Properties

Label 1216.2.c.i.609.1
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.1
Root \(-0.656712 - 2.13746i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.i.609.7

$q$-expansion

\(f(q)\) \(=\) \(q-3.04547i q^{5} -0.418627 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-3.04547i q^{5} -0.418627 q^{7} +3.00000 q^{9} -5.27492i q^{11} +2.62685i q^{13} +3.27492 q^{17} -1.00000i q^{19} -3.46410 q^{23} -4.27492 q^{25} +2.62685i q^{29} -6.09095 q^{31} +1.27492i q^{35} -9.55505i q^{37} +4.54983 q^{41} -2.72508i q^{43} -9.13642i q^{45} +0.418627 q^{47} -6.82475 q^{49} -10.3923i q^{53} -16.0646 q^{55} +6.54983i q^{59} -3.04547i q^{61} -1.25588 q^{63} +8.00000 q^{65} +6.54983i q^{67} +11.3446 q^{71} -3.27492 q^{73} +2.20822i q^{77} -13.0192 q^{79} +9.00000 q^{81} -17.0997i q^{83} -9.97368i q^{85} -10.0000 q^{89} -1.09967i q^{91} -3.04547 q^{95} +16.5498 q^{97} -15.8248i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 4q^{17} - 4q^{25} - 24q^{41} + 36q^{49} + 64q^{65} + 4q^{73} + 72q^{81} - 80q^{89} + 72q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) − 3.04547i − 1.36198i −0.732294 0.680989i \(-0.761550\pi\)
0.732294 0.680989i \(-0.238450\pi\)
\(6\) 0 0
\(7\) −0.418627 −0.158226 −0.0791130 0.996866i \(-0.525209\pi\)
−0.0791130 + 0.996866i \(0.525209\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) − 5.27492i − 1.59045i −0.606316 0.795224i \(-0.707353\pi\)
0.606316 0.795224i \(-0.292647\pi\)
\(12\) 0 0
\(13\) 2.62685i 0.728557i 0.931290 + 0.364278i \(0.118684\pi\)
−0.931290 + 0.364278i \(0.881316\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.27492 0.794284 0.397142 0.917757i \(-0.370002\pi\)
0.397142 + 0.917757i \(0.370002\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −4.27492 −0.854983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.62685i 0.487793i 0.969801 + 0.243897i \(0.0784258\pi\)
−0.969801 + 0.243897i \(0.921574\pi\)
\(30\) 0 0
\(31\) −6.09095 −1.09397 −0.546983 0.837143i \(-0.684224\pi\)
−0.546983 + 0.837143i \(0.684224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.27492i 0.215500i
\(36\) 0 0
\(37\) − 9.55505i − 1.57084i −0.618963 0.785420i \(-0.712447\pi\)
0.618963 0.785420i \(-0.287553\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.54983 0.710565 0.355282 0.934759i \(-0.384385\pi\)
0.355282 + 0.934759i \(0.384385\pi\)
\(42\) 0 0
\(43\) − 2.72508i − 0.415571i −0.978174 0.207786i \(-0.933374\pi\)
0.978174 0.207786i \(-0.0666256\pi\)
\(44\) 0 0
\(45\) − 9.13642i − 1.36198i
\(46\) 0 0
\(47\) 0.418627 0.0610630 0.0305315 0.999534i \(-0.490280\pi\)
0.0305315 + 0.999534i \(0.490280\pi\)
\(48\) 0 0
\(49\) −6.82475 −0.974965
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.3923i − 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) −16.0646 −2.16615
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.54983i 0.852716i 0.904555 + 0.426358i \(0.140204\pi\)
−0.904555 + 0.426358i \(0.859796\pi\)
\(60\) 0 0
\(61\) − 3.04547i − 0.389933i −0.980810 0.194967i \(-0.937540\pi\)
0.980810 0.194967i \(-0.0624598\pi\)
\(62\) 0 0
\(63\) −1.25588 −0.158226
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 6.54983i 0.800190i 0.916474 + 0.400095i \(0.131023\pi\)
−0.916474 + 0.400095i \(0.868977\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3446 1.34636 0.673181 0.739478i \(-0.264928\pi\)
0.673181 + 0.739478i \(0.264928\pi\)
\(72\) 0 0
\(73\) −3.27492 −0.383300 −0.191650 0.981463i \(-0.561384\pi\)
−0.191650 + 0.981463i \(0.561384\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.20822i 0.251650i
\(78\) 0 0
\(79\) −13.0192 −1.46477 −0.732385 0.680891i \(-0.761593\pi\)
−0.732385 + 0.680891i \(0.761593\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 17.0997i − 1.87693i −0.345371 0.938466i \(-0.612247\pi\)
0.345371 0.938466i \(-0.387753\pi\)
\(84\) 0 0
\(85\) − 9.97368i − 1.08180i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) − 1.09967i − 0.115277i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.04547 −0.312459
\(96\) 0 0
\(97\) 16.5498 1.68038 0.840191 0.542291i \(-0.182443\pi\)
0.840191 + 0.542291i \(0.182443\pi\)
\(98\) 0 0
\(99\) − 15.8248i − 1.59045i
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) −6.09095 −0.600159 −0.300080 0.953914i \(-0.597013\pi\)
−0.300080 + 0.953914i \(0.597013\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.45017i − 0.140193i −0.997540 0.0700964i \(-0.977669\pi\)
0.997540 0.0700964i \(-0.0223307\pi\)
\(108\) 0 0
\(109\) 3.46410i 0.331801i 0.986143 + 0.165900i \(0.0530530\pi\)
−0.986143 + 0.165900i \(0.946947\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.5498 1.18059 0.590295 0.807188i \(-0.299012\pi\)
0.590295 + 0.807188i \(0.299012\pi\)
\(114\) 0 0
\(115\) 10.5498i 0.983777i
\(116\) 0 0
\(117\) 7.88054i 0.728557i
\(118\) 0 0
\(119\) −1.37097 −0.125676
\(120\) 0 0
\(121\) −16.8248 −1.52952
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.20822i − 0.197509i
\(126\) 0 0
\(127\) 17.4356 1.54716 0.773579 0.633699i \(-0.218464\pi\)
0.773579 + 0.633699i \(0.218464\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 5.27492i − 0.460872i −0.973088 0.230436i \(-0.925985\pi\)
0.973088 0.230436i \(-0.0740152\pi\)
\(132\) 0 0
\(133\) 0.418627i 0.0362995i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.7251 −1.08718 −0.543589 0.839352i \(-0.682935\pi\)
−0.543589 + 0.839352i \(0.682935\pi\)
\(138\) 0 0
\(139\) − 2.72508i − 0.231139i −0.993299 0.115569i \(-0.963131\pi\)
0.993299 0.115569i \(-0.0368692\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.0646i 1.31607i 0.752989 + 0.658033i \(0.228611\pi\)
−0.752989 + 0.658033i \(0.771389\pi\)
\(150\) 0 0
\(151\) 0.837253 0.0681347 0.0340674 0.999420i \(-0.489154\pi\)
0.0340674 + 0.999420i \(0.489154\pi\)
\(152\) 0 0
\(153\) 9.82475 0.794284
\(154\) 0 0
\(155\) 18.5498i 1.48996i
\(156\) 0 0
\(157\) − 12.1819i − 0.972221i −0.873897 0.486111i \(-0.838415\pi\)
0.873897 0.486111i \(-0.161585\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.45017 0.114289
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.1101 1.47878 0.739392 0.673275i \(-0.235113\pi\)
0.739392 + 0.673275i \(0.235113\pi\)
\(168\) 0 0
\(169\) 6.09967 0.469205
\(170\) 0 0
\(171\) − 3.00000i − 0.229416i
\(172\) 0 0
\(173\) 8.71780i 0.662802i 0.943490 + 0.331401i \(0.107521\pi\)
−0.943490 + 0.331401i \(0.892479\pi\)
\(174\) 0 0
\(175\) 1.78959 0.135281
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.0997i 1.57706i 0.614994 + 0.788532i \(0.289158\pi\)
−0.614994 + 0.788532i \(0.710842\pi\)
\(180\) 0 0
\(181\) − 3.46410i − 0.257485i −0.991678 0.128742i \(-0.958906\pi\)
0.991678 0.128742i \(-0.0410940\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −29.0997 −2.13945
\(186\) 0 0
\(187\) − 17.2749i − 1.26327i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.7633 −0.851161 −0.425580 0.904921i \(-0.639930\pi\)
−0.425580 + 0.904921i \(0.639930\pi\)
\(192\) 0 0
\(193\) 11.0997 0.798972 0.399486 0.916739i \(-0.369189\pi\)
0.399486 + 0.916739i \(0.369189\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 5.25370i − 0.374310i −0.982330 0.187155i \(-0.940073\pi\)
0.982330 0.187155i \(-0.0599267\pi\)
\(198\) 0 0
\(199\) 14.2750 1.01193 0.505965 0.862554i \(-0.331136\pi\)
0.505965 + 0.862554i \(0.331136\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.09967i − 0.0771816i
\(204\) 0 0
\(205\) − 13.8564i − 0.967773i
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) −5.27492 −0.364874
\(210\) 0 0
\(211\) 26.5498i 1.82777i 0.405978 + 0.913883i \(0.366931\pi\)
−0.405978 + 0.913883i \(0.633069\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.29917 −0.565999
\(216\) 0 0
\(217\) 2.54983 0.173094
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.60271i 0.578681i
\(222\) 0 0
\(223\) −17.4356 −1.16757 −0.583787 0.811907i \(-0.698430\pi\)
−0.583787 + 0.811907i \(0.698430\pi\)
\(224\) 0 0
\(225\) −12.8248 −0.854983
\(226\) 0 0
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) 17.7391i 1.17224i 0.810226 + 0.586118i \(0.199344\pi\)
−0.810226 + 0.586118i \(0.800656\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20.3746 −1.33478 −0.667392 0.744707i \(-0.732589\pi\)
−0.667392 + 0.744707i \(0.732589\pi\)
\(234\) 0 0
\(235\) − 1.27492i − 0.0831664i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 25.6197 1.65720 0.828600 0.559842i \(-0.189138\pi\)
0.828600 + 0.559842i \(0.189138\pi\)
\(240\) 0 0
\(241\) −12.5498 −0.808406 −0.404203 0.914669i \(-0.632451\pi\)
−0.404203 + 0.914669i \(0.632451\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.7846i 1.32788i
\(246\) 0 0
\(247\) 2.62685 0.167142
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.7251i 0.676961i 0.940973 + 0.338481i \(0.109913\pi\)
−0.940973 + 0.338481i \(0.890087\pi\)
\(252\) 0 0
\(253\) 18.2728i 1.14880i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.45017 0.464729 0.232364 0.972629i \(-0.425354\pi\)
0.232364 + 0.972629i \(0.425354\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 7.88054i 0.487793i
\(262\) 0 0
\(263\) −1.25588 −0.0774409 −0.0387204 0.999250i \(-0.512328\pi\)
−0.0387204 + 0.999250i \(0.512328\pi\)
\(264\) 0 0
\(265\) −31.6495 −1.94421
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.13861i 0.313306i 0.987654 + 0.156653i \(0.0500705\pi\)
−0.987654 + 0.156653i \(0.949929\pi\)
\(270\) 0 0
\(271\) 27.8279 1.69042 0.845212 0.534431i \(-0.179474\pi\)
0.845212 + 0.534431i \(0.179474\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.5498i 1.35981i
\(276\) 0 0
\(277\) − 16.0646i − 0.965230i −0.875833 0.482615i \(-0.839687\pi\)
0.875833 0.482615i \(-0.160313\pi\)
\(278\) 0 0
\(279\) −18.2728 −1.09397
\(280\) 0 0
\(281\) 11.0997 0.662151 0.331075 0.943604i \(-0.392589\pi\)
0.331075 + 0.943604i \(0.392589\pi\)
\(282\) 0 0
\(283\) 26.3746i 1.56781i 0.620883 + 0.783903i \(0.286774\pi\)
−0.620883 + 0.783903i \(0.713226\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.90468 −0.112430
\(288\) 0 0
\(289\) −6.27492 −0.369113
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.3397i 1.77246i 0.463244 + 0.886231i \(0.346685\pi\)
−0.463244 + 0.886231i \(0.653315\pi\)
\(294\) 0 0
\(295\) 19.9474 1.16138
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 9.09967i − 0.526247i
\(300\) 0 0
\(301\) 1.14079i 0.0657542i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.27492 −0.531080
\(306\) 0 0
\(307\) 1.45017i 0.0827653i 0.999143 + 0.0413827i \(0.0131763\pi\)
−0.999143 + 0.0413827i \(0.986824\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.7824 1.40528 0.702641 0.711544i \(-0.252004\pi\)
0.702641 + 0.711544i \(0.252004\pi\)
\(312\) 0 0
\(313\) 15.0997 0.853484 0.426742 0.904373i \(-0.359661\pi\)
0.426742 + 0.904373i \(0.359661\pi\)
\(314\) 0 0
\(315\) 3.82475i 0.215500i
\(316\) 0 0
\(317\) − 8.71780i − 0.489640i −0.969569 0.244820i \(-0.921271\pi\)
0.969569 0.244820i \(-0.0787289\pi\)
\(318\) 0 0
\(319\) 13.8564 0.775810
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 3.27492i − 0.182221i
\(324\) 0 0
\(325\) − 11.2296i − 0.622904i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.175248 −0.00966175
\(330\) 0 0
\(331\) − 9.45017i − 0.519428i −0.965686 0.259714i \(-0.916372\pi\)
0.965686 0.259714i \(-0.0836283\pi\)
\(332\) 0 0
\(333\) − 28.6652i − 1.57084i
\(334\) 0 0
\(335\) 19.9474 1.08984
\(336\) 0 0
\(337\) 33.6495 1.83301 0.916503 0.400029i \(-0.131000\pi\)
0.916503 + 0.400029i \(0.131000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 32.1293i 1.73990i
\(342\) 0 0
\(343\) 5.78741 0.312491
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.82475i − 0.420055i −0.977696 0.210027i \(-0.932645\pi\)
0.977696 0.210027i \(-0.0673553\pi\)
\(348\) 0 0
\(349\) 12.7156i 0.680651i 0.940308 + 0.340326i \(0.110537\pi\)
−0.940308 + 0.340326i \(0.889463\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) − 34.5498i − 1.83371i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.6005 0.665030 0.332515 0.943098i \(-0.392103\pi\)
0.332515 + 0.943098i \(0.392103\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.97368i 0.522046i
\(366\) 0 0
\(367\) −15.6460 −0.816715 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(368\) 0 0
\(369\) 13.6495 0.710565
\(370\) 0 0
\(371\) 4.35050i 0.225867i
\(372\) 0 0
\(373\) 0.952341i 0.0493104i 0.999696 + 0.0246552i \(0.00784878\pi\)
−0.999696 + 0.0246552i \(0.992151\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.90033 −0.355385
\(378\) 0 0
\(379\) − 23.6495i − 1.21479i −0.794399 0.607397i \(-0.792214\pi\)
0.794399 0.607397i \(-0.207786\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.1293 1.64173 0.820864 0.571124i \(-0.193492\pi\)
0.820864 + 0.571124i \(0.193492\pi\)
\(384\) 0 0
\(385\) 6.72508 0.342742
\(386\) 0 0
\(387\) − 8.17525i − 0.415571i
\(388\) 0 0
\(389\) − 4.71998i − 0.239313i −0.992815 0.119656i \(-0.961821\pi\)
0.992815 0.119656i \(-0.0381793\pi\)
\(390\) 0 0
\(391\) −11.3446 −0.573723
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 39.6495i 1.99498i
\(396\) 0 0
\(397\) 32.6630i 1.63931i 0.572859 + 0.819654i \(0.305834\pi\)
−0.572859 + 0.819654i \(0.694166\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.0997 1.75279 0.876397 0.481590i \(-0.159940\pi\)
0.876397 + 0.481590i \(0.159940\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) 0 0
\(405\) − 27.4093i − 1.36198i
\(406\) 0 0
\(407\) −50.4021 −2.49834
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 2.74194i − 0.134922i
\(414\) 0 0
\(415\) −52.0766 −2.55634
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 12.0000i − 0.586238i −0.956076 0.293119i \(-0.905307\pi\)
0.956076 0.293119i \(-0.0946933\pi\)
\(420\) 0 0
\(421\) − 24.2487i − 1.18181i −0.806741 0.590905i \(-0.798771\pi\)
0.806741 0.590905i \(-0.201229\pi\)
\(422\) 0 0
\(423\) 1.25588 0.0610630
\(424\) 0 0
\(425\) −14.0000 −0.679100
\(426\) 0 0
\(427\) 1.27492i 0.0616976i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.67451 −0.0806582 −0.0403291 0.999186i \(-0.512841\pi\)
−0.0403291 + 0.999186i \(0.512841\pi\)
\(432\) 0 0
\(433\) −17.6495 −0.848181 −0.424091 0.905620i \(-0.639406\pi\)
−0.424091 + 0.905620i \(0.639406\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410i 0.165710i
\(438\) 0 0
\(439\) 31.2920 1.49349 0.746743 0.665113i \(-0.231617\pi\)
0.746743 + 0.665113i \(0.231617\pi\)
\(440\) 0 0
\(441\) −20.4743 −0.974965
\(442\) 0 0
\(443\) 10.3746i 0.492911i 0.969154 + 0.246456i \(0.0792660\pi\)
−0.969154 + 0.246456i \(0.920734\pi\)
\(444\) 0 0
\(445\) 30.4547i 1.44369i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −20.5498 −0.969807 −0.484903 0.874568i \(-0.661145\pi\)
−0.484903 + 0.874568i \(0.661145\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.34901 −0.157004
\(456\) 0 0
\(457\) −15.2749 −0.714530 −0.357265 0.934003i \(-0.616291\pi\)
−0.357265 + 0.934003i \(0.616291\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7156i 0.592225i 0.955153 + 0.296113i \(0.0956904\pi\)
−0.955153 + 0.296113i \(0.904310\pi\)
\(462\) 0 0
\(463\) 2.93039 0.136187 0.0680933 0.997679i \(-0.478308\pi\)
0.0680933 + 0.997679i \(0.478308\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.17525i 0.378305i 0.981948 + 0.189153i \(0.0605741\pi\)
−0.981948 + 0.189153i \(0.939426\pi\)
\(468\) 0 0
\(469\) − 2.74194i − 0.126611i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14.3746 −0.660944
\(474\) 0 0
\(475\) 4.27492i 0.196147i
\(476\) 0 0
\(477\) − 31.1769i − 1.42749i
\(478\) 0 0
\(479\) 17.3205 0.791394 0.395697 0.918381i \(-0.370503\pi\)
0.395697 + 0.918381i \(0.370503\pi\)
\(480\) 0 0
\(481\) 25.0997 1.14445
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 50.4021i − 2.28864i
\(486\) 0 0
\(487\) −39.0575 −1.76986 −0.884931 0.465722i \(-0.845795\pi\)
−0.884931 + 0.465722i \(0.845795\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) 8.60271i 0.387447i
\(494\) 0 0
\(495\) −48.1939 −2.16615
\(496\) 0 0
\(497\) −4.74917 −0.213029
\(498\) 0 0
\(499\) 26.7251i 1.19638i 0.801355 + 0.598190i \(0.204113\pi\)
−0.801355 + 0.598190i \(0.795887\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 41.6843 1.85861 0.929306 0.369311i \(-0.120406\pi\)
0.929306 + 0.369311i \(0.120406\pi\)
\(504\) 0 0
\(505\) 42.1993 1.87785
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 4.30136i − 0.190654i −0.995446 0.0953271i \(-0.969610\pi\)
0.995446 0.0953271i \(-0.0303897\pi\)
\(510\) 0 0
\(511\) 1.37097 0.0606480
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.5498i 0.817403i
\(516\) 0 0
\(517\) − 2.20822i − 0.0971175i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0997 −1.18726 −0.593629 0.804739i \(-0.702305\pi\)
−0.593629 + 0.804739i \(0.702305\pi\)
\(522\) 0 0
\(523\) 5.45017i 0.238319i 0.992875 + 0.119160i \(0.0380200\pi\)
−0.992875 + 0.119160i \(0.961980\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.9474 −0.868920
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 19.6495i 0.852716i
\(532\) 0 0
\(533\) 11.9517i 0.517687i
\(534\) 0 0
\(535\) −4.41644 −0.190939
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000i 1.55063i
\(540\) 0 0
\(541\) − 29.0838i − 1.25041i −0.780461 0.625205i \(-0.785015\pi\)
0.780461 0.625205i \(-0.214985\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.5498 0.451905
\(546\) 0 0
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) 0 0
\(549\) − 9.13642i − 0.389933i
\(550\) 0 0
\(551\) 2.62685 0.111907
\(552\) 0 0
\(553\) 5.45017 0.231765
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 29.9210i − 1.26779i −0.773417 0.633897i \(-0.781454\pi\)
0.773417 0.633897i \(-0.218546\pi\)
\(558\) 0 0
\(559\) 7.15838 0.302767
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) − 38.2202i − 1.60794i
\(566\) 0 0
\(567\) −3.76764 −0.158226
\(568\) 0 0
\(569\) −19.4502 −0.815393 −0.407697 0.913117i \(-0.633668\pi\)
−0.407697 + 0.913117i \(0.633668\pi\)
\(570\) 0 0
\(571\) − 41.0997i − 1.71997i −0.510321 0.859984i \(-0.670474\pi\)
0.510321 0.859984i \(-0.329526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.8087 0.617567
\(576\) 0 0
\(577\) 23.2749 0.968947 0.484474 0.874806i \(-0.339011\pi\)
0.484474 + 0.874806i \(0.339011\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.15838i 0.296980i
\(582\) 0 0
\(583\) −54.8185 −2.27035
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) − 37.2749i − 1.53850i −0.638948 0.769250i \(-0.720630\pi\)
0.638948 0.769250i \(-0.279370\pi\)
\(588\) 0 0
\(589\) 6.09095i 0.250973i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.0997 0.784329 0.392165 0.919895i \(-0.371726\pi\)
0.392165 + 0.919895i \(0.371726\pi\)
\(594\) 0 0
\(595\) 4.17525i 0.171168i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.43996 0.385706 0.192853 0.981228i \(-0.438226\pi\)
0.192853 + 0.981228i \(0.438226\pi\)
\(600\) 0 0
\(601\) 11.0997 0.452765 0.226382 0.974038i \(-0.427310\pi\)
0.226382 + 0.974038i \(0.427310\pi\)
\(602\) 0 0
\(603\) 19.6495i 0.800190i
\(604\) 0 0
\(605\) 51.2394i 2.08318i
\(606\) 0 0
\(607\) −29.3873 −1.19279 −0.596397 0.802689i \(-0.703402\pi\)
−0.596397 + 0.802689i \(0.703402\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.09967i 0.0444878i
\(612\) 0 0
\(613\) − 29.9210i − 1.20850i −0.796795 0.604250i \(-0.793473\pi\)
0.796795 0.604250i \(-0.206527\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7251 0.512293 0.256146 0.966638i \(-0.417547\pi\)
0.256146 + 0.966638i \(0.417547\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.18627 0.167719
\(624\) 0 0
\(625\) −28.0997 −1.12399
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 31.2920i − 1.24769i
\(630\) 0 0
\(631\) −13.4378 −0.534950 −0.267475 0.963565i \(-0.586189\pi\)
−0.267475 + 0.963565i \(0.586189\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 53.0997i − 2.10720i
\(636\) 0 0
\(637\) − 17.9276i − 0.710317i
\(638\) 0 0
\(639\) 34.0339 1.34636
\(640\) 0 0
\(641\) −12.5498 −0.495689 −0.247844 0.968800i \(-0.579722\pi\)
−0.247844 + 0.968800i \(0.579722\pi\)
\(642\) 0 0
\(643\) − 44.9244i − 1.77165i −0.464023 0.885823i \(-0.653595\pi\)
0.464023 0.885823i \(-0.346405\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.6197 −1.00721 −0.503607 0.863933i \(-0.667994\pi\)
−0.503607 + 0.863933i \(0.667994\pi\)
\(648\) 0 0
\(649\) 34.5498 1.35620
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 16.0646i − 0.628657i −0.949314 0.314329i \(-0.898221\pi\)
0.949314 0.314329i \(-0.101779\pi\)
\(654\) 0 0
\(655\) −16.0646 −0.627697
\(656\) 0 0
\(657\) −9.82475 −0.383300
\(658\) 0 0
\(659\) − 13.4502i − 0.523944i −0.965075 0.261972i \(-0.915627\pi\)
0.965075 0.261972i \(-0.0843728\pi\)
\(660\) 0 0
\(661\) − 9.55505i − 0.371648i −0.982583 0.185824i \(-0.940505\pi\)
0.982583 0.185824i \(-0.0594955\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.27492 0.0494392
\(666\) 0 0
\(667\) − 9.09967i − 0.352341i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16.0646 −0.620168
\(672\) 0 0
\(673\) −47.0997 −1.81556 −0.907779 0.419448i \(-0.862224\pi\)
−0.907779 + 0.419448i \(0.862224\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 0.722166i − 0.0277551i −0.999904 0.0138775i \(-0.995582\pi\)
0.999904 0.0138775i \(-0.00441750\pi\)
\(678\) 0 0
\(679\) −6.92820 −0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.1993i 0.849434i 0.905326 + 0.424717i \(0.139626\pi\)
−0.905326 + 0.424717i \(0.860374\pi\)
\(684\) 0 0
\(685\) 38.7539i 1.48071i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 27.2990 1.04001
\(690\) 0 0
\(691\) 23.4743i 0.893003i 0.894783 + 0.446501i \(0.147330\pi\)
−0.894783 + 0.446501i \(0.852670\pi\)
\(692\) 0 0
\(693\) 6.62466i 0.251650i
\(694\) 0 0
\(695\) −8.29917 −0.314806
\(696\) 0 0
\(697\) 14.9003 0.564390
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 21.0148i − 0.793717i −0.917880 0.396859i \(-0.870100\pi\)
0.917880 0.396859i \(-0.129900\pi\)
\(702\) 0 0
\(703\) −9.55505 −0.360376
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 5.80066i − 0.218156i
\(708\) 0 0
\(709\) 38.2202i 1.43539i 0.696358 + 0.717695i \(0.254803\pi\)
−0.696358 + 0.717695i \(0.745197\pi\)
\(710\) 0 0
\(711\) −39.0575 −1.46477
\(712\) 0 0
\(713\) 21.0997 0.790189
\(714\) 0 0
\(715\) − 42.1993i − 1.57817i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −51.6580 −1.92652 −0.963259 0.268575i \(-0.913447\pi\)
−0.963259 + 0.268575i \(0.913447\pi\)
\(720\) 0 0
\(721\) 2.54983 0.0949608
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 11.2296i − 0.417055i
\(726\) 0 0
\(727\) 10.9260 0.405224 0.202612 0.979259i \(-0.435057\pi\)
0.202612 + 0.979259i \(0.435057\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) − 8.92442i − 0.330082i
\(732\) 0 0
\(733\) − 19.1101i − 0.705848i −0.935652 0.352924i \(-0.885187\pi\)
0.935652 0.352924i \(-0.114813\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.5498 1.27266
\(738\) 0 0
\(739\) 7.82475i 0.287838i 0.989589 + 0.143919i \(0.0459705\pi\)
−0.989589 + 0.143919i \(0.954029\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 52.9139 1.94122 0.970611 0.240655i \(-0.0773622\pi\)
0.970611 + 0.240655i \(0.0773622\pi\)
\(744\) 0 0
\(745\) 48.9244 1.79245
\(746\) 0 0
\(747\) − 51.2990i − 1.87693i
\(748\) 0 0
\(749\) 0.607078i 0.0221822i
\(750\) 0 0
\(751\) 36.3155 1.32517 0.662586 0.748986i \(-0.269459\pi\)
0.662586 + 0.748986i \(0.269459\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2.54983i − 0.0927980i
\(756\) 0 0
\(757\) 3.04547i 0.110690i 0.998467 + 0.0553448i \(0.0176258\pi\)
−0.998467 + 0.0553448i \(0.982374\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.27492 0.118716 0.0593578 0.998237i \(-0.481095\pi\)
0.0593578 + 0.998237i \(0.481095\pi\)
\(762\) 0 0
\(763\) − 1.45017i − 0.0524995i
\(764\) 0 0
\(765\) − 29.9210i − 1.08180i
\(766\) 0 0
\(767\) −17.2054 −0.621252
\(768\) 0 0
\(769\) −13.8248 −0.498533 −0.249267 0.968435i \(-0.580190\pi\)
−0.249267 + 0.968435i \(0.580190\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.7561i 1.25009i 0.780589 + 0.625045i \(0.214919\pi\)
−0.780589 + 0.625045i \(0.785081\pi\)
\(774\) 0 0
\(775\) 26.0383 0.935324
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 4.54983i − 0.163015i
\(780\) 0 0
\(781\) − 59.8421i − 2.14132i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −37.0997 −1.32414
\(786\) 0 0
\(787\) − 14.1993i − 0.506152i −0.967446 0.253076i \(-0.918558\pi\)
0.967446 0.253076i \(-0.0814422\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.25370 −0.186800
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.2487i 0.858933i 0.903083 + 0.429467i \(0.141298\pi\)
−0.903083 + 0.429467i \(0.858702\pi\)
\(798\) 0 0
\(799\) 1.37097 0.0485014
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 17.2749i 0.609619i
\(804\) 0 0
\(805\) − 4.41644i − 0.155659i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.82475 −0.204787 −0.102394 0.994744i \(-0.532650\pi\)
−0.102394 + 0.994744i \(0.532650\pi\)
\(810\) 0 0
\(811\) 47.2990i 1.66089i 0.557099 + 0.830446i \(0.311915\pi\)
−0.557099 + 0.830446i \(0.688085\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.1819 −0.426713
\(816\) 0 0
\(817\) −2.72508 −0.0953386
\(818\) 0 0
\(819\) − 3.29901i − 0.115277i
\(820\) 0 0
\(821\) 26.5720i 0.927370i 0.886000 + 0.463685i \(0.153473\pi\)
−0.886000 + 0.463685i \(0.846527\pi\)
\(822\) 0 0
\(823\) 21.4334 0.747122 0.373561 0.927606i \(-0.378137\pi\)
0.373561 + 0.927606i \(0.378137\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 47.6495i 1.65694i 0.560037 + 0.828468i \(0.310787\pi\)
−0.560037 + 0.828468i \(0.689213\pi\)
\(828\) 0 0
\(829\) − 25.3161i − 0.879266i −0.898178 0.439633i \(-0.855109\pi\)
0.898178 0.439633i \(-0.144891\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −22.3505 −0.774399
\(834\) 0 0
\(835\) − 58.1993i − 2.01407i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 47.6602 1.64541 0.822706 0.568467i \(-0.192463\pi\)
0.822706 + 0.568467i \(0.192463\pi\)
\(840\) 0 0
\(841\) 22.0997 0.762058
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 18.5764i − 0.639047i
\(846\) 0 0
\(847\) 7.04329 0.242010
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.0997i 1.13464i
\(852\) 0 0
\(853\) 32.9665i 1.12875i 0.825518 + 0.564376i \(0.190883\pi\)
−0.825518 + 0.564376i \(0.809117\pi\)
\(854\) 0 0
\(855\) −9.13642 −0.312459
\(856\) 0 0
\(857\) −16.1993 −0.553359 −0.276679 0.960962i \(-0.589234\pi\)
−0.276679 + 0.960962i \(0.589234\pi\)
\(858\) 0 0
\(859\) 55.4743i 1.89276i 0.323060 + 0.946379i \(0.395289\pi\)
−0.323060 + 0.946379i \(0.604711\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.7994 1.42287 0.711434 0.702753i \(-0.248046\pi\)
0.711434 + 0.702753i \(0.248046\pi\)
\(864\) 0 0
\(865\) 26.5498 0.902721
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 68.6750i 2.32964i
\(870\) 0 0
\(871\) −17.2054 −0.582983
\(872\) 0 0
\(873\) 49.6495 1.68038
\(874\) 0 0
\(875\) 0.924421i 0.0312511i
\(876\) 0 0
\(877\) − 31.4071i − 1.06054i −0.847828 0.530271i \(-0.822090\pi\)
0.847828 0.530271i \(-0.177910\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.82475 0.0614774 0.0307387 0.999527i \(-0.490214\pi\)
0.0307387 + 0.999527i \(0.490214\pi\)
\(882\) 0 0
\(883\) 13.6254i 0.458532i 0.973364 + 0.229266i \(0.0736325\pi\)
−0.973364 + 0.229266i \(0.926367\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.837253 −0.0281122 −0.0140561 0.999901i \(-0.504474\pi\)
−0.0140561 + 0.999901i \(0.504474\pi\)
\(888\) 0 0
\(889\) −7.29901 −0.244801
\(890\) 0 0
\(891\) − 47.4743i − 1.59045i
\(892\) 0 0
\(893\) − 0.418627i − 0.0140088i
\(894\) 0 0
\(895\) 64.2585 2.14793
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 16.0000i − 0.533630i
\(900\) 0 0
\(901\) − 34.0339i − 1.13383i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.5498 −0.350688
\(906\) 0 0
\(907\) 23.6495i 0.785269i 0.919695 + 0.392634i \(0.128436\pi\)
−0.919695 + 0.392634i \(0.871564\pi\)
\(908\) 0 0
\(909\) 41.5692i 1.37876i
\(910\) 0 0
\(911\) −34.0339 −1.12759 −0.563797 0.825913i \(-0.690660\pi\)
−0.563797 + 0.825913i \(0.690660\pi\)
\(912\) 0 0
\(913\) −90.1993 −2.98516
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.20822i 0.0729219i
\(918\) 0 0
\(919\) −0.115088 −0.00379639 −0.00189820 0.999998i \(-0.500604\pi\)
−0.00189820 + 0.999998i \(0.500604\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29.8007i 0.980901i
\(924\) 0 0
\(925\) 40.8471i 1.34304i
\(926\) 0 0
\(927\) −18.2728 −0.600159
\(928\) 0 0
\(929\) −7.09967 −0.232933 −0.116466 0.993195i \(-0.537157\pi\)
−0.116466 + 0.993195i \(0.537157\pi\)
\(930\) 0 0
\(931\) 6.82475i 0.223672i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −52.6103 −1.72054
\(936\) 0 0
\(937\) −14.9244 −0.487560 −0.243780 0.969831i \(-0.578387\pi\)
−0.243780 + 0.969831i \(0.578387\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 31.1769i − 1.01634i −0.861257 0.508169i \(-0.830322\pi\)
0.861257 0.508169i \(-0.169678\pi\)
\(942\) 0 0
\(943\) −15.7611 −0.513252
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.1993i 0.981347i 0.871344 + 0.490673i \(0.163249\pi\)
−0.871344 + 0.490673i \(0.836751\pi\)
\(948\) 0 0
\(949\) − 8.60271i − 0.279256i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45.2990 −1.46738 −0.733689 0.679485i \(-0.762203\pi\)
−0.733689 + 0.679485i \(0.762203\pi\)
\(954\) 0 0
\(955\) 35.8248i 1.15926i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.32706 0.172020
\(960\) 0 0
\(961\) 6.09967 0.196764
\(962\) 0 0
\(963\) − 4.35050i − 0.140193i
\(964\) 0 0
\(965\) − 33.8038i − 1.08818i
\(966\) 0 0
\(967\) −55.5407 −1.78607 −0.893034 0.449988i \(-0.851428\pi\)
−0.893034 + 0.449988i \(0.851428\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 44.0000i − 1.41203i −0.708198 0.706014i \(-0.750492\pi\)
0.708198 0.706014i \(-0.249508\pi\)
\(972\) 0 0
\(973\) 1.14079i 0.0365721i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.1993 −0.902177 −0.451088 0.892479i \(-0.648964\pi\)
−0.451088 + 0.892479i \(0.648964\pi\)
\(978\) 0 0
\(979\) 52.7492i 1.68587i
\(980\) 0 0
\(981\) 10.3923i 0.331801i
\(982\) 0 0
\(983\) 22.6893 0.723676 0.361838 0.932241i \(-0.382149\pi\)
0.361838 + 0.932241i \(0.382149\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.43996i 0.300173i
\(990\) 0 0
\(991\) −47.0531 −1.49469 −0.747345 0.664436i \(-0.768672\pi\)
−0.747345 + 0.664436i \(0.768672\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 43.4743i − 1.37823i
\(996\) 0 0
\(997\) − 42.1029i − 1.33341i −0.745320 0.666707i \(-0.767703\pi\)
0.745320 0.666707i \(-0.232297\pi\)
\(998\) 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 1216.2.c.i.609.1 8
4.3 odd 2 inner 1216.2.c.i.609.2 yes 8
8.3 odd 2 inner 1216.2.c.i.609.8 yes 8
8.5 even 2 inner 1216.2.c.i.609.7 yes 8
16.3 odd 4 4864.2.a.bk.1.1 4
16.5 even 4 4864.2.a.bk.1.4 4
16.11 odd 4 4864.2.a.bj.1.4 4
16.13 even 4 4864.2.a.bj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.1 8 1.1 even 1 trivial
1216.2.c.i.609.2 yes 8 4.3 odd 2 inner
1216.2.c.i.609.7 yes 8 8.5 even 2 inner
1216.2.c.i.609.8 yes 8 8.3 odd 2 inner
4864.2.a.bj.1.1 4 16.13 even 4
4864.2.a.bj.1.4 4 16.11 odd 4
4864.2.a.bk.1.1 4 16.3 odd 4
4864.2.a.bk.1.4 4 16.5 even 4