pi\)
−0.925350 + 0.379113i \(0.876229\pi\)
\(692\) 0 0
\(693\) − 28.0000i − 1.06363i
\(694\) 0 0
\(695\) 65.5139 2.48508
\(696\) 0 0
\(697\) −11.3137 −0.428537
\(698\) 0 0
\(699\) 39.6452i 1.49952i
\(700\) 0 0
\(701\) 28.8701i 1.09041i 0.838304 + 0.545204i \(0.183548\pi\)
−0.838304 + 0.545204i \(0.816452\pi\)
\(702\) 0 0
\(703\) 2.27357 0.0857493
\(704\) 0 0
\(705\) −65.9411 −2.48349
\(706\) 0 0
\(707\) − 4.48342i − 0.168616i
\(708\) 0 0
\(709\) − 35.6985i − 1.34068i −0.742052 0.670342i \(-0.766147\pi\)
0.742052 0.670342i \(-0.233853\pi\)
\(710\) 0 0
\(711\) 23.4412 0.879114
\(712\) 0 0
\(713\) 43.3137 1.62211
\(714\) 0 0
\(715\) − 9.55582i − 0.357367i
\(716\) 0 0
\(717\) − 77.2548i − 2.88513i
\(718\) 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
\(723\) 49.2011i 1.82981i
\(724\) 0 0
\(725\) 30.5269i 1.13374i
\(726\) 0 0
\(727\) −42.8155 −1.58794 −0.793969 0.607958i \(-0.791989\pi\)
−0.793969 + 0.607958i \(0.791989\pi\)
\(728\) 0 0
\(729\) 39.2843 1.45497
\(730\) 0 0
\(731\) 7.39104i 0.273367i
\(732\) 0 0
\(733\) − 46.5269i − 1.71851i −0.511547 0.859255i \(-0.670927\pi\)
0.511547 0.859255i \(-0.329073\pi\)
\(734\) 0 0
\(735\) −41.5474 −1.53250
\(736\) 0 0
\(737\) 47.7990 1.76070
\(738\) 0 0
\(739\) − 31.4344i − 1.15633i −0.815918 0.578167i \(-0.803768\pi\)
0.815918 0.578167i \(-0.196232\pi\)
\(740\) 0 0
\(741\) − 0.686292i − 0.0252115i
\(742\) 0 0
\(743\) −28.5587 −1.04772 −0.523858 0.851806i \(-0.675508\pi\)
−0.523858 + 0.851806i \(0.675508\pi\)
\(744\) 0 0
\(745\) −34.9706 −1.28122
\(746\) 0 0
\(747\) 13.4370i 0.491636i
\(748\) 0 0
\(749\) − 15.3137i − 0.559551i
\(750\) 0 0
\(751\) 6.12293 0.223429 0.111715 0.993740i \(-0.464366\pi\)
0.111715 + 0.993740i \(0.464366\pi\)
\(752\) 0 0
\(753\) −36.4853 −1.32960
\(754\) 0 0
\(755\) 30.4608i 1.10858i
\(756\) 0 0
\(757\) − 33.0711i − 1.20199i −0.799253 0.600994i \(-0.794771\pi\)
0.799253 0.600994i \(-0.205229\pi\)
\(758\) 0 0
\(759\) 73.1675 2.65581
\(760\) 0 0
\(761\) 10.6274 0.385244 0.192622 0.981273i \(-0.438301\pi\)
0.192622 + 0.981273i \(0.438301\pi\)
\(762\) 0 0
\(763\) 1.63952i 0.0593546i
\(764\) 0 0
\(765\) 36.9706i 1.33667i
\(766\) 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
\(771\) − 49.5725i − 1.78531i
\(772\) 0 0
\(773\) − 19.6985i − 0.708505i −0.935150 0.354253i \(-0.884735\pi\)
0.935150 0.354253i \(-0.115265\pi\)
\(774\) 0 0
\(775\) 49.2011 1.76735
\(776\) 0 0
\(777\) 20.2843 0.727694
\(778\) 0 0
\(779\) 1.79337i 0.0642540i
\(780\) 0 0
\(781\) − 57.2548i − 2.04874i
\(782\) 0 0
\(783\) 9.92724 0.354771
\(784\) 0 0
\(785\) −21.3137 −0.760719
\(786\) 0 0
\(787\) 48.7527i 1.73785i 0.494947 + 0.868923i \(0.335187\pi\)
−0.494947 + 0.868923i \(0.664813\pi\)
\(788\) 0 0
\(789\) − 53.9411i − 1.92035i
\(790\) 0 0
\(791\) −5.59767 −0.199030
\(792\) 0 0
\(793\) −7.65685 −0.271903
\(794\) 0 0
\(795\) 66.1479i 2.34602i
\(796\) 0 0
\(797\) − 19.6985i − 0.697756i −0.937168 0.348878i \(-0.886563\pi\)
0.937168 0.348878i \(-0.113437\pi\)
\(798\) 0 0
\(799\) 20.9050 0.739566
\(800\) 0 0
\(801\) −3.17157 −0.112062
\(802\) 0 0
\(803\) 50.0977i 1.76791i
\(804\) 0 0
\(805\) 30.6274i 1.07947i
\(806\) 0 0
\(807\) 14.5194 0.511108
\(808\) 0 0
\(809\) 47.3137 1.66346 0.831731 0.555179i \(-0.187350\pi\)
0.831731 + 0.555179i \(0.187350\pi\)
\(810\) 0 0
\(811\) 20.4567i 0.718331i 0.933274 + 0.359165i \(0.116939\pi\)
−0.933274 + 0.359165i \(0.883061\pi\)
\(812\) 0 0
\(813\) 73.9411i 2.59323i
\(814\) 0 0
\(815\) 44.6088 1.56258
\(816\) 0 0
\(817\) 1.17157 0.0409881
\(818\) 0 0
\(819\) − 3.43289i − 0.119955i
\(820\) 0 0
\(821\) − 26.5269i − 0.925796i −0.886412 0.462898i \(-0.846810\pi\)
0.886412 0.462898i \(-0.153190\pi\)
\(822\) 0 0
\(823\) −42.8155 −1.49245 −0.746227 0.665692i \(-0.768137\pi\)
−0.746227 + 0.665692i \(0.768137\pi\)
\(824\) 0 0
\(825\) 83.1127 2.89361
\(826\) 0 0
\(827\) − 29.1158i − 1.01246i −0.862400 0.506228i \(-0.831039\pi\)
0.862400 0.506228i \(-0.168961\pi\)
\(828\) 0 0
\(829\) − 24.3848i − 0.846918i −0.905915 0.423459i \(-0.860816\pi\)
0.905915 0.423459i \(-0.139184\pi\)
\(830\) 0 0
\(831\) −1.53073 −0.0531006
\(832\) 0 0
\(833\) 13.1716 0.456368
\(834\) 0 0
\(835\) − 5.22625i − 0.180862i
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) −39.7540 −1.37246 −0.686231 0.727384i \(-0.740736\pi\)
−0.686231 + 0.727384i \(0.740736\pi\)
\(840\) 0 0
\(841\) 7.97056 0.274847
\(842\) 0 0
\(843\) − 56.9635i − 1.96193i
\(844\) 0 0
\(845\) 43.2132i 1.48658i
\(846\) 0 0
\(847\) 18.1062 0.622135
\(848\) 0 0
\(849\) 23.7990 0.816779
\(850\) 0 0
\(851\) 29.7180i 1.01872i
\(852\) 0 0
\(853\) − 51.8995i − 1.77700i −0.458872 0.888502i \(-0.651746\pi\)
0.458872 0.888502i \(-0.348254\pi\)
\(854\) 0 0
\(855\) 5.86030 0.200418
\(856\) 0 0
\(857\) 32.2843 1.10281 0.551405 0.834238i \(-0.314092\pi\)
0.551405 + 0.834238i \(0.314092\pi\)
\(858\) 0 0
\(859\) − 28.7444i − 0.980746i −0.871513 0.490373i \(-0.836861\pi\)
0.871513 0.490373i \(-0.163139\pi\)
\(860\) 0 0
\(861\) 16.0000i 0.545279i
\(862\) 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
\(867\) 23.5181i 0.798718i
\(868\) 0 0
\(869\) 29.2548i 0.992402i
\(870\) 0 0
\(871\) 5.86030 0.198569
\(872\) 0 0
\(873\) −41.4558 −1.40307
\(874\) 0 0
\(875\) 8.65914i 0.292732i
\(876\) 0 0
\(877\) − 18.4437i − 0.622798i −0.950279 0.311399i \(-0.899202\pi\)
0.950279 0.311399i \(-0.100798\pi\)
\(878\) 0 0
\(879\) 8.92177 0.300924
\(880\) 0 0
\(881\) 27.9411 0.941360 0.470680 0.882304i \(-0.344009\pi\)
0.470680 + 0.882304i \(0.344009\pi\)
\(882\) 0 0
\(883\) 2.98454i 0.100438i 0.998738 + 0.0502190i \(0.0159919\pi\)
−0.998738 + 0.0502190i \(0.984008\pi\)
\(884\) 0 0
\(885\) 23.3137i 0.783682i
\(886\) 0 0
\(887\) 10.1899 0.342142 0.171071 0.985259i \(-0.445277\pi\)
0.171071 + 0.985259i \(0.445277\pi\)
\(888\) 0 0
\(889\) −20.6863 −0.693796
\(890\) 0 0
\(891\) 27.8477i 0.932933i
\(892\) 0 0
\(893\) − 3.31371i − 0.110889i
\(894\) 0 0
\(895\) 5.86030 0.195888
\(896\) 0 0
\(897\) 8.97056 0.299518
\(898\) 0 0
\(899\) 33.8937i 1.13042i
\(900\) 0 0
\(901\) − 20.9706i − 0.698631i
\(902\) 0 0
\(903\) 10.4525 0.347838
\(904\) 0 0
\(905\) 19.6569 0.653416
\(906\) 0 0
\(907\) − 9.47890i − 0.314742i −0.987540 0.157371i \(-0.949698\pi\)
0.987540 0.157371i \(-0.0503018\pi\)
\(908\) 0 0
\(909\) − 11.2132i − 0.371918i
\(910\) 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
\(915\) − 116.617i − 3.85524i
\(916\) 0 0
\(917\) 13.9411i 0.460377i
\(918\) 0 0
\(919\) −35.4244 −1.16854 −0.584272 0.811558i \(-0.698620\pi\)
−0.584272 + 0.811558i \(0.698620\pi\)
\(920\) 0 0
\(921\) 67.1127 2.21144
\(922\) 0 0
\(923\) − 7.01962i − 0.231054i
\(924\) 0 0
\(925\) 33.7574i 1.10994i
\(926\) 0 0
\(927\) 10.7151 0.351931
\(928\) 0 0
\(929\) −37.1716 −1.21956 −0.609780 0.792571i \(-0.708742\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(930\) 0 0
\(931\) − 2.08786i − 0.0684269i
\(932\) 0 0
\(933\) 18.6274i 0.609834i
\(934\) 0 0
\(935\) −46.1396 −1.50893
\(936\) 0 0
\(937\) 7.45584 0.243572 0.121786 0.992556i \(-0.461138\pi\)
0.121786 + 0.992556i \(0.461138\pi\)
\(938\) 0 0
\(939\) − 22.6984i − 0.740733i
\(940\) 0 0
\(941\) − 17.0711i − 0.556501i −0.960509 0.278250i \(-0.910245\pi\)
0.960509 0.278250i \(-0.0897545\pi\)
\(942\) 0 0
\(943\) −23.4412 −0.763351
\(944\) 0 0
\(945\) 11.3137 0.368035
\(946\) 0 0
\(947\) − 37.0321i − 1.20338i −0.798729 0.601691i \(-0.794494\pi\)
0.798729 0.601691i \(-0.205506\pi\)
\(948\) 0 0
\(949\) 6.14214i 0.199382i
\(950\) 0 0
\(951\) −32.8882 −1.06647
\(952\) 0 0
\(953\) −44.9706 −1.45674 −0.728370 0.685184i \(-0.759722\pi\)
−0.728370 + 0.685184i \(0.759722\pi\)
\(954\) 0 0
\(955\) − 46.1396i − 1.49304i
\(956\) 0 0
\(957\) 57.2548i 1.85079i
\(958\) 0 0
\(959\) −32.1003 −1.03657
\(960\) 0 0
\(961\) 23.6274 0.762175
\(962\) 0 0
\(963\) − 38.3002i − 1.23421i
\(964\) 0 0
\(965\) − 56.2843i − 1.81185i
\(966\) 0 0
\(967\) 4.59220 0.147675 0.0738376 0.997270i \(-0.476475\pi\)
0.0738376 + 0.997270i \(0.476475\pi\)
\(968\) 0 0
\(969\) −3.31371 −0.106452
\(970\) 0 0
\(971\) 8.58221i 0.275416i 0.990473 + 0.137708i \(0.0439736\pi\)
−0.990473 + 0.137708i \(0.956026\pi\)
\(972\) 0 0
\(973\) 29.3726i 0.941642i
\(974\) 0 0
\(975\) 10.1899 0.326337
\(976\) 0 0
\(977\) 15.1127 0.483498 0.241749 0.970339i \(-0.422279\pi\)
0.241749 + 0.970339i \(0.422279\pi\)
\(978\) 0 0
\(979\) − 3.95815i − 0.126503i
\(980\) 0 0
\(981\) 4.10051i 0.130919i
\(982\) 0 0
\(983\) −2.05600 −0.0655762 −0.0327881 0.999462i \(-0.510439\pi\)
−0.0327881 + 0.999462i \(0.510439\pi\)
\(984\) 0 0
\(985\) −50.2843 −1.60219
\(986\) 0 0
\(987\) − 29.5641i − 0.941037i
\(988\) 0 0
\(989\) 15.3137i 0.486948i
\(990\) 0 0
\(991\) 23.4412 0.744635 0.372317 0.928106i \(-0.378563\pi\)
0.372317 + 0.928106i \(0.378563\pi\)
\(992\) 0 0
\(993\) 14.8284 0.470566
\(994\) 0 0
\(995\) − 76.6004i − 2.42840i
\(996\) 0 0
\(997\) − 8.38478i − 0.265549i −0.991146 0.132774i \(-0.957611\pi\)
0.991146 0.132774i \(-0.0423885\pi\)
\(998\) 0 0
\(999\) 10.9778 0.347321
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 1024.2.b.g.513.2 8
4.3 odd 2 inner 1024.2.b.g.513.8 8
8.3 odd 2 inner 1024.2.b.g.513.1 8
8.5 even 2 inner 1024.2.b.g.513.7 8
16.3 odd 4 1024.2.a.i.1.1 4
16.5 even 4 1024.2.a.h.1.1 4
16.11 odd 4 1024.2.a.h.1.4 4
16.13 even 4 1024.2.a.i.1.4 4
32.3 odd 8 512.2.e.i.385.1 yes 8
32.5 even 8 512.2.e.i.129.4 yes 8
32.11 odd 8 512.2.e.j.129.4 yes 8
32.13 even 8 512.2.e.j.385.1 yes 8
32.19 odd 8 512.2.e.j.385.4 yes 8
32.21 even 8 512.2.e.j.129.1 yes 8
32.27 odd 8 512.2.e.i.129.1 8
32.29 even 8 512.2.e.i.385.4 yes 8
48.5 odd 4 9216.2.a.bp.1.4 4
48.11 even 4 9216.2.a.bp.1.3 4
48.29 odd 4 9216.2.a.w.1.2 4
48.35 even 4 9216.2.a.w.1.1 4
96.5 odd 8 4608.2.k.bi.1153.3 8
96.11 even 8 4608.2.k.bd.1153.2 8
96.29 odd 8 4608.2.k.bi.3457.4 8
96.35 even 8 4608.2.k.bi.3457.3 8
96.53 odd 8 4608.2.k.bd.1153.1 8
96.59 even 8 4608.2.k.bi.1153.4 8
96.77 odd 8 4608.2.k.bd.3457.2 8
96.83 even 8 4608.2.k.bd.3457.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.i.129.1 8 32.27 odd 8
512.2.e.i.129.4 yes 8 32.5 even 8
512.2.e.i.385.1 yes 8 32.3 odd 8
512.2.e.i.385.4 yes 8 32.29 even 8
512.2.e.j.129.1 yes 8 32.21 even 8
512.2.e.j.129.4 yes 8 32.11 odd 8
512.2.e.j.385.1 yes 8 32.13 even 8
512.2.e.j.385.4 yes 8 32.19 odd 8
1024.2.a.h.1.1 4 16.5 even 4
1024.2.a.h.1.4 4 16.11 odd 4
1024.2.a.i.1.1 4 16.3 odd 4
1024.2.a.i.1.4 4 16.13 even 4
1024.2.b.g.513.1 8 8.3 odd 2 inner
1024.2.b.g.513.2 8 1.1 even 1 trivial
1024.2.b.g.513.7 8 8.5 even 2 inner
1024.2.b.g.513.8 8 4.3 odd 2 inner
4608.2.k.bd.1153.1 8 96.53 odd 8
4608.2.k.bd.1153.2 8 96.11 even 8
4608.2.k.bd.3457.1 8 96.83 even 8
4608.2.k.bd.3457.2 8 96.77 odd 8
4608.2.k.bi.1153.3 8 96.5 odd 8
4608.2.k.bi.1153.4 8 96.59 even 8
4608.2.k.bi.3457.3 8 96.35 even 8
4608.2.k.bi.3457.4 8 96.29 odd 8
9216.2.a.w.1.1 4 48.35 even 4
9216.2.a.w.1.2 4 48.29 odd 4
9216.2.a.bp.1.3 4 48.11 even 4
9216.2.a.bp.1.4 4 48.5 odd 4