Properties

Label 384.7.b.c.319.3
Level $384$
Weight $7$
Character 384.319
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.3
Root \(-1.23205 + 3.79090i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.7.b.c.319.2

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −15.5885 −0.577350
\(4\) 0 0
\(5\) 85.3891i 0.683112i 0.939861 + 0.341556i \(0.110954\pi\)
−0.939861 + 0.341556i \(0.889046\pi\)
\(6\) 0 0
\(7\) 511.824i 1.49220i 0.665834 + 0.746100i \(0.268076\pi\)
−0.665834 + 0.746100i \(0.731924\pi\)
\(8\) 0 0
\(9\) 243.000 0.333333
\(10\) 0 0
\(11\) −1212.89 −0.911261 −0.455631 0.890169i \(-0.650586\pi\)
−0.455631 + 0.890169i \(0.650586\pi\)
\(12\) 0 0
\(13\) − 2744.91i − 1.24939i −0.780869 0.624694i \(-0.785224\pi\)
0.780869 0.624694i \(-0.214776\pi\)
\(14\) 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
\(19\) 10429.2 1.52051 0.760257 0.649622i \(-0.225073\pi\)
0.760257 + 0.649622i \(0.225073\pi\)
\(20\) 0 0
\(21\) − 7978.55i − 0.861522i
\(22\) 0 0
\(23\) − 4300.69i − 0.353471i −0.984258 0.176736i \(-0.943446\pi\)
0.984258 0.176736i \(-0.0565538\pi\)
\(24\) 0 0
\(25\) 8333.71 0.533357
\(26\) 0 0
\(27\) −3788.00 −0.192450
\(28\) 0 0
\(29\) − 42442.0i − 1.74021i −0.492866 0.870105i \(-0.664051\pi\)
0.492866 0.870105i \(-0.335949\pi\)
\(30\) 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
\(35\) −43704.2 −1.01934
\(36\) 0 0
\(37\) 83505.3i 1.64858i 0.566171 + 0.824288i \(0.308424\pi\)
−0.566171 + 0.824288i \(0.691576\pi\)
\(38\) 0 0
\(39\) 42788.9i 0.721335i
\(40\) 0 0
\(41\) −70152.4 −1.01787 −0.508933 0.860806i \(-0.669960\pi\)
−0.508933 + 0.860806i \(0.669960\pi\)
\(42\) 0 0
\(43\) 28701.0 0.360987 0.180494 0.983576i \(-0.442230\pi\)
0.180494 + 0.983576i \(0.442230\pi\)
\(44\) 0 0
\(45\) 20749.5i 0.227704i
\(46\) 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. 1.00174
\(52\) 0 0
\(53\) 111109.i 0.746316i 0.927768 + 0.373158i \(0.121725\pi\)
−0.927768 + 0.373158i \(0.878275\pi\)
\(54\) 0 0
\(55\) − 103567.i − 0.622494i
\(56\) 0 0
\(57\) −162575. −0.877870
\(58\) 0 0
\(59\) 118257. 0.575798 0.287899 0.957661i \(-0.407043\pi\)
0.287899 + 0.957661i \(0.407043\pi\)
\(60\) 0 0
\(61\) 53174.3i 0.234268i 0.993116 + 0.117134i \(0.0373706\pi\)
−0.993116 + 0.117134i \(0.962629\pi\)
\(62\) 0 0
\(63\) 124373.i 0.497400i
\(64\) 0 0
\(65\) 234385. 0.853473
\(66\) 0 0
\(67\) −42166.1 −0.140197 −0.0700986 0.997540i \(-0.522331\pi\)
−0.0700986 + 0.997540i \(0.522331\pi\)
\(68\) 0 0
\(69\) 67041.1i 0.204077i
\(70\) 0 0
\(71\) 351599.i 0.982363i 0.871057 + 0.491182i \(0.163435\pi\)
−0.871057 + 0.491182i \(0.836565\pi\)
\(72\) 0 0
\(73\) −91202.9 −0.234445 −0.117222 0.993106i \(-0.537399\pi\)
−0.117222 + 0.993106i \(0.537399\pi\)
\(74\) 0 0
\(75\) −129910. −0.307934
\(76\) 0 0
\(77\) − 620786.i − 1.35978i
\(78\) 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. 1.17170 0.585852 0.810418i \(-0.300760\pi\)
0.585852 + 0.810418i \(0.300760\pi\)
\(84\) 0 0
\(85\) − 727887.i − 1.18524i
\(86\) 0 0
\(87\) 661605.i 1.00471i
\(88\) 0 0
\(89\) −504715. −0.715940 −0.357970 0.933733i \(-0.616531\pi\)
−0.357970 + 0.933733i \(0.616531\pi\)
\(90\) 0 0
\(91\) 1.40491e6 1.86434
\(92\) 0 0
\(93\) 901683.i 1.12100i
\(94\) 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. −0.303754
\(100\) 0 0
\(101\) − 927592.i − 0.900312i −0.892950 0.450156i \(-0.851368\pi\)
0.892950 0.450156i \(-0.148632\pi\)
\(102\) 0 0
\(103\) 357299.i 0.326979i 0.986545 + 0.163489i \(0.0522750\pi\)
−0.986545 + 0.163489i \(0.947725\pi\)
\(104\) 0 0
\(105\) 681281. 0.588516
\(106\) 0 0
\(107\) 1.16196e6 0.948508 0.474254 0.880388i \(-0.342718\pi\)
0.474254 + 0.880388i \(0.342718\pi\)
\(108\) 0 0
\(109\) 741394.i 0.572492i 0.958156 + 0.286246i \(0.0924075\pi\)
−0.958156 + 0.286246i \(0.907593\pi\)
\(110\) 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. 0.241461
\(116\) 0 0
\(117\) − 667012.i − 0.416463i
\(118\) 0 0
\(119\) − 4.36297e6i − 2.58906i
\(120\) 0 0
\(121\) −300462. −0.169603
\(122\) 0 0
\(123\) 1.09357e6 0.587665
\(124\) 0 0
\(125\) 2.04581e6i 1.04746i
\(126\) 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
\(131\) 3.76018e6 1.67261 0.836306 0.548263i \(-0.184711\pi\)
0.836306 + 0.548263i \(0.184711\pi\)
\(132\) 0 0
\(133\) 5.33793e6i 2.26891i
\(134\) 0 0
\(135\) − 323453.i − 0.131465i
\(136\) 0 0
\(137\) 1.86568e6 0.725563 0.362782 0.931874i \(-0.381827\pi\)
0.362782 + 0.931874i \(0.381827\pi\)
\(138\) 0 0
\(139\) 3.37311e6 1.25599 0.627995 0.778218i \(-0.283876\pi\)
0.627995 + 0.778218i \(0.283876\pi\)
\(140\) 0 0
\(141\) − 915768.i − 0.326684i
\(142\) 0 0
\(143\) 3.32927e6i 1.13852i
\(144\) 0 0
\(145\) 3.62408e6 1.18876
\(146\) 0 0
\(147\) 2.24965e6 0.708212
\(148\) 0 0
\(149\) − 3.35713e6i − 1.01487i −0.861691 0.507434i \(-0.830594\pi\)
0.861691 0.507434i \(-0.169406\pi\)
\(150\) 0 0
\(151\) 3.78813e6i 1.10026i 0.835081 + 0.550128i \(0.185421\pi\)
−0.835081 + 0.550128i \(0.814579\pi\)
\(152\) 0 0
\(153\) −2.07142e6 −0.578354
\(154\) 0 0
\(155\) 4.93916e6 1.32635
\(156\) 0 0
\(157\) − 5.81264e6i − 1.50202i −0.660293 0.751008i \(-0.729568\pi\)
0.660293 0.751008i \(-0.270432\pi\)
\(158\) 0 0
\(159\) − 1.73202e6i − 0.430886i
\(160\) 0 0
\(161\) 2.20120e6 0.527450
\(162\) 0 0
\(163\) 5.12374e6 1.18311 0.591554 0.806266i \(-0.298515\pi\)
0.591554 + 0.806266i \(0.298515\pi\)
\(164\) 0 0
\(165\) 1.61446e6i 0.359397i
\(166\) 0 0
\(167\) − 1.27729e6i − 0.274247i −0.990554 0.137123i \(-0.956214\pi\)
0.990554 0.137123i \(-0.0437857\pi\)
\(168\) 0 0
\(169\) −2.70770e6 −0.560972
\(170\) 0 0
\(171\) 2.53430e6 0.506838
\(172\) 0 0
\(173\) − 2.87446e6i − 0.555160i −0.960703 0.277580i \(-0.910468\pi\)
0.960703 0.277580i \(-0.0895324\pi\)
\(174\) 0 0
\(175\) 4.26540e6i 0.795876i
\(176\) 0 0
\(177\) −1.84344e6 −0.332437
\(178\) 0 0
\(179\) −8.99970e6 −1.56917 −0.784583 0.620024i \(-0.787123\pi\)
−0.784583 + 0.620024i \(0.787123\pi\)
\(180\) 0 0
\(181\) − 3.33395e6i − 0.562242i −0.959672 0.281121i \(-0.909294\pi\)
0.959672 0.281121i \(-0.0907062\pi\)
\(182\) 0 0
\(183\) − 828905.i − 0.135254i
\(184\) 0 0
\(185\) −7.13044e6 −1.12616
\(186\) 0 0
\(187\) 1.03391e7 1.58109
\(188\) 0 0
\(189\) − 1.93879e6i − 0.287174i
\(190\) 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
\(195\) −3.65370e6 −0.492753
\(196\) 0 0
\(197\) 1.04264e7i 1.36375i 0.731470 + 0.681874i \(0.238835\pi\)
−0.731470 + 0.681874i \(0.761165\pi\)
\(198\) 0 0
\(199\) 4.34498e6i 0.551352i 0.961251 + 0.275676i \(0.0889017\pi\)
−0.961251 + 0.275676i \(0.911098\pi\)
\(200\) 0 0
\(201\) 657305. 0.0809429
\(202\) 0 0
\(203\) 2.17228e7 2.59674
\(204\) 0 0
\(205\) − 5.99024e6i − 0.695317i
\(206\) 0 0
\(207\) − 1.04507e6i − 0.117824i
\(208\) 0 0
\(209\) −1.26495e7 −1.38559
\(210\) 0 0
\(211\) 5.90463e6 0.628558 0.314279 0.949331i \(-0.398237\pi\)
0.314279 + 0.949331i \(0.398237\pi\)
\(212\) 0 0
\(213\) − 5.48088e6i − 0.567168i
\(214\) 0 0
\(215\) 2.45075e6i 0.246595i
\(216\) 0 0
\(217\) 2.96054e7 2.89729
\(218\) 0 0
\(219\) 1.42171e6 0.135357
\(220\) 0 0
\(221\) 2.33986e7i 2.16777i
\(222\) 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
\(227\) 3.99262e6 0.341334 0.170667 0.985329i \(-0.445408\pi\)
0.170667 + 0.985329i \(0.445408\pi\)
\(228\) 0 0
\(229\) 3.71868e6i 0.309658i 0.987941 + 0.154829i \(0.0494826\pi\)
−0.987941 + 0.154829i \(0.950517\pi\)
\(230\) 0 0
\(231\) 9.67710e6i 0.785071i
\(232\) 0 0
\(233\) −5.25478e6 −0.415419 −0.207710 0.978191i \(-0.566601\pi\)
−0.207710 + 0.978191i \(0.566601\pi\)
\(234\) 0 0
\(235\) −5.01631e6 −0.386528
\(236\) 0 0
\(237\) 1.20843e7i 0.907771i
\(238\) 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. −0.0641500
\(244\) 0 0
\(245\) − 1.23229e7i − 0.837946i
\(246\) 0 0
\(247\) − 2.86272e7i − 1.89971i
\(248\) 0 0
\(249\) −1.04437e7 −0.676484
\(250\) 0 0
\(251\) 920323. 0.0581995 0.0290998 0.999577i \(-0.490736\pi\)
0.0290998 + 0.999577i \(0.490736\pi\)
\(252\) 0 0
\(253\) 5.21625e6i 0.322105i
\(254\) 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
\(259\) −4.27400e7 −2.46000
\(260\) 0 0
\(261\) − 1.03134e7i − 0.580070i
\(262\) 0 0
\(263\) 1.36973e7i 0.752952i 0.926426 + 0.376476i \(0.122864\pi\)
−0.926426 + 0.376476i \(0.877136\pi\)
\(264\) 0 0
\(265\) −9.48752e6 −0.509818
\(266\) 0 0
\(267\) 7.86774e6 0.413348
\(268\) 0 0
\(269\) − 1.95267e7i − 1.00317i −0.865110 0.501583i \(-0.832751\pi\)
0.865110 0.501583i \(-0.167249\pi\)
\(270\) 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
\(275\) −1.01079e7 −0.486028
\(276\) 0 0
\(277\) 1.32017e6i 0.0621142i 0.999518 + 0.0310571i \(0.00988737\pi\)
−0.999518 + 0.0310571i \(0.990113\pi\)
\(278\) 0 0
\(279\) − 1.40558e7i − 0.647208i
\(280\) 0 0
\(281\) 1.77259e7 0.798893 0.399446 0.916757i \(-0.369202\pi\)
0.399446 + 0.916757i \(0.369202\pi\)
\(282\) 0 0
\(283\) 4.14391e7 1.82832 0.914158 0.405358i \(-0.132853\pi\)
0.914158 + 0.405358i \(0.132853\pi\)
\(284\) 0 0
\(285\) − 1.38822e7i − 0.599684i
\(286\) 0 0
\(287\) − 3.59057e7i − 1.51886i
\(288\) 0 0
\(289\) 4.85271e7 2.01044
\(290\) 0 0
\(291\) −1.60742e7 −0.652303
\(292\) 0 0
\(293\) − 4.93196e6i − 0.196072i −0.995183 0.0980362i \(-0.968744\pi\)
0.995183 0.0980362i \(-0.0312561\pi\)
\(294\) 0 0
\(295\) 1.00978e7i 0.393335i
\(296\) 0 0
\(297\) 4.59442e6 0.175372
\(298\) 0 0
\(299\) −1.18050e7 −0.441623
\(300\) 0 0
\(301\) 1.46899e7i 0.538665i
\(302\) 0 0
\(303\) 1.44597e7i 0.519795i
\(304\) 0 0
\(305\) −4.54050e6 −0.160031
\(306\) 0 0
\(307\) −4.35700e7 −1.50582 −0.752910 0.658124i \(-0.771350\pi\)
−0.752910 + 0.658124i \(0.771350\pi\)
\(308\) 0 0
\(309\) − 5.56974e6i − 0.188781i
\(310\) 0 0
\(311\) 5.31388e7i 1.76657i 0.468837 + 0.883285i \(0.344673\pi\)
−0.468837 + 0.883285i \(0.655327\pi\)
\(312\) 0 0
\(313\) 3.18652e7 1.03916 0.519582 0.854421i \(-0.326088\pi\)
0.519582 + 0.854421i \(0.326088\pi\)
\(314\) 0 0
\(315\) −1.06201e7 −0.339780
\(316\) 0 0
\(317\) − 3.33162e7i − 1.04587i −0.852372 0.522936i \(-0.824837\pi\)
0.852372 0.522936i \(-0.175163\pi\)
\(318\) 0 0
\(319\) 5.14774e7i 1.58579i
\(320\) 0 0
\(321\) −1.81132e7 −0.547621
\(322\) 0 0
\(323\) −8.89023e7 −2.63819
\(324\) 0 0
\(325\) − 2.28753e7i − 0.666371i
\(326\) 0 0
\(327\) − 1.15572e7i − 0.330528i
\(328\) 0 0
\(329\) −3.00679e7 −0.844337
\(330\) 0 0
\(331\) 2.35633e6 0.0649758 0.0324879 0.999472i \(-0.489657\pi\)
0.0324879 + 0.999472i \(0.489657\pi\)
\(332\) 0 0
\(333\) 2.02918e7i 0.549525i
\(334\) 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
\(339\) 2.97330e6 0.0763202
\(340\) 0 0
\(341\) 7.01571e7i 1.76933i
\(342\) 0 0
\(343\) − 1.36484e7i − 0.338221i
\(344\) 0 0
\(345\) −5.72457e6 −0.139407
\(346\) 0 0
\(347\) −1.76051e7 −0.421357 −0.210679 0.977555i \(-0.567567\pi\)
−0.210679 + 0.977555i \(0.567567\pi\)
\(348\) 0 0
\(349\) − 3.02782e6i − 0.0712285i −0.999366 0.0356143i \(-0.988661\pi\)
0.999366 0.0356143i \(-0.0113388\pi\)
\(350\) 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
\(355\) −3.00227e7 −0.671064
\(356\) 0 0
\(357\) 6.80120e7i 1.49479i
\(358\) 0 0
\(359\) − 5.14739e7i − 1.11251i −0.831012 0.556254i \(-0.812238\pi\)
0.831012 0.556254i \(-0.187762\pi\)
\(360\) 0 0
\(361\) 6.17226e7 1.31197
\(362\) 0 0
\(363\) 4.68374e6 0.0979203
\(364\) 0 0
\(365\) − 7.78773e6i − 0.160152i
\(366\) 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
\(371\) −5.68684e7 −1.11365
\(372\) 0 0
\(373\) 6.19317e7i 1.19340i 0.802464 + 0.596701i \(0.203522\pi\)
−0.802464 + 0.596701i \(0.796478\pi\)
\(374\) 0 0
\(375\) − 3.18910e7i − 0.604749i
\(376\) 0 0
\(377\) −1.16499e8 −2.17420
\(378\) 0 0
\(379\) −3.43446e7 −0.630870 −0.315435 0.948947i \(-0.602151\pi\)
−0.315435 + 0.948947i \(0.602151\pi\)
\(380\) 0 0
\(381\) − 1.27733e7i − 0.230956i
\(382\) 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
\(387\) 6.97435e6 0.120329
\(388\) 0 0
\(389\) − 1.13679e8i − 1.93122i −0.259993 0.965610i \(-0.583720\pi\)
0.259993 0.965610i \(-0.416280\pi\)
\(390\) 0 0
\(391\) 3.66606e7i 0.613294i
\(392\) 0 0
\(393\) −5.86155e7 −0.965683
\(394\) 0 0
\(395\) 6.61943e7 1.07406
\(396\) 0 0
\(397\) − 3.96900e6i − 0.0634322i −0.999497 0.0317161i \(-0.989903\pi\)
0.999497 0.0317161i \(-0.0100972\pi\)
\(398\) 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
\(403\) −1.58774e8 −2.42584
\(404\) 0 0
\(405\) 5.04214e6i 0.0759014i
\(406\) 0 0
\(407\) − 1.01283e8i − 1.50228i
\(408\) 0 0
\(409\) 4.23651e7 0.619211 0.309606 0.950865i \(-0.399803\pi\)
0.309606 + 0.950865i \(0.399803\pi\)
\(410\) 0 0
\(411\) −2.90831e7 −0.418904
\(412\) 0 0
\(413\) 6.05267e7i 0.859206i
\(414\) 0 0
\(415\) 5.72077e7i 0.800406i
\(416\) 0 0
\(417\) −5.25816e7 −0.725146
\(418\) 0 0
\(419\) −8.16747e7 −1.11031 −0.555157 0.831746i \(-0.687342\pi\)
−0.555157 + 0.831746i \(0.687342\pi\)
\(420\) 0 0
\(421\) − 7.99635e7i − 1.07163i −0.844335 0.535816i \(-0.820004\pi\)
0.844335 0.535816i \(-0.179996\pi\)
\(422\) 0 0
\(423\) 1.42754e7i 0.188611i
\(424\) 0 0
\(425\) −7.10395e7 −0.925408
\(426\) 0 0
\(427\) −2.72159e7 −0.349574
\(428\) 0 0
\(429\) − 5.18981e7i − 0.657325i
\(430\) 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
\(435\) −5.64938e7 −0.686330
\(436\) 0 0
\(437\) − 4.48528e7i − 0.537458i
\(438\) 0 0
\(439\) 3.84937e7i 0.454984i 0.973780 + 0.227492i \(0.0730525\pi\)
−0.973780 + 0.227492i \(0.926947\pi\)
\(440\) 0 0
\(441\) −3.50686e7 −0.408886
\(442\) 0 0
\(443\) 4.20100e7 0.483216 0.241608 0.970374i \(-0.422325\pi\)
0.241608 + 0.970374i \(0.422325\pi\)
\(444\) 0 0
\(445\) − 4.30972e7i − 0.489067i
\(446\) 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
\(451\) 8.50870e7 0.927542
\(452\) 0 0
\(453\) − 5.90510e7i − 0.635233i
\(454\) 0 0
\(455\) 1.19964e8i 1.27355i
\(456\) 0 0
\(457\) 1.64317e8 1.72160 0.860802 0.508941i \(-0.169963\pi\)
0.860802 + 0.508941i \(0.169963\pi\)
\(458\) 0 0
\(459\) 3.22902e7 0.333913
\(460\) 0 0
\(461\) 1.01392e7i 0.103491i 0.998660 + 0.0517455i \(0.0164785\pi\)
−0.998660 + 0.0517455i \(0.983522\pi\)
\(462\) 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
\(467\) 1.30526e8 1.28159 0.640793 0.767713i \(-0.278606\pi\)
0.640793 + 0.767713i \(0.278606\pi\)
\(468\) 0 0
\(469\) − 2.15817e7i − 0.209202i
\(470\) 0 0
\(471\) 9.06101e7i 0.867189i
\(472\) 0 0
\(473\) −3.48111e7 −0.328954
\(474\) 0 0
\(475\) 8.69140e7 0.810978
\(476\) 0 0
\(477\) 2.69996e7i 0.248772i
\(478\) 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
\(483\) −3.43132e7 −0.304523
\(484\) 0 0
\(485\) 8.80496e7i 0.771795i
\(486\) 0 0
\(487\) − 6.38316e7i − 0.552648i −0.961065 0.276324i \(-0.910884\pi\)
0.961065 0.276324i \(-0.0891163\pi\)
\(488\) 0 0
\(489\) −7.98712e7 −0.683067
\(490\) 0 0
\(491\) −1.36774e8 −1.15547 −0.577735 0.816224i \(-0.696063\pi\)
−0.577735 + 0.816224i \(0.696063\pi\)
\(492\) 0 0
\(493\) 3.61791e8i 3.01937i
\(494\) 0 0
\(495\) − 2.51669e7i − 0.207498i
\(496\) 0 0
\(497\) −1.79957e8 −1.46588
\(498\) 0 0
\(499\) 1.94124e8 1.56235 0.781175 0.624312i \(-0.214621\pi\)
0.781175 + 0.624312i \(0.214621\pi\)
\(500\) 0 0
\(501\) 1.99110e7i 0.158336i
\(502\) 0 0
\(503\) 9.34518e7i 0.734317i 0.930158 + 0.367159i \(0.119669\pi\)
−0.930158 + 0.367159i \(0.880331\pi\)
\(504\) 0 0
\(505\) 7.92062e7 0.615014
\(506\) 0 0
\(507\) 4.22089e7 0.323877
\(508\) 0 0
\(509\) − 6.32104e7i − 0.479331i −0.970856 0.239665i \(-0.922962\pi\)
0.970856 0.239665i \(-0.0770377\pi\)
\(510\) 0 0
\(511\) − 4.66799e7i − 0.349838i
\(512\) 0 0
\(513\) −3.95058e7 −0.292623
\(514\) 0 0
\(515\) −3.05094e7 −0.223363
\(516\) 0 0
\(517\) − 7.12530e7i − 0.515622i
\(518\) 0 0
\(519\) 4.48084e7i 0.320522i
\(520\) 0 0
\(521\) −2.37718e8 −1.68092 −0.840462 0.541871i \(-0.817716\pi\)
−0.840462 + 0.541871i \(0.817716\pi\)
\(522\) 0 0
\(523\) −5.13720e7 −0.359105 −0.179553 0.983748i \(-0.557465\pi\)
−0.179553 + 0.983748i \(0.557465\pi\)
\(524\) 0 0
\(525\) − 6.64909e7i − 0.459499i
\(526\) 0 0
\(527\) 4.93074e8i 3.36884i
\(528\) 0 0
\(529\) 1.29540e8 0.875058
\(530\) 0 0
\(531\) 2.87364e7 0.191933
\(532\) 0 0
\(533\) 1.92562e8i 1.27171i
\(534\) 0 0
\(535\) 9.92189e7i 0.647938i
\(536\) 0 0
\(537\) 1.40291e8 0.905958
\(538\) 0 0
\(539\) 1.75038e8 1.11781
\(540\) 0 0
\(541\) − 1.82128e8i − 1.15023i −0.818072 0.575116i \(-0.804957\pi\)
0.818072 0.575116i \(-0.195043\pi\)
\(542\) 0 0
\(543\) 5.19711e7i 0.324610i
\(544\) 0 0
\(545\) −6.33069e7 −0.391076
\(546\) 0 0
\(547\) −6.68154e7 −0.408239 −0.204120 0.978946i \(-0.565433\pi\)
−0.204120 + 0.978946i \(0.565433\pi\)
\(548\) 0 0
\(549\) 1.29213e7i 0.0780892i
\(550\) 0 0
\(551\) − 4.42636e8i − 2.64602i
\(552\) 0 0
\(553\) 3.96770e8 2.34619
\(554\) 0 0
\(555\) 1.11153e8 0.650190
\(556\) 0 0
\(557\) 606923.i 0.00351211i 0.999998 + 0.00175606i \(0.000558970\pi\)
−0.999998 + 0.00175606i \(0.999441\pi\)
\(558\) 0 0
\(559\) − 7.87816e7i − 0.451013i
\(560\) 0 0
\(561\) −1.61171e8 −0.912845
\(562\) 0 0
\(563\) 2.47309e8 1.38584 0.692922 0.721012i \(-0.256323\pi\)
0.692922 + 0.721012i \(0.256323\pi\)
\(564\) 0 0
\(565\) − 1.62869e7i − 0.0903009i
\(566\) 0 0
\(567\) 3.02227e7i 0.165800i
\(568\) 0 0
\(569\) 1.83163e8 0.994262 0.497131 0.867675i \(-0.334387\pi\)
0.497131 + 0.867675i \(0.334387\pi\)
\(570\) 0 0
\(571\) 1.30136e8 0.699021 0.349510 0.936933i \(-0.386348\pi\)
0.349510 + 0.936933i \(0.386348\pi\)
\(572\) 0 0
\(573\) 5.71365e7i 0.303703i
\(574\) 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
\(579\) 1.33773e8 0.689180
\(580\) 0 0
\(581\) 3.42905e8i 1.74842i
\(582\) 0 0
\(583\) − 1.34763e8i − 0.680089i
\(584\) 0 0
\(585\) 5.69556e7 0.284491
\(586\) 0 0
\(587\) 4.56349e7 0.225623 0.112811 0.993616i \(-0.464014\pi\)
0.112811 + 0.993616i \(0.464014\pi\)
\(588\) 0 0
\(589\) − 6.03257e8i − 2.95227i
\(590\) 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
\(595\) 3.72550e8 1.76862
\(596\) 0 0
\(597\) − 6.77316e7i − 0.318323i
\(598\) 0 0
\(599\) 2.34911e8i 1.09301i 0.837457 + 0.546504i \(0.184042\pi\)
−0.837457 + 0.546504i \(0.815958\pi\)
\(600\) 0 0
\(601\) 6.40513e7 0.295056 0.147528 0.989058i \(-0.452868\pi\)
0.147528 + 0.989058i \(0.452868\pi\)
\(602\) 0 0
\(603\) −1.02464e7 −0.0467324
\(604\) 0 0
\(605\) − 2.56562e7i − 0.115858i
\(606\) 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
\(611\) 1.61254e8 0.706946
\(612\) 0 0
\(613\) − 2.62634e7i − 0.114017i −0.998374 0.0570085i \(-0.981844\pi\)
0.998374 0.0570085i \(-0.0181562\pi\)
\(614\) 0 0
\(615\) 9.33786e7i 0.401441i
\(616\) 0 0
\(617\) −2.98936e8 −1.27269 −0.636345 0.771404i \(-0.719555\pi\)
−0.636345 + 0.771404i \(0.719555\pi\)
\(618\) 0 0
\(619\) 1.32500e8 0.558657 0.279328 0.960196i \(-0.409888\pi\)
0.279328 + 0.960196i \(0.409888\pi\)
\(620\) 0 0
\(621\) 1.62910e7i 0.0680256i
\(622\) 0 0
\(623\) − 2.58326e8i − 1.06833i
\(624\) 0 0
\(625\) −4.44757e7 −0.182172
\(626\) 0 0
\(627\) 1.97186e8 0.799969
\(628\) 0 0
\(629\) − 7.11829e8i − 2.86038i
\(630\) 0 0
\(631\) − 1.19185e8i − 0.474390i −0.971462 0.237195i \(-0.923772\pi\)
0.971462 0.237195i \(-0.0762280\pi\)
\(632\) 0 0
\(633\) −9.20440e7 −0.362898
\(634\) 0 0
\(635\) −6.99687e7 −0.273264
\(636\) 0 0
\(637\) 3.96132e8i 1.53257i
\(638\) 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
\(643\) 1.05651e8 0.397410 0.198705 0.980059i \(-0.436326\pi\)
0.198705 + 0.980059i \(0.436326\pi\)
\(644\) 0 0
\(645\) − 3.82035e7i − 0.142372i
\(646\) 0 0
\(647\) − 2.37697e8i − 0.877628i −0.898578 0.438814i \(-0.855399\pi\)
0.898578 0.438814i \(-0.144601\pi\)
\(648\) 0 0
\(649\) −1.43432e8 −0.524703
\(650\) 0 0
\(651\) −4.61503e8 −1.67275
\(652\) 0 0
\(653\) − 1.71315e7i − 0.0615257i −0.999527 0.0307628i \(-0.990206\pi\)
0.999527 0.0307628i \(-0.00979366\pi\)
\(654\) 0 0
\(655\) 3.21079e8i 1.14258i
\(656\) 0 0
\(657\) −2.21623e7 −0.0781482
\(658\) 0 0
\(659\) 3.62898e8 1.26803 0.634013 0.773323i \(-0.281407\pi\)
0.634013 + 0.773323i \(0.281407\pi\)
\(660\) 0 0
\(661\) 4.90659e8i 1.69893i 0.527645 + 0.849465i \(0.323075\pi\)
−0.527645 + 0.849465i \(0.676925\pi\)
\(662\) 0 0
\(663\) − 3.64747e8i − 1.25156i
\(664\) 0 0
\(665\) −4.55800e8 −1.54992
\(666\) 0 0
\(667\) −1.82530e8 −0.615114
\(668\) 0 0
\(669\) 2.30950e8i 0.771330i
\(670\) 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
\(675\) −3.15681e7 −0.102645
\(676\) 0 0
\(677\) 1.98782e8i 0.640635i 0.947310 + 0.320318i \(0.103790\pi\)
−0.947310 + 0.320318i \(0.896210\pi\)
\(678\) 0 0
\(679\) 5.27772e8i 1.68592i
\(680\) 0 0
\(681\) −6.22387e7 −0.197070
\(682\) 0 0
\(683\) −795875. −0.00249794 −0.00124897 0.999999i \(-0.500398\pi\)
−0.00124897 + 0.999999i \(0.500398\pi\)
\(684\) 0 0
\(685\) 1.59309e8i 0.495641i
\(686\) 0 0
\(687\) − 5.79684e7i − 0.178781i
\(688\) 0 0
\(689\) 3.04985e8 0.932439
\(690\) 0 0
\(691\) 6.26115e8 1.89767 0.948833 0.315778i \(-0.102266\pi\)
0.948833 + 0.315778i \(0.102266\pi\)
\(692\) 0 0
\(693\) − 1.50851e8i − 0.453261i
\(694\) 0 0
\(695\) 2.88027e8i 0.857982i
\(696\) 0 0
\(697\) 5.98004e8 1.76606
\(698\) 0 0
\(699\) 8.19139e7 0.239843
\(700\) 0 0
\(701\) 7.92815e7i 0.230154i 0.993357 + 0.115077i \(0.0367114\pi\)
−0.993357 + 0.115077i \(0.963289\pi\)
\(702\) 0 0
\(703\) 8.70894e8i 2.50668i
\(704\) 0 0
\(705\) 7.81966e7 0.223162
\(706\) 0 0
\(707\) 4.74764e8 1.34344
\(708\) 0 0
\(709\) 6.27826e7i 0.176157i 0.996114 + 0.0880786i \(0.0280727\pi\)
−0.996114 + 0.0880786i \(0.971927\pi\)
\(710\) 0 0
\(711\) − 1.88376e8i − 0.524102i
\(712\) 0 0
\(713\) −2.48764e8 −0.686309
\(714\) 0 0
\(715\) −2.84283e8 −0.777737
\(716\) 0 0
\(717\) − 1.24126e8i − 0.336747i
\(718\) 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
\(723\) −2.25190e7 −0.0595845
\(724\) 0 0
\(725\) − 3.53699e8i − 0.928154i
\(726\) 0 0
\(727\) − 1.75113e8i − 0.455737i −0.973692 0.227869i \(-0.926824\pi\)
0.973692 0.227869i \(-0.0731757\pi\)
\(728\) 0 0
\(729\) 1.43489e7 0.0370370
\(730\) 0 0
\(731\) −2.44658e8 −0.626335
\(732\) 0 0
\(733\) 2.79408e8i 0.709458i 0.934969 + 0.354729i \(0.115427\pi\)
−0.934969 + 0.354729i \(0.884573\pi\)
\(734\) 0 0
\(735\) 1.92096e8i 0.483789i
\(736\) 0 0
\(737\) 5.11428e7 0.127756
\(738\) 0 0
\(739\) −5.00369e8 −1.23981 −0.619907 0.784675i \(-0.712830\pi\)
−0.619907 + 0.784675i \(0.712830\pi\)
\(740\) 0 0
\(741\) 4.46254e8i 1.09680i
\(742\) 0 0
\(743\) 5.12068e8i 1.24842i 0.781256 + 0.624211i \(0.214579\pi\)
−0.781256 + 0.624211i \(0.785421\pi\)
\(744\) 0 0
\(745\) 2.86662e8 0.693269
\(746\) 0 0
\(747\) 1.62802e8 0.390568
\(748\) 0 0
\(749\) 5.94721e8i 1.41536i
\(750\) 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
\(755\) −3.23464e8 −0.751598
\(756\) 0 0
\(757\) − 3.55132e8i − 0.818657i −0.912387 0.409329i \(-0.865763\pi\)
0.912387 0.409329i \(-0.134237\pi\)
\(758\) 0 0
\(759\) − 8.13133e7i − 0.185967i
\(760\) 0 0
\(761\) −7.07962e8 −1.60641 −0.803204 0.595704i \(-0.796873\pi\)
−0.803204 + 0.595704i \(0.796873\pi\)
\(762\) 0 0
\(763\) −3.79463e8 −0.854272
\(764\) 0 0
\(765\) − 1.76876e8i − 0.395081i
\(766\) 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
\(771\) −8.93856e7 −0.195031
\(772\) 0 0
\(773\) − 5.71149e8i − 1.23655i −0.785963 0.618274i \(-0.787832\pi\)
0.785963 0.618274i \(-0.212168\pi\)
\(774\) 0 0
\(775\) − 4.82046e8i − 1.03558i
\(776\) 0 0
\(777\) 6.66251e8 1.42028
\(778\) 0 0
\(779\) −7.31634e8 −1.54768
\(780\) 0 0
\(781\) − 4.26450e8i − 0.895189i
\(782\) 0 0
\(783\) 1.60770e8i 0.334904i
\(784\) 0 0
\(785\) 4.96336e8 1.02605
\(786\) 0 0
\(787\) −1.56385e8 −0.320828 −0.160414 0.987050i \(-0.551283\pi\)
−0.160414 + 0.987050i \(0.551283\pi\)
\(788\) 0 0
\(789\) − 2.13519e8i − 0.434717i
\(790\) 0 0
\(791\) − 9.76239e7i − 0.197254i
\(792\) 0 0
\(793\) 1.45958e8 0.292691
\(794\) 0 0
\(795\) 1.47896e8 0.294343
\(796\) 0 0
\(797\) − 6.02539e8i − 1.19017i −0.803661 0.595087i \(-0.797118\pi\)
0.803661 0.595087i \(-0.202882\pi\)
\(798\) 0 0
\(799\) − 5.00777e8i − 0.981756i
\(800\) 0 0
\(801\) −1.22646e8 −0.238647
\(802\) 0 0
\(803\) 1.10619e8 0.213640
\(804\) 0 0
\(805\) 1.87958e8i 0.360307i
\(806\) 0 0
\(807\) 3.04392e8i 0.579178i
\(808\) 0 0
\(809\) 2.30938e8 0.436164 0.218082 0.975930i \(-0.430020\pi\)
0.218082 + 0.975930i \(0.430020\pi\)
\(810\) 0 0
\(811\) −1.15316e8 −0.216185 −0.108093 0.994141i \(-0.534474\pi\)
−0.108093 + 0.994141i \(0.534474\pi\)
\(812\) 0 0
\(813\) 1.37099e8i 0.255131i
\(814\) 0 0
\(815\) 4.37511e8i 0.808195i
\(816\) 0 0
\(817\) 2.99329e8 0.548887
\(818\) 0 0
\(819\) 3.41393e8 0.621446
\(820\) 0 0
\(821\) − 8.70428e7i − 0.157291i −0.996903 0.0786454i \(-0.974941\pi\)
0.996903 0.0786454i \(-0.0250595\pi\)
\(822\) 0 0
\(823\) 1.58502e8i 0.284337i 0.989842 + 0.142169i \(0.0454076\pi\)
−0.989842 + 0.142169i \(0.954592\pi\)
\(824\) 0 0
\(825\) 1.57566e8 0.280608
\(826\) 0 0
\(827\) −6.20614e8 −1.09725 −0.548624 0.836069i \(-0.684848\pi\)
−0.548624 + 0.836069i \(0.684848\pi\)
\(828\) 0 0
\(829\) − 1.05341e9i − 1.84898i −0.381206 0.924490i \(-0.624491\pi\)
0.381206 0.924490i \(-0.375509\pi\)
\(830\) 0 0
\(831\) − 2.05794e7i − 0.0358617i
\(832\) 0 0
\(833\) 1.23019e9 2.12833
\(834\) 0 0
\(835\) 1.09067e8 0.187341
\(836\) 0 0
\(837\) 2.19109e8i 0.373666i
\(838\) 0 0
\(839\) 1.80870e8i 0.306254i 0.988207 + 0.153127i \(0.0489343\pi\)
−0.988207 + 0.153127i \(0.951066\pi\)
\(840\) 0 0
\(841\) −1.20650e9 −2.02833
\(842\) 0 0
\(843\) −2.76319e8 −0.461241
\(844\) 0 0
\(845\) − 2.31208e8i − 0.383207i
\(846\) 0 0
\(847\) − 1.53784e8i − 0.253081i
\(848\) 0 0
\(849\) −6.45972e8 −1.05558
\(850\) 0 0
\(851\) 3.59130e8 0.582724
\(852\) 0 0
\(853\) 8.06034e8i 1.29869i 0.760493 + 0.649346i \(0.224957\pi\)
−0.760493 + 0.649346i \(0.775043\pi\)
\(854\) 0 0
\(855\) 2.16401e8i 0.346228i
\(856\) 0 0
\(857\) −4.09751e8 −0.650995 −0.325497 0.945543i \(-0.605532\pi\)
−0.325497 + 0.945543i \(0.605532\pi\)
\(858\) 0 0
\(859\) −8.81567e8 −1.39084 −0.695418 0.718606i \(-0.744781\pi\)
−0.695418 + 0.718606i \(0.744781\pi\)
\(860\) 0 0
\(861\) 5.59714e8i 0.876914i
\(862\) 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
\(867\) −7.56462e8 −1.16073
\(868\) 0 0
\(869\) 9.40241e8i 1.43278i
\(870\) 0 0
\(871\) 1.15742e8i 0.175161i
\(872\) 0 0
\(873\) 2.50571e8 0.376607
\(874\) 0 0
\(875\) −1.04710e9 −1.56301
\(876\) 0 0
\(877\) 1.12822e8i 0.167262i 0.996497 + 0.0836308i \(0.0266516\pi\)
−0.996497 + 0.0836308i \(0.973348\pi\)
\(878\) 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
\(883\) 8.09446e8 1.17573 0.587863 0.808961i \(-0.299970\pi\)
0.587863 + 0.808961i \(0.299970\pi\)
\(884\) 0 0
\(885\) − 1.57410e8i − 0.227092i
\(886\) 0 0
\(887\) − 3.59419e6i − 0.00515027i −0.999997 0.00257514i \(-0.999180\pi\)
0.999997 0.00257514i \(-0.000819692\pi\)
\(888\) 0 0
\(889\) −4.19394e8 −0.596922
\(890\) 0 0
\(891\) −7.16199e7 −0.101251
\(892\) 0 0
\(893\) 6.12680e8i 0.860359i
\(894\) 0 0
\(895\) − 7.68476e8i − 1.07192i
\(896\) 0 0
\(897\) 1.84021e8 0.254971
\(898\) 0 0
\(899\) −2.45497e9 −3.37884
\(900\) 0 0
\(901\) − 9.47135e8i − 1.29490i
\(902\) 0 0
\(903\) − 2.28993e8i − 0.310998i
\(904\) 0 0
\(905\) 2.84683e8 0.384074
\(906\) 0 0
\(907\) −7.63921e8 −1.02383 −0.511913 0.859037i \(-0.671063\pi\)
−0.511913 + 0.859037i \(0.671063\pi\)
\(908\) 0 0
\(909\) − 2.25405e8i − 0.300104i
\(910\) 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
\(915\) 7.07794e7 0.0923940
\(916\) 0 0
\(917\) 1.92455e9i 2.49587i
\(918\) 0 0
\(919\) − 1.18709e9i − 1.52945i −0.644356 0.764726i \(-0.722874\pi\)
0.644356 0.764726i \(-0.277126\pi\)
\(920\) 0 0
\(921\) 6.79190e8 0.869385
\(922\) 0 0
\(923\) 9.65105e8 1.22735
\(924\) 0 0
\(925\) 6.95909e8i 0.879280i
\(926\) 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
\(931\) −1.50509e9 −1.86515
\(932\) 0 0
\(933\) − 8.28352e8i − 1.01993i
\(934\) 0 0
\(935\) 8.82845e8i 1.08006i
\(936\) 0 0
\(937\) 7.69038e8 0.934822 0.467411 0.884040i \(-0.345187\pi\)
0.467411 + 0.884040i \(0.345187\pi\)
\(938\) 0 0
\(939\) −4.96730e8 −0.599961
\(940\) 0 0
\(941\) − 6.80625e8i − 0.816843i −0.912793 0.408422i \(-0.866079\pi\)
0.912793 0.408422i \(-0.133921\pi\)
\(942\) 0 0
\(943\) 3.01703e8i 0.359787i
\(944\) 0 0
\(945\) 1.65551e8 0.196172
\(946\) 0 0
\(947\) 4.54633e8 0.535317 0.267659 0.963514i \(-0.413750\pi\)
0.267659 + 0.963514i \(0.413750\pi\)
\(948\) 0 0
\(949\) 2.50343e8i 0.292912i
\(950\) 0 0
\(951\) 5.19349e8i 0.603834i
\(952\) 0 0
\(953\) −6.97601e8 −0.805987 −0.402994 0.915203i \(-0.632030\pi\)
−0.402994 + 0.915203i \(0.632030\pi\)
\(954\) 0 0
\(955\) 3.12977e8 0.359337
\(956\) 0 0
\(957\) − 8.02453e8i − 0.915554i
\(958\) 0 0
\(959\) 9.54900e8i 1.08268i
\(960\) 0 0
\(961\) −2.45830e9 −2.76991
\(962\) 0 0
\(963\) 2.82357e8 0.316169
\(964\) 0 0
\(965\) − 7.32769e8i − 0.815428i
\(966\) 0 0
\(967\) − 3.55787e7i − 0.0393469i −0.999806 0.0196735i \(-0.993737\pi\)
0.999806 0.0196735i \(-0.00626266\pi\)
\(968\) 0 0
\(969\) 1.38585e9 1.52316
\(970\) 0 0
\(971\) 8.23470e8 0.899477 0.449738 0.893160i \(-0.351517\pi\)
0.449738 + 0.893160i \(0.351517\pi\)
\(972\) 0 0
\(973\) 1.72644e9i 1.87419i
\(974\) 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
\(979\) 6.12164e8 0.652408
\(980\) 0 0
\(981\) 1.80159e8i 0.190831i
\(982\) 0 0
\(983\) − 1.45686e9i − 1.53376i −0.641793 0.766878i \(-0.721809\pi\)
0.641793 0.766878i \(-0.278191\pi\)
\(984\) 0 0
\(985\) −8.90297e8 −0.931593
\(986\) 0 0
\(987\) 4.68713e8 0.487478
\(988\) 0 0
\(989\) − 1.23434e8i − 0.127599i
\(990\) 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
\(995\) −3.71014e8 −0.376635
\(996\) 0 0
\(997\) − 8.99928e8i − 0.908076i −0.890982 0.454038i \(-0.849983\pi\)
0.890982 0.454038i \(-0.150017\pi\)
\(998\) 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 384.7.b.c.319.3 yes 8
4.3 odd 2 inner 384.7.b.c.319.7 yes 8
8.3 odd 2 inner 384.7.b.c.319.2 8
8.5 even 2 inner 384.7.b.c.319.6 yes 8
16.3 odd 4 768.7.g.g.511.7 8
16.5 even 4 768.7.g.g.511.6 8
16.11 odd 4 768.7.g.g.511.2 8
16.13 even 4 768.7.g.g.511.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.c.319.2 8 8.3 odd 2 inner
384.7.b.c.319.3 yes 8 1.1 even 1 trivial
384.7.b.c.319.6 yes 8 8.5 even 2 inner
384.7.b.c.319.7 yes 8 4.3 odd 2 inner
768.7.g.g.511.2 8 16.11 odd 4
768.7.g.g.511.3 8 16.13 even 4
768.7.g.g.511.6 8 16.5 even 4
768.7.g.g.511.7 8 16.3 odd 4