Properties

Label 768.5.g.c.511.2
Level $768$
Weight $5$
Character 768.511
Analytic conductor $79.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.3881316484\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 511.2
Root \(-1.32288 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 768.511
Dual form 768.5.g.c.511.4

$q$-expansion

\(f(q)\) \(=\) \(q-5.19615i q^{3} +23.7490 q^{5} +9.38251i q^{7} -27.0000 q^{9} +O(q^{10})\) \(q-5.19615i q^{3} +23.7490 q^{5} +9.38251i q^{7} -27.0000 q^{9} +112.001i q^{11} -56.5020 q^{13} -123.404i q^{15} -79.9921 q^{17} -211.297i q^{19} +48.7530 q^{21} -217.355i q^{23} -60.9843 q^{25} +140.296i q^{27} -616.753 q^{29} -1111.81i q^{31} +581.976 q^{33} +222.825i q^{35} +802.980 q^{37} +293.593i q^{39} +2411.93 q^{41} -2130.38i q^{43} -641.223 q^{45} -3596.58i q^{47} +2312.97 q^{49} +415.651i q^{51} -833.725 q^{53} +2659.92i q^{55} -1097.93 q^{57} -1309.43i q^{59} +4785.91 q^{61} -253.328i q^{63} -1341.87 q^{65} -4025.23i q^{67} -1129.41 q^{69} +9487.13i q^{71} +266.031 q^{73} +316.883i q^{75} -1050.85 q^{77} -5756.11i q^{79} +729.000 q^{81} -7287.16i q^{83} -1899.73 q^{85} +3204.74i q^{87} +1414.08 q^{89} -530.130i q^{91} -5777.13 q^{93} -5018.09i q^{95} -3110.08 q^{97} -3024.04i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{5} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97}+O(q^{100}) \) 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
\(3\) − 5.19615i − 0.577350i
\(4\) 0 0
\(5\) 23.7490 0.949961 0.474980 0.879996i \(-0.342455\pi\)
0.474980 + 0.879996i \(0.342455\pi\)
\(6\) 0 0
\(7\) 9.38251i 0.191480i 0.995406 + 0.0957399i \(0.0305217\pi\)
−0.995406 + 0.0957399i \(0.969478\pi\)
\(8\) 0 0
\(9\) −27.0000 −0.333333
\(10\) 0 0
\(11\) 112.001i 0.925631i 0.886455 + 0.462816i \(0.153161\pi\)
−0.886455 + 0.462816i \(0.846839\pi\)
\(12\) 0 0
\(13\) −56.5020 −0.334331 −0.167166 0.985929i \(-0.553461\pi\)
−0.167166 + 0.985929i \(0.553461\pi\)
\(14\) 0 0
\(15\) − 123.404i − 0.548460i
\(16\) 0 0
\(17\) −79.9921 −0.276789 −0.138395 0.990377i \(-0.544194\pi\)
−0.138395 + 0.990377i \(0.544194\pi\)
\(18\) 0 0
\(19\) − 211.297i − 0.585309i −0.956218 0.292655i \(-0.905461\pi\)
0.956218 0.292655i \(-0.0945386\pi\)
\(20\) 0 0
\(21\) 48.7530 0.110551
\(22\) 0 0
\(23\) − 217.355i − 0.410880i −0.978670 0.205440i \(-0.934138\pi\)
0.978670 0.205440i \(-0.0658625\pi\)
\(24\) 0 0
\(25\) −60.9843 −0.0975748
\(26\) 0 0
\(27\) 140.296i 0.192450i
\(28\) 0 0
\(29\) −616.753 −0.733357 −0.366678 0.930348i \(-0.619505\pi\)
−0.366678 + 0.930348i \(0.619505\pi\)
\(30\) 0 0
\(31\) − 1111.81i − 1.15693i −0.815707 0.578465i \(-0.803652\pi\)
0.815707 0.578465i \(-0.196348\pi\)
\(32\) 0 0
\(33\) 581.976 0.534414
\(34\) 0 0
\(35\) 222.825i 0.181898i
\(36\) 0 0
\(37\) 802.980 0.586545 0.293273 0.956029i \(-0.405256\pi\)
0.293273 + 0.956029i \(0.405256\pi\)
\(38\) 0 0
\(39\) 293.593i 0.193026i
\(40\) 0 0
\(41\) 2411.93 1.43482 0.717409 0.696652i \(-0.245328\pi\)
0.717409 + 0.696652i \(0.245328\pi\)
\(42\) 0 0
\(43\) − 2130.38i − 1.15218i −0.817386 0.576090i \(-0.804578\pi\)
0.817386 0.576090i \(-0.195422\pi\)
\(44\) 0 0
\(45\) −641.223 −0.316654
\(46\) 0 0
\(47\) − 3596.58i − 1.62815i −0.580761 0.814074i \(-0.697245\pi\)
0.580761 0.814074i \(-0.302755\pi\)
\(48\) 0 0
\(49\) 2312.97 0.963335
\(50\) 0 0
\(51\) 415.651i 0.159804i
\(52\) 0 0
\(53\) −833.725 −0.296805 −0.148403 0.988927i \(-0.547413\pi\)
−0.148403 + 0.988927i \(0.547413\pi\)
\(54\) 0 0
\(55\) 2659.92i 0.879313i
\(56\) 0 0
\(57\) −1097.93 −0.337928
\(58\) 0 0
\(59\) − 1309.43i − 0.376165i −0.982153 0.188083i \(-0.939773\pi\)
0.982153 0.188083i \(-0.0602272\pi\)
\(60\) 0 0
\(61\) 4785.91 1.28619 0.643095 0.765786i \(-0.277650\pi\)
0.643095 + 0.765786i \(0.277650\pi\)
\(62\) 0 0
\(63\) − 253.328i − 0.0638266i
\(64\) 0 0
\(65\) −1341.87 −0.317601
\(66\) 0 0
\(67\) − 4025.23i − 0.896688i −0.893861 0.448344i \(-0.852014\pi\)
0.893861 0.448344i \(-0.147986\pi\)
\(68\) 0 0
\(69\) −1129.41 −0.237221
\(70\) 0 0
\(71\) 9487.13i 1.88199i 0.338416 + 0.940997i \(0.390109\pi\)
−0.338416 + 0.940997i \(0.609891\pi\)
\(72\) 0 0
\(73\) 266.031 0.0499215 0.0249607 0.999688i \(-0.492054\pi\)
0.0249607 + 0.999688i \(0.492054\pi\)
\(74\) 0 0
\(75\) 316.883i 0.0563348i
\(76\) 0 0
\(77\) −1050.85 −0.177240
\(78\) 0 0
\(79\) − 5756.11i − 0.922306i −0.887321 0.461153i \(-0.847436\pi\)
0.887321 0.461153i \(-0.152564\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) − 7287.16i − 1.05780i −0.848685 0.528898i \(-0.822605\pi\)
0.848685 0.528898i \(-0.177395\pi\)
\(84\) 0 0
\(85\) −1899.73 −0.262939
\(86\) 0 0
\(87\) 3204.74i 0.423404i
\(88\) 0 0
\(89\) 1414.08 0.178523 0.0892614 0.996008i \(-0.471549\pi\)
0.0892614 + 0.996008i \(0.471549\pi\)
\(90\) 0 0
\(91\) − 530.130i − 0.0640177i
\(92\) 0 0
\(93\) −5777.13 −0.667953
\(94\) 0 0
\(95\) − 5018.09i − 0.556021i
\(96\) 0 0
\(97\) −3110.08 −0.330543 −0.165271 0.986248i \(-0.552850\pi\)
−0.165271 + 0.986248i \(0.552850\pi\)
\(98\) 0 0
\(99\) − 3024.04i − 0.308544i
\(100\) 0 0
\(101\) −488.501 −0.0478876 −0.0239438 0.999713i \(-0.507622\pi\)
−0.0239438 + 0.999713i \(0.507622\pi\)
\(102\) 0 0
\(103\) 20097.0i 1.89434i 0.320739 + 0.947168i \(0.396069\pi\)
−0.320739 + 0.947168i \(0.603931\pi\)
\(104\) 0 0
\(105\) 1157.83 0.105019
\(106\) 0 0
\(107\) − 12383.0i − 1.08158i −0.841158 0.540790i \(-0.818126\pi\)
0.841158 0.540790i \(-0.181874\pi\)
\(108\) 0 0
\(109\) 15542.1 1.30814 0.654072 0.756433i \(-0.273059\pi\)
0.654072 + 0.756433i \(0.273059\pi\)
\(110\) 0 0
\(111\) − 4172.41i − 0.338642i
\(112\) 0 0
\(113\) 11653.6 0.912651 0.456325 0.889813i \(-0.349165\pi\)
0.456325 + 0.889813i \(0.349165\pi\)
\(114\) 0 0
\(115\) − 5161.98i − 0.390319i
\(116\) 0 0
\(117\) 1525.55 0.111444
\(118\) 0 0
\(119\) − 750.527i − 0.0529996i
\(120\) 0 0
\(121\) 2096.69 0.143206
\(122\) 0 0
\(123\) − 12532.8i − 0.828393i
\(124\) 0 0
\(125\) −16291.5 −1.04265
\(126\) 0 0
\(127\) − 4724.04i − 0.292891i −0.989219 0.146445i \(-0.953217\pi\)
0.989219 0.146445i \(-0.0467833\pi\)
\(128\) 0 0
\(129\) −11069.8 −0.665212
\(130\) 0 0
\(131\) − 9632.86i − 0.561323i −0.959807 0.280661i \(-0.909446\pi\)
0.959807 0.280661i \(-0.0905538\pi\)
\(132\) 0 0
\(133\) 1982.49 0.112075
\(134\) 0 0
\(135\) 3331.89i 0.182820i
\(136\) 0 0
\(137\) −16407.6 −0.874185 −0.437093 0.899417i \(-0.643992\pi\)
−0.437093 + 0.899417i \(0.643992\pi\)
\(138\) 0 0
\(139\) − 25544.1i − 1.32209i −0.750346 0.661046i \(-0.770113\pi\)
0.750346 0.661046i \(-0.229887\pi\)
\(140\) 0 0
\(141\) −18688.4 −0.940012
\(142\) 0 0
\(143\) − 6328.30i − 0.309467i
\(144\) 0 0
\(145\) −14647.3 −0.696660
\(146\) 0 0
\(147\) − 12018.5i − 0.556182i
\(148\) 0 0
\(149\) 18716.6 0.843054 0.421527 0.906816i \(-0.361494\pi\)
0.421527 + 0.906816i \(0.361494\pi\)
\(150\) 0 0
\(151\) − 28860.0i − 1.26574i −0.774260 0.632868i \(-0.781878\pi\)
0.774260 0.632868i \(-0.218122\pi\)
\(152\) 0 0
\(153\) 2159.79 0.0922631
\(154\) 0 0
\(155\) − 26404.4i − 1.09904i
\(156\) 0 0
\(157\) 23963.1 0.972172 0.486086 0.873911i \(-0.338424\pi\)
0.486086 + 0.873911i \(0.338424\pi\)
\(158\) 0 0
\(159\) 4332.16i 0.171360i
\(160\) 0 0
\(161\) 2039.34 0.0786752
\(162\) 0 0
\(163\) − 24424.8i − 0.919297i −0.888101 0.459648i \(-0.847975\pi\)
0.888101 0.459648i \(-0.152025\pi\)
\(164\) 0 0
\(165\) 13821.4 0.507672
\(166\) 0 0
\(167\) − 3088.40i − 0.110739i −0.998466 0.0553696i \(-0.982366\pi\)
0.998466 0.0553696i \(-0.0176337\pi\)
\(168\) 0 0
\(169\) −25368.5 −0.888223
\(170\) 0 0
\(171\) 5705.01i 0.195103i
\(172\) 0 0
\(173\) −18621.0 −0.622171 −0.311085 0.950382i \(-0.600693\pi\)
−0.311085 + 0.950382i \(0.600693\pi\)
\(174\) 0 0
\(175\) − 572.185i − 0.0186836i
\(176\) 0 0
\(177\) −6804.00 −0.217179
\(178\) 0 0
\(179\) − 50739.3i − 1.58357i −0.610797 0.791787i \(-0.709151\pi\)
0.610797 0.791787i \(-0.290849\pi\)
\(180\) 0 0
\(181\) −58050.5 −1.77194 −0.885970 0.463742i \(-0.846506\pi\)
−0.885970 + 0.463742i \(0.846506\pi\)
\(182\) 0 0
\(183\) − 24868.3i − 0.742582i
\(184\) 0 0
\(185\) 19070.0 0.557195
\(186\) 0 0
\(187\) − 8959.23i − 0.256205i
\(188\) 0 0
\(189\) −1316.33 −0.0368503
\(190\) 0 0
\(191\) 21285.6i 0.583471i 0.956499 + 0.291736i \(0.0942327\pi\)
−0.956499 + 0.291736i \(0.905767\pi\)
\(192\) 0 0
\(193\) −26074.3 −0.700001 −0.350001 0.936749i \(-0.613819\pi\)
−0.350001 + 0.936749i \(0.613819\pi\)
\(194\) 0 0
\(195\) 6972.54i 0.183367i
\(196\) 0 0
\(197\) 4731.26 0.121911 0.0609557 0.998140i \(-0.480585\pi\)
0.0609557 + 0.998140i \(0.480585\pi\)
\(198\) 0 0
\(199\) − 38561.7i − 0.973756i −0.873470 0.486878i \(-0.838136\pi\)
0.873470 0.486878i \(-0.161864\pi\)
\(200\) 0 0
\(201\) −20915.7 −0.517703
\(202\) 0 0
\(203\) − 5786.69i − 0.140423i
\(204\) 0 0
\(205\) 57280.9 1.36302
\(206\) 0 0
\(207\) 5868.59i 0.136960i
\(208\) 0 0
\(209\) 23665.5 0.541780
\(210\) 0 0
\(211\) − 48656.4i − 1.09289i −0.837496 0.546444i \(-0.815981\pi\)
0.837496 0.546444i \(-0.184019\pi\)
\(212\) 0 0
\(213\) 49296.6 1.08657
\(214\) 0 0
\(215\) − 50594.5i − 1.09453i
\(216\) 0 0
\(217\) 10431.6 0.221529
\(218\) 0 0
\(219\) − 1382.34i − 0.0288222i
\(220\) 0 0
\(221\) 4519.71 0.0925393
\(222\) 0 0
\(223\) 73299.7i 1.47398i 0.675902 + 0.736992i \(0.263754\pi\)
−0.675902 + 0.736992i \(0.736246\pi\)
\(224\) 0 0
\(225\) 1646.57 0.0325249
\(226\) 0 0
\(227\) 14996.0i 0.291020i 0.989357 + 0.145510i \(0.0464823\pi\)
−0.989357 + 0.145510i \(0.953518\pi\)
\(228\) 0 0
\(229\) −74694.9 −1.42436 −0.712181 0.701996i \(-0.752292\pi\)
−0.712181 + 0.701996i \(0.752292\pi\)
\(230\) 0 0
\(231\) 5460.40i 0.102329i
\(232\) 0 0
\(233\) 69476.6 1.27975 0.639877 0.768477i \(-0.278985\pi\)
0.639877 + 0.768477i \(0.278985\pi\)
\(234\) 0 0
\(235\) − 85415.2i − 1.54668i
\(236\) 0 0
\(237\) −29909.6 −0.532494
\(238\) 0 0
\(239\) 87478.2i 1.53145i 0.643166 + 0.765727i \(0.277620\pi\)
−0.643166 + 0.765727i \(0.722380\pi\)
\(240\) 0 0
\(241\) 74951.4 1.29046 0.645232 0.763987i \(-0.276761\pi\)
0.645232 + 0.763987i \(0.276761\pi\)
\(242\) 0 0
\(243\) − 3788.00i − 0.0641500i
\(244\) 0 0
\(245\) 54930.7 0.915131
\(246\) 0 0
\(247\) 11938.7i 0.195687i
\(248\) 0 0
\(249\) −37865.2 −0.610719
\(250\) 0 0
\(251\) − 43677.8i − 0.693287i −0.937997 0.346643i \(-0.887321\pi\)
0.937997 0.346643i \(-0.112679\pi\)
\(252\) 0 0
\(253\) 24344.1 0.380323
\(254\) 0 0
\(255\) 9871.31i 0.151808i
\(256\) 0 0
\(257\) −1164.55 −0.0176316 −0.00881581 0.999961i \(-0.502806\pi\)
−0.00881581 + 0.999961i \(0.502806\pi\)
\(258\) 0 0
\(259\) 7533.97i 0.112312i
\(260\) 0 0
\(261\) 16652.3 0.244452
\(262\) 0 0
\(263\) − 35001.0i − 0.506021i −0.967464 0.253010i \(-0.918579\pi\)
0.967464 0.253010i \(-0.0814207\pi\)
\(264\) 0 0
\(265\) −19800.2 −0.281953
\(266\) 0 0
\(267\) − 7347.77i − 0.103070i
\(268\) 0 0
\(269\) −41077.9 −0.567681 −0.283840 0.958872i \(-0.591609\pi\)
−0.283840 + 0.958872i \(0.591609\pi\)
\(270\) 0 0
\(271\) 46419.1i 0.632060i 0.948749 + 0.316030i \(0.102350\pi\)
−0.948749 + 0.316030i \(0.897650\pi\)
\(272\) 0 0
\(273\) −2754.64 −0.0369606
\(274\) 0 0
\(275\) − 6830.32i − 0.0903183i
\(276\) 0 0
\(277\) −150297. −1.95881 −0.979405 0.201907i \(-0.935286\pi\)
−0.979405 + 0.201907i \(0.935286\pi\)
\(278\) 0 0
\(279\) 30018.8i 0.385643i
\(280\) 0 0
\(281\) 117793. 1.49179 0.745895 0.666063i \(-0.232022\pi\)
0.745895 + 0.666063i \(0.232022\pi\)
\(282\) 0 0
\(283\) 41243.6i 0.514973i 0.966282 + 0.257486i \(0.0828942\pi\)
−0.966282 + 0.257486i \(0.917106\pi\)
\(284\) 0 0
\(285\) −26074.7 −0.321019
\(286\) 0 0
\(287\) 22629.9i 0.274739i
\(288\) 0 0
\(289\) −77122.3 −0.923388
\(290\) 0 0
\(291\) 16160.4i 0.190839i
\(292\) 0 0
\(293\) 71937.8 0.837957 0.418979 0.907996i \(-0.362388\pi\)
0.418979 + 0.907996i \(0.362388\pi\)
\(294\) 0 0
\(295\) − 31097.7i − 0.357342i
\(296\) 0 0
\(297\) −15713.4 −0.178138
\(298\) 0 0
\(299\) 12281.0i 0.137370i
\(300\) 0 0
\(301\) 19988.3 0.220619
\(302\) 0 0
\(303\) 2538.33i 0.0276479i
\(304\) 0 0
\(305\) 113661. 1.22183
\(306\) 0 0
\(307\) − 54572.7i − 0.579027i −0.957174 0.289514i \(-0.906506\pi\)
0.957174 0.289514i \(-0.0934936\pi\)
\(308\) 0 0
\(309\) 104427. 1.09370
\(310\) 0 0
\(311\) 60945.8i 0.630119i 0.949072 + 0.315060i \(0.102025\pi\)
−0.949072 + 0.315060i \(0.897975\pi\)
\(312\) 0 0
\(313\) −102774. −1.04904 −0.524521 0.851397i \(-0.675756\pi\)
−0.524521 + 0.851397i \(0.675756\pi\)
\(314\) 0 0
\(315\) − 6016.29i − 0.0606328i
\(316\) 0 0
\(317\) −39779.2 −0.395857 −0.197928 0.980217i \(-0.563421\pi\)
−0.197928 + 0.980217i \(0.563421\pi\)
\(318\) 0 0
\(319\) − 69077.2i − 0.678818i
\(320\) 0 0
\(321\) −64344.0 −0.624450
\(322\) 0 0
\(323\) 16902.1i 0.162007i
\(324\) 0 0
\(325\) 3445.73 0.0326223
\(326\) 0 0
\(327\) − 80758.9i − 0.755257i
\(328\) 0 0
\(329\) 33744.9 0.311758
\(330\) 0 0
\(331\) 87804.6i 0.801422i 0.916204 + 0.400711i \(0.131237\pi\)
−0.916204 + 0.400711i \(0.868763\pi\)
\(332\) 0 0
\(333\) −21680.5 −0.195515
\(334\) 0 0
\(335\) − 95595.3i − 0.851818i
\(336\) 0 0
\(337\) −57972.0 −0.510456 −0.255228 0.966881i \(-0.582151\pi\)
−0.255228 + 0.966881i \(0.582151\pi\)
\(338\) 0 0
\(339\) − 60554.1i − 0.526919i
\(340\) 0 0
\(341\) 124524. 1.07089
\(342\) 0 0
\(343\) 44228.9i 0.375939i
\(344\) 0 0
\(345\) −26822.4 −0.225351
\(346\) 0 0
\(347\) 159188.i 1.32206i 0.750359 + 0.661031i \(0.229881\pi\)
−0.750359 + 0.661031i \(0.770119\pi\)
\(348\) 0 0
\(349\) 17396.8 0.142830 0.0714149 0.997447i \(-0.477249\pi\)
0.0714149 + 0.997447i \(0.477249\pi\)
\(350\) 0 0
\(351\) − 7927.01i − 0.0643421i
\(352\) 0 0
\(353\) 150532. 1.20803 0.604017 0.796972i \(-0.293566\pi\)
0.604017 + 0.796972i \(0.293566\pi\)
\(354\) 0 0
\(355\) 225310.i 1.78782i
\(356\) 0 0
\(357\) −3899.85 −0.0305993
\(358\) 0 0
\(359\) 24818.5i 0.192569i 0.995354 + 0.0962846i \(0.0306959\pi\)
−0.995354 + 0.0962846i \(0.969304\pi\)
\(360\) 0 0
\(361\) 85674.8 0.657413
\(362\) 0 0
\(363\) − 10894.7i − 0.0826803i
\(364\) 0 0
\(365\) 6317.99 0.0474234
\(366\) 0 0
\(367\) − 89484.6i − 0.664379i −0.943213 0.332190i \(-0.892213\pi\)
0.943213 0.332190i \(-0.107787\pi\)
\(368\) 0 0
\(369\) −65122.1 −0.478273
\(370\) 0 0
\(371\) − 7822.44i − 0.0568322i
\(372\) 0 0
\(373\) −168536. −1.21137 −0.605684 0.795705i \(-0.707100\pi\)
−0.605684 + 0.795705i \(0.707100\pi\)
\(374\) 0 0
\(375\) 84652.9i 0.601976i
\(376\) 0 0
\(377\) 34847.8 0.245184
\(378\) 0 0
\(379\) 126865.i 0.883208i 0.897210 + 0.441604i \(0.145590\pi\)
−0.897210 + 0.441604i \(0.854410\pi\)
\(380\) 0 0
\(381\) −24546.8 −0.169101
\(382\) 0 0
\(383\) − 257277.i − 1.75389i −0.480587 0.876947i \(-0.659576\pi\)
0.480587 0.876947i \(-0.340424\pi\)
\(384\) 0 0
\(385\) −24956.8 −0.168371
\(386\) 0 0
\(387\) 57520.3i 0.384060i
\(388\) 0 0
\(389\) −150562. −0.994983 −0.497492 0.867469i \(-0.665745\pi\)
−0.497492 + 0.867469i \(0.665745\pi\)
\(390\) 0 0
\(391\) 17386.7i 0.113727i
\(392\) 0 0
\(393\) −50053.8 −0.324080
\(394\) 0 0
\(395\) − 136702.i − 0.876155i
\(396\) 0 0
\(397\) 185402. 1.17634 0.588171 0.808736i \(-0.299848\pi\)
0.588171 + 0.808736i \(0.299848\pi\)
\(398\) 0 0
\(399\) − 10301.3i − 0.0647064i
\(400\) 0 0
\(401\) 167681. 1.04279 0.521393 0.853317i \(-0.325413\pi\)
0.521393 + 0.853317i \(0.325413\pi\)
\(402\) 0 0
\(403\) 62819.4i 0.386798i
\(404\) 0 0
\(405\) 17313.0 0.105551
\(406\) 0 0
\(407\) 89934.9i 0.542925i
\(408\) 0 0
\(409\) −60544.6 −0.361934 −0.180967 0.983489i \(-0.557923\pi\)
−0.180967 + 0.983489i \(0.557923\pi\)
\(410\) 0 0
\(411\) 85256.3i 0.504711i
\(412\) 0 0
\(413\) 12285.7 0.0720280
\(414\) 0 0
\(415\) − 173063.i − 1.00486i
\(416\) 0 0
\(417\) −132731. −0.763310
\(418\) 0 0
\(419\) 307306.i 1.75042i 0.483741 + 0.875211i \(0.339277\pi\)
−0.483741 + 0.875211i \(0.660723\pi\)
\(420\) 0 0
\(421\) 172189. 0.971496 0.485748 0.874099i \(-0.338547\pi\)
0.485748 + 0.874099i \(0.338547\pi\)
\(422\) 0 0
\(423\) 97107.7i 0.542716i
\(424\) 0 0
\(425\) 4878.26 0.0270077
\(426\) 0 0
\(427\) 44903.9i 0.246279i
\(428\) 0 0
\(429\) −32882.8 −0.178671
\(430\) 0 0
\(431\) − 28153.8i − 0.151559i −0.997125 0.0757797i \(-0.975855\pi\)
0.997125 0.0757797i \(-0.0241446\pi\)
\(432\) 0 0
\(433\) −322037. −1.71763 −0.858815 0.512285i \(-0.828799\pi\)
−0.858815 + 0.512285i \(0.828799\pi\)
\(434\) 0 0
\(435\) 76109.5i 0.402217i
\(436\) 0 0
\(437\) −45926.4 −0.240492
\(438\) 0 0
\(439\) 292438.i 1.51742i 0.651430 + 0.758709i \(0.274170\pi\)
−0.651430 + 0.758709i \(0.725830\pi\)
\(440\) 0 0
\(441\) −62450.1 −0.321112
\(442\) 0 0
\(443\) 227681.i 1.16016i 0.814558 + 0.580082i \(0.196980\pi\)
−0.814558 + 0.580082i \(0.803020\pi\)
\(444\) 0 0
\(445\) 33583.0 0.169590
\(446\) 0 0
\(447\) − 97254.5i − 0.486737i
\(448\) 0 0
\(449\) 159097. 0.789169 0.394585 0.918860i \(-0.370889\pi\)
0.394585 + 0.918860i \(0.370889\pi\)
\(450\) 0 0
\(451\) 270139.i 1.32811i
\(452\) 0 0
\(453\) −149961. −0.730773
\(454\) 0 0
\(455\) − 12590.1i − 0.0608143i
\(456\) 0 0
\(457\) −340590. −1.63079 −0.815397 0.578902i \(-0.803481\pi\)
−0.815397 + 0.578902i \(0.803481\pi\)
\(458\) 0 0
\(459\) − 11222.6i − 0.0532681i
\(460\) 0 0
\(461\) −178561. −0.840204 −0.420102 0.907477i \(-0.638006\pi\)
−0.420102 + 0.907477i \(0.638006\pi\)
\(462\) 0 0
\(463\) 4179.11i 0.0194950i 0.999952 + 0.00974748i \(0.00310277\pi\)
−0.999952 + 0.00974748i \(0.996897\pi\)
\(464\) 0 0
\(465\) −137201. −0.634529
\(466\) 0 0
\(467\) − 160583.i − 0.736318i −0.929763 0.368159i \(-0.879988\pi\)
0.929763 0.368159i \(-0.120012\pi\)
\(468\) 0 0
\(469\) 37766.8 0.171698
\(470\) 0 0
\(471\) − 124516.i − 0.561284i
\(472\) 0 0
\(473\) 238606. 1.06649
\(474\) 0 0
\(475\) 12885.8i 0.0571114i
\(476\) 0 0
\(477\) 22510.6 0.0989350
\(478\) 0 0
\(479\) 153260.i 0.667970i 0.942578 + 0.333985i \(0.108393\pi\)
−0.942578 + 0.333985i \(0.891607\pi\)
\(480\) 0 0
\(481\) −45370.0 −0.196100
\(482\) 0 0
\(483\) − 10596.7i − 0.0454231i
\(484\) 0 0
\(485\) −73861.3 −0.314003
\(486\) 0 0
\(487\) 113347.i 0.477917i 0.971030 + 0.238958i \(0.0768059\pi\)
−0.971030 + 0.238958i \(0.923194\pi\)
\(488\) 0 0
\(489\) −126915. −0.530756
\(490\) 0 0
\(491\) 248336.i 1.03009i 0.857162 + 0.515046i \(0.172225\pi\)
−0.857162 + 0.515046i \(0.827775\pi\)
\(492\) 0 0
\(493\) 49335.4 0.202985
\(494\) 0 0
\(495\) − 71817.9i − 0.293104i
\(496\) 0 0
\(497\) −89013.1 −0.360364
\(498\) 0 0
\(499\) − 378612.i − 1.52052i −0.649617 0.760262i \(-0.725071\pi\)
0.649617 0.760262i \(-0.274929\pi\)
\(500\) 0 0
\(501\) −16047.8 −0.0639353
\(502\) 0 0
\(503\) 255639.i 1.01040i 0.863004 + 0.505198i \(0.168580\pi\)
−0.863004 + 0.505198i \(0.831420\pi\)
\(504\) 0 0
\(505\) −11601.4 −0.0454913
\(506\) 0 0
\(507\) 131819.i 0.512816i
\(508\) 0 0
\(509\) 204106. 0.787809 0.393904 0.919151i \(-0.371124\pi\)
0.393904 + 0.919151i \(0.371124\pi\)
\(510\) 0 0
\(511\) 2496.04i 0.00955895i
\(512\) 0 0
\(513\) 29644.1 0.112643
\(514\) 0 0
\(515\) 477284.i 1.79954i
\(516\) 0 0
\(517\) 402822. 1.50707
\(518\) 0 0
\(519\) 96757.3i 0.359211i
\(520\) 0 0
\(521\) 295131. 1.08727 0.543637 0.839320i \(-0.317047\pi\)
0.543637 + 0.839320i \(0.317047\pi\)
\(522\) 0 0
\(523\) 212950.i 0.778527i 0.921127 + 0.389263i \(0.127270\pi\)
−0.921127 + 0.389263i \(0.872730\pi\)
\(524\) 0 0
\(525\) −2973.16 −0.0107870
\(526\) 0 0
\(527\) 88936.0i 0.320226i
\(528\) 0 0
\(529\) 232598. 0.831178
\(530\) 0 0
\(531\) 35354.6i 0.125388i
\(532\) 0 0
\(533\) −136279. −0.479704
\(534\) 0 0
\(535\) − 294084.i − 1.02746i
\(536\) 0 0
\(537\) −263649. −0.914277
\(538\) 0 0
\(539\) 259056.i 0.891694i
\(540\) 0 0
\(541\) −387756. −1.32484 −0.662421 0.749132i \(-0.730471\pi\)
−0.662421 + 0.749132i \(0.730471\pi\)
\(542\) 0 0
\(543\) 301639.i 1.02303i
\(544\) 0 0
\(545\) 369108. 1.24268
\(546\) 0 0
\(547\) 241227.i 0.806215i 0.915153 + 0.403107i \(0.132070\pi\)
−0.915153 + 0.403107i \(0.867930\pi\)
\(548\) 0 0
\(549\) −129220. −0.428730
\(550\) 0 0
\(551\) 130318.i 0.429240i
\(552\) 0 0
\(553\) 54006.8 0.176603
\(554\) 0 0
\(555\) − 99090.6i − 0.321697i
\(556\) 0 0
\(557\) −86454.9 −0.278663 −0.139331 0.990246i \(-0.544495\pi\)
−0.139331 + 0.990246i \(0.544495\pi\)
\(558\) 0 0
\(559\) 120371.i 0.385210i
\(560\) 0 0
\(561\) −46553.5 −0.147920
\(562\) 0 0
\(563\) − 99585.2i − 0.314179i −0.987584 0.157090i \(-0.949789\pi\)
0.987584 0.157090i \(-0.0502112\pi\)
\(564\) 0 0
\(565\) 276762. 0.866982
\(566\) 0 0
\(567\) 6839.85i 0.0212755i
\(568\) 0 0
\(569\) 524085. 1.61874 0.809371 0.587298i \(-0.199808\pi\)
0.809371 + 0.587298i \(0.199808\pi\)
\(570\) 0 0
\(571\) 32749.2i 0.100445i 0.998738 + 0.0502225i \(0.0159931\pi\)
−0.998738 + 0.0502225i \(0.984007\pi\)
\(572\) 0 0
\(573\) 110603. 0.336867
\(574\) 0 0
\(575\) 13255.3i 0.0400915i
\(576\) 0 0
\(577\) −140276. −0.421338 −0.210669 0.977557i \(-0.567564\pi\)
−0.210669 + 0.977557i \(0.567564\pi\)
\(578\) 0 0
\(579\) 135486.i 0.404146i
\(580\) 0 0
\(581\) 68371.8 0.202547
\(582\) 0 0
\(583\) − 93378.4i − 0.274732i
\(584\) 0 0
\(585\) 36230.4 0.105867
\(586\) 0 0
\(587\) − 303094.i − 0.879634i −0.898087 0.439817i \(-0.855043\pi\)
0.898087 0.439817i \(-0.144957\pi\)
\(588\) 0 0
\(589\) −234921. −0.677161
\(590\) 0 0
\(591\) − 24584.4i − 0.0703856i
\(592\) 0 0
\(593\) 117438. 0.333963 0.166981 0.985960i \(-0.446598\pi\)
0.166981 + 0.985960i \(0.446598\pi\)
\(594\) 0 0
\(595\) − 17824.3i − 0.0503475i
\(596\) 0 0
\(597\) −200373. −0.562198
\(598\) 0 0
\(599\) − 537187.i − 1.49717i −0.663037 0.748587i \(-0.730733\pi\)
0.663037 0.748587i \(-0.269267\pi\)
\(600\) 0 0
\(601\) −382705. −1.05953 −0.529767 0.848143i \(-0.677721\pi\)
−0.529767 + 0.848143i \(0.677721\pi\)
\(602\) 0 0
\(603\) 108681.i 0.298896i
\(604\) 0 0
\(605\) 49794.2 0.136040
\(606\) 0 0
\(607\) 578934.i 1.57127i 0.618688 + 0.785637i \(0.287664\pi\)
−0.618688 + 0.785637i \(0.712336\pi\)
\(608\) 0 0
\(609\) −30068.5 −0.0810732
\(610\) 0 0
\(611\) 203214.i 0.544341i
\(612\) 0 0
\(613\) 287542. 0.765209 0.382605 0.923912i \(-0.375027\pi\)
0.382605 + 0.923912i \(0.375027\pi\)
\(614\) 0 0
\(615\) − 297641.i − 0.786940i
\(616\) 0 0
\(617\) 400806. 1.05284 0.526422 0.850223i \(-0.323533\pi\)
0.526422 + 0.850223i \(0.323533\pi\)
\(618\) 0 0
\(619\) − 233469.i − 0.609323i −0.952461 0.304661i \(-0.901457\pi\)
0.952461 0.304661i \(-0.0985433\pi\)
\(620\) 0 0
\(621\) 30494.1 0.0790738
\(622\) 0 0
\(623\) 13267.6i 0.0341835i
\(624\) 0 0
\(625\) −348791. −0.892904
\(626\) 0 0
\(627\) − 122970.i − 0.312797i
\(628\) 0 0
\(629\) −64232.1 −0.162349
\(630\) 0 0
\(631\) − 499538.i − 1.25461i −0.778773 0.627306i \(-0.784157\pi\)
0.778773 0.627306i \(-0.215843\pi\)
\(632\) 0 0
\(633\) −252826. −0.630979
\(634\) 0 0
\(635\) − 112191.i − 0.278235i
\(636\) 0 0
\(637\) −130687. −0.322073
\(638\) 0 0
\(639\) − 256152.i − 0.627331i
\(640\) 0 0
\(641\) 405774. 0.987571 0.493786 0.869584i \(-0.335613\pi\)
0.493786 + 0.869584i \(0.335613\pi\)
\(642\) 0 0
\(643\) 550571.i 1.33165i 0.746106 + 0.665827i \(0.231921\pi\)
−0.746106 + 0.665827i \(0.768079\pi\)
\(644\) 0 0
\(645\) −262897. −0.631925
\(646\) 0 0
\(647\) − 428220.i − 1.02296i −0.859295 0.511480i \(-0.829097\pi\)
0.859295 0.511480i \(-0.170903\pi\)
\(648\) 0 0
\(649\) 146658. 0.348190
\(650\) 0 0
\(651\) − 54204.0i − 0.127900i
\(652\) 0 0
\(653\) 802147. 1.88117 0.940584 0.339561i \(-0.110279\pi\)
0.940584 + 0.339561i \(0.110279\pi\)
\(654\) 0 0
\(655\) − 228771.i − 0.533234i
\(656\) 0 0
\(657\) −7182.85 −0.0166405
\(658\) 0 0
\(659\) − 662958.i − 1.52656i −0.646065 0.763282i \(-0.723586\pi\)
0.646065 0.763282i \(-0.276414\pi\)
\(660\) 0 0
\(661\) 231396. 0.529607 0.264804 0.964302i \(-0.414693\pi\)
0.264804 + 0.964302i \(0.414693\pi\)
\(662\) 0 0
\(663\) − 23485.1i − 0.0534276i
\(664\) 0 0
\(665\) 47082.2 0.106467
\(666\) 0 0
\(667\) 134055.i 0.301321i
\(668\) 0 0
\(669\) 380877. 0.851005
\(670\) 0 0
\(671\) 536029.i 1.19054i
\(672\) 0 0
\(673\) 311867. 0.688557 0.344278 0.938868i \(-0.388124\pi\)
0.344278 + 0.938868i \(0.388124\pi\)
\(674\) 0 0
\(675\) − 8555.85i − 0.0187783i
\(676\) 0 0
\(677\) −655173. −1.42948 −0.714741 0.699389i \(-0.753455\pi\)
−0.714741 + 0.699389i \(0.753455\pi\)
\(678\) 0 0
\(679\) − 29180.3i − 0.0632923i
\(680\) 0 0
\(681\) 77921.4 0.168021
\(682\) 0 0
\(683\) − 358002.i − 0.767439i −0.923450 0.383719i \(-0.874643\pi\)
0.923450 0.383719i \(-0.125357\pi\)
\(684\) 0 0
\(685\) −389664. −0.830442
\(686\) 0 0
\(687\) 388126.i 0.822355i
\(688\) 0 0
\(689\) 47107.1 0.0992312
\(690\) 0 0
\(691\) − 826006.i − 1.72992i −0.501838 0.864962i \(-0.667343\pi\)
0.501838 0.864962i \(-0.332657\pi\)
\(692\) 0 0
\(693\) 28373.1 0.0590799
\(694\) 0 0
\(695\) − 606648.i − 1.25593i
\(696\) 0 0
\(697\) −192935. −0.397142
\(698\) 0 0
\(699\) − 361011.i − 0.738866i
\(700\) 0 0
\(701\) −229057. −0.466130 −0.233065 0.972461i \(-0.574876\pi\)
−0.233065 + 0.972461i \(0.574876\pi\)
\(702\) 0 0
\(703\) − 169667.i − 0.343310i
\(704\) 0 0
\(705\) −443831. −0.892974
\(706\) 0 0
\(707\) − 4583.37i − 0.00916950i
\(708\) 0 0
\(709\) 6312.88 0.0125584 0.00627921 0.999980i \(-0.498001\pi\)
0.00627921 + 0.999980i \(0.498001\pi\)
\(710\) 0 0
\(711\) 155415.i 0.307435i
\(712\) 0 0
\(713\) −241658. −0.475359
\(714\) 0 0
\(715\) − 150291.i − 0.293982i
\(716\) 0 0
\(717\) 454550. 0.884185
\(718\) 0 0
\(719\) 179483.i 0.347189i 0.984817 + 0.173594i \(0.0555382\pi\)
−0.984817 + 0.173594i \(0.944462\pi\)
\(720\) 0 0
\(721\) −188560. −0.362727
\(722\) 0 0
\(723\) − 389459.i − 0.745049i
\(724\) 0 0
\(725\) 37612.2 0.0715571
\(726\) 0 0
\(727\) − 310931.i − 0.588295i −0.955760 0.294147i \(-0.904964\pi\)
0.955760 0.294147i \(-0.0950357\pi\)
\(728\) 0 0
\(729\) −19683.0 −0.0370370
\(730\) 0 0
\(731\) 170414.i 0.318911i
\(732\) 0 0
\(733\) −602271. −1.12094 −0.560472 0.828173i \(-0.689380\pi\)
−0.560472 + 0.828173i \(0.689380\pi\)
\(734\) 0 0
\(735\) − 285428.i − 0.528351i
\(736\) 0 0
\(737\) 450832. 0.830002
\(738\) 0 0
\(739\) 64180.3i 0.117520i 0.998272 + 0.0587602i \(0.0187147\pi\)
−0.998272 + 0.0587602i \(0.981285\pi\)
\(740\) 0 0
\(741\) 62035.2 0.112980
\(742\) 0 0
\(743\) 525793.i 0.952440i 0.879326 + 0.476220i \(0.157993\pi\)
−0.879326 + 0.476220i \(0.842007\pi\)
\(744\) 0 0
\(745\) 444502. 0.800868
\(746\) 0 0
\(747\) 196753.i 0.352599i
\(748\) 0 0
\(749\) 116184. 0.207101
\(750\) 0 0
\(751\) 405381.i 0.718759i 0.933191 + 0.359380i \(0.117012\pi\)
−0.933191 + 0.359380i \(0.882988\pi\)
\(752\) 0 0
\(753\) −226956. −0.400269
\(754\) 0 0
\(755\) − 685398.i − 1.20240i
\(756\) 0 0
\(757\) 5071.58 0.00885016 0.00442508 0.999990i \(-0.498591\pi\)
0.00442508 + 0.999990i \(0.498591\pi\)
\(758\) 0 0
\(759\) − 126496.i − 0.219580i
\(760\) 0 0
\(761\) 29104.4 0.0502561 0.0251280 0.999684i \(-0.492001\pi\)
0.0251280 + 0.999684i \(0.492001\pi\)
\(762\) 0 0
\(763\) 145823.i 0.250483i
\(764\) 0 0
\(765\) 51292.8 0.0876463
\(766\) 0 0
\(767\) 73985.4i 0.125764i
\(768\) 0 0
\(769\) −235095. −0.397548 −0.198774 0.980045i \(-0.563696\pi\)
−0.198774 + 0.980045i \(0.563696\pi\)
\(770\) 0 0
\(771\) 6051.19i 0.0101796i
\(772\) 0 0
\(773\) 89776.8 0.150247 0.0751234 0.997174i \(-0.476065\pi\)
0.0751234 + 0.997174i \(0.476065\pi\)
\(774\) 0 0
\(775\) 67802.8i 0.112887i
\(776\) 0 0
\(777\) 39147.7 0.0648431
\(778\) 0 0
\(779\) − 509632.i − 0.839812i
\(780\) 0 0
\(781\) −1.06257e6 −1.74203
\(782\) 0 0
\(783\) − 86528.0i − 0.141135i
\(784\) 0 0
\(785\) 569099. 0.923525
\(786\) 0 0
\(787\) − 406715.i − 0.656660i −0.944563 0.328330i \(-0.893514\pi\)
0.944563 0.328330i \(-0.106486\pi\)
\(788\) 0 0
\(789\) −181870. −0.292151
\(790\) 0 0
\(791\) 109340.i 0.174754i
\(792\) 0 0
\(793\) −270414. −0.430013
\(794\) 0 0
\(795\) 102885.i 0.162786i
\(796\) 0 0
\(797\) 1.02634e6 1.61576 0.807878 0.589350i \(-0.200616\pi\)
0.807878 + 0.589350i \(0.200616\pi\)
\(798\) 0 0
\(799\) 287698.i 0.450654i
\(800\) 0 0
\(801\) −38180.1 −0.0595076
\(802\) 0 0
\(803\) 29795.9i 0.0462089i
\(804\) 0 0
\(805\) 48432.3 0.0747383
\(806\) 0 0
\(807\) 213447.i 0.327751i
\(808\) 0 0
\(809\) −245306. −0.374810 −0.187405 0.982283i \(-0.560008\pi\)
−0.187405 + 0.982283i \(0.560008\pi\)
\(810\) 0 0
\(811\) − 1.26760e6i − 1.92726i −0.267246 0.963628i \(-0.586114\pi\)
0.267246 0.963628i \(-0.413886\pi\)
\(812\) 0 0
\(813\) 241201. 0.364920
\(814\) 0 0
\(815\) − 580065.i − 0.873296i
\(816\) 0 0
\(817\) −450142. −0.674382
\(818\) 0 0
\(819\) 14313.5i 0.0213392i
\(820\) 0 0
\(821\) −163484. −0.242542 −0.121271 0.992619i \(-0.538697\pi\)
−0.121271 + 0.992619i \(0.538697\pi\)
\(822\) 0 0
\(823\) 913000.i 1.34794i 0.738758 + 0.673971i \(0.235413\pi\)
−0.738758 + 0.673971i \(0.764587\pi\)
\(824\) 0 0
\(825\) −35491.4 −0.0521453
\(826\) 0 0
\(827\) − 378235.i − 0.553032i −0.961009 0.276516i \(-0.910820\pi\)
0.961009 0.276516i \(-0.0891799\pi\)
\(828\) 0 0
\(829\) 70195.3 0.102141 0.0510704 0.998695i \(-0.483737\pi\)
0.0510704 + 0.998695i \(0.483737\pi\)
\(830\) 0 0
\(831\) 780969.i 1.13092i
\(832\) 0 0
\(833\) −185019. −0.266641
\(834\) 0 0
\(835\) − 73346.5i − 0.105198i
\(836\) 0 0
\(837\) 155982. 0.222651
\(838\) 0 0
\(839\) 410513.i 0.583180i 0.956543 + 0.291590i \(0.0941843\pi\)
−0.956543 + 0.291590i \(0.905816\pi\)
\(840\) 0 0
\(841\) −326897. −0.462188
\(842\) 0 0
\(843\) − 612072.i − 0.861286i
\(844\) 0 0
\(845\) −602478. −0.843777
\(846\) 0 0
\(847\) 19672.2i 0.0274211i
\(848\) 0 0
\(849\) 214308. 0.297320
\(850\) 0 0
\(851\) − 174532.i − 0.240999i
\(852\) 0 0
\(853\) 577725. 0.794005 0.397003 0.917817i \(-0.370050\pi\)
0.397003 + 0.917817i \(0.370050\pi\)
\(854\) 0 0
\(855\) 135488.i 0.185340i
\(856\) 0 0
\(857\) 388294. 0.528687 0.264344 0.964429i \(-0.414845\pi\)
0.264344 + 0.964429i \(0.414845\pi\)
\(858\) 0 0
\(859\) − 124443.i − 0.168650i −0.996438 0.0843248i \(-0.973127\pi\)
0.996438 0.0843248i \(-0.0268733\pi\)
\(860\) 0 0
\(861\) 117589. 0.158620
\(862\) 0 0
\(863\) − 652166.i − 0.875662i −0.899057 0.437831i \(-0.855747\pi\)
0.899057 0.437831i \(-0.144253\pi\)
\(864\) 0 0
\(865\) −442229. −0.591038
\(866\) 0 0
\(867\) 400739.i 0.533118i
\(868\) 0 0
\(869\) 644693. 0.853716
\(870\) 0 0
\(871\) 227434.i 0.299791i
\(872\) 0 0
\(873\) 83972.1 0.110181
\(874\) 0 0
\(875\) − 152855.i − 0.199647i
\(876\) 0 0
\(877\) −75235.9 −0.0978196 −0.0489098 0.998803i \(-0.515575\pi\)
−0.0489098 + 0.998803i \(0.515575\pi\)
\(878\) 0 0
\(879\) − 373800.i − 0.483795i
\(880\) 0 0
\(881\) −916209. −1.18044 −0.590218 0.807244i \(-0.700958\pi\)
−0.590218 + 0.807244i \(0.700958\pi\)
\(882\) 0 0
\(883\) − 1.14981e6i − 1.47471i −0.675506 0.737354i \(-0.736075\pi\)
0.675506 0.737354i \(-0.263925\pi\)
\(884\) 0 0
\(885\) −161588. −0.206311
\(886\) 0 0
\(887\) 1.01630e6i 1.29174i 0.763448 + 0.645870i \(0.223505\pi\)
−0.763448 + 0.645870i \(0.776495\pi\)
\(888\) 0 0
\(889\) 44323.3 0.0560827
\(890\) 0 0
\(891\) 81649.0i 0.102848i
\(892\) 0 0
\(893\) −759945. −0.952970
\(894\) 0 0
\(895\) − 1.20501e6i − 1.50433i
\(896\) 0 0
\(897\) 63814.0 0.0793105
\(898\) 0 0
\(899\) 685711.i 0.848442i
\(900\) 0 0
\(901\) 66691.5 0.0821525
\(902\) 0 0
\(903\) − 103862.i − 0.127375i
\(904\) 0 0
\(905\) −1.37864e6 −1.68327
\(906\) 0 0
\(907\) − 1.00415e6i − 1.22063i −0.792159 0.610315i \(-0.791043\pi\)
0.792159 0.610315i \(-0.208957\pi\)
\(908\) 0 0
\(909\) 13189.5 0.0159625
\(910\) 0 0
\(911\) − 231087.i − 0.278444i −0.990261 0.139222i \(-0.955540\pi\)
0.990261 0.139222i \(-0.0444602\pi\)
\(912\) 0 0
\(913\) 816172. 0.979129
\(914\) 0 0
\(915\) − 590598.i − 0.705424i
\(916\) 0 0
\(917\) 90380.4 0.107482
\(918\) 0 0
\(919\) 545658.i 0.646084i 0.946385 + 0.323042i \(0.104706\pi\)
−0.946385 + 0.323042i \(0.895294\pi\)
\(920\) 0 0
\(921\) −283568. −0.334302
\(922\) 0 0
\(923\) − 536041.i − 0.629209i
\(924\) 0 0
\(925\) −48969.2 −0.0572320
\(926\) 0 0
\(927\) − 542619.i − 0.631445i
\(928\) 0 0
\(929\) −243550. −0.282200 −0.141100 0.989995i \(-0.545064\pi\)
−0.141100 + 0.989995i \(0.545064\pi\)
\(930\) 0 0
\(931\) − 488722.i − 0.563849i
\(932\) 0 0
\(933\) 316684. 0.363800
\(934\) 0 0
\(935\) − 212773.i − 0.243385i
\(936\) 0 0
\(937\) −479076. −0.545664 −0.272832 0.962062i \(-0.587960\pi\)
−0.272832 + 0.962062i \(0.587960\pi\)
\(938\) 0 0
\(939\) 534027.i 0.605665i
\(940\) 0 0
\(941\) 1.25655e6 1.41906 0.709530 0.704675i \(-0.248907\pi\)
0.709530 + 0.704675i \(0.248907\pi\)
\(942\) 0 0
\(943\) − 524246.i − 0.589538i
\(944\) 0 0
\(945\) −31261.5 −0.0350063
\(946\) 0 0
\(947\) − 1.22304e6i − 1.36377i −0.731459 0.681886i \(-0.761160\pi\)
0.731459 0.681886i \(-0.238840\pi\)
\(948\) 0 0
\(949\) −15031.3 −0.0166903
\(950\) 0 0
\(951\) 206699.i 0.228548i
\(952\) 0 0
\(953\) 1.12329e6 1.23682 0.618409 0.785856i \(-0.287777\pi\)
0.618409 + 0.785856i \(0.287777\pi\)
\(954\) 0 0
\(955\) 505512.i 0.554275i
\(956\) 0 0
\(957\) −358936. −0.391916
\(958\) 0 0
\(959\) − 153944.i − 0.167389i
\(960\) 0 0
\(961\) −312598. −0.338485
\(962\) 0 0
\(963\) 334341.i 0.360526i
\(964\) 0 0
\(965\) −619240. −0.664974
\(966\) 0 0
\(967\) 310108.i 0.331635i 0.986156 + 0.165818i \(0.0530263\pi\)
−0.986156 + 0.165818i \(0.946974\pi\)
\(968\) 0 0
\(969\) 87825.7 0.0935350
\(970\) 0 0
\(971\) 459519.i 0.487376i 0.969854 + 0.243688i \(0.0783574\pi\)
−0.969854 + 0.243688i \(0.921643\pi\)
\(972\) 0 0
\(973\) 239668. 0.253154
\(974\) 0 0
\(975\) − 17904.5i − 0.0188345i
\(976\) 0 0
\(977\) −645770. −0.676532 −0.338266 0.941050i \(-0.609840\pi\)
−0.338266 + 0.941050i \(0.609840\pi\)
\(978\) 0 0
\(979\) 158379.i 0.165246i
\(980\) 0 0
\(981\) −419635. −0.436048
\(982\) 0 0
\(983\) 828941.i 0.857860i 0.903338 + 0.428930i \(0.141109\pi\)
−0.903338 + 0.428930i \(0.858891\pi\)
\(984\) 0 0
\(985\) 112363. 0.115811
\(986\) 0 0
\(987\) − 175344.i − 0.179993i
\(988\) 0 0
\(989\) −463050. −0.473407
\(990\) 0 0
\(991\) − 460245.i − 0.468642i −0.972159 0.234321i \(-0.924713\pi\)
0.972159 0.234321i \(-0.0752867\pi\)
\(992\) 0 0
\(993\) 456246. 0.462701
\(994\) 0 0
\(995\) − 915803.i − 0.925030i
\(996\) 0 0
\(997\) −1.82730e6 −1.83831 −0.919156 0.393894i \(-0.871128\pi\)
−0.919156 + 0.393894i \(0.871128\pi\)
\(998\) 0 0
\(999\) 112655.i 0.112881i
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 768.5.g.c.511.2 4
4.3 odd 2 inner 768.5.g.c.511.4 4
8.3 odd 2 768.5.g.g.511.1 4
8.5 even 2 768.5.g.g.511.3 4
16.3 odd 4 384.5.b.c.319.2 8
16.5 even 4 384.5.b.c.319.3 yes 8
16.11 odd 4 384.5.b.c.319.7 yes 8
16.13 even 4 384.5.b.c.319.6 yes 8
48.5 odd 4 1152.5.b.k.703.3 8
48.11 even 4 1152.5.b.k.703.4 8
48.29 odd 4 1152.5.b.k.703.5 8
48.35 even 4 1152.5.b.k.703.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.2 8 16.3 odd 4
384.5.b.c.319.3 yes 8 16.5 even 4
384.5.b.c.319.6 yes 8 16.13 even 4
384.5.b.c.319.7 yes 8 16.11 odd 4
768.5.g.c.511.2 4 1.1 even 1 trivial
768.5.g.c.511.4 4 4.3 odd 2 inner
768.5.g.g.511.1 4 8.3 odd 2
768.5.g.g.511.3 4 8.5 even 2
1152.5.b.k.703.3 8 48.5 odd 4
1152.5.b.k.703.4 8 48.11 even 4
1152.5.b.k.703.5 8 48.29 odd 4
1152.5.b.k.703.6 8 48.35 even 4