Properties

Label 2304.3.h.g.2177.1
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.g.2177.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{5} +8.00000 q^{7} +O(q^{10})\) \(q-1.41421 q^{5} +8.00000 q^{7} +11.3137 q^{11} -8.00000i q^{13} +12.7279i q^{17} -32.0000i q^{19} -33.9411i q^{23} -23.0000 q^{25} +43.8406 q^{29} -40.0000 q^{31} -11.3137 q^{35} +26.0000i q^{37} -66.4680i q^{41} +16.0000i q^{43} -11.3137i q^{47} +15.0000 q^{49} -32.5269 q^{53} -16.0000 q^{55} +22.6274 q^{59} -54.0000i q^{61} +11.3137i q^{65} +80.0000i q^{67} +79.1960i q^{71} -96.0000 q^{73} +90.5097 q^{77} +104.000 q^{79} +101.823 q^{83} -18.0000i q^{85} -77.7817i q^{89} -64.0000i q^{91} +45.2548i q^{95} -80.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{7} + O(q^{10}) \) \( 4 q + 32 q^{7} - 92 q^{25} - 160 q^{31} + 60 q^{49} - 64 q^{55} - 384 q^{73} + 416 q^{79} - 320 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.41421 −0.282843 −0.141421 0.989949i \(-0.545167\pi\)
−0.141421 + 0.989949i \(0.545167\pi\)
\(6\) 0 0
\(7\) 8.00000 1.14286 0.571429 0.820652i \(-0.306389\pi\)
0.571429 + 0.820652i \(0.306389\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 11.3137 1.02852 0.514259 0.857635i \(-0.328067\pi\)
0.514259 + 0.857635i \(0.328067\pi\)
\(12\) 0 0
\(13\) − 8.00000i − 0.615385i −0.951486 0.307692i \(-0.900443\pi\)
0.951486 0.307692i \(-0.0995567\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.7279i 0.748701i 0.927287 + 0.374351i \(0.122134\pi\)
−0.927287 + 0.374351i \(0.877866\pi\)
\(18\) 0 0
\(19\) − 32.0000i − 1.68421i −0.539313 0.842105i \(-0.681316\pi\)
0.539313 0.842105i \(-0.318684\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 33.9411i − 1.47570i −0.674964 0.737851i \(-0.735841\pi\)
0.674964 0.737851i \(-0.264159\pi\)
\(24\) 0 0
\(25\) −23.0000 −0.920000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 43.8406 1.51175 0.755873 0.654719i \(-0.227213\pi\)
0.755873 + 0.654719i \(0.227213\pi\)
\(30\) 0 0
\(31\) −40.0000 −1.29032 −0.645161 0.764046i \(-0.723210\pi\)
−0.645161 + 0.764046i \(0.723210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −11.3137 −0.323249
\(36\) 0 0
\(37\) 26.0000i 0.702703i 0.936244 + 0.351351i \(0.114278\pi\)
−0.936244 + 0.351351i \(0.885722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 66.4680i − 1.62117i −0.585620 0.810586i \(-0.699149\pi\)
0.585620 0.810586i \(-0.300851\pi\)
\(42\) 0 0
\(43\) 16.0000i 0.372093i 0.982541 + 0.186047i \(0.0595675\pi\)
−0.982541 + 0.186047i \(0.940432\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.3137i − 0.240717i −0.992730 0.120359i \(-0.961596\pi\)
0.992730 0.120359i \(-0.0384044\pi\)
\(48\) 0 0
\(49\) 15.0000 0.306122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −32.5269 −0.613715 −0.306858 0.951755i \(-0.599278\pi\)
−0.306858 + 0.951755i \(0.599278\pi\)
\(54\) 0 0
\(55\) −16.0000 −0.290909
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 22.6274 0.383516 0.191758 0.981442i \(-0.438581\pi\)
0.191758 + 0.981442i \(0.438581\pi\)
\(60\) 0 0
\(61\) − 54.0000i − 0.885246i −0.896708 0.442623i \(-0.854048\pi\)
0.896708 0.442623i \(-0.145952\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.3137i 0.174057i
\(66\) 0 0
\(67\) 80.0000i 1.19403i 0.802230 + 0.597015i \(0.203647\pi\)
−0.802230 + 0.597015i \(0.796353\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 79.1960i 1.11544i 0.830030 + 0.557718i \(0.188323\pi\)
−0.830030 + 0.557718i \(0.811677\pi\)
\(72\) 0 0
\(73\) −96.0000 −1.31507 −0.657534 0.753425i \(-0.728401\pi\)
−0.657534 + 0.753425i \(0.728401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 90.5097 1.17545
\(78\) 0 0
\(79\) 104.000 1.31646 0.658228 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 101.823 1.22679 0.613394 0.789777i \(-0.289804\pi\)
0.613394 + 0.789777i \(0.289804\pi\)
\(84\) 0 0
\(85\) − 18.0000i − 0.211765i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 77.7817i − 0.873952i −0.899473 0.436976i \(-0.856049\pi\)
0.899473 0.436976i \(-0.143951\pi\)
\(90\) 0 0
\(91\) − 64.0000i − 0.703297i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 45.2548i 0.476367i
\(96\) 0 0
\(97\) −80.0000 −0.824742 −0.412371 0.911016i \(-0.635299\pi\)
−0.412371 + 0.911016i \(0.635299\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 123.037 1.21818 0.609092 0.793100i \(-0.291534\pi\)
0.609092 + 0.793100i \(0.291534\pi\)
\(102\) 0 0
\(103\) −72.0000 −0.699029 −0.349515 0.936931i \(-0.613653\pi\)
−0.349515 + 0.936931i \(0.613653\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −181.019 −1.69177 −0.845885 0.533366i \(-0.820927\pi\)
−0.845885 + 0.533366i \(0.820927\pi\)
\(108\) 0 0
\(109\) − 88.0000i − 0.807339i −0.914905 0.403670i \(-0.867734\pi\)
0.914905 0.403670i \(-0.132266\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 137.179i − 1.21397i −0.794713 0.606985i \(-0.792379\pi\)
0.794713 0.606985i \(-0.207621\pi\)
\(114\) 0 0
\(115\) 48.0000i 0.417391i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 101.823i 0.855659i
\(120\) 0 0
\(121\) 7.00000 0.0578512
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 67.8823 0.543058
\(126\) 0 0
\(127\) 56.0000 0.440945 0.220472 0.975393i \(-0.429240\pi\)
0.220472 + 0.975393i \(0.429240\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −248.902 −1.90001 −0.950006 0.312231i \(-0.898924\pi\)
−0.950006 + 0.312231i \(0.898924\pi\)
\(132\) 0 0
\(133\) − 256.000i − 1.92481i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 46.6690i − 0.340650i −0.985388 0.170325i \(-0.945518\pi\)
0.985388 0.170325i \(-0.0544818\pi\)
\(138\) 0 0
\(139\) 16.0000i 0.115108i 0.998342 + 0.0575540i \(0.0183301\pi\)
−0.998342 + 0.0575540i \(0.981670\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 90.5097i − 0.632935i
\(144\) 0 0
\(145\) −62.0000 −0.427586
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 182.434 1.22439 0.612193 0.790708i \(-0.290288\pi\)
0.612193 + 0.790708i \(0.290288\pi\)
\(150\) 0 0
\(151\) 168.000 1.11258 0.556291 0.830987i \(-0.312224\pi\)
0.556291 + 0.830987i \(0.312224\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 56.5685 0.364958
\(156\) 0 0
\(157\) − 10.0000i − 0.0636943i −0.999493 0.0318471i \(-0.989861\pi\)
0.999493 0.0318471i \(-0.0101390\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 271.529i − 1.68652i
\(162\) 0 0
\(163\) − 80.0000i − 0.490798i −0.969422 0.245399i \(-0.921081\pi\)
0.969422 0.245399i \(-0.0789189\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 294.156i − 1.76142i −0.473660 0.880708i \(-0.657067\pi\)
0.473660 0.880708i \(-0.342933\pi\)
\(168\) 0 0
\(169\) 105.000 0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −55.1543 −0.318811 −0.159406 0.987213i \(-0.550958\pi\)
−0.159406 + 0.987213i \(0.550958\pi\)
\(174\) 0 0
\(175\) −184.000 −1.05143
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 135.765 0.758461 0.379230 0.925302i \(-0.376189\pi\)
0.379230 + 0.925302i \(0.376189\pi\)
\(180\) 0 0
\(181\) − 8.00000i − 0.0441989i −0.999756 0.0220994i \(-0.992965\pi\)
0.999756 0.0220994i \(-0.00703505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 36.7696i − 0.198754i
\(186\) 0 0
\(187\) 144.000i 0.770053i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 67.8823i 0.355404i 0.984084 + 0.177702i \(0.0568664\pi\)
−0.984084 + 0.177702i \(0.943134\pi\)
\(192\) 0 0
\(193\) −258.000 −1.33679 −0.668394 0.743808i \(-0.733018\pi\)
−0.668394 + 0.743808i \(0.733018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 371.938 1.88801 0.944005 0.329930i \(-0.107025\pi\)
0.944005 + 0.329930i \(0.107025\pi\)
\(198\) 0 0
\(199\) 88.0000 0.442211 0.221106 0.975250i \(-0.429033\pi\)
0.221106 + 0.975250i \(0.429033\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 350.725 1.72771
\(204\) 0 0
\(205\) 94.0000i 0.458537i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 362.039i − 1.73224i
\(210\) 0 0
\(211\) − 368.000i − 1.74408i −0.489438 0.872038i \(-0.662798\pi\)
0.489438 0.872038i \(-0.337202\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 22.6274i − 0.105244i
\(216\) 0 0
\(217\) −320.000 −1.47465
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 101.823 0.460739
\(222\) 0 0
\(223\) −104.000 −0.466368 −0.233184 0.972433i \(-0.574914\pi\)
−0.233184 + 0.972433i \(0.574914\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 169.706 0.747602 0.373801 0.927509i \(-0.378054\pi\)
0.373801 + 0.927509i \(0.378054\pi\)
\(228\) 0 0
\(229\) − 344.000i − 1.50218i −0.660198 0.751092i \(-0.729528\pi\)
0.660198 0.751092i \(-0.270472\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 80.6102i − 0.345966i −0.984925 0.172983i \(-0.944659\pi\)
0.984925 0.172983i \(-0.0553406\pi\)
\(234\) 0 0
\(235\) 16.0000i 0.0680851i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 271.529i 1.13610i 0.822992 + 0.568052i \(0.192303\pi\)
−0.822992 + 0.568052i \(0.807697\pi\)
\(240\) 0 0
\(241\) −272.000 −1.12863 −0.564315 0.825559i \(-0.690860\pi\)
−0.564315 + 0.825559i \(0.690860\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −21.2132 −0.0865845
\(246\) 0 0
\(247\) −256.000 −1.03644
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 33.9411 0.135224 0.0676118 0.997712i \(-0.478462\pi\)
0.0676118 + 0.997712i \(0.478462\pi\)
\(252\) 0 0
\(253\) − 384.000i − 1.51779i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 43.8406i − 0.170586i −0.996356 0.0852930i \(-0.972817\pi\)
0.996356 0.0852930i \(-0.0271826\pi\)
\(258\) 0 0
\(259\) 208.000i 0.803089i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 203.647i 0.774322i 0.922012 + 0.387161i \(0.126544\pi\)
−0.922012 + 0.387161i \(0.873456\pi\)
\(264\) 0 0
\(265\) 46.0000 0.173585
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 46.6690 0.173491 0.0867454 0.996231i \(-0.472353\pi\)
0.0867454 + 0.996231i \(0.472353\pi\)
\(270\) 0 0
\(271\) −264.000 −0.974170 −0.487085 0.873355i \(-0.661940\pi\)
−0.487085 + 0.873355i \(0.661940\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −260.215 −0.946237
\(276\) 0 0
\(277\) − 40.0000i − 0.144404i −0.997390 0.0722022i \(-0.976997\pi\)
0.997390 0.0722022i \(-0.0230027\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 190.919i − 0.679426i −0.940529 0.339713i \(-0.889670\pi\)
0.940529 0.339713i \(-0.110330\pi\)
\(282\) 0 0
\(283\) − 224.000i − 0.791519i −0.918354 0.395760i \(-0.870481\pi\)
0.918354 0.395760i \(-0.129519\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 531.744i − 1.85277i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 352.139 1.20184 0.600920 0.799309i \(-0.294801\pi\)
0.600920 + 0.799309i \(0.294801\pi\)
\(294\) 0 0
\(295\) −32.0000 −0.108475
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −271.529 −0.908124
\(300\) 0 0
\(301\) 128.000i 0.425249i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 76.3675i 0.250385i
\(306\) 0 0
\(307\) 432.000i 1.40717i 0.710613 + 0.703583i \(0.248418\pi\)
−0.710613 + 0.703583i \(0.751582\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 203.647i − 0.654813i −0.944884 0.327406i \(-0.893825\pi\)
0.944884 0.327406i \(-0.106175\pi\)
\(312\) 0 0
\(313\) 14.0000 0.0447284 0.0223642 0.999750i \(-0.492881\pi\)
0.0223642 + 0.999750i \(0.492881\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 589.727 1.86034 0.930169 0.367132i \(-0.119660\pi\)
0.930169 + 0.367132i \(0.119660\pi\)
\(318\) 0 0
\(319\) 496.000 1.55486
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 407.294 1.26097
\(324\) 0 0
\(325\) 184.000i 0.566154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 90.5097i − 0.275105i
\(330\) 0 0
\(331\) 16.0000i 0.0483384i 0.999708 + 0.0241692i \(0.00769404\pi\)
−0.999708 + 0.0241692i \(0.992306\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 113.137i − 0.337723i
\(336\) 0 0
\(337\) 128.000 0.379822 0.189911 0.981801i \(-0.439180\pi\)
0.189911 + 0.981801i \(0.439180\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −452.548 −1.32712
\(342\) 0 0
\(343\) −272.000 −0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −350.725 −1.01073 −0.505367 0.862904i \(-0.668643\pi\)
−0.505367 + 0.862904i \(0.668643\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.0286533i 0.999897 + 0.0143266i \(0.00456047\pi\)
−0.999897 + 0.0143266i \(0.995440\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 247.487i 0.701097i 0.936545 + 0.350549i \(0.114005\pi\)
−0.936545 + 0.350549i \(0.885995\pi\)
\(354\) 0 0
\(355\) − 112.000i − 0.315493i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 328.098i 0.913921i 0.889487 + 0.456960i \(0.151062\pi\)
−0.889487 + 0.456960i \(0.848938\pi\)
\(360\) 0 0
\(361\) −663.000 −1.83657
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 135.765 0.371958
\(366\) 0 0
\(367\) 696.000 1.89646 0.948229 0.317588i \(-0.102873\pi\)
0.948229 + 0.317588i \(0.102873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −260.215 −0.701389
\(372\) 0 0
\(373\) 454.000i 1.21716i 0.793493 + 0.608579i \(0.208260\pi\)
−0.793493 + 0.608579i \(0.791740\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 350.725i − 0.930305i
\(378\) 0 0
\(379\) − 64.0000i − 0.168865i −0.996429 0.0844327i \(-0.973092\pi\)
0.996429 0.0844327i \(-0.0269078\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 362.039i 0.945271i 0.881258 + 0.472635i \(0.156697\pi\)
−0.881258 + 0.472635i \(0.843303\pi\)
\(384\) 0 0
\(385\) −128.000 −0.332468
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 386.080 0.992494 0.496247 0.868181i \(-0.334711\pi\)
0.496247 + 0.868181i \(0.334711\pi\)
\(390\) 0 0
\(391\) 432.000 1.10486
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −147.078 −0.372350
\(396\) 0 0
\(397\) − 662.000i − 1.66751i −0.552137 0.833753i \(-0.686188\pi\)
0.552137 0.833753i \(-0.313812\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 171.120i 0.426733i 0.976972 + 0.213366i \(0.0684428\pi\)
−0.976972 + 0.213366i \(0.931557\pi\)
\(402\) 0 0
\(403\) 320.000i 0.794045i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 294.156i 0.722743i
\(408\) 0 0
\(409\) 176.000 0.430318 0.215159 0.976579i \(-0.430973\pi\)
0.215159 + 0.976579i \(0.430973\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 181.019 0.438303
\(414\) 0 0
\(415\) −144.000 −0.346988
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −79.1960 −0.189012 −0.0945059 0.995524i \(-0.530127\pi\)
−0.0945059 + 0.995524i \(0.530127\pi\)
\(420\) 0 0
\(421\) − 488.000i − 1.15914i −0.814921 0.579572i \(-0.803220\pi\)
0.814921 0.579572i \(-0.196780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 292.742i − 0.688805i
\(426\) 0 0
\(427\) − 432.000i − 1.01171i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 305.470i 0.708747i 0.935104 + 0.354374i \(0.115306\pi\)
−0.935104 + 0.354374i \(0.884694\pi\)
\(432\) 0 0
\(433\) 478.000 1.10393 0.551963 0.833869i \(-0.313879\pi\)
0.551963 + 0.833869i \(0.313879\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1086.12 −2.48539
\(438\) 0 0
\(439\) 392.000 0.892938 0.446469 0.894799i \(-0.352681\pi\)
0.446469 + 0.894799i \(0.352681\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.9411 0.0766165 0.0383083 0.999266i \(-0.487803\pi\)
0.0383083 + 0.999266i \(0.487803\pi\)
\(444\) 0 0
\(445\) 110.000i 0.247191i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 125.865i − 0.280323i −0.990129 0.140161i \(-0.955238\pi\)
0.990129 0.140161i \(-0.0447622\pi\)
\(450\) 0 0
\(451\) − 752.000i − 1.66741i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 90.5097i 0.198922i
\(456\) 0 0
\(457\) 16.0000 0.0350109 0.0175055 0.999847i \(-0.494428\pi\)
0.0175055 + 0.999847i \(0.494428\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −284.257 −0.616609 −0.308305 0.951288i \(-0.599762\pi\)
−0.308305 + 0.951288i \(0.599762\pi\)
\(462\) 0 0
\(463\) −568.000 −1.22678 −0.613391 0.789779i \(-0.710195\pi\)
−0.613391 + 0.789779i \(0.710195\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −418.607 −0.896375 −0.448188 0.893940i \(-0.647930\pi\)
−0.448188 + 0.893940i \(0.647930\pi\)
\(468\) 0 0
\(469\) 640.000i 1.36461i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 181.019i 0.382705i
\(474\) 0 0
\(475\) 736.000i 1.54947i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.9411i 0.0708583i 0.999372 + 0.0354291i \(0.0112798\pi\)
−0.999372 + 0.0354291i \(0.988720\pi\)
\(480\) 0 0
\(481\) 208.000 0.432432
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 113.137 0.233272
\(486\) 0 0
\(487\) −424.000 −0.870637 −0.435318 0.900277i \(-0.643364\pi\)
−0.435318 + 0.900277i \(0.643364\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 724.077 1.47470 0.737350 0.675511i \(-0.236077\pi\)
0.737350 + 0.675511i \(0.236077\pi\)
\(492\) 0 0
\(493\) 558.000i 1.13185i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 633.568i 1.27478i
\(498\) 0 0
\(499\) 192.000i 0.384770i 0.981320 + 0.192385i \(0.0616222\pi\)
−0.981320 + 0.192385i \(0.938378\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 441.235i − 0.877206i −0.898681 0.438603i \(-0.855473\pi\)
0.898681 0.438603i \(-0.144527\pi\)
\(504\) 0 0
\(505\) −174.000 −0.344554
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −250.316 −0.491780 −0.245890 0.969298i \(-0.579080\pi\)
−0.245890 + 0.969298i \(0.579080\pi\)
\(510\) 0 0
\(511\) −768.000 −1.50294
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 101.823 0.197715
\(516\) 0 0
\(517\) − 128.000i − 0.247582i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 156.978i 0.301301i 0.988587 + 0.150650i \(0.0481368\pi\)
−0.988587 + 0.150650i \(0.951863\pi\)
\(522\) 0 0
\(523\) 576.000i 1.10134i 0.834724 + 0.550669i \(0.185627\pi\)
−0.834724 + 0.550669i \(0.814373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 509.117i − 0.966066i
\(528\) 0 0
\(529\) −623.000 −1.17769
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −531.744 −0.997644
\(534\) 0 0
\(535\) 256.000 0.478505
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 169.706 0.314853
\(540\) 0 0
\(541\) − 536.000i − 0.990758i −0.868677 0.495379i \(-0.835029\pi\)
0.868677 0.495379i \(-0.164971\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 124.451i 0.228350i
\(546\) 0 0
\(547\) − 144.000i − 0.263254i −0.991299 0.131627i \(-0.957980\pi\)
0.991299 0.131627i \(-0.0420201\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1402.90i − 2.54610i
\(552\) 0 0
\(553\) 832.000 1.50452
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −533.159 −0.957197 −0.478598 0.878034i \(-0.658855\pi\)
−0.478598 + 0.878034i \(0.658855\pi\)
\(558\) 0 0
\(559\) 128.000 0.228980
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 576.999 1.02487 0.512433 0.858727i \(-0.328744\pi\)
0.512433 + 0.858727i \(0.328744\pi\)
\(564\) 0 0
\(565\) 194.000i 0.343363i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 994.192i 1.74726i 0.486589 + 0.873631i \(0.338241\pi\)
−0.486589 + 0.873631i \(0.661759\pi\)
\(570\) 0 0
\(571\) 416.000i 0.728546i 0.931292 + 0.364273i \(0.118683\pi\)
−0.931292 + 0.364273i \(0.881317\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 780.646i 1.35765i
\(576\) 0 0
\(577\) 834.000 1.44541 0.722704 0.691158i \(-0.242899\pi\)
0.722704 + 0.691158i \(0.242899\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 814.587 1.40204
\(582\) 0 0
\(583\) −368.000 −0.631218
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 67.8823 0.115643 0.0578213 0.998327i \(-0.481585\pi\)
0.0578213 + 0.998327i \(0.481585\pi\)
\(588\) 0 0
\(589\) 1280.00i 2.17317i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 224.860i − 0.379190i −0.981862 0.189595i \(-0.939282\pi\)
0.981862 0.189595i \(-0.0607176\pi\)
\(594\) 0 0
\(595\) − 144.000i − 0.242017i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 690.136i 1.15215i 0.817398 + 0.576074i \(0.195416\pi\)
−0.817398 + 0.576074i \(0.804584\pi\)
\(600\) 0 0
\(601\) 626.000 1.04160 0.520799 0.853680i \(-0.325634\pi\)
0.520799 + 0.853680i \(0.325634\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.89949 −0.0163628
\(606\) 0 0
\(607\) −232.000 −0.382208 −0.191104 0.981570i \(-0.561207\pi\)
−0.191104 + 0.981570i \(0.561207\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −90.5097 −0.148134
\(612\) 0 0
\(613\) − 666.000i − 1.08646i −0.839584 0.543230i \(-0.817201\pi\)
0.839584 0.543230i \(-0.182799\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 284.257i 0.460708i 0.973107 + 0.230354i \(0.0739884\pi\)
−0.973107 + 0.230354i \(0.926012\pi\)
\(618\) 0 0
\(619\) − 768.000i − 1.24071i −0.784321 0.620355i \(-0.786988\pi\)
0.784321 0.620355i \(-0.213012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 622.254i − 0.998803i
\(624\) 0 0
\(625\) 479.000 0.766400
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −330.926 −0.526114
\(630\) 0 0
\(631\) 472.000 0.748019 0.374010 0.927425i \(-0.377983\pi\)
0.374010 + 0.927425i \(0.377983\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −79.1960 −0.124718
\(636\) 0 0
\(637\) − 120.000i − 0.188383i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 371.938i − 0.580247i −0.956989 0.290123i \(-0.906304\pi\)
0.956989 0.290123i \(-0.0936963\pi\)
\(642\) 0 0
\(643\) 240.000i 0.373250i 0.982431 + 0.186625i \(0.0597550\pi\)
−0.982431 + 0.186625i \(0.940245\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 712.764i 1.10164i 0.834623 + 0.550822i \(0.185686\pi\)
−0.834623 + 0.550822i \(0.814314\pi\)
\(648\) 0 0
\(649\) 256.000 0.394453
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1163.90 −1.78239 −0.891193 0.453625i \(-0.850131\pi\)
−0.891193 + 0.453625i \(0.850131\pi\)
\(654\) 0 0
\(655\) 352.000 0.537405
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −203.647 −0.309024 −0.154512 0.987991i \(-0.549381\pi\)
−0.154512 + 0.987991i \(0.549381\pi\)
\(660\) 0 0
\(661\) 1018.00i 1.54009i 0.637989 + 0.770045i \(0.279766\pi\)
−0.637989 + 0.770045i \(0.720234\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 362.039i 0.544419i
\(666\) 0 0
\(667\) − 1488.00i − 2.23088i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 610.940i − 0.910492i
\(672\) 0 0
\(673\) −382.000 −0.567608 −0.283804 0.958882i \(-0.591596\pi\)
−0.283804 + 0.958882i \(0.591596\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 247.487 0.365565 0.182782 0.983153i \(-0.441490\pi\)
0.182782 + 0.983153i \(0.441490\pi\)
\(678\) 0 0
\(679\) −640.000 −0.942563
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 169.706 0.248471 0.124235 0.992253i \(-0.460352\pi\)
0.124235 + 0.992253i \(0.460352\pi\)
\(684\) 0 0
\(685\) 66.0000i 0.0963504i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 260.215i 0.377671i
\(690\) 0 0
\(691\) − 1040.00i − 1.50507i −0.658555 0.752533i \(-0.728832\pi\)
0.658555 0.752533i \(-0.271168\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 22.6274i − 0.0325574i
\(696\) 0 0
\(697\) 846.000 1.21377
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −657.609 −0.938102 −0.469051 0.883171i \(-0.655404\pi\)
−0.469051 + 0.883171i \(0.655404\pi\)
\(702\) 0 0
\(703\) 832.000 1.18350
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 984.293 1.39221
\(708\) 0 0
\(709\) − 952.000i − 1.34274i −0.741124 0.671368i \(-0.765707\pi\)
0.741124 0.671368i \(-0.234293\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1357.65i 1.90413i
\(714\) 0 0
\(715\) 128.000i 0.179021i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 192.333i 0.267501i 0.991015 + 0.133750i \(0.0427020\pi\)
−0.991015 + 0.133750i \(0.957298\pi\)
\(720\) 0 0
\(721\) −576.000 −0.798890
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1008.33 −1.39081
\(726\) 0 0
\(727\) −648.000 −0.891334 −0.445667 0.895199i \(-0.647033\pi\)
−0.445667 + 0.895199i \(0.647033\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −203.647 −0.278587
\(732\) 0 0
\(733\) 1208.00i 1.64802i 0.566574 + 0.824011i \(0.308269\pi\)
−0.566574 + 0.824011i \(0.691731\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 905.097i 1.22808i
\(738\) 0 0
\(739\) 1312.00i 1.77537i 0.460449 + 0.887686i \(0.347688\pi\)
−0.460449 + 0.887686i \(0.652312\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 610.940i 0.822261i 0.911576 + 0.411131i \(0.134866\pi\)
−0.911576 + 0.411131i \(0.865134\pi\)
\(744\) 0 0
\(745\) −258.000 −0.346309
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1448.15 −1.93345
\(750\) 0 0
\(751\) 632.000 0.841545 0.420772 0.907166i \(-0.361759\pi\)
0.420772 + 0.907166i \(0.361759\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −237.588 −0.314686
\(756\) 0 0
\(757\) 840.000i 1.10964i 0.831969 + 0.554822i \(0.187214\pi\)
−0.831969 + 0.554822i \(0.812786\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 521.845i 0.685736i 0.939384 + 0.342868i \(0.111398\pi\)
−0.939384 + 0.342868i \(0.888602\pi\)
\(762\) 0 0
\(763\) − 704.000i − 0.922674i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 181.019i − 0.236010i
\(768\) 0 0
\(769\) 130.000 0.169051 0.0845254 0.996421i \(-0.473063\pi\)
0.0845254 + 0.996421i \(0.473063\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 284.257 0.367732 0.183866 0.982951i \(-0.441139\pi\)
0.183866 + 0.982951i \(0.441139\pi\)
\(774\) 0 0
\(775\) 920.000 1.18710
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2126.98 −2.73039
\(780\) 0 0
\(781\) 896.000i 1.14725i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.1421i 0.0180155i
\(786\) 0 0
\(787\) 864.000i 1.09784i 0.835875 + 0.548920i \(0.184961\pi\)
−0.835875 + 0.548920i \(0.815039\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1097.43i − 1.38740i
\(792\) 0 0
\(793\) −432.000 −0.544767
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 759.433 0.952864 0.476432 0.879211i \(-0.341930\pi\)
0.476432 + 0.879211i \(0.341930\pi\)
\(798\) 0 0
\(799\) 144.000 0.180225
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1086.12 −1.35257
\(804\) 0 0
\(805\) 384.000i 0.477019i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 906.511i 1.12053i 0.828313 + 0.560266i \(0.189301\pi\)
−0.828313 + 0.560266i \(0.810699\pi\)
\(810\) 0 0
\(811\) 1472.00i 1.81504i 0.420005 + 0.907522i \(0.362028\pi\)
−0.420005 + 0.907522i \(0.637972\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 113.137i 0.138819i
\(816\) 0 0
\(817\) 512.000 0.626683
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1189.35 −1.44866 −0.724332 0.689451i \(-0.757852\pi\)
−0.724332 + 0.689451i \(0.757852\pi\)
\(822\) 0 0
\(823\) −664.000 −0.806804 −0.403402 0.915023i \(-0.632172\pi\)
−0.403402 + 0.915023i \(0.632172\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 248.902 0.300969 0.150485 0.988612i \(-0.451917\pi\)
0.150485 + 0.988612i \(0.451917\pi\)
\(828\) 0 0
\(829\) − 280.000i − 0.337756i −0.985637 0.168878i \(-0.945986\pi\)
0.985637 0.168878i \(-0.0540144\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 190.919i 0.229194i
\(834\) 0 0
\(835\) 416.000i 0.498204i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 554.372i 0.660753i 0.943849 + 0.330376i \(0.107176\pi\)
−0.943849 + 0.330376i \(0.892824\pi\)
\(840\) 0 0
\(841\) 1081.00 1.28537
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −148.492 −0.175731
\(846\) 0 0
\(847\) 56.0000 0.0661157
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 882.469 1.03698
\(852\) 0 0
\(853\) 762.000i 0.893318i 0.894704 + 0.446659i \(0.147386\pi\)
−0.894704 + 0.446659i \(0.852614\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1537.25i − 1.79376i −0.442277 0.896879i \(-0.645829\pi\)
0.442277 0.896879i \(-0.354171\pi\)
\(858\) 0 0
\(859\) 1552.00i 1.80675i 0.428849 + 0.903376i \(0.358919\pi\)
−0.428849 + 0.903376i \(0.641081\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.6274i 0.0262195i 0.999914 + 0.0131097i \(0.00417308\pi\)
−0.999914 + 0.0131097i \(0.995827\pi\)
\(864\) 0 0
\(865\) 78.0000 0.0901734
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1176.63 1.35400
\(870\) 0 0
\(871\) 640.000 0.734788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 543.058 0.620638
\(876\) 0 0
\(877\) − 86.0000i − 0.0980616i −0.998797 0.0490308i \(-0.984387\pi\)
0.998797 0.0490308i \(-0.0156132\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 657.609i 0.746435i 0.927744 + 0.373218i \(0.121745\pi\)
−0.927744 + 0.373218i \(0.878255\pi\)
\(882\) 0 0
\(883\) − 496.000i − 0.561721i −0.959749 0.280861i \(-0.909380\pi\)
0.959749 0.280861i \(-0.0906199\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1018.23i 1.14795i 0.818872 + 0.573976i \(0.194600\pi\)
−0.818872 + 0.573976i \(0.805400\pi\)
\(888\) 0 0
\(889\) 448.000 0.503937
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −362.039 −0.405418
\(894\) 0 0
\(895\) −192.000 −0.214525
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1753.62 −1.95064
\(900\) 0 0
\(901\) − 414.000i − 0.459489i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.3137i 0.0125013i
\(906\) 0 0
\(907\) − 880.000i − 0.970232i −0.874450 0.485116i \(-0.838777\pi\)
0.874450 0.485116i \(-0.161223\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1561.29i 1.71382i 0.515464 + 0.856911i \(0.327619\pi\)
−0.515464 + 0.856911i \(0.672381\pi\)
\(912\) 0 0
\(913\) 1152.00 1.26177
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1991.21 −2.17144
\(918\) 0 0
\(919\) 264.000 0.287269 0.143634 0.989631i \(-0.454121\pi\)
0.143634 + 0.989631i \(0.454121\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 633.568 0.686422
\(924\) 0 0
\(925\) − 598.000i − 0.646486i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 284.257i 0.305982i 0.988228 + 0.152991i \(0.0488905\pi\)
−0.988228 + 0.152991i \(0.951110\pi\)
\(930\) 0 0
\(931\) − 480.000i − 0.515575i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 203.647i − 0.217804i
\(936\) 0 0
\(937\) −1070.00 −1.14194 −0.570971 0.820970i \(-0.693433\pi\)
−0.570971 + 0.820970i \(0.693433\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 575.585 0.611674 0.305837 0.952084i \(-0.401064\pi\)
0.305837 + 0.952084i \(0.401064\pi\)
\(942\) 0 0
\(943\) −2256.00 −2.39236
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 610.940 0.645132 0.322566 0.946547i \(-0.395455\pi\)
0.322566 + 0.946547i \(0.395455\pi\)
\(948\) 0 0
\(949\) 768.000i 0.809273i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1039.45i 1.09071i 0.838205 + 0.545355i \(0.183605\pi\)
−0.838205 + 0.545355i \(0.816395\pi\)
\(954\) 0 0
\(955\) − 96.0000i − 0.100524i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 373.352i − 0.389314i
\(960\) 0 0
\(961\) 639.000 0.664932
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 364.867 0.378101
\(966\) 0 0
\(967\) 696.000 0.719752 0.359876 0.933000i \(-0.382819\pi\)
0.359876 + 0.933000i \(0.382819\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1572.61 1.61957 0.809787 0.586725i \(-0.199583\pi\)
0.809787 + 0.586725i \(0.199583\pi\)
\(972\) 0 0
\(973\) 128.000i 0.131552i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 507.703i 0.519655i 0.965655 + 0.259827i \(0.0836657\pi\)
−0.965655 + 0.259827i \(0.916334\pi\)
\(978\) 0 0
\(979\) − 880.000i − 0.898876i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 475.176i − 0.483393i −0.970352 0.241697i \(-0.922296\pi\)
0.970352 0.241697i \(-0.0777039\pi\)
\(984\) 0 0
\(985\) −526.000 −0.534010
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 543.058 0.549098
\(990\) 0 0
\(991\) −520.000 −0.524723 −0.262361 0.964970i \(-0.584501\pi\)
−0.262361 + 0.964970i \(0.584501\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −124.451 −0.125076
\(996\) 0 0
\(997\) 486.000i 0.487462i 0.969843 + 0.243731i \(0.0783715\pi\)
−0.969843 + 0.243731i \(0.921629\pi\)
\(998\) 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 2304.3.h.g.2177.1 4
3.2 odd 2 inner 2304.3.h.g.2177.3 4
4.3 odd 2 2304.3.h.b.2177.1 4
8.3 odd 2 2304.3.h.b.2177.4 4
8.5 even 2 inner 2304.3.h.g.2177.4 4
12.11 even 2 2304.3.h.b.2177.3 4
16.3 odd 4 576.3.e.g.449.2 2
16.5 even 4 288.3.e.a.161.1 2
16.11 odd 4 288.3.e.d.161.1 yes 2
16.13 even 4 576.3.e.b.449.2 2
24.5 odd 2 inner 2304.3.h.g.2177.2 4
24.11 even 2 2304.3.h.b.2177.2 4
48.5 odd 4 288.3.e.a.161.2 yes 2
48.11 even 4 288.3.e.d.161.2 yes 2
48.29 odd 4 576.3.e.b.449.1 2
48.35 even 4 576.3.e.g.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.a.161.1 2 16.5 even 4
288.3.e.a.161.2 yes 2 48.5 odd 4
288.3.e.d.161.1 yes 2 16.11 odd 4
288.3.e.d.161.2 yes 2 48.11 even 4
576.3.e.b.449.1 2 48.29 odd 4
576.3.e.b.449.2 2 16.13 even 4
576.3.e.g.449.1 2 48.35 even 4
576.3.e.g.449.2 2 16.3 odd 4
2304.3.h.b.2177.1 4 4.3 odd 2
2304.3.h.b.2177.2 4 24.11 even 2
2304.3.h.b.2177.3 4 12.11 even 2
2304.3.h.b.2177.4 4 8.3 odd 2
2304.3.h.g.2177.1 4 1.1 even 1 trivial
2304.3.h.g.2177.2 4 24.5 odd 2 inner
2304.3.h.g.2177.3 4 3.2 odd 2 inner
2304.3.h.g.2177.4 4 8.5 even 2 inner