Properties

Label 1216.4.a.i.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{93}) \)
Defining polynomial: \(x^{2} - x - 23\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.32183\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -17.2873 q^{5} -12.3563 q^{7} -26.0000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -17.2873 q^{5} -12.3563 q^{7} -26.0000 q^{9} +39.2873 q^{11} -18.9310 q^{13} +17.2873 q^{15} +100.149 q^{17} +19.0000 q^{19} +12.3563 q^{21} +158.218 q^{23} +173.851 q^{25} +53.0000 q^{27} -141.229 q^{29} +215.586 q^{31} -39.2873 q^{33} +213.608 q^{35} -209.862 q^{37} +18.9310 q^{39} +276.873 q^{41} -166.713 q^{43} +449.470 q^{45} -60.5524 q^{47} -190.321 q^{49} -100.149 q^{51} +76.2405 q^{53} -679.171 q^{55} -19.0000 q^{57} -575.575 q^{59} +91.0111 q^{61} +321.265 q^{63} +327.265 q^{65} +444.978 q^{67} -158.218 q^{69} -943.287 q^{71} -569.851 q^{73} -173.851 q^{75} -485.448 q^{77} +336.205 q^{79} +649.000 q^{81} -603.929 q^{83} -1731.31 q^{85} +141.229 q^{87} +1635.63 q^{89} +233.917 q^{91} -215.586 q^{93} -328.459 q^{95} +319.470 q^{97} -1021.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + 40q^{11} + 20q^{13} - 4q^{15} + 46q^{17} + 38q^{19} + 44q^{21} + 220q^{23} + 502q^{25} + 106q^{27} + 84q^{29} + 84q^{31} - 40q^{33} - 460q^{35} - 304q^{37} - 20q^{39} + 168q^{41} - 372q^{43} - 104q^{45} - 584q^{47} + 468q^{49} - 46q^{51} - 484q^{53} - 664q^{55} - 38q^{57} - 1074q^{59} - 88q^{61} + 1144q^{63} + 1156q^{65} + 1430q^{67} - 220q^{69} - 1848q^{71} - 1294q^{73} - 502q^{75} - 508q^{77} - 832q^{79} + 1298q^{81} + 528q^{83} - 2884q^{85} - 84q^{87} + 1844q^{89} - 998q^{91} - 84q^{93} + 76q^{95} - 364q^{97} - 1040q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.192450 −0.0962250 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 0 0
\(5\) −17.2873 −1.54622 −0.773112 0.634270i \(-0.781301\pi\)
−0.773112 + 0.634270i \(0.781301\pi\)
\(6\) 0 0
\(7\) −12.3563 −0.667180 −0.333590 0.942718i \(-0.608260\pi\)
−0.333590 + 0.942718i \(0.608260\pi\)
\(8\) 0 0
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) 39.2873 1.07687 0.538435 0.842667i \(-0.319016\pi\)
0.538435 + 0.842667i \(0.319016\pi\)
\(12\) 0 0
\(13\) −18.9310 −0.403885 −0.201942 0.979397i \(-0.564725\pi\)
−0.201942 + 0.979397i \(0.564725\pi\)
\(14\) 0 0
\(15\) 17.2873 0.297571
\(16\) 0 0
\(17\) 100.149 1.42881 0.714404 0.699733i \(-0.246698\pi\)
0.714404 + 0.699733i \(0.246698\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 12.3563 0.128399
\(22\) 0 0
\(23\) 158.218 1.43438 0.717191 0.696877i \(-0.245428\pi\)
0.717191 + 0.696877i \(0.245428\pi\)
\(24\) 0 0
\(25\) 173.851 1.39081
\(26\) 0 0
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) −141.229 −0.904332 −0.452166 0.891934i \(-0.649349\pi\)
−0.452166 + 0.891934i \(0.649349\pi\)
\(30\) 0 0
\(31\) 215.586 1.24904 0.624522 0.781008i \(-0.285294\pi\)
0.624522 + 0.781008i \(0.285294\pi\)
\(32\) 0 0
\(33\) −39.2873 −0.207244
\(34\) 0 0
\(35\) 213.608 1.03161
\(36\) 0 0
\(37\) −209.862 −0.932462 −0.466231 0.884663i \(-0.654388\pi\)
−0.466231 + 0.884663i \(0.654388\pi\)
\(38\) 0 0
\(39\) 18.9310 0.0777277
\(40\) 0 0
\(41\) 276.873 1.05464 0.527321 0.849666i \(-0.323197\pi\)
0.527321 + 0.849666i \(0.323197\pi\)
\(42\) 0 0
\(43\) −166.713 −0.591243 −0.295621 0.955305i \(-0.595527\pi\)
−0.295621 + 0.955305i \(0.595527\pi\)
\(44\) 0 0
\(45\) 449.470 1.48896
\(46\) 0 0
\(47\) −60.5524 −0.187925 −0.0939625 0.995576i \(-0.529953\pi\)
−0.0939625 + 0.995576i \(0.529953\pi\)
\(48\) 0 0
\(49\) −190.321 −0.554871
\(50\) 0 0
\(51\) −100.149 −0.274974
\(52\) 0 0
\(53\) 76.2405 0.197593 0.0987966 0.995108i \(-0.468501\pi\)
0.0987966 + 0.995108i \(0.468501\pi\)
\(54\) 0 0
\(55\) −679.171 −1.66508
\(56\) 0 0
\(57\) −19.0000 −0.0441511
\(58\) 0 0
\(59\) −575.575 −1.27006 −0.635029 0.772488i \(-0.719012\pi\)
−0.635029 + 0.772488i \(0.719012\pi\)
\(60\) 0 0
\(61\) 91.0111 0.191029 0.0955146 0.995428i \(-0.469550\pi\)
0.0955146 + 0.995428i \(0.469550\pi\)
\(62\) 0 0
\(63\) 321.265 0.642470
\(64\) 0 0
\(65\) 327.265 0.624496
\(66\) 0 0
\(67\) 444.978 0.811383 0.405692 0.914010i \(-0.367031\pi\)
0.405692 + 0.914010i \(0.367031\pi\)
\(68\) 0 0
\(69\) −158.218 −0.276047
\(70\) 0 0
\(71\) −943.287 −1.57673 −0.788363 0.615210i \(-0.789071\pi\)
−0.788363 + 0.615210i \(0.789071\pi\)
\(72\) 0 0
\(73\) −569.851 −0.913644 −0.456822 0.889558i \(-0.651012\pi\)
−0.456822 + 0.889558i \(0.651012\pi\)
\(74\) 0 0
\(75\) −173.851 −0.267661
\(76\) 0 0
\(77\) −485.448 −0.718466
\(78\) 0 0
\(79\) 336.205 0.478810 0.239405 0.970920i \(-0.423048\pi\)
0.239405 + 0.970920i \(0.423048\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) −603.929 −0.798672 −0.399336 0.916805i \(-0.630759\pi\)
−0.399336 + 0.916805i \(0.630759\pi\)
\(84\) 0 0
\(85\) −1731.31 −2.20926
\(86\) 0 0
\(87\) 141.229 0.174039
\(88\) 0 0
\(89\) 1635.63 1.94805 0.974025 0.226441i \(-0.0727089\pi\)
0.974025 + 0.226441i \(0.0727089\pi\)
\(90\) 0 0
\(91\) 233.917 0.269464
\(92\) 0 0
\(93\) −215.586 −0.240378
\(94\) 0 0
\(95\) −328.459 −0.354728
\(96\) 0 0
\(97\) 319.470 0.334405 0.167202 0.985923i \(-0.446527\pi\)
0.167202 + 0.985923i \(0.446527\pi\)
\(98\) 0 0
\(99\) −1021.47 −1.03699
\(100\) 0 0
\(101\) −1595.19 −1.57156 −0.785781 0.618505i \(-0.787739\pi\)
−0.785781 + 0.618505i \(0.787739\pi\)
\(102\) 0 0
\(103\) −1927.51 −1.84392 −0.921959 0.387287i \(-0.873412\pi\)
−0.921959 + 0.387287i \(0.873412\pi\)
\(104\) 0 0
\(105\) −213.608 −0.198533
\(106\) 0 0
\(107\) 717.089 0.647884 0.323942 0.946077i \(-0.394992\pi\)
0.323942 + 0.946077i \(0.394992\pi\)
\(108\) 0 0
\(109\) −383.047 −0.336598 −0.168299 0.985736i \(-0.553827\pi\)
−0.168299 + 0.985736i \(0.553827\pi\)
\(110\) 0 0
\(111\) 209.862 0.179452
\(112\) 0 0
\(113\) 622.917 0.518577 0.259288 0.965800i \(-0.416512\pi\)
0.259288 + 0.965800i \(0.416512\pi\)
\(114\) 0 0
\(115\) −2735.17 −2.21787
\(116\) 0 0
\(117\) 492.205 0.388926
\(118\) 0 0
\(119\) −1237.48 −0.953273
\(120\) 0 0
\(121\) 212.492 0.159648
\(122\) 0 0
\(123\) −276.873 −0.202966
\(124\) 0 0
\(125\) −844.498 −0.604274
\(126\) 0 0
\(127\) −89.3095 −0.0624011 −0.0312005 0.999513i \(-0.509933\pi\)
−0.0312005 + 0.999513i \(0.509933\pi\)
\(128\) 0 0
\(129\) 166.713 0.113785
\(130\) 0 0
\(131\) 348.690 0.232559 0.116279 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(132\) 0 0
\(133\) −234.771 −0.153062
\(134\) 0 0
\(135\) −916.227 −0.584120
\(136\) 0 0
\(137\) 1356.98 0.846237 0.423118 0.906074i \(-0.360935\pi\)
0.423118 + 0.906074i \(0.360935\pi\)
\(138\) 0 0
\(139\) 629.287 0.383996 0.191998 0.981395i \(-0.438503\pi\)
0.191998 + 0.981395i \(0.438503\pi\)
\(140\) 0 0
\(141\) 60.5524 0.0361662
\(142\) 0 0
\(143\) −743.746 −0.434931
\(144\) 0 0
\(145\) 2441.47 1.39830
\(146\) 0 0
\(147\) 190.321 0.106785
\(148\) 0 0
\(149\) 872.641 0.479796 0.239898 0.970798i \(-0.422886\pi\)
0.239898 + 0.970798i \(0.422886\pi\)
\(150\) 0 0
\(151\) 1892.71 1.02004 0.510021 0.860162i \(-0.329638\pi\)
0.510021 + 0.860162i \(0.329638\pi\)
\(152\) 0 0
\(153\) −2603.88 −1.37589
\(154\) 0 0
\(155\) −3726.90 −1.93130
\(156\) 0 0
\(157\) 2939.51 1.49426 0.747129 0.664679i \(-0.231432\pi\)
0.747129 + 0.664679i \(0.231432\pi\)
\(158\) 0 0
\(159\) −76.2405 −0.0380268
\(160\) 0 0
\(161\) −1955.00 −0.956991
\(162\) 0 0
\(163\) −3301.38 −1.58640 −0.793202 0.608959i \(-0.791587\pi\)
−0.793202 + 0.608959i \(0.791587\pi\)
\(164\) 0 0
\(165\) 679.171 0.320445
\(166\) 0 0
\(167\) −2630.39 −1.21884 −0.609418 0.792849i \(-0.708597\pi\)
−0.609418 + 0.792849i \(0.708597\pi\)
\(168\) 0 0
\(169\) −1838.62 −0.836877
\(170\) 0 0
\(171\) −494.000 −0.220919
\(172\) 0 0
\(173\) 1482.41 0.651479 0.325740 0.945460i \(-0.394387\pi\)
0.325740 + 0.945460i \(0.394387\pi\)
\(174\) 0 0
\(175\) −2148.16 −0.927918
\(176\) 0 0
\(177\) 575.575 0.244423
\(178\) 0 0
\(179\) −1048.00 −0.437604 −0.218802 0.975769i \(-0.570215\pi\)
−0.218802 + 0.975769i \(0.570215\pi\)
\(180\) 0 0
\(181\) 761.354 0.312657 0.156329 0.987705i \(-0.450034\pi\)
0.156329 + 0.987705i \(0.450034\pi\)
\(182\) 0 0
\(183\) −91.0111 −0.0367636
\(184\) 0 0
\(185\) 3627.95 1.44179
\(186\) 0 0
\(187\) 3934.59 1.53864
\(188\) 0 0
\(189\) −654.887 −0.252042
\(190\) 0 0
\(191\) 4005.98 1.51760 0.758802 0.651322i \(-0.225785\pi\)
0.758802 + 0.651322i \(0.225785\pi\)
\(192\) 0 0
\(193\) −3855.87 −1.43809 −0.719046 0.694962i \(-0.755421\pi\)
−0.719046 + 0.694962i \(0.755421\pi\)
\(194\) 0 0
\(195\) −327.265 −0.120184
\(196\) 0 0
\(197\) 2719.72 0.983616 0.491808 0.870704i \(-0.336336\pi\)
0.491808 + 0.870704i \(0.336336\pi\)
\(198\) 0 0
\(199\) −4454.56 −1.58681 −0.793405 0.608694i \(-0.791694\pi\)
−0.793405 + 0.608694i \(0.791694\pi\)
\(200\) 0 0
\(201\) −444.978 −0.156151
\(202\) 0 0
\(203\) 1745.08 0.603353
\(204\) 0 0
\(205\) −4786.39 −1.63071
\(206\) 0 0
\(207\) −4113.67 −1.38126
\(208\) 0 0
\(209\) 746.459 0.247051
\(210\) 0 0
\(211\) 3481.06 1.13576 0.567881 0.823111i \(-0.307763\pi\)
0.567881 + 0.823111i \(0.307763\pi\)
\(212\) 0 0
\(213\) 943.287 0.303441
\(214\) 0 0
\(215\) 2882.01 0.914194
\(216\) 0 0
\(217\) −2663.85 −0.833337
\(218\) 0 0
\(219\) 569.851 0.175831
\(220\) 0 0
\(221\) −1895.92 −0.577074
\(222\) 0 0
\(223\) 3193.81 0.959074 0.479537 0.877522i \(-0.340805\pi\)
0.479537 + 0.877522i \(0.340805\pi\)
\(224\) 0 0
\(225\) −4520.12 −1.33930
\(226\) 0 0
\(227\) −5139.70 −1.50279 −0.751396 0.659852i \(-0.770619\pi\)
−0.751396 + 0.659852i \(0.770619\pi\)
\(228\) 0 0
\(229\) −4310.87 −1.24397 −0.621987 0.783027i \(-0.713674\pi\)
−0.621987 + 0.783027i \(0.713674\pi\)
\(230\) 0 0
\(231\) 485.448 0.138269
\(232\) 0 0
\(233\) 175.879 0.0494517 0.0247258 0.999694i \(-0.492129\pi\)
0.0247258 + 0.999694i \(0.492129\pi\)
\(234\) 0 0
\(235\) 1046.79 0.290574
\(236\) 0 0
\(237\) −336.205 −0.0921470
\(238\) 0 0
\(239\) 3811.82 1.03166 0.515829 0.856692i \(-0.327484\pi\)
0.515829 + 0.856692i \(0.327484\pi\)
\(240\) 0 0
\(241\) −1356.97 −0.362698 −0.181349 0.983419i \(-0.558046\pi\)
−0.181349 + 0.983419i \(0.558046\pi\)
\(242\) 0 0
\(243\) −2080.00 −0.549103
\(244\) 0 0
\(245\) 3290.13 0.857954
\(246\) 0 0
\(247\) −359.688 −0.0926575
\(248\) 0 0
\(249\) 603.929 0.153704
\(250\) 0 0
\(251\) −5465.11 −1.37432 −0.687160 0.726506i \(-0.741143\pi\)
−0.687160 + 0.726506i \(0.741143\pi\)
\(252\) 0 0
\(253\) 6215.97 1.54464
\(254\) 0 0
\(255\) 1731.31 0.425172
\(256\) 0 0
\(257\) 1249.60 0.303299 0.151649 0.988434i \(-0.451542\pi\)
0.151649 + 0.988434i \(0.451542\pi\)
\(258\) 0 0
\(259\) 2593.13 0.622120
\(260\) 0 0
\(261\) 3671.96 0.870838
\(262\) 0 0
\(263\) −8043.06 −1.88577 −0.942883 0.333124i \(-0.891897\pi\)
−0.942883 + 0.333124i \(0.891897\pi\)
\(264\) 0 0
\(265\) −1317.99 −0.305523
\(266\) 0 0
\(267\) −1635.63 −0.374902
\(268\) 0 0
\(269\) −3051.13 −0.691565 −0.345782 0.938315i \(-0.612386\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(270\) 0 0
\(271\) 1967.96 0.441127 0.220563 0.975373i \(-0.429210\pi\)
0.220563 + 0.975373i \(0.429210\pi\)
\(272\) 0 0
\(273\) −233.917 −0.0518583
\(274\) 0 0
\(275\) 6830.13 1.49772
\(276\) 0 0
\(277\) 4077.51 0.884454 0.442227 0.896903i \(-0.354189\pi\)
0.442227 + 0.896903i \(0.354189\pi\)
\(278\) 0 0
\(279\) −5605.23 −1.20278
\(280\) 0 0
\(281\) −7982.51 −1.69465 −0.847325 0.531075i \(-0.821788\pi\)
−0.847325 + 0.531075i \(0.821788\pi\)
\(282\) 0 0
\(283\) −4137.36 −0.869049 −0.434524 0.900660i \(-0.643083\pi\)
−0.434524 + 0.900660i \(0.643083\pi\)
\(284\) 0 0
\(285\) 328.459 0.0682674
\(286\) 0 0
\(287\) −3421.14 −0.703636
\(288\) 0 0
\(289\) 5116.86 1.04149
\(290\) 0 0
\(291\) −319.470 −0.0643562
\(292\) 0 0
\(293\) −6807.50 −1.35733 −0.678666 0.734447i \(-0.737442\pi\)
−0.678666 + 0.734447i \(0.737442\pi\)
\(294\) 0 0
\(295\) 9950.13 1.96379
\(296\) 0 0
\(297\) 2082.23 0.406812
\(298\) 0 0
\(299\) −2995.22 −0.579325
\(300\) 0 0
\(301\) 2059.96 0.394466
\(302\) 0 0
\(303\) 1595.19 0.302447
\(304\) 0 0
\(305\) −1573.34 −0.295374
\(306\) 0 0
\(307\) −2822.75 −0.524765 −0.262383 0.964964i \(-0.584508\pi\)
−0.262383 + 0.964964i \(0.584508\pi\)
\(308\) 0 0
\(309\) 1927.51 0.354862
\(310\) 0 0
\(311\) −8221.36 −1.49901 −0.749503 0.662001i \(-0.769707\pi\)
−0.749503 + 0.662001i \(0.769707\pi\)
\(312\) 0 0
\(313\) −1896.42 −0.342466 −0.171233 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(314\) 0 0
\(315\) −5553.81 −0.993402
\(316\) 0 0
\(317\) 2875.66 0.509505 0.254753 0.967006i \(-0.418006\pi\)
0.254753 + 0.967006i \(0.418006\pi\)
\(318\) 0 0
\(319\) −5548.52 −0.973848
\(320\) 0 0
\(321\) −717.089 −0.124685
\(322\) 0 0
\(323\) 1902.83 0.327791
\(324\) 0 0
\(325\) −3291.16 −0.561725
\(326\) 0 0
\(327\) 383.047 0.0647784
\(328\) 0 0
\(329\) 748.206 0.125380
\(330\) 0 0
\(331\) 9263.48 1.53827 0.769134 0.639088i \(-0.220688\pi\)
0.769134 + 0.639088i \(0.220688\pi\)
\(332\) 0 0
\(333\) 5456.41 0.897926
\(334\) 0 0
\(335\) −7692.47 −1.25458
\(336\) 0 0
\(337\) −2071.30 −0.334810 −0.167405 0.985888i \(-0.553539\pi\)
−0.167405 + 0.985888i \(0.553539\pi\)
\(338\) 0 0
\(339\) −622.917 −0.0998001
\(340\) 0 0
\(341\) 8469.78 1.34506
\(342\) 0 0
\(343\) 6589.90 1.03738
\(344\) 0 0
\(345\) 2735.17 0.426830
\(346\) 0 0
\(347\) −8255.77 −1.27721 −0.638607 0.769533i \(-0.720489\pi\)
−0.638607 + 0.769533i \(0.720489\pi\)
\(348\) 0 0
\(349\) 6600.97 1.01244 0.506220 0.862404i \(-0.331042\pi\)
0.506220 + 0.862404i \(0.331042\pi\)
\(350\) 0 0
\(351\) −1003.34 −0.152577
\(352\) 0 0
\(353\) −6136.89 −0.925308 −0.462654 0.886539i \(-0.653103\pi\)
−0.462654 + 0.886539i \(0.653103\pi\)
\(354\) 0 0
\(355\) 16306.9 2.43797
\(356\) 0 0
\(357\) 1237.48 0.183457
\(358\) 0 0
\(359\) −4314.86 −0.634344 −0.317172 0.948368i \(-0.602733\pi\)
−0.317172 + 0.948368i \(0.602733\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −212.492 −0.0307244
\(364\) 0 0
\(365\) 9851.18 1.41270
\(366\) 0 0
\(367\) 2643.59 0.376006 0.188003 0.982168i \(-0.439798\pi\)
0.188003 + 0.982168i \(0.439798\pi\)
\(368\) 0 0
\(369\) −7198.70 −1.01558
\(370\) 0 0
\(371\) −942.054 −0.131830
\(372\) 0 0
\(373\) −3643.51 −0.505774 −0.252887 0.967496i \(-0.581380\pi\)
−0.252887 + 0.967496i \(0.581380\pi\)
\(374\) 0 0
\(375\) 844.498 0.116293
\(376\) 0 0
\(377\) 2673.61 0.365246
\(378\) 0 0
\(379\) −9731.09 −1.31887 −0.659436 0.751761i \(-0.729205\pi\)
−0.659436 + 0.751761i \(0.729205\pi\)
\(380\) 0 0
\(381\) 89.3095 0.0120091
\(382\) 0 0
\(383\) −4167.84 −0.556049 −0.278025 0.960574i \(-0.589680\pi\)
−0.278025 + 0.960574i \(0.589680\pi\)
\(384\) 0 0
\(385\) 8392.08 1.11091
\(386\) 0 0
\(387\) 4334.53 0.569345
\(388\) 0 0
\(389\) 4267.52 0.556226 0.278113 0.960548i \(-0.410291\pi\)
0.278113 + 0.960548i \(0.410291\pi\)
\(390\) 0 0
\(391\) 15845.4 2.04946
\(392\) 0 0
\(393\) −348.690 −0.0447560
\(394\) 0 0
\(395\) −5812.07 −0.740347
\(396\) 0 0
\(397\) −7610.34 −0.962096 −0.481048 0.876694i \(-0.659744\pi\)
−0.481048 + 0.876694i \(0.659744\pi\)
\(398\) 0 0
\(399\) 234.771 0.0294567
\(400\) 0 0
\(401\) −15827.1 −1.97099 −0.985495 0.169704i \(-0.945719\pi\)
−0.985495 + 0.169704i \(0.945719\pi\)
\(402\) 0 0
\(403\) −4081.24 −0.504469
\(404\) 0 0
\(405\) −11219.5 −1.37654
\(406\) 0 0
\(407\) −8244.91 −1.00414
\(408\) 0 0
\(409\) 5134.26 0.620716 0.310358 0.950620i \(-0.399551\pi\)
0.310358 + 0.950620i \(0.399551\pi\)
\(410\) 0 0
\(411\) −1356.98 −0.162858
\(412\) 0 0
\(413\) 7112.00 0.847358
\(414\) 0 0
\(415\) 10440.3 1.23493
\(416\) 0 0
\(417\) −629.287 −0.0739001
\(418\) 0 0
\(419\) −5437.64 −0.634001 −0.317000 0.948425i \(-0.602676\pi\)
−0.317000 + 0.948425i \(0.602676\pi\)
\(420\) 0 0
\(421\) 1552.15 0.179685 0.0898423 0.995956i \(-0.471364\pi\)
0.0898423 + 0.995956i \(0.471364\pi\)
\(422\) 0 0
\(423\) 1574.36 0.180965
\(424\) 0 0
\(425\) 17411.0 1.98720
\(426\) 0 0
\(427\) −1124.57 −0.127451
\(428\) 0 0
\(429\) 743.746 0.0837026
\(430\) 0 0
\(431\) −5106.26 −0.570673 −0.285336 0.958427i \(-0.592105\pi\)
−0.285336 + 0.958427i \(0.592105\pi\)
\(432\) 0 0
\(433\) 17422.7 1.93367 0.966835 0.255401i \(-0.0822076\pi\)
0.966835 + 0.255401i \(0.0822076\pi\)
\(434\) 0 0
\(435\) −2441.47 −0.269103
\(436\) 0 0
\(437\) 3006.15 0.329070
\(438\) 0 0
\(439\) 2682.09 0.291592 0.145796 0.989315i \(-0.453426\pi\)
0.145796 + 0.989315i \(0.453426\pi\)
\(440\) 0 0
\(441\) 4948.34 0.534320
\(442\) 0 0
\(443\) −7407.64 −0.794464 −0.397232 0.917718i \(-0.630029\pi\)
−0.397232 + 0.917718i \(0.630029\pi\)
\(444\) 0 0
\(445\) −28275.6 −3.01212
\(446\) 0 0
\(447\) −872.641 −0.0923367
\(448\) 0 0
\(449\) −15930.7 −1.67442 −0.837212 0.546878i \(-0.815816\pi\)
−0.837212 + 0.546878i \(0.815816\pi\)
\(450\) 0 0
\(451\) 10877.6 1.13571
\(452\) 0 0
\(453\) −1892.71 −0.196307
\(454\) 0 0
\(455\) −4043.80 −0.416651
\(456\) 0 0
\(457\) −6952.91 −0.711693 −0.355846 0.934544i \(-0.615807\pi\)
−0.355846 + 0.934544i \(0.615807\pi\)
\(458\) 0 0
\(459\) 5307.91 0.539765
\(460\) 0 0
\(461\) −13281.0 −1.34177 −0.670886 0.741561i \(-0.734086\pi\)
−0.670886 + 0.741561i \(0.734086\pi\)
\(462\) 0 0
\(463\) −6220.21 −0.624358 −0.312179 0.950023i \(-0.601059\pi\)
−0.312179 + 0.950023i \(0.601059\pi\)
\(464\) 0 0
\(465\) 3726.90 0.371679
\(466\) 0 0
\(467\) 9875.06 0.978508 0.489254 0.872141i \(-0.337269\pi\)
0.489254 + 0.872141i \(0.337269\pi\)
\(468\) 0 0
\(469\) −5498.30 −0.541339
\(470\) 0 0
\(471\) −2939.51 −0.287570
\(472\) 0 0
\(473\) −6549.69 −0.636692
\(474\) 0 0
\(475\) 3303.17 0.319073
\(476\) 0 0
\(477\) −1982.25 −0.190275
\(478\) 0 0
\(479\) 19014.0 1.81372 0.906861 0.421429i \(-0.138471\pi\)
0.906861 + 0.421429i \(0.138471\pi\)
\(480\) 0 0
\(481\) 3972.89 0.376607
\(482\) 0 0
\(483\) 1955.00 0.184173
\(484\) 0 0
\(485\) −5522.77 −0.517064
\(486\) 0 0
\(487\) 8907.07 0.828784 0.414392 0.910099i \(-0.363994\pi\)
0.414392 + 0.910099i \(0.363994\pi\)
\(488\) 0 0
\(489\) 3301.38 0.305303
\(490\) 0 0
\(491\) 18451.8 1.69596 0.847981 0.530027i \(-0.177818\pi\)
0.847981 + 0.530027i \(0.177818\pi\)
\(492\) 0 0
\(493\) −14144.0 −1.29212
\(494\) 0 0
\(495\) 17658.5 1.60341
\(496\) 0 0
\(497\) 11655.6 1.05196
\(498\) 0 0
\(499\) 14771.6 1.32519 0.662594 0.748979i \(-0.269456\pi\)
0.662594 + 0.748979i \(0.269456\pi\)
\(500\) 0 0
\(501\) 2630.39 0.234565
\(502\) 0 0
\(503\) 18175.4 1.61114 0.805570 0.592501i \(-0.201859\pi\)
0.805570 + 0.592501i \(0.201859\pi\)
\(504\) 0 0
\(505\) 27576.6 2.42998
\(506\) 0 0
\(507\) 1838.62 0.161057
\(508\) 0 0
\(509\) 5172.14 0.450395 0.225198 0.974313i \(-0.427697\pi\)
0.225198 + 0.974313i \(0.427697\pi\)
\(510\) 0 0
\(511\) 7041.28 0.609565
\(512\) 0 0
\(513\) 1007.00 0.0866669
\(514\) 0 0
\(515\) 33321.5 2.85111
\(516\) 0 0
\(517\) −2378.94 −0.202371
\(518\) 0 0
\(519\) −1482.41 −0.125377
\(520\) 0 0
\(521\) −12920.1 −1.08645 −0.543225 0.839587i \(-0.682797\pi\)
−0.543225 + 0.839587i \(0.682797\pi\)
\(522\) 0 0
\(523\) −2486.41 −0.207884 −0.103942 0.994583i \(-0.533146\pi\)
−0.103942 + 0.994583i \(0.533146\pi\)
\(524\) 0 0
\(525\) 2148.16 0.178578
\(526\) 0 0
\(527\) 21590.7 1.78464
\(528\) 0 0
\(529\) 12866.0 1.05745
\(530\) 0 0
\(531\) 14964.9 1.22302
\(532\) 0 0
\(533\) −5241.47 −0.425954
\(534\) 0 0
\(535\) −12396.5 −1.00177
\(536\) 0 0
\(537\) 1048.00 0.0842170
\(538\) 0 0
\(539\) −7477.18 −0.597523
\(540\) 0 0
\(541\) −4553.45 −0.361863 −0.180932 0.983496i \(-0.557911\pi\)
−0.180932 + 0.983496i \(0.557911\pi\)
\(542\) 0 0
\(543\) −761.354 −0.0601710
\(544\) 0 0
\(545\) 6621.85 0.520456
\(546\) 0 0
\(547\) 18289.1 1.42959 0.714796 0.699333i \(-0.246520\pi\)
0.714796 + 0.699333i \(0.246520\pi\)
\(548\) 0 0
\(549\) −2366.29 −0.183954
\(550\) 0 0
\(551\) −2683.36 −0.207468
\(552\) 0 0
\(553\) −4154.26 −0.319453
\(554\) 0 0
\(555\) −3627.95 −0.277473
\(556\) 0 0
\(557\) 21732.8 1.65323 0.826613 0.562771i \(-0.190265\pi\)
0.826613 + 0.562771i \(0.190265\pi\)
\(558\) 0 0
\(559\) 3156.03 0.238794
\(560\) 0 0
\(561\) −3934.59 −0.296112
\(562\) 0 0
\(563\) −6231.21 −0.466455 −0.233227 0.972422i \(-0.574929\pi\)
−0.233227 + 0.972422i \(0.574929\pi\)
\(564\) 0 0
\(565\) −10768.6 −0.801835
\(566\) 0 0
\(567\) −8019.27 −0.593964
\(568\) 0 0
\(569\) −24670.7 −1.81766 −0.908829 0.417168i \(-0.863023\pi\)
−0.908829 + 0.417168i \(0.863023\pi\)
\(570\) 0 0
\(571\) −20912.5 −1.53268 −0.766342 0.642433i \(-0.777925\pi\)
−0.766342 + 0.642433i \(0.777925\pi\)
\(572\) 0 0
\(573\) −4005.98 −0.292063
\(574\) 0 0
\(575\) 27506.4 1.99495
\(576\) 0 0
\(577\) 20604.1 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(578\) 0 0
\(579\) 3855.87 0.276761
\(580\) 0 0
\(581\) 7462.35 0.532858
\(582\) 0 0
\(583\) 2995.28 0.212782
\(584\) 0 0
\(585\) −8508.89 −0.601366
\(586\) 0 0
\(587\) −5511.88 −0.387563 −0.193781 0.981045i \(-0.562075\pi\)
−0.193781 + 0.981045i \(0.562075\pi\)
\(588\) 0 0
\(589\) 4096.13 0.286550
\(590\) 0 0
\(591\) −2719.72 −0.189297
\(592\) 0 0
\(593\) 11836.1 0.819648 0.409824 0.912165i \(-0.365590\pi\)
0.409824 + 0.912165i \(0.365590\pi\)
\(594\) 0 0
\(595\) 21392.7 1.47397
\(596\) 0 0
\(597\) 4454.56 0.305382
\(598\) 0 0
\(599\) −8407.13 −0.573466 −0.286733 0.958011i \(-0.592569\pi\)
−0.286733 + 0.958011i \(0.592569\pi\)
\(600\) 0 0
\(601\) 11313.6 0.767872 0.383936 0.923360i \(-0.374568\pi\)
0.383936 + 0.923360i \(0.374568\pi\)
\(602\) 0 0
\(603\) −11569.4 −0.781332
\(604\) 0 0
\(605\) −3673.41 −0.246852
\(606\) 0 0
\(607\) 1581.43 0.105747 0.0528733 0.998601i \(-0.483162\pi\)
0.0528733 + 0.998601i \(0.483162\pi\)
\(608\) 0 0
\(609\) −1745.08 −0.116115
\(610\) 0 0
\(611\) 1146.31 0.0759000
\(612\) 0 0
\(613\) 2299.35 0.151500 0.0757502 0.997127i \(-0.475865\pi\)
0.0757502 + 0.997127i \(0.475865\pi\)
\(614\) 0 0
\(615\) 4786.39 0.313831
\(616\) 0 0
\(617\) −16144.5 −1.05341 −0.526703 0.850049i \(-0.676572\pi\)
−0.526703 + 0.850049i \(0.676572\pi\)
\(618\) 0 0
\(619\) 2734.58 0.177564 0.0887819 0.996051i \(-0.471703\pi\)
0.0887819 + 0.996051i \(0.471703\pi\)
\(620\) 0 0
\(621\) 8385.57 0.541870
\(622\) 0 0
\(623\) −20210.4 −1.29970
\(624\) 0 0
\(625\) −7132.25 −0.456464
\(626\) 0 0
\(627\) −746.459 −0.0475450
\(628\) 0 0
\(629\) −21017.5 −1.33231
\(630\) 0 0
\(631\) −15419.2 −0.972787 −0.486393 0.873740i \(-0.661688\pi\)
−0.486393 + 0.873740i \(0.661688\pi\)
\(632\) 0 0
\(633\) −3481.06 −0.218578
\(634\) 0 0
\(635\) 1543.92 0.0964860
\(636\) 0 0
\(637\) 3602.95 0.224104
\(638\) 0 0
\(639\) 24525.5 1.51833
\(640\) 0 0
\(641\) −11905.5 −0.733600 −0.366800 0.930300i \(-0.619547\pi\)
−0.366800 + 0.930300i \(0.619547\pi\)
\(642\) 0 0
\(643\) −3525.47 −0.216223 −0.108111 0.994139i \(-0.534480\pi\)
−0.108111 + 0.994139i \(0.534480\pi\)
\(644\) 0 0
\(645\) −2882.01 −0.175937
\(646\) 0 0
\(647\) −7502.61 −0.455886 −0.227943 0.973675i \(-0.573200\pi\)
−0.227943 + 0.973675i \(0.573200\pi\)
\(648\) 0 0
\(649\) −22612.8 −1.36769
\(650\) 0 0
\(651\) 2663.85 0.160376
\(652\) 0 0
\(653\) 6539.35 0.391891 0.195945 0.980615i \(-0.437222\pi\)
0.195945 + 0.980615i \(0.437222\pi\)
\(654\) 0 0
\(655\) −6027.92 −0.359588
\(656\) 0 0
\(657\) 14816.1 0.879805
\(658\) 0 0
\(659\) −22229.4 −1.31401 −0.657007 0.753884i \(-0.728178\pi\)
−0.657007 + 0.753884i \(0.728178\pi\)
\(660\) 0 0
\(661\) 19497.3 1.14729 0.573643 0.819105i \(-0.305530\pi\)
0.573643 + 0.819105i \(0.305530\pi\)
\(662\) 0 0
\(663\) 1895.92 0.111058
\(664\) 0 0
\(665\) 4058.55 0.236667
\(666\) 0 0
\(667\) −22345.1 −1.29716
\(668\) 0 0
\(669\) −3193.81 −0.184574
\(670\) 0 0
\(671\) 3575.58 0.205714
\(672\) 0 0
\(673\) −6952.91 −0.398239 −0.199120 0.979975i \(-0.563808\pi\)
−0.199120 + 0.979975i \(0.563808\pi\)
\(674\) 0 0
\(675\) 9214.09 0.525408
\(676\) 0 0
\(677\) −2010.38 −0.114129 −0.0570643 0.998371i \(-0.518174\pi\)
−0.0570643 + 0.998371i \(0.518174\pi\)
\(678\) 0 0
\(679\) −3947.48 −0.223108
\(680\) 0 0
\(681\) 5139.70 0.289212
\(682\) 0 0
\(683\) 16401.7 0.918877 0.459439 0.888209i \(-0.348051\pi\)
0.459439 + 0.888209i \(0.348051\pi\)
\(684\) 0 0
\(685\) −23458.5 −1.30847
\(686\) 0 0
\(687\) 4310.87 0.239403
\(688\) 0 0
\(689\) −1443.30 −0.0798048
\(690\) 0 0
\(691\) −15967.5 −0.879064 −0.439532 0.898227i \(-0.644856\pi\)
−0.439532 + 0.898227i \(0.644856\pi\)
\(692\) 0 0
\(693\) 12621.6 0.691856
\(694\) 0 0
\(695\) −10878.7 −0.593744
\(696\) 0 0
\(697\) 27728.6 1.50688
\(698\) 0 0
\(699\) −175.879 −0.00951698
\(700\) 0 0
\(701\) −22503.7 −1.21248 −0.606242 0.795280i \(-0.707324\pi\)
−0.606242 + 0.795280i \(0.707324\pi\)
\(702\) 0 0
\(703\) −3987.38 −0.213921
\(704\) 0 0
\(705\) −1046.79 −0.0559210
\(706\) 0 0
\(707\) 19710.8 1.04851
\(708\) 0 0
\(709\) −14922.1 −0.790426 −0.395213 0.918590i \(-0.629329\pi\)
−0.395213 + 0.918590i \(0.629329\pi\)
\(710\) 0 0
\(711\) −8741.32 −0.461076
\(712\) 0 0
\(713\) 34109.6 1.79161
\(714\) 0 0
\(715\) 12857.4 0.672501
\(716\) 0 0
\(717\) −3811.82 −0.198543
\(718\) 0 0
\(719\) 3483.14 0.180667 0.0903333 0.995912i \(-0.471207\pi\)
0.0903333 + 0.995912i \(0.471207\pi\)
\(720\) 0 0
\(721\) 23817.0 1.23023
\(722\) 0 0
\(723\) 1356.97 0.0698013
\(724\) 0 0
\(725\) −24552.8 −1.25775
\(726\) 0 0
\(727\) −6572.23 −0.335283 −0.167641 0.985848i \(-0.553615\pi\)
−0.167641 + 0.985848i \(0.553615\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 0 0
\(731\) −16696.1 −0.844773
\(732\) 0 0
\(733\) 27149.4 1.36806 0.684030 0.729454i \(-0.260226\pi\)
0.684030 + 0.729454i \(0.260226\pi\)
\(734\) 0 0
\(735\) −3290.13 −0.165113
\(736\) 0 0
\(737\) 17482.0 0.873754
\(738\) 0 0
\(739\) −26251.0 −1.30671 −0.653355 0.757051i \(-0.726639\pi\)
−0.653355 + 0.757051i \(0.726639\pi\)
\(740\) 0 0
\(741\) 359.688 0.0178319
\(742\) 0 0
\(743\) −10583.0 −0.522545 −0.261273 0.965265i \(-0.584142\pi\)
−0.261273 + 0.965265i \(0.584142\pi\)
\(744\) 0 0
\(745\) −15085.6 −0.741871
\(746\) 0 0
\(747\) 15702.1 0.769092
\(748\) 0 0
\(749\) −8860.60 −0.432255
\(750\) 0 0
\(751\) −9812.03 −0.476759 −0.238380 0.971172i \(-0.576616\pi\)
−0.238380 + 0.971172i \(0.576616\pi\)
\(752\) 0 0
\(753\) 5465.11 0.264488
\(754\) 0 0
\(755\) −32719.8 −1.57721
\(756\) 0 0
\(757\) −10954.1 −0.525938 −0.262969 0.964804i \(-0.584702\pi\)
−0.262969 + 0.964804i \(0.584702\pi\)
\(758\) 0 0
\(759\) −6215.97 −0.297267
\(760\) 0 0
\(761\) −19350.2 −0.921739 −0.460869 0.887468i \(-0.652462\pi\)
−0.460869 + 0.887468i \(0.652462\pi\)
\(762\) 0 0
\(763\) 4733.06 0.224572
\(764\) 0 0
\(765\) 45014.0 2.12743
\(766\) 0 0
\(767\) 10896.2 0.512957
\(768\) 0 0
\(769\) 17844.5 0.836786 0.418393 0.908266i \(-0.362593\pi\)
0.418393 + 0.908266i \(0.362593\pi\)
\(770\) 0 0
\(771\) −1249.60 −0.0583699
\(772\) 0 0
\(773\) 38019.3 1.76903 0.884514 0.466513i \(-0.154490\pi\)
0.884514 + 0.466513i \(0.154490\pi\)
\(774\) 0 0
\(775\) 37479.7 1.73718
\(776\) 0 0
\(777\) −2593.13 −0.119727
\(778\) 0 0
\(779\) 5260.59 0.241951
\(780\) 0 0
\(781\) −37059.2 −1.69793
\(782\) 0 0
\(783\) −7485.16 −0.341632
\(784\) 0 0
\(785\) −50816.2 −2.31046
\(786\) 0 0
\(787\) 32884.2 1.48945 0.744724 0.667373i \(-0.232581\pi\)
0.744724 + 0.667373i \(0.232581\pi\)
\(788\) 0 0
\(789\) 8043.06 0.362916
\(790\) 0 0
\(791\) −7696.99 −0.345984
\(792\) 0 0
\(793\) −1722.93 −0.0771538
\(794\) 0 0
\(795\) 1317.99 0.0587979
\(796\) 0 0
\(797\) −4141.59 −0.184069 −0.0920344 0.995756i \(-0.529337\pi\)
−0.0920344 + 0.995756i \(0.529337\pi\)
\(798\) 0 0
\(799\) −6064.27 −0.268509
\(800\) 0 0
\(801\) −42526.4 −1.87590
\(802\) 0 0
\(803\) −22387.9 −0.983875
\(804\) 0 0
\(805\) 33796.7 1.47972
\(806\) 0 0
\(807\) 3051.13 0.133092
\(808\) 0 0
\(809\) 17007.2 0.739111 0.369555 0.929209i \(-0.379510\pi\)
0.369555 + 0.929209i \(0.379510\pi\)
\(810\) 0 0
\(811\) −18814.0 −0.814611 −0.407306 0.913292i \(-0.633532\pi\)
−0.407306 + 0.913292i \(0.633532\pi\)
\(812\) 0 0
\(813\) −1967.96 −0.0848949
\(814\) 0 0
\(815\) 57071.9 2.45293
\(816\) 0 0
\(817\) −3167.54 −0.135640
\(818\) 0 0
\(819\) −6081.85 −0.259484
\(820\) 0 0
\(821\) 30735.2 1.30653 0.653267 0.757128i \(-0.273398\pi\)
0.653267 + 0.757128i \(0.273398\pi\)
\(822\) 0 0
\(823\) −11638.0 −0.492923 −0.246461 0.969153i \(-0.579268\pi\)
−0.246461 + 0.969153i \(0.579268\pi\)
\(824\) 0 0
\(825\) −6830.13 −0.288236
\(826\) 0 0
\(827\) 4691.30 0.197258 0.0986291 0.995124i \(-0.468554\pi\)
0.0986291 + 0.995124i \(0.468554\pi\)
\(828\) 0 0
\(829\) 12990.2 0.544232 0.272116 0.962264i \(-0.412276\pi\)
0.272116 + 0.962264i \(0.412276\pi\)
\(830\) 0 0
\(831\) −4077.51 −0.170213
\(832\) 0 0
\(833\) −19060.5 −0.792804
\(834\) 0 0
\(835\) 45472.3 1.88459
\(836\) 0 0
\(837\) 11426.0 0.471854
\(838\) 0 0
\(839\) −22152.7 −0.911558 −0.455779 0.890093i \(-0.650639\pi\)
−0.455779 + 0.890093i \(0.650639\pi\)
\(840\) 0 0
\(841\) −4443.27 −0.182183
\(842\) 0 0
\(843\) 7982.51 0.326136
\(844\) 0 0
\(845\) 31784.8 1.29400
\(846\) 0 0
\(847\) −2625.63 −0.106514
\(848\) 0 0
\(849\) 4137.36 0.167248
\(850\) 0 0
\(851\) −33204.0 −1.33751
\(852\) 0 0
\(853\) −27556.6 −1.10612 −0.553060 0.833141i \(-0.686540\pi\)
−0.553060 + 0.833141i \(0.686540\pi\)
\(854\) 0 0
\(855\) 8539.93 0.341590
\(856\) 0 0
\(857\) −2728.49 −0.108755 −0.0543777 0.998520i \(-0.517318\pi\)
−0.0543777 + 0.998520i \(0.517318\pi\)
\(858\) 0 0
\(859\) 24012.7 0.953786 0.476893 0.878961i \(-0.341763\pi\)
0.476893 + 0.878961i \(0.341763\pi\)
\(860\) 0 0
\(861\) 3421.14 0.135415
\(862\) 0 0
\(863\) −23821.0 −0.939603 −0.469802 0.882772i \(-0.655675\pi\)
−0.469802 + 0.882772i \(0.655675\pi\)
\(864\) 0 0
\(865\) −25626.9 −1.00733
\(866\) 0 0
\(867\) −5116.86 −0.200436
\(868\) 0 0
\(869\) 13208.6 0.515616
\(870\) 0 0
\(871\) −8423.85 −0.327705
\(872\) 0 0
\(873\) −8306.22 −0.322019
\(874\) 0 0
\(875\) 10434.9 0.403160
\(876\) 0 0
\(877\) −19.8882 −0.000765765 0 −0.000382882 1.00000i \(-0.500122\pi\)
−0.000382882 1.00000i \(0.500122\pi\)
\(878\) 0 0
\(879\) 6807.50 0.261219
\(880\) 0 0
\(881\) −846.013 −0.0323529 −0.0161764 0.999869i \(-0.505149\pi\)
−0.0161764 + 0.999869i \(0.505149\pi\)
\(882\) 0 0
\(883\) 14028.9 0.534666 0.267333 0.963604i \(-0.413858\pi\)
0.267333 + 0.963604i \(0.413858\pi\)
\(884\) 0 0
\(885\) −9950.13 −0.377932
\(886\) 0 0
\(887\) 6993.33 0.264727 0.132364 0.991201i \(-0.457743\pi\)
0.132364 + 0.991201i \(0.457743\pi\)
\(888\) 0 0
\(889\) 1103.54 0.0416328
\(890\) 0 0
\(891\) 25497.5 0.958695
\(892\) 0 0
\(893\) −1150.50 −0.0431129
\(894\) 0 0
\(895\) 18117.1 0.676634
\(896\) 0 0
\(897\) 2995.22 0.111491
\(898\) 0 0
\(899\) −30447.0 −1.12955
\(900\) 0 0
\(901\) 7635.42 0.282323
\(902\) 0 0
\(903\) −2059.96 −0.0759149
\(904\) 0 0
\(905\) −13161.8 −0.483438
\(906\) 0 0
\(907\) −26569.7 −0.972693 −0.486347 0.873766i \(-0.661671\pi\)
−0.486347 + 0.873766i \(0.661671\pi\)
\(908\) 0 0
\(909\) 41475.0 1.51336
\(910\) 0 0
\(911\) −3037.05 −0.110452 −0.0552260 0.998474i \(-0.517588\pi\)
−0.0552260 + 0.998474i \(0.517588\pi\)
\(912\) 0 0
\(913\) −23726.7 −0.860066
\(914\) 0 0
\(915\) 1573.34 0.0568447
\(916\) 0 0
\(917\) −4308.54 −0.155159
\(918\) 0 0
\(919\) −30480.3 −1.09407 −0.547036 0.837109i \(-0.684244\pi\)
−0.547036 + 0.837109i \(0.684244\pi\)
\(920\) 0 0
\(921\) 2822.75 0.100991
\(922\) 0 0
\(923\) 17857.3 0.636816
\(924\) 0 0
\(925\) −36484.7 −1.29687
\(926\) 0 0
\(927\) 50115.4 1.77563
\(928\) 0 0
\(929\) −12218.1 −0.431498 −0.215749 0.976449i \(-0.569219\pi\)
−0.215749 + 0.976449i \(0.569219\pi\)
\(930\) 0 0
\(931\) −3616.09 −0.127296
\(932\) 0 0
\(933\) 8221.36 0.288484
\(934\) 0 0
\(935\) −68018.5 −2.37908
\(936\) 0 0
\(937\) 29036.4 1.01236 0.506178 0.862429i \(-0.331058\pi\)
0.506178 + 0.862429i \(0.331058\pi\)
\(938\) 0 0
\(939\) 1896.42 0.0659076
\(940\) 0 0
\(941\) 15188.6 0.526179 0.263090 0.964771i \(-0.415259\pi\)
0.263090 + 0.964771i \(0.415259\pi\)
\(942\) 0 0
\(943\) 43806.4 1.51276
\(944\) 0 0
\(945\) 11321.2 0.389714
\(946\) 0 0
\(947\) −38267.7 −1.31313 −0.656565 0.754270i \(-0.727991\pi\)
−0.656565 + 0.754270i \(0.727991\pi\)
\(948\) 0 0
\(949\) 10787.8 0.369007
\(950\) 0 0
\(951\) −2875.66 −0.0980544
\(952\) 0 0
\(953\) −30092.6 −1.02287 −0.511434 0.859322i \(-0.670886\pi\)
−0.511434 + 0.859322i \(0.670886\pi\)
\(954\) 0 0
\(955\) −69252.5 −2.34655
\(956\) 0 0
\(957\) 5548.52 0.187417
\(958\) 0 0
\(959\) −16767.3 −0.564592
\(960\) 0 0
\(961\) 16686.2 0.560109
\(962\) 0 0
\(963\) −18644.3 −0.623888
\(964\) 0 0
\(965\) 66657.7 2.22361
\(966\) 0 0
\(967\) 22521.1 0.748943 0.374472 0.927238i \(-0.377824\pi\)
0.374472 + 0.927238i \(0.377824\pi\)
\(968\) 0 0
\(969\) −1902.83 −0.0630835
\(970\) 0 0
\(971\) −16791.8 −0.554969 −0.277485 0.960730i \(-0.589501\pi\)
−0.277485 + 0.960730i \(0.589501\pi\)
\(972\) 0 0
\(973\) −7775.69 −0.256195
\(974\) 0 0
\(975\) 3291.16 0.108104
\(976\) 0 0
\(977\) −28654.3 −0.938312 −0.469156 0.883115i \(-0.655442\pi\)
−0.469156 + 0.883115i \(0.655442\pi\)
\(978\) 0 0
\(979\) 64259.5 2.09780
\(980\) 0 0
\(981\) 9959.22 0.324132
\(982\) 0 0
\(983\) −43981.3 −1.42704 −0.713522 0.700633i \(-0.752901\pi\)
−0.713522 + 0.700633i \(0.752901\pi\)
\(984\) 0 0
\(985\) −47016.7 −1.52089
\(986\) 0 0
\(987\) −748.206 −0.0241294
\(988\) 0 0
\(989\) −26377.0 −0.848068
\(990\) 0 0
\(991\) −10308.7 −0.330442 −0.165221 0.986257i \(-0.552834\pi\)
−0.165221 + 0.986257i \(0.552834\pi\)
\(992\) 0 0
\(993\) −9263.48 −0.296040
\(994\) 0 0
\(995\) 77007.3 2.45356
\(996\) 0 0
\(997\) −56141.7 −1.78337 −0.891687 0.452652i \(-0.850478\pi\)
−0.891687 + 0.452652i \(0.850478\pi\)
\(998\) 0 0
\(999\) −11122.7 −0.352258
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1216.4.a.i.1.1 2
4.3 odd 2 1216.4.a.n.1.1 2
8.3 odd 2 608.4.a.c.1.2 2
8.5 even 2 608.4.a.d.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.c.1.2 2 8.3 odd 2
608.4.a.d.1.2 yes 2 8.5 even 2
1216.4.a.i.1.1 2 1.1 even 1 trivial
1216.4.a.n.1.1 2 4.3 odd 2