Properties

Label 768.7.g.g.511.3
Level $768$
Weight $7$
Character 768.511
Analytic conductor $176.682$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(176.681536220\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 53x^{6} - 2x^{5} + 2532x^{4} - 772x^{3} - 31349x^{2} - 33880x + 366025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 511.3
Root \(-3.03301 - 2.32846i\) of defining polynomial
Character \(\chi\) \(=\) 768.511
Dual form 768.7.g.g.511.7

$q$-expansion

\(f(q)\) \(=\) \(q-15.5885i q^{3} +85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} +O(q^{10})\) \(q-15.5885i q^{3} +85.3891 q^{5} -511.824i q^{7} -243.000 q^{9} +1212.89i q^{11} +2744.91 q^{13} -1331.08i q^{15} -8524.35 q^{17} +10429.2i q^{19} -7978.55 q^{21} +4300.69i q^{23} -8333.71 q^{25} +3788.00i q^{27} +42442.0 q^{29} -57843.0i q^{31} +18907.1 q^{33} -43704.2i q^{35} +83505.3 q^{37} -42788.9i q^{39} +70152.4 q^{41} -28701.0i q^{43} -20749.5 q^{45} +58746.6i q^{47} -144315. q^{49} +132882. i q^{51} +111109. q^{53} +103567. i q^{55} +162575. q^{57} -118257. i q^{59} -53174.3 q^{61} +124373. i q^{63} +234385. q^{65} -42166.1i q^{67} +67041.1 q^{69} -351599. i q^{71} +91202.9 q^{73} +129910. i q^{75} +620786. q^{77} -775208. i q^{79} +59049.0 q^{81} +669966. i q^{83} -727887. q^{85} -661605. i q^{87} +504715. q^{89} -1.40491e6i q^{91} -901683. q^{93} +890541. i q^{95} +1.03116e6 q^{97} -294732. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1944 q^{9} - 6544 q^{17} + 56632 q^{25} - 33696 q^{33} + 499568 q^{41} - 414712 q^{49} + 375840 q^{57} - 36096 q^{65} + 1962640 q^{73} + 472392 q^{81} + 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 15.5885i − 0.577350i
\(4\) 0 0
\(5\) 85.3891 0.683112 0.341556 0.939861i \(-0.389046\pi\)
0.341556 + 0.939861i \(0.389046\pi\)
\(6\) 0 0
\(7\) − 511.824i − 1.49220i −0.665834 0.746100i \(-0.731924\pi\)
0.665834 0.746100i \(-0.268076\pi\)
\(8\) 0 0
\(9\) −243.000 −0.333333
\(10\) 0 0
\(11\) 1212.89i 0.911261i 0.890169 + 0.455631i \(0.150586\pi\)
−0.890169 + 0.455631i \(0.849414\pi\)
\(12\) 0 0
\(13\) 2744.91 1.24939 0.624694 0.780869i \(-0.285224\pi\)
0.624694 + 0.780869i \(0.285224\pi\)
\(14\) 0 0
\(15\) − 1331.08i − 0.394395i
\(16\) 0 0
\(17\) −8524.35 −1.73506 −0.867531 0.497384i \(-0.834294\pi\)
−0.867531 + 0.497384i \(0.834294\pi\)
\(18\) 0 0
\(19\) 10429.2i 1.52051i 0.649622 + 0.760257i \(0.274927\pi\)
−0.649622 + 0.760257i \(0.725073\pi\)
\(20\) 0 0
\(21\) −7978.55 −0.861522
\(22\) 0 0
\(23\) 4300.69i 0.353471i 0.984258 + 0.176736i \(0.0565538\pi\)
−0.984258 + 0.176736i \(0.943446\pi\)
\(24\) 0 0
\(25\) −8333.71 −0.533357
\(26\) 0 0
\(27\) 3788.00i 0.192450i
\(28\) 0 0
\(29\) 42442.0 1.74021 0.870105 0.492866i \(-0.164051\pi\)
0.870105 + 0.492866i \(0.164051\pi\)
\(30\) 0 0
\(31\) − 57843.0i − 1.94163i −0.239839 0.970813i \(-0.577095\pi\)
0.239839 0.970813i \(-0.422905\pi\)
\(32\) 0 0
\(33\) 18907.1 0.526117
\(34\) 0 0
\(35\) − 43704.2i − 1.01934i
\(36\) 0 0
\(37\) 83505.3 1.64858 0.824288 0.566171i \(-0.191576\pi\)
0.824288 + 0.566171i \(0.191576\pi\)
\(38\) 0 0
\(39\) − 42788.9i − 0.721335i
\(40\) 0 0
\(41\) 70152.4 1.01787 0.508933 0.860806i \(-0.330040\pi\)
0.508933 + 0.860806i \(0.330040\pi\)
\(42\) 0 0
\(43\) − 28701.0i − 0.360987i −0.983576 0.180494i \(-0.942230\pi\)
0.983576 0.180494i \(-0.0577695\pi\)
\(44\) 0 0
\(45\) −20749.5 −0.227704
\(46\) 0 0
\(47\) 58746.6i 0.565834i 0.959144 + 0.282917i \(0.0913021\pi\)
−0.959144 + 0.282917i \(0.908698\pi\)
\(48\) 0 0
\(49\) −144315. −1.22666
\(50\) 0 0
\(51\) 132882.i 1.00174i
\(52\) 0 0
\(53\) 111109. 0.746316 0.373158 0.927768i \(-0.378275\pi\)
0.373158 + 0.927768i \(0.378275\pi\)
\(54\) 0 0
\(55\) 103567.i 0.622494i
\(56\) 0 0
\(57\) 162575. 0.877870
\(58\) 0 0
\(59\) − 118257.i − 0.575798i −0.957661 0.287899i \(-0.907043\pi\)
0.957661 0.287899i \(-0.0929568\pi\)
\(60\) 0 0
\(61\) −53174.3 −0.234268 −0.117134 0.993116i \(-0.537371\pi\)
−0.117134 + 0.993116i \(0.537371\pi\)
\(62\) 0 0
\(63\) 124373.i 0.497400i
\(64\) 0 0
\(65\) 234385. 0.853473
\(66\) 0 0
\(67\) − 42166.1i − 0.140197i −0.997540 0.0700986i \(-0.977669\pi\)
0.997540 0.0700986i \(-0.0223314\pi\)
\(68\) 0 0
\(69\) 67041.1 0.204077
\(70\) 0 0
\(71\) − 351599.i − 0.982363i −0.871057 0.491182i \(-0.836565\pi\)
0.871057 0.491182i \(-0.163435\pi\)
\(72\) 0 0
\(73\) 91202.9 0.234445 0.117222 0.993106i \(-0.462601\pi\)
0.117222 + 0.993106i \(0.462601\pi\)
\(74\) 0 0
\(75\) 129910.i 0.307934i
\(76\) 0 0
\(77\) 620786. 1.35978
\(78\) 0 0
\(79\) − 775208.i − 1.57231i −0.618032 0.786153i \(-0.712070\pi\)
0.618032 0.786153i \(-0.287930\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) 669966.i 1.17170i 0.810418 + 0.585852i \(0.199240\pi\)
−0.810418 + 0.585852i \(0.800760\pi\)
\(84\) 0 0
\(85\) −727887. −1.18524
\(86\) 0 0
\(87\) − 661605.i − 1.00471i
\(88\) 0 0
\(89\) 504715. 0.715940 0.357970 0.933733i \(-0.383469\pi\)
0.357970 + 0.933733i \(0.383469\pi\)
\(90\) 0 0
\(91\) − 1.40491e6i − 1.86434i
\(92\) 0 0
\(93\) −901683. −1.12100
\(94\) 0 0
\(95\) 890541.i 1.03868i
\(96\) 0 0
\(97\) 1.03116e6 1.12982 0.564911 0.825152i \(-0.308911\pi\)
0.564911 + 0.825152i \(0.308911\pi\)
\(98\) 0 0
\(99\) − 294732.i − 0.303754i
\(100\) 0 0
\(101\) −927592. −0.900312 −0.450156 0.892950i \(-0.648632\pi\)
−0.450156 + 0.892950i \(0.648632\pi\)
\(102\) 0 0
\(103\) − 357299.i − 0.326979i −0.986545 0.163489i \(-0.947725\pi\)
0.986545 0.163489i \(-0.0522750\pi\)
\(104\) 0 0
\(105\) −681281. −0.588516
\(106\) 0 0
\(107\) − 1.16196e6i − 0.948508i −0.880388 0.474254i \(-0.842718\pi\)
0.880388 0.474254i \(-0.157282\pi\)
\(108\) 0 0
\(109\) −741394. −0.572492 −0.286246 0.958156i \(-0.592407\pi\)
−0.286246 + 0.958156i \(0.592407\pi\)
\(110\) 0 0
\(111\) − 1.30172e6i − 0.951805i
\(112\) 0 0
\(113\) −190737. −0.132190 −0.0660952 0.997813i \(-0.521054\pi\)
−0.0660952 + 0.997813i \(0.521054\pi\)
\(114\) 0 0
\(115\) 367231.i 0.241461i
\(116\) 0 0
\(117\) −667012. −0.416463
\(118\) 0 0
\(119\) 4.36297e6i 2.58906i
\(120\) 0 0
\(121\) 300462. 0.169603
\(122\) 0 0
\(123\) − 1.09357e6i − 0.587665i
\(124\) 0 0
\(125\) −2.04581e6 −1.04746
\(126\) 0 0
\(127\) 819411.i 0.400028i 0.979793 + 0.200014i \(0.0640988\pi\)
−0.979793 + 0.200014i \(0.935901\pi\)
\(128\) 0 0
\(129\) −447405. −0.208416
\(130\) 0 0
\(131\) 3.76018e6i 1.67261i 0.548263 + 0.836306i \(0.315289\pi\)
−0.548263 + 0.836306i \(0.684711\pi\)
\(132\) 0 0
\(133\) 5.33793e6 2.26891
\(134\) 0 0
\(135\) 323453.i 0.131465i
\(136\) 0 0
\(137\) −1.86568e6 −0.725563 −0.362782 0.931874i \(-0.618173\pi\)
−0.362782 + 0.931874i \(0.618173\pi\)
\(138\) 0 0
\(139\) − 3.37311e6i − 1.25599i −0.778218 0.627995i \(-0.783876\pi\)
0.778218 0.627995i \(-0.216124\pi\)
\(140\) 0 0
\(141\) 915768. 0.326684
\(142\) 0 0
\(143\) 3.32927e6i 1.13852i
\(144\) 0 0
\(145\) 3.62408e6 1.18876
\(146\) 0 0
\(147\) 2.24965e6i 0.708212i
\(148\) 0 0
\(149\) −3.35713e6 −1.01487 −0.507434 0.861691i \(-0.669406\pi\)
−0.507434 + 0.861691i \(0.669406\pi\)
\(150\) 0 0
\(151\) − 3.78813e6i − 1.10026i −0.835081 0.550128i \(-0.814579\pi\)
0.835081 0.550128i \(-0.185421\pi\)
\(152\) 0 0
\(153\) 2.07142e6 0.578354
\(154\) 0 0
\(155\) − 4.93916e6i − 1.32635i
\(156\) 0 0
\(157\) 5.81264e6 1.50202 0.751008 0.660293i \(-0.229568\pi\)
0.751008 + 0.660293i \(0.229568\pi\)
\(158\) 0 0
\(159\) − 1.73202e6i − 0.430886i
\(160\) 0 0
\(161\) 2.20120e6 0.527450
\(162\) 0 0
\(163\) 5.12374e6i 1.18311i 0.806266 + 0.591554i \(0.201485\pi\)
−0.806266 + 0.591554i \(0.798515\pi\)
\(164\) 0 0
\(165\) 1.61446e6 0.359397
\(166\) 0 0
\(167\) 1.27729e6i 0.274247i 0.990554 + 0.137123i \(0.0437857\pi\)
−0.990554 + 0.137123i \(0.956214\pi\)
\(168\) 0 0
\(169\) 2.70770e6 0.560972
\(170\) 0 0
\(171\) − 2.53430e6i − 0.506838i
\(172\) 0 0
\(173\) 2.87446e6 0.555160 0.277580 0.960703i \(-0.410468\pi\)
0.277580 + 0.960703i \(0.410468\pi\)
\(174\) 0 0
\(175\) 4.26540e6i 0.795876i
\(176\) 0 0
\(177\) −1.84344e6 −0.332437
\(178\) 0 0
\(179\) − 8.99970e6i − 1.56917i −0.620024 0.784583i \(-0.712877\pi\)
0.620024 0.784583i \(-0.287123\pi\)
\(180\) 0 0
\(181\) −3.33395e6 −0.562242 −0.281121 0.959672i \(-0.590706\pi\)
−0.281121 + 0.959672i \(0.590706\pi\)
\(182\) 0 0
\(183\) 828905.i 0.135254i
\(184\) 0 0
\(185\) 7.13044e6 1.12616
\(186\) 0 0
\(187\) − 1.03391e7i − 1.58109i
\(188\) 0 0
\(189\) 1.93879e6 0.287174
\(190\) 0 0
\(191\) − 3.66531e6i − 0.526030i −0.964792 0.263015i \(-0.915283\pi\)
0.964792 0.263015i \(-0.0847168\pi\)
\(192\) 0 0
\(193\) −8.58154e6 −1.19369 −0.596847 0.802355i \(-0.703580\pi\)
−0.596847 + 0.802355i \(0.703580\pi\)
\(194\) 0 0
\(195\) − 3.65370e6i − 0.492753i
\(196\) 0 0
\(197\) 1.04264e7 1.36375 0.681874 0.731470i \(-0.261165\pi\)
0.681874 + 0.731470i \(0.261165\pi\)
\(198\) 0 0
\(199\) − 4.34498e6i − 0.551352i −0.961251 0.275676i \(-0.911098\pi\)
0.961251 0.275676i \(-0.0889017\pi\)
\(200\) 0 0
\(201\) −657305. −0.0809429
\(202\) 0 0
\(203\) − 2.17228e7i − 2.59674i
\(204\) 0 0
\(205\) 5.99024e6 0.695317
\(206\) 0 0
\(207\) − 1.04507e6i − 0.117824i
\(208\) 0 0
\(209\) −1.26495e7 −1.38559
\(210\) 0 0
\(211\) 5.90463e6i 0.628558i 0.949331 + 0.314279i \(0.101763\pi\)
−0.949331 + 0.314279i \(0.898237\pi\)
\(212\) 0 0
\(213\) −5.48088e6 −0.567168
\(214\) 0 0
\(215\) − 2.45075e6i − 0.246595i
\(216\) 0 0
\(217\) −2.96054e7 −2.89729
\(218\) 0 0
\(219\) − 1.42171e6i − 0.135357i
\(220\) 0 0
\(221\) −2.33986e7 −2.16777
\(222\) 0 0
\(223\) − 1.48155e7i − 1.33598i −0.744169 0.667992i \(-0.767154\pi\)
0.744169 0.667992i \(-0.232846\pi\)
\(224\) 0 0
\(225\) 2.02509e6 0.177786
\(226\) 0 0
\(227\) 3.99262e6i 0.341334i 0.985329 + 0.170667i \(0.0545923\pi\)
−0.985329 + 0.170667i \(0.945408\pi\)
\(228\) 0 0
\(229\) 3.71868e6 0.309658 0.154829 0.987941i \(-0.450517\pi\)
0.154829 + 0.987941i \(0.450517\pi\)
\(230\) 0 0
\(231\) − 9.67710e6i − 0.785071i
\(232\) 0 0
\(233\) 5.25478e6 0.415419 0.207710 0.978191i \(-0.433399\pi\)
0.207710 + 0.978191i \(0.433399\pi\)
\(234\) 0 0
\(235\) 5.01631e6i 0.386528i
\(236\) 0 0
\(237\) −1.20843e7 −0.907771
\(238\) 0 0
\(239\) 7.96267e6i 0.583264i 0.956531 + 0.291632i \(0.0941982\pi\)
−0.956531 + 0.291632i \(0.905802\pi\)
\(240\) 0 0
\(241\) 1.44459e6 0.103203 0.0516017 0.998668i \(-0.483567\pi\)
0.0516017 + 0.998668i \(0.483567\pi\)
\(242\) 0 0
\(243\) − 920483.i − 0.0641500i
\(244\) 0 0
\(245\) −1.23229e7 −0.837946
\(246\) 0 0
\(247\) 2.86272e7i 1.89971i
\(248\) 0 0
\(249\) 1.04437e7 0.676484
\(250\) 0 0
\(251\) − 920323.i − 0.0581995i −0.999577 0.0290998i \(-0.990736\pi\)
0.999577 0.0290998i \(-0.00926405\pi\)
\(252\) 0 0
\(253\) −5.21625e6 −0.322105
\(254\) 0 0
\(255\) 1.13466e7i 0.684300i
\(256\) 0 0
\(257\) 5.73409e6 0.337804 0.168902 0.985633i \(-0.445978\pi\)
0.168902 + 0.985633i \(0.445978\pi\)
\(258\) 0 0
\(259\) − 4.27400e7i − 2.46000i
\(260\) 0 0
\(261\) −1.03134e7 −0.580070
\(262\) 0 0
\(263\) − 1.36973e7i − 0.752952i −0.926426 0.376476i \(-0.877136\pi\)
0.926426 0.376476i \(-0.122864\pi\)
\(264\) 0 0
\(265\) 9.48752e6 0.509818
\(266\) 0 0
\(267\) − 7.86774e6i − 0.413348i
\(268\) 0 0
\(269\) 1.95267e7 1.00317 0.501583 0.865110i \(-0.332751\pi\)
0.501583 + 0.865110i \(0.332751\pi\)
\(270\) 0 0
\(271\) − 8.79492e6i − 0.441900i −0.975285 0.220950i \(-0.929084\pi\)
0.975285 0.220950i \(-0.0709158\pi\)
\(272\) 0 0
\(273\) −2.19004e7 −1.07638
\(274\) 0 0
\(275\) − 1.01079e7i − 0.486028i
\(276\) 0 0
\(277\) 1.32017e6 0.0621142 0.0310571 0.999518i \(-0.490113\pi\)
0.0310571 + 0.999518i \(0.490113\pi\)
\(278\) 0 0
\(279\) 1.40558e7i 0.647208i
\(280\) 0 0
\(281\) −1.77259e7 −0.798893 −0.399446 0.916757i \(-0.630798\pi\)
−0.399446 + 0.916757i \(0.630798\pi\)
\(282\) 0 0
\(283\) − 4.14391e7i − 1.82832i −0.405358 0.914158i \(-0.632853\pi\)
0.405358 0.914158i \(-0.367147\pi\)
\(284\) 0 0
\(285\) 1.38822e7 0.599684
\(286\) 0 0
\(287\) − 3.59057e7i − 1.51886i
\(288\) 0 0
\(289\) 4.85271e7 2.01044
\(290\) 0 0
\(291\) − 1.60742e7i − 0.652303i
\(292\) 0 0
\(293\) −4.93196e6 −0.196072 −0.0980362 0.995183i \(-0.531256\pi\)
−0.0980362 + 0.995183i \(0.531256\pi\)
\(294\) 0 0
\(295\) − 1.00978e7i − 0.393335i
\(296\) 0 0
\(297\) −4.59442e6 −0.175372
\(298\) 0 0
\(299\) 1.18050e7i 0.441623i
\(300\) 0 0
\(301\) −1.46899e7 −0.538665
\(302\) 0 0
\(303\) 1.44597e7i 0.519795i
\(304\) 0 0
\(305\) −4.54050e6 −0.160031
\(306\) 0 0
\(307\) − 4.35700e7i − 1.50582i −0.658124 0.752910i \(-0.728650\pi\)
0.658124 0.752910i \(-0.271350\pi\)
\(308\) 0 0
\(309\) −5.56974e6 −0.188781
\(310\) 0 0
\(311\) − 5.31388e7i − 1.76657i −0.468837 0.883285i \(-0.655327\pi\)
0.468837 0.883285i \(-0.344673\pi\)
\(312\) 0 0
\(313\) −3.18652e7 −1.03916 −0.519582 0.854421i \(-0.673912\pi\)
−0.519582 + 0.854421i \(0.673912\pi\)
\(314\) 0 0
\(315\) 1.06201e7i 0.339780i
\(316\) 0 0
\(317\) 3.33162e7 1.04587 0.522936 0.852372i \(-0.324837\pi\)
0.522936 + 0.852372i \(0.324837\pi\)
\(318\) 0 0
\(319\) 5.14774e7i 1.58579i
\(320\) 0 0
\(321\) −1.81132e7 −0.547621
\(322\) 0 0
\(323\) − 8.89023e7i − 2.63819i
\(324\) 0 0
\(325\) −2.28753e7 −0.666371
\(326\) 0 0
\(327\) 1.15572e7i 0.330528i
\(328\) 0 0
\(329\) 3.00679e7 0.844337
\(330\) 0 0
\(331\) − 2.35633e6i − 0.0649758i −0.999472 0.0324879i \(-0.989657\pi\)
0.999472 0.0324879i \(-0.0103430\pi\)
\(332\) 0 0
\(333\) −2.02918e7 −0.549525
\(334\) 0 0
\(335\) − 3.60053e6i − 0.0957705i
\(336\) 0 0
\(337\) 3.61254e7 0.943894 0.471947 0.881627i \(-0.343551\pi\)
0.471947 + 0.881627i \(0.343551\pi\)
\(338\) 0 0
\(339\) 2.97330e6i 0.0763202i
\(340\) 0 0
\(341\) 7.01571e7 1.76933
\(342\) 0 0
\(343\) 1.36484e7i 0.338221i
\(344\) 0 0
\(345\) 5.72457e6 0.139407
\(346\) 0 0
\(347\) 1.76051e7i 0.421357i 0.977555 + 0.210679i \(0.0675674\pi\)
−0.977555 + 0.210679i \(0.932433\pi\)
\(348\) 0 0
\(349\) 3.02782e6 0.0712285 0.0356143 0.999366i \(-0.488661\pi\)
0.0356143 + 0.999366i \(0.488661\pi\)
\(350\) 0 0
\(351\) 1.03977e7i 0.240445i
\(352\) 0 0
\(353\) −3.74661e7 −0.851755 −0.425877 0.904781i \(-0.640034\pi\)
−0.425877 + 0.904781i \(0.640034\pi\)
\(354\) 0 0
\(355\) − 3.00227e7i − 0.671064i
\(356\) 0 0
\(357\) 6.80120e7 1.49479
\(358\) 0 0
\(359\) 5.14739e7i 1.11251i 0.831012 + 0.556254i \(0.187762\pi\)
−0.831012 + 0.556254i \(0.812238\pi\)
\(360\) 0 0
\(361\) −6.17226e7 −1.31197
\(362\) 0 0
\(363\) − 4.68374e6i − 0.0979203i
\(364\) 0 0
\(365\) 7.78773e6 0.160152
\(366\) 0 0
\(367\) 1.52811e7i 0.309142i 0.987982 + 0.154571i \(0.0493995\pi\)
−0.987982 + 0.154571i \(0.950601\pi\)
\(368\) 0 0
\(369\) −1.70470e7 −0.339289
\(370\) 0 0
\(371\) − 5.68684e7i − 1.11365i
\(372\) 0 0
\(373\) 6.19317e7 1.19340 0.596701 0.802464i \(-0.296478\pi\)
0.596701 + 0.802464i \(0.296478\pi\)
\(374\) 0 0
\(375\) 3.18910e7i 0.604749i
\(376\) 0 0
\(377\) 1.16499e8 2.17420
\(378\) 0 0
\(379\) 3.43446e7i 0.630870i 0.948947 + 0.315435i \(0.102151\pi\)
−0.948947 + 0.315435i \(0.897849\pi\)
\(380\) 0 0
\(381\) 1.27733e7 0.230956
\(382\) 0 0
\(383\) − 1.34734e7i − 0.239817i −0.992785 0.119908i \(-0.961740\pi\)
0.992785 0.119908i \(-0.0382601\pi\)
\(384\) 0 0
\(385\) 5.30083e7 0.928885
\(386\) 0 0
\(387\) 6.97435e6i 0.120329i
\(388\) 0 0
\(389\) −1.13679e8 −1.93122 −0.965610 0.259993i \(-0.916280\pi\)
−0.965610 + 0.259993i \(0.916280\pi\)
\(390\) 0 0
\(391\) − 3.66606e7i − 0.613294i
\(392\) 0 0
\(393\) 5.86155e7 0.965683
\(394\) 0 0
\(395\) − 6.61943e7i − 1.07406i
\(396\) 0 0
\(397\) 3.96900e6 0.0634322 0.0317161 0.999497i \(-0.489903\pi\)
0.0317161 + 0.999497i \(0.489903\pi\)
\(398\) 0 0
\(399\) − 8.32100e7i − 1.30996i
\(400\) 0 0
\(401\) 4.54308e6 0.0704558 0.0352279 0.999379i \(-0.488784\pi\)
0.0352279 + 0.999379i \(0.488784\pi\)
\(402\) 0 0
\(403\) − 1.58774e8i − 2.42584i
\(404\) 0 0
\(405\) 5.04214e6 0.0759014
\(406\) 0 0
\(407\) 1.01283e8i 1.50228i
\(408\) 0 0
\(409\) −4.23651e7 −0.619211 −0.309606 0.950865i \(-0.600197\pi\)
−0.309606 + 0.950865i \(0.600197\pi\)
\(410\) 0 0
\(411\) 2.90831e7i 0.418904i
\(412\) 0 0
\(413\) −6.05267e7 −0.859206
\(414\) 0 0
\(415\) 5.72077e7i 0.800406i
\(416\) 0 0
\(417\) −5.25816e7 −0.725146
\(418\) 0 0
\(419\) − 8.16747e7i − 1.11031i −0.831746 0.555157i \(-0.812658\pi\)
0.831746 0.555157i \(-0.187342\pi\)
\(420\) 0 0
\(421\) −7.99635e7 −1.07163 −0.535816 0.844335i \(-0.679996\pi\)
−0.535816 + 0.844335i \(0.679996\pi\)
\(422\) 0 0
\(423\) − 1.42754e7i − 0.188611i
\(424\) 0 0
\(425\) 7.10395e7 0.925408
\(426\) 0 0
\(427\) 2.72159e7i 0.349574i
\(428\) 0 0
\(429\) 5.18981e7 0.657325
\(430\) 0 0
\(431\) 7.45512e7i 0.931157i 0.885007 + 0.465578i \(0.154154\pi\)
−0.885007 + 0.465578i \(0.845846\pi\)
\(432\) 0 0
\(433\) −5.35595e7 −0.659740 −0.329870 0.944026i \(-0.607005\pi\)
−0.329870 + 0.944026i \(0.607005\pi\)
\(434\) 0 0
\(435\) − 5.64938e7i − 0.686330i
\(436\) 0 0
\(437\) −4.48528e7 −0.537458
\(438\) 0 0
\(439\) − 3.84937e7i − 0.454984i −0.973780 0.227492i \(-0.926947\pi\)
0.973780 0.227492i \(-0.0730525\pi\)
\(440\) 0 0
\(441\) 3.50686e7 0.408886
\(442\) 0 0
\(443\) − 4.20100e7i − 0.483216i −0.970374 0.241608i \(-0.922325\pi\)
0.970374 0.241608i \(-0.0776749\pi\)
\(444\) 0 0
\(445\) 4.30972e7 0.489067
\(446\) 0 0
\(447\) 5.23325e7i 0.585934i
\(448\) 0 0
\(449\) 1.46094e8 1.61396 0.806981 0.590577i \(-0.201100\pi\)
0.806981 + 0.590577i \(0.201100\pi\)
\(450\) 0 0
\(451\) 8.50870e7i 0.927542i
\(452\) 0 0
\(453\) −5.90510e7 −0.635233
\(454\) 0 0
\(455\) − 1.19964e8i − 1.27355i
\(456\) 0 0
\(457\) −1.64317e8 −1.72160 −0.860802 0.508941i \(-0.830037\pi\)
−0.860802 + 0.508941i \(0.830037\pi\)
\(458\) 0 0
\(459\) − 3.22902e7i − 0.333913i
\(460\) 0 0
\(461\) −1.01392e7 −0.103491 −0.0517455 0.998660i \(-0.516478\pi\)
−0.0517455 + 0.998660i \(0.516478\pi\)
\(462\) 0 0
\(463\) − 1.91081e8i − 1.92519i −0.270937 0.962597i \(-0.587333\pi\)
0.270937 0.962597i \(-0.412667\pi\)
\(464\) 0 0
\(465\) −7.69938e7 −0.765768
\(466\) 0 0
\(467\) 1.30526e8i 1.28159i 0.767713 + 0.640793i \(0.221394\pi\)
−0.767713 + 0.640793i \(0.778606\pi\)
\(468\) 0 0
\(469\) −2.15817e7 −0.209202
\(470\) 0 0
\(471\) − 9.06101e7i − 0.867189i
\(472\) 0 0
\(473\) 3.48111e7 0.328954
\(474\) 0 0
\(475\) − 8.69140e7i − 0.810978i
\(476\) 0 0
\(477\) −2.69996e7 −0.248772
\(478\) 0 0
\(479\) − 1.71231e8i − 1.55803i −0.627003 0.779017i \(-0.715719\pi\)
0.627003 0.779017i \(-0.284281\pi\)
\(480\) 0 0
\(481\) 2.29214e8 2.05971
\(482\) 0 0
\(483\) − 3.43132e7i − 0.304523i
\(484\) 0 0
\(485\) 8.80496e7 0.771795
\(486\) 0 0
\(487\) 6.38316e7i 0.552648i 0.961065 + 0.276324i \(0.0891163\pi\)
−0.961065 + 0.276324i \(0.910884\pi\)
\(488\) 0 0
\(489\) 7.98712e7 0.683067
\(490\) 0 0
\(491\) 1.36774e8i 1.15547i 0.816224 + 0.577735i \(0.196063\pi\)
−0.816224 + 0.577735i \(0.803937\pi\)
\(492\) 0 0
\(493\) −3.61791e8 −3.01937
\(494\) 0 0
\(495\) − 2.51669e7i − 0.207498i
\(496\) 0 0
\(497\) −1.79957e8 −1.46588
\(498\) 0 0
\(499\) 1.94124e8i 1.56235i 0.624312 + 0.781175i \(0.285379\pi\)
−0.624312 + 0.781175i \(0.714621\pi\)
\(500\) 0 0
\(501\) 1.99110e7 0.158336
\(502\) 0 0
\(503\) − 9.34518e7i − 0.734317i −0.930158 0.367159i \(-0.880331\pi\)
0.930158 0.367159i \(-0.119669\pi\)
\(504\) 0 0
\(505\) −7.92062e7 −0.615014
\(506\) 0 0
\(507\) − 4.22089e7i − 0.323877i
\(508\) 0 0
\(509\) 6.32104e7 0.479331 0.239665 0.970856i \(-0.422962\pi\)
0.239665 + 0.970856i \(0.422962\pi\)
\(510\) 0 0
\(511\) − 4.66799e7i − 0.349838i
\(512\) 0 0
\(513\) −3.95058e7 −0.292623
\(514\) 0 0
\(515\) − 3.05094e7i − 0.223363i
\(516\) 0 0
\(517\) −7.12530e7 −0.515622
\(518\) 0 0
\(519\) − 4.48084e7i − 0.320522i
\(520\) 0 0
\(521\) 2.37718e8 1.68092 0.840462 0.541871i \(-0.182284\pi\)
0.840462 + 0.541871i \(0.182284\pi\)
\(522\) 0 0
\(523\) 5.13720e7i 0.359105i 0.983748 + 0.179553i \(0.0574650\pi\)
−0.983748 + 0.179553i \(0.942535\pi\)
\(524\) 0 0
\(525\) 6.64909e7 0.459499
\(526\) 0 0
\(527\) 4.93074e8i 3.36884i
\(528\) 0 0
\(529\) 1.29540e8 0.875058
\(530\) 0 0
\(531\) 2.87364e7i 0.191933i
\(532\) 0 0
\(533\) 1.92562e8 1.27171
\(534\) 0 0
\(535\) − 9.92189e7i − 0.647938i
\(536\) 0 0
\(537\) −1.40291e8 −0.905958
\(538\) 0 0
\(539\) − 1.75038e8i − 1.11781i
\(540\) 0 0
\(541\) 1.82128e8 1.15023 0.575116 0.818072i \(-0.304957\pi\)
0.575116 + 0.818072i \(0.304957\pi\)
\(542\) 0 0
\(543\) 5.19711e7i 0.324610i
\(544\) 0 0
\(545\) −6.33069e7 −0.391076
\(546\) 0 0
\(547\) − 6.68154e7i − 0.408239i −0.978946 0.204120i \(-0.934567\pi\)
0.978946 0.204120i \(-0.0654332\pi\)
\(548\) 0 0
\(549\) 1.29213e7 0.0780892
\(550\) 0 0
\(551\) 4.42636e8i 2.64602i
\(552\) 0 0
\(553\) −3.96770e8 −2.34619
\(554\) 0 0
\(555\) − 1.11153e8i − 0.650190i
\(556\) 0 0
\(557\) −606923. −0.00351211 −0.00175606 0.999998i \(-0.500559\pi\)
−0.00175606 + 0.999998i \(0.500559\pi\)
\(558\) 0 0
\(559\) − 7.87816e7i − 0.451013i
\(560\) 0 0
\(561\) −1.61171e8 −0.912845
\(562\) 0 0
\(563\) 2.47309e8i 1.38584i 0.721012 + 0.692922i \(0.243677\pi\)
−0.721012 + 0.692922i \(0.756323\pi\)
\(564\) 0 0
\(565\) −1.62869e7 −0.0903009
\(566\) 0 0
\(567\) − 3.02227e7i − 0.165800i
\(568\) 0 0
\(569\) −1.83163e8 −0.994262 −0.497131 0.867675i \(-0.665613\pi\)
−0.497131 + 0.867675i \(0.665613\pi\)
\(570\) 0 0
\(571\) − 1.30136e8i − 0.699021i −0.936933 0.349510i \(-0.886348\pi\)
0.936933 0.349510i \(-0.113652\pi\)
\(572\) 0 0
\(573\) −5.71365e7 −0.303703
\(574\) 0 0
\(575\) − 3.58407e7i − 0.188527i
\(576\) 0 0
\(577\) 2.46187e7 0.128156 0.0640778 0.997945i \(-0.479589\pi\)
0.0640778 + 0.997945i \(0.479589\pi\)
\(578\) 0 0
\(579\) 1.33773e8i 0.689180i
\(580\) 0 0
\(581\) 3.42905e8 1.74842
\(582\) 0 0
\(583\) 1.34763e8i 0.680089i
\(584\) 0 0
\(585\) −5.69556e7 −0.284491
\(586\) 0 0
\(587\) − 4.56349e7i − 0.225623i −0.993616 0.112811i \(-0.964014\pi\)
0.993616 0.112811i \(-0.0359856\pi\)
\(588\) 0 0
\(589\) 6.03257e8 2.95227
\(590\) 0 0
\(591\) − 1.62531e8i − 0.787360i
\(592\) 0 0
\(593\) 1.58370e8 0.759466 0.379733 0.925096i \(-0.376016\pi\)
0.379733 + 0.925096i \(0.376016\pi\)
\(594\) 0 0
\(595\) 3.72550e8i 1.76862i
\(596\) 0 0
\(597\) −6.77316e7 −0.318323
\(598\) 0 0
\(599\) − 2.34911e8i − 1.09301i −0.837457 0.546504i \(-0.815958\pi\)
0.837457 0.546504i \(-0.184042\pi\)
\(600\) 0 0
\(601\) −6.40513e7 −0.295056 −0.147528 0.989058i \(-0.547132\pi\)
−0.147528 + 0.989058i \(0.547132\pi\)
\(602\) 0 0
\(603\) 1.02464e7i 0.0467324i
\(604\) 0 0
\(605\) 2.56562e7 0.115858
\(606\) 0 0
\(607\) − 6.98586e7i − 0.312359i −0.987729 0.156179i \(-0.950082\pi\)
0.987729 0.156179i \(-0.0499178\pi\)
\(608\) 0 0
\(609\) −3.38626e8 −1.49923
\(610\) 0 0
\(611\) 1.61254e8i 0.706946i
\(612\) 0 0
\(613\) −2.62634e7 −0.114017 −0.0570085 0.998374i \(-0.518156\pi\)
−0.0570085 + 0.998374i \(0.518156\pi\)
\(614\) 0 0
\(615\) − 9.33786e7i − 0.401441i
\(616\) 0 0
\(617\) 2.98936e8 1.27269 0.636345 0.771404i \(-0.280445\pi\)
0.636345 + 0.771404i \(0.280445\pi\)
\(618\) 0 0
\(619\) − 1.32500e8i − 0.558657i −0.960196 0.279328i \(-0.909888\pi\)
0.960196 0.279328i \(-0.0901118\pi\)
\(620\) 0 0
\(621\) −1.62910e7 −0.0680256
\(622\) 0 0
\(623\) − 2.58326e8i − 1.06833i
\(624\) 0 0
\(625\) −4.44757e7 −0.182172
\(626\) 0 0
\(627\) 1.97186e8i 0.799969i
\(628\) 0 0
\(629\) −7.11829e8 −2.86038
\(630\) 0 0
\(631\) 1.19185e8i 0.474390i 0.971462 + 0.237195i \(0.0762280\pi\)
−0.971462 + 0.237195i \(0.923772\pi\)
\(632\) 0 0
\(633\) 9.20440e7 0.362898
\(634\) 0 0
\(635\) 6.99687e7i 0.273264i
\(636\) 0 0
\(637\) −3.96132e8 −1.53257
\(638\) 0 0
\(639\) 8.54385e7i 0.327454i
\(640\) 0 0
\(641\) 7.84005e7 0.297677 0.148838 0.988862i \(-0.452447\pi\)
0.148838 + 0.988862i \(0.452447\pi\)
\(642\) 0 0
\(643\) 1.05651e8i 0.397410i 0.980059 + 0.198705i \(0.0636736\pi\)
−0.980059 + 0.198705i \(0.936326\pi\)
\(644\) 0 0
\(645\) −3.82035e7 −0.142372
\(646\) 0 0
\(647\) 2.37697e8i 0.877628i 0.898578 + 0.438814i \(0.144601\pi\)
−0.898578 + 0.438814i \(0.855399\pi\)
\(648\) 0 0
\(649\) 1.43432e8 0.524703
\(650\) 0 0
\(651\) 4.61503e8i 1.67275i
\(652\) 0 0
\(653\) 1.71315e7 0.0615257 0.0307628 0.999527i \(-0.490206\pi\)
0.0307628 + 0.999527i \(0.490206\pi\)
\(654\) 0 0
\(655\) 3.21079e8i 1.14258i
\(656\) 0 0
\(657\) −2.21623e7 −0.0781482
\(658\) 0 0
\(659\) 3.62898e8i 1.26803i 0.773323 + 0.634013i \(0.218593\pi\)
−0.773323 + 0.634013i \(0.781407\pi\)
\(660\) 0 0
\(661\) 4.90659e8 1.69893 0.849465 0.527645i \(-0.176925\pi\)
0.849465 + 0.527645i \(0.176925\pi\)
\(662\) 0 0
\(663\) 3.64747e8i 1.25156i
\(664\) 0 0
\(665\) 4.55800e8 1.54992
\(666\) 0 0
\(667\) 1.82530e8i 0.615114i
\(668\) 0 0
\(669\) −2.30950e8 −0.771330
\(670\) 0 0
\(671\) − 6.44945e7i − 0.213479i
\(672\) 0 0
\(673\) −1.66234e8 −0.545350 −0.272675 0.962106i \(-0.587908\pi\)
−0.272675 + 0.962106i \(0.587908\pi\)
\(674\) 0 0
\(675\) − 3.15681e7i − 0.102645i
\(676\) 0 0
\(677\) 1.98782e8 0.640635 0.320318 0.947310i \(-0.396210\pi\)
0.320318 + 0.947310i \(0.396210\pi\)
\(678\) 0 0
\(679\) − 5.27772e8i − 1.68592i
\(680\) 0 0
\(681\) 6.22387e7 0.197070
\(682\) 0 0
\(683\) 795875.i 0.00249794i 0.999999 + 0.00124897i \(0.000397560\pi\)
−0.999999 + 0.00124897i \(0.999602\pi\)
\(684\) 0 0
\(685\) −1.59309e8 −0.495641
\(686\) 0 0
\(687\) − 5.79684e7i − 0.178781i
\(688\) 0 0
\(689\) 3.04985e8 0.932439
\(690\) 0 0
\(691\) 6.26115e8i 1.89767i 0.315778 + 0.948833i \(0.397734\pi\)
−0.315778 + 0.948833i \(0.602266\pi\)
\(692\) 0 0
\(693\) −1.50851e8 −0.453261
\(694\) 0 0
\(695\) − 2.88027e8i − 0.857982i
\(696\) 0 0
\(697\) −5.98004e8 −1.76606
\(698\) 0 0
\(699\) − 8.19139e7i − 0.239843i
\(700\) 0 0
\(701\) −7.92815e7 −0.230154 −0.115077 0.993357i \(-0.536711\pi\)
−0.115077 + 0.993357i \(0.536711\pi\)
\(702\) 0 0
\(703\) 8.70894e8i 2.50668i
\(704\) 0 0
\(705\) 7.81966e7 0.223162
\(706\) 0 0
\(707\) 4.74764e8i 1.34344i
\(708\) 0 0
\(709\) 6.27826e7 0.176157 0.0880786 0.996114i \(-0.471927\pi\)
0.0880786 + 0.996114i \(0.471927\pi\)
\(710\) 0 0
\(711\) 1.88376e8i 0.524102i
\(712\) 0 0
\(713\) 2.48764e8 0.686309
\(714\) 0 0
\(715\) 2.84283e8i 0.777737i
\(716\) 0 0
\(717\) 1.24126e8 0.336747
\(718\) 0 0
\(719\) − 3.24833e8i − 0.873924i −0.899480 0.436962i \(-0.856054\pi\)
0.899480 0.436962i \(-0.143946\pi\)
\(720\) 0 0
\(721\) −1.82874e8 −0.487918
\(722\) 0 0
\(723\) − 2.25190e7i − 0.0595845i
\(724\) 0 0
\(725\) −3.53699e8 −0.928154
\(726\) 0 0
\(727\) 1.75113e8i 0.455737i 0.973692 + 0.227869i \(0.0731757\pi\)
−0.973692 + 0.227869i \(0.926824\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −0.0370370
\(730\) 0 0
\(731\) 2.44658e8i 0.626335i
\(732\) 0 0
\(733\) −2.79408e8 −0.709458 −0.354729 0.934969i \(-0.615427\pi\)
−0.354729 + 0.934969i \(0.615427\pi\)
\(734\) 0 0
\(735\) 1.92096e8i 0.483789i
\(736\) 0 0
\(737\) 5.11428e7 0.127756
\(738\) 0 0
\(739\) − 5.00369e8i − 1.23981i −0.784675 0.619907i \(-0.787170\pi\)
0.784675 0.619907i \(-0.212830\pi\)
\(740\) 0 0
\(741\) 4.46254e8 1.09680
\(742\) 0 0
\(743\) − 5.12068e8i − 1.24842i −0.781256 0.624211i \(-0.785421\pi\)
0.781256 0.624211i \(-0.214579\pi\)
\(744\) 0 0
\(745\) −2.86662e8 −0.693269
\(746\) 0 0
\(747\) − 1.62802e8i − 0.390568i
\(748\) 0 0
\(749\) −5.94721e8 −1.41536
\(750\) 0 0
\(751\) − 4.54568e8i − 1.07319i −0.843838 0.536597i \(-0.819709\pi\)
0.843838 0.536597i \(-0.180291\pi\)
\(752\) 0 0
\(753\) −1.43464e7 −0.0336015
\(754\) 0 0
\(755\) − 3.23464e8i − 0.751598i
\(756\) 0 0
\(757\) −3.55132e8 −0.818657 −0.409329 0.912387i \(-0.634237\pi\)
−0.409329 + 0.912387i \(0.634237\pi\)
\(758\) 0 0
\(759\) 8.13133e7i 0.185967i
\(760\) 0 0
\(761\) 7.07962e8 1.60641 0.803204 0.595704i \(-0.203127\pi\)
0.803204 + 0.595704i \(0.203127\pi\)
\(762\) 0 0
\(763\) 3.79463e8i 0.854272i
\(764\) 0 0
\(765\) 1.76876e8 0.395081
\(766\) 0 0
\(767\) − 3.24604e8i − 0.719396i
\(768\) 0 0
\(769\) −5.29332e8 −1.16399 −0.581995 0.813193i \(-0.697728\pi\)
−0.581995 + 0.813193i \(0.697728\pi\)
\(770\) 0 0
\(771\) − 8.93856e7i − 0.195031i
\(772\) 0 0
\(773\) −5.71149e8 −1.23655 −0.618274 0.785963i \(-0.712168\pi\)
−0.618274 + 0.785963i \(0.712168\pi\)
\(774\) 0 0
\(775\) 4.82046e8i 1.03558i
\(776\) 0 0
\(777\) −6.66251e8 −1.42028
\(778\) 0 0
\(779\) 7.31634e8i 1.54768i
\(780\) 0 0
\(781\) 4.26450e8 0.895189
\(782\) 0 0
\(783\) 1.60770e8i 0.334904i
\(784\) 0 0
\(785\) 4.96336e8 1.02605
\(786\) 0 0
\(787\) − 1.56385e8i − 0.320828i −0.987050 0.160414i \(-0.948717\pi\)
0.987050 0.160414i \(-0.0512829\pi\)
\(788\) 0 0
\(789\) −2.13519e8 −0.434717
\(790\) 0 0
\(791\) 9.76239e7i 0.197254i
\(792\) 0 0
\(793\) −1.45958e8 −0.292691
\(794\) 0 0
\(795\) − 1.47896e8i − 0.294343i
\(796\) 0 0
\(797\) 6.02539e8 1.19017 0.595087 0.803661i \(-0.297118\pi\)
0.595087 + 0.803661i \(0.297118\pi\)
\(798\) 0 0
\(799\) − 5.00777e8i − 0.981756i
\(800\) 0 0
\(801\) −1.22646e8 −0.238647
\(802\) 0 0
\(803\) 1.10619e8i 0.213640i
\(804\) 0 0
\(805\) 1.87958e8 0.360307
\(806\) 0 0
\(807\) − 3.04392e8i − 0.579178i
\(808\) 0 0
\(809\) −2.30938e8 −0.436164 −0.218082 0.975930i \(-0.569980\pi\)
−0.218082 + 0.975930i \(0.569980\pi\)
\(810\) 0 0
\(811\) 1.15316e8i 0.216185i 0.994141 + 0.108093i \(0.0344743\pi\)
−0.994141 + 0.108093i \(0.965526\pi\)
\(812\) 0 0
\(813\) −1.37099e8 −0.255131
\(814\) 0 0
\(815\) 4.37511e8i 0.808195i
\(816\) 0 0
\(817\) 2.99329e8 0.548887
\(818\) 0 0
\(819\) 3.41393e8i 0.621446i
\(820\) 0 0
\(821\) −8.70428e7 −0.157291 −0.0786454 0.996903i \(-0.525059\pi\)
−0.0786454 + 0.996903i \(0.525059\pi\)
\(822\) 0 0
\(823\) − 1.58502e8i − 0.284337i −0.989842 0.142169i \(-0.954592\pi\)
0.989842 0.142169i \(-0.0454076\pi\)
\(824\) 0 0
\(825\) −1.57566e8 −0.280608
\(826\) 0 0
\(827\) 6.20614e8i 1.09725i 0.836069 + 0.548624i \(0.184848\pi\)
−0.836069 + 0.548624i \(0.815152\pi\)
\(828\) 0 0
\(829\) 1.05341e9 1.84898 0.924490 0.381206i \(-0.124491\pi\)
0.924490 + 0.381206i \(0.124491\pi\)
\(830\) 0 0
\(831\) − 2.05794e7i − 0.0358617i
\(832\) 0 0
\(833\) 1.23019e9 2.12833
\(834\) 0 0
\(835\) 1.09067e8i 0.187341i
\(836\) 0 0
\(837\) 2.19109e8 0.373666
\(838\) 0 0
\(839\) − 1.80870e8i − 0.306254i −0.988207 0.153127i \(-0.951066\pi\)
0.988207 0.153127i \(-0.0489343\pi\)
\(840\) 0 0
\(841\) 1.20650e9 2.02833
\(842\) 0 0
\(843\) 2.76319e8i 0.461241i
\(844\) 0 0
\(845\) 2.31208e8 0.383207
\(846\) 0 0
\(847\) − 1.53784e8i − 0.253081i
\(848\) 0 0
\(849\) −6.45972e8 −1.05558
\(850\) 0 0
\(851\) 3.59130e8i 0.582724i
\(852\) 0 0
\(853\) 8.06034e8 1.29869 0.649346 0.760493i \(-0.275043\pi\)
0.649346 + 0.760493i \(0.275043\pi\)
\(854\) 0 0
\(855\) − 2.16401e8i − 0.346228i
\(856\) 0 0
\(857\) 4.09751e8 0.650995 0.325497 0.945543i \(-0.394468\pi\)
0.325497 + 0.945543i \(0.394468\pi\)
\(858\) 0 0
\(859\) 8.81567e8i 1.39084i 0.718606 + 0.695418i \(0.244781\pi\)
−0.718606 + 0.695418i \(0.755219\pi\)
\(860\) 0 0
\(861\) −5.59714e8 −0.876914
\(862\) 0 0
\(863\) − 5.12988e8i − 0.798132i −0.916922 0.399066i \(-0.869334\pi\)
0.916922 0.399066i \(-0.130666\pi\)
\(864\) 0 0
\(865\) 2.45448e8 0.379237
\(866\) 0 0
\(867\) − 7.56462e8i − 1.16073i
\(868\) 0 0
\(869\) 9.40241e8 1.43278
\(870\) 0 0
\(871\) − 1.15742e8i − 0.175161i
\(872\) 0 0
\(873\) −2.50571e8 −0.376607
\(874\) 0 0
\(875\) 1.04710e9i 1.56301i
\(876\) 0 0
\(877\) −1.12822e8 −0.167262 −0.0836308 0.996497i \(-0.526652\pi\)
−0.0836308 + 0.996497i \(0.526652\pi\)
\(878\) 0 0
\(879\) 7.68816e7i 0.113202i
\(880\) 0 0
\(881\) 1.06050e9 1.55090 0.775448 0.631412i \(-0.217524\pi\)
0.775448 + 0.631412i \(0.217524\pi\)
\(882\) 0 0
\(883\) 8.09446e8i 1.17573i 0.808961 + 0.587863i \(0.200030\pi\)
−0.808961 + 0.587863i \(0.799970\pi\)
\(884\) 0 0
\(885\) −1.57410e8 −0.227092
\(886\) 0 0
\(887\) 3.59419e6i 0.00515027i 0.999997 + 0.00257514i \(0.000819692\pi\)
−0.999997 + 0.00257514i \(0.999180\pi\)
\(888\) 0 0
\(889\) 4.19394e8 0.596922
\(890\) 0 0
\(891\) 7.16199e7i 0.101251i
\(892\) 0 0
\(893\) −6.12680e8 −0.860359
\(894\) 0 0
\(895\) − 7.68476e8i − 1.07192i
\(896\) 0 0
\(897\) 1.84021e8 0.254971
\(898\) 0 0
\(899\) − 2.45497e9i − 3.37884i
\(900\) 0 0
\(901\) −9.47135e8 −1.29490
\(902\) 0 0
\(903\) 2.28993e8i 0.310998i
\(904\) 0 0
\(905\) −2.84683e8 −0.384074
\(906\) 0 0
\(907\) 7.63921e8i 1.02383i 0.859037 + 0.511913i \(0.171063\pi\)
−0.859037 + 0.511913i \(0.828937\pi\)
\(908\) 0 0
\(909\) 2.25405e8 0.300104
\(910\) 0 0
\(911\) 9.16841e7i 0.121266i 0.998160 + 0.0606330i \(0.0193119\pi\)
−0.998160 + 0.0606330i \(0.980688\pi\)
\(912\) 0 0
\(913\) −8.12594e8 −1.06773
\(914\) 0 0
\(915\) 7.07794e7i 0.0923940i
\(916\) 0 0
\(917\) 1.92455e9 2.49587
\(918\) 0 0
\(919\) 1.18709e9i 1.52945i 0.644356 + 0.764726i \(0.277126\pi\)
−0.644356 + 0.764726i \(0.722874\pi\)
\(920\) 0 0
\(921\) −6.79190e8 −0.869385
\(922\) 0 0
\(923\) − 9.65105e8i − 1.22735i
\(924\) 0 0
\(925\) −6.95909e8 −0.879280
\(926\) 0 0
\(927\) 8.68236e7i 0.108993i
\(928\) 0 0
\(929\) 3.88302e8 0.484309 0.242155 0.970238i \(-0.422146\pi\)
0.242155 + 0.970238i \(0.422146\pi\)
\(930\) 0 0
\(931\) − 1.50509e9i − 1.86515i
\(932\) 0 0
\(933\) −8.28352e8 −1.01993
\(934\) 0 0
\(935\) − 8.82845e8i − 1.08006i
\(936\) 0 0
\(937\) −7.69038e8 −0.934822 −0.467411 0.884040i \(-0.654813\pi\)
−0.467411 + 0.884040i \(0.654813\pi\)
\(938\) 0 0
\(939\) 4.96730e8i 0.599961i
\(940\) 0 0
\(941\) 6.80625e8 0.816843 0.408422 0.912793i \(-0.366079\pi\)
0.408422 + 0.912793i \(0.366079\pi\)
\(942\) 0 0
\(943\) 3.01703e8i 0.359787i
\(944\) 0 0
\(945\) 1.65551e8 0.196172
\(946\) 0 0
\(947\) 4.54633e8i 0.535317i 0.963514 + 0.267659i \(0.0862499\pi\)
−0.963514 + 0.267659i \(0.913750\pi\)
\(948\) 0 0
\(949\) 2.50343e8 0.292912
\(950\) 0 0
\(951\) − 5.19349e8i − 0.603834i
\(952\) 0 0
\(953\) 6.97601e8 0.805987 0.402994 0.915203i \(-0.367970\pi\)
0.402994 + 0.915203i \(0.367970\pi\)
\(954\) 0 0
\(955\) − 3.12977e8i − 0.359337i
\(956\) 0 0
\(957\) 8.02453e8 0.915554
\(958\) 0 0
\(959\) 9.54900e8i 1.08268i
\(960\) 0 0
\(961\) −2.45830e9 −2.76991
\(962\) 0 0
\(963\) 2.82357e8i 0.316169i
\(964\) 0 0
\(965\) −7.32769e8 −0.815428
\(966\) 0 0
\(967\) 3.55787e7i 0.0393469i 0.999806 + 0.0196735i \(0.00626266\pi\)
−0.999806 + 0.0196735i \(0.993737\pi\)
\(968\) 0 0
\(969\) −1.38585e9 −1.52316
\(970\) 0 0
\(971\) − 8.23470e8i − 0.899477i −0.893160 0.449738i \(-0.851517\pi\)
0.893160 0.449738i \(-0.148483\pi\)
\(972\) 0 0
\(973\) −1.72644e9 −1.87419
\(974\) 0 0
\(975\) 3.56590e8i 0.384729i
\(976\) 0 0
\(977\) 4.25779e7 0.0456563 0.0228281 0.999739i \(-0.492733\pi\)
0.0228281 + 0.999739i \(0.492733\pi\)
\(978\) 0 0
\(979\) 6.12164e8i 0.652408i
\(980\) 0 0
\(981\) 1.80159e8 0.190831
\(982\) 0 0
\(983\) 1.45686e9i 1.53376i 0.641793 + 0.766878i \(0.278191\pi\)
−0.641793 + 0.766878i \(0.721809\pi\)
\(984\) 0 0
\(985\) 8.90297e8 0.931593
\(986\) 0 0
\(987\) − 4.68713e8i − 0.487478i
\(988\) 0 0
\(989\) 1.23434e8 0.127599
\(990\) 0 0
\(991\) 1.26328e9i 1.29801i 0.760784 + 0.649005i \(0.224814\pi\)
−0.760784 + 0.649005i \(0.775186\pi\)
\(992\) 0 0
\(993\) −3.67315e7 −0.0375138
\(994\) 0 0
\(995\) − 3.71014e8i − 0.376635i
\(996\) 0 0
\(997\) −8.99928e8 −0.908076 −0.454038 0.890982i \(-0.650017\pi\)
−0.454038 + 0.890982i \(0.650017\pi\)
\(998\) 0 0
\(999\) 3.16318e8i 0.317268i
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 768.7.g.g.511.3 8
4.3 odd 2 inner 768.7.g.g.511.7 8
8.3 odd 2 inner 768.7.g.g.511.2 8
8.5 even 2 inner 768.7.g.g.511.6 8
16.3 odd 4 384.7.b.c.319.2 8
16.5 even 4 384.7.b.c.319.3 yes 8
16.11 odd 4 384.7.b.c.319.7 yes 8
16.13 even 4 384.7.b.c.319.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.c.319.2 8 16.3 odd 4
384.7.b.c.319.3 yes 8 16.5 even 4
384.7.b.c.319.6 yes 8 16.13 even 4
384.7.b.c.319.7 yes 8 16.11 odd 4
768.7.g.g.511.2 8 8.3 odd 2 inner
768.7.g.g.511.3 8 1.1 even 1 trivial
768.7.g.g.511.6 8 8.5 even 2 inner
768.7.g.g.511.7 8 4.3 odd 2 inner