Properties

Label 324.6.a.d.1.4
Level 324
Weight 6
Character 324.1
Self dual yes
Analytic conductor 51.964
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.9643576194\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 110 x^{3} + 39 x^{2} + 2214 x - 1944\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.900358\) of defining polynomial
Character \(\chi\) \(=\) 324.1

$q$-expansion

\(f(q)\) \(=\) \(q+26.3205 q^{5} +63.2575 q^{7} +O(q^{10})\) \(q+26.3205 q^{5} +63.2575 q^{7} -98.2387 q^{11} -738.285 q^{13} -250.060 q^{17} +1102.41 q^{19} -4409.41 q^{23} -2432.23 q^{25} +7882.12 q^{29} -4611.29 q^{31} +1664.97 q^{35} +11896.3 q^{37} -10081.2 q^{41} +7037.98 q^{43} -14918.7 q^{47} -12805.5 q^{49} -22451.7 q^{53} -2585.69 q^{55} -10810.5 q^{59} -1189.29 q^{61} -19432.0 q^{65} +59180.5 q^{67} -14326.6 q^{71} -53098.2 q^{73} -6214.33 q^{77} -37391.1 q^{79} -120879. q^{83} -6581.71 q^{85} -97873.2 q^{89} -46702.1 q^{91} +29015.9 q^{95} +106713. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 21q^{5} - 29q^{7} + O(q^{10}) \) \( 5q - 21q^{5} - 29q^{7} + 177q^{11} + 181q^{13} - 1140q^{17} - 416q^{19} + 399q^{23} + 4778q^{25} - 6033q^{29} - 2759q^{31} - 18573q^{35} - 7586q^{37} - 18435q^{41} - 1469q^{43} - 25155q^{47} + 4056q^{49} - 58422q^{53} + 7389q^{55} - 90537q^{59} - 1403q^{61} - 148407q^{65} - 13907q^{67} - 114684q^{71} + 7600q^{73} - 211983q^{77} - 29993q^{79} - 228951q^{83} + 49662q^{85} - 299166q^{89} + 62465q^{91} - 394764q^{95} - 40541q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 26.3205 0.470835 0.235418 0.971894i \(-0.424354\pi\)
0.235418 + 0.971894i \(0.424354\pi\)
\(6\) 0 0
\(7\) 63.2575 0.487940 0.243970 0.969783i \(-0.421550\pi\)
0.243970 + 0.969783i \(0.421550\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −98.2387 −0.244794 −0.122397 0.992481i \(-0.539058\pi\)
−0.122397 + 0.992481i \(0.539058\pi\)
\(12\) 0 0
\(13\) −738.285 −1.21162 −0.605809 0.795610i \(-0.707151\pi\)
−0.605809 + 0.795610i \(0.707151\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −250.060 −0.209856 −0.104928 0.994480i \(-0.533461\pi\)
−0.104928 + 0.994480i \(0.533461\pi\)
\(18\) 0 0
\(19\) 1102.41 0.700579 0.350290 0.936641i \(-0.386083\pi\)
0.350290 + 0.936641i \(0.386083\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4409.41 −1.73804 −0.869022 0.494774i \(-0.835251\pi\)
−0.869022 + 0.494774i \(0.835251\pi\)
\(24\) 0 0
\(25\) −2432.23 −0.778314
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7882.12 1.74039 0.870197 0.492703i \(-0.163991\pi\)
0.870197 + 0.492703i \(0.163991\pi\)
\(30\) 0 0
\(31\) −4611.29 −0.861823 −0.430912 0.902394i \(-0.641808\pi\)
−0.430912 + 0.902394i \(0.641808\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1664.97 0.229740
\(36\) 0 0
\(37\) 11896.3 1.42859 0.714297 0.699843i \(-0.246747\pi\)
0.714297 + 0.699843i \(0.246747\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10081.2 −0.936594 −0.468297 0.883571i \(-0.655132\pi\)
−0.468297 + 0.883571i \(0.655132\pi\)
\(42\) 0 0
\(43\) 7037.98 0.580466 0.290233 0.956956i \(-0.406267\pi\)
0.290233 + 0.956956i \(0.406267\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −14918.7 −0.985111 −0.492555 0.870281i \(-0.663937\pi\)
−0.492555 + 0.870281i \(0.663937\pi\)
\(48\) 0 0
\(49\) −12805.5 −0.761914
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −22451.7 −1.09789 −0.548947 0.835857i \(-0.684971\pi\)
−0.548947 + 0.835857i \(0.684971\pi\)
\(54\) 0 0
\(55\) −2585.69 −0.115258
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10810.5 −0.404311 −0.202156 0.979353i \(-0.564795\pi\)
−0.202156 + 0.979353i \(0.564795\pi\)
\(60\) 0 0
\(61\) −1189.29 −0.0409227 −0.0204614 0.999791i \(-0.506514\pi\)
−0.0204614 + 0.999791i \(0.506514\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −19432.0 −0.570473
\(66\) 0 0
\(67\) 59180.5 1.61061 0.805307 0.592858i \(-0.202000\pi\)
0.805307 + 0.592858i \(0.202000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14326.6 −0.337284 −0.168642 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(72\) 0 0
\(73\) −53098.2 −1.16620 −0.583099 0.812401i \(-0.698160\pi\)
−0.583099 + 0.812401i \(0.698160\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6214.33 −0.119445
\(78\) 0 0
\(79\) −37391.1 −0.674063 −0.337032 0.941493i \(-0.609423\pi\)
−0.337032 + 0.941493i \(0.609423\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −120879. −1.92600 −0.962998 0.269509i \(-0.913139\pi\)
−0.962998 + 0.269509i \(0.913139\pi\)
\(84\) 0 0
\(85\) −6581.71 −0.0988078
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −97873.2 −1.30975 −0.654875 0.755737i \(-0.727279\pi\)
−0.654875 + 0.755737i \(0.727279\pi\)
\(90\) 0 0
\(91\) −46702.1 −0.591198
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 29015.9 0.329858
\(96\) 0 0
\(97\) 106713. 1.15157 0.575784 0.817602i \(-0.304697\pi\)
0.575784 + 0.817602i \(0.304697\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −96472.1 −0.941019 −0.470510 0.882395i \(-0.655930\pi\)
−0.470510 + 0.882395i \(0.655930\pi\)
\(102\) 0 0
\(103\) 29944.4 0.278114 0.139057 0.990284i \(-0.455593\pi\)
0.139057 + 0.990284i \(0.455593\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −22758.9 −0.192173 −0.0960865 0.995373i \(-0.530633\pi\)
−0.0960865 + 0.995373i \(0.530633\pi\)
\(108\) 0 0
\(109\) −2671.93 −0.0215407 −0.0107703 0.999942i \(-0.503428\pi\)
−0.0107703 + 0.999942i \(0.503428\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −131339. −0.967606 −0.483803 0.875177i \(-0.660745\pi\)
−0.483803 + 0.875177i \(0.660745\pi\)
\(114\) 0 0
\(115\) −116058. −0.818332
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15818.2 −0.102397
\(120\) 0 0
\(121\) −151400. −0.940076
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −146269. −0.837293
\(126\) 0 0
\(127\) 236012. 1.29845 0.649223 0.760598i \(-0.275094\pi\)
0.649223 + 0.760598i \(0.275094\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −360366. −1.83470 −0.917352 0.398078i \(-0.869677\pi\)
−0.917352 + 0.398078i \(0.869677\pi\)
\(132\) 0 0
\(133\) 69735.4 0.341841
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 237666. 1.08185 0.540923 0.841072i \(-0.318075\pi\)
0.540923 + 0.841072i \(0.318075\pi\)
\(138\) 0 0
\(139\) 163347. 0.717090 0.358545 0.933513i \(-0.383273\pi\)
0.358545 + 0.933513i \(0.383273\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 72528.2 0.296597
\(144\) 0 0
\(145\) 207461. 0.819440
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −335607. −1.23841 −0.619205 0.785229i \(-0.712545\pi\)
−0.619205 + 0.785229i \(0.712545\pi\)
\(150\) 0 0
\(151\) −135577. −0.483886 −0.241943 0.970290i \(-0.577785\pi\)
−0.241943 + 0.970290i \(0.577785\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −121371. −0.405777
\(156\) 0 0
\(157\) −425432. −1.37747 −0.688733 0.725016i \(-0.741833\pi\)
−0.688733 + 0.725016i \(0.741833\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −278928. −0.848062
\(162\) 0 0
\(163\) −158679. −0.467789 −0.233895 0.972262i \(-0.575147\pi\)
−0.233895 + 0.972262i \(0.575147\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 518446. 1.43851 0.719254 0.694747i \(-0.244484\pi\)
0.719254 + 0.694747i \(0.244484\pi\)
\(168\) 0 0
\(169\) 173772. 0.468019
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 153178. 0.389118 0.194559 0.980891i \(-0.437672\pi\)
0.194559 + 0.980891i \(0.437672\pi\)
\(174\) 0 0
\(175\) −153857. −0.379771
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 736845. 1.71887 0.859437 0.511242i \(-0.170815\pi\)
0.859437 + 0.511242i \(0.170815\pi\)
\(180\) 0 0
\(181\) 28183.8 0.0639445 0.0319722 0.999489i \(-0.489821\pi\)
0.0319722 + 0.999489i \(0.489821\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 313118. 0.672633
\(186\) 0 0
\(187\) 24565.6 0.0513716
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −99321.0 −0.196996 −0.0984981 0.995137i \(-0.531404\pi\)
−0.0984981 + 0.995137i \(0.531404\pi\)
\(192\) 0 0
\(193\) 417272. 0.806355 0.403177 0.915122i \(-0.367906\pi\)
0.403177 + 0.915122i \(0.367906\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −254565. −0.467340 −0.233670 0.972316i \(-0.575073\pi\)
−0.233670 + 0.972316i \(0.575073\pi\)
\(198\) 0 0
\(199\) −599702. −1.07350 −0.536751 0.843741i \(-0.680348\pi\)
−0.536751 + 0.843741i \(0.680348\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 498603. 0.849209
\(204\) 0 0
\(205\) −265341. −0.440982
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −108299. −0.171498
\(210\) 0 0
\(211\) 756584. 1.16991 0.584953 0.811067i \(-0.301113\pi\)
0.584953 + 0.811067i \(0.301113\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 185243. 0.273304
\(216\) 0 0
\(217\) −291699. −0.420518
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 184616. 0.254266
\(222\) 0 0
\(223\) −111205. −0.149748 −0.0748741 0.997193i \(-0.523855\pi\)
−0.0748741 + 0.997193i \(0.523855\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 490760. 0.632127 0.316063 0.948738i \(-0.397639\pi\)
0.316063 + 0.948738i \(0.397639\pi\)
\(228\) 0 0
\(229\) 761021. 0.958977 0.479488 0.877548i \(-0.340822\pi\)
0.479488 + 0.877548i \(0.340822\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 356267. 0.429918 0.214959 0.976623i \(-0.431038\pi\)
0.214959 + 0.976623i \(0.431038\pi\)
\(234\) 0 0
\(235\) −392667. −0.463825
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 906239. 1.02624 0.513119 0.858318i \(-0.328490\pi\)
0.513119 + 0.858318i \(0.328490\pi\)
\(240\) 0 0
\(241\) 522576. 0.579571 0.289785 0.957092i \(-0.406416\pi\)
0.289785 + 0.957092i \(0.406416\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −337047. −0.358736
\(246\) 0 0
\(247\) −813890. −0.848835
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 625287. 0.626462 0.313231 0.949677i \(-0.398589\pi\)
0.313231 + 0.949677i \(0.398589\pi\)
\(252\) 0 0
\(253\) 433175. 0.425463
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.30221e6 1.22984 0.614921 0.788589i \(-0.289188\pi\)
0.614921 + 0.788589i \(0.289188\pi\)
\(258\) 0 0
\(259\) 752532. 0.697069
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.43352e6 1.27795 0.638977 0.769226i \(-0.279358\pi\)
0.638977 + 0.769226i \(0.279358\pi\)
\(264\) 0 0
\(265\) −590941. −0.516927
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.52329e6 1.28352 0.641758 0.766907i \(-0.278205\pi\)
0.641758 + 0.766907i \(0.278205\pi\)
\(270\) 0 0
\(271\) 1.60134e6 1.32453 0.662263 0.749271i \(-0.269596\pi\)
0.662263 + 0.749271i \(0.269596\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 238939. 0.190527
\(276\) 0 0
\(277\) −893943. −0.700020 −0.350010 0.936746i \(-0.613822\pi\)
−0.350010 + 0.936746i \(0.613822\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28400.1 0.0214563 0.0107281 0.999942i \(-0.496585\pi\)
0.0107281 + 0.999942i \(0.496585\pi\)
\(282\) 0 0
\(283\) −1.10957e6 −0.823544 −0.411772 0.911287i \(-0.635090\pi\)
−0.411772 + 0.911287i \(0.635090\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −637709. −0.457002
\(288\) 0 0
\(289\) −1.35733e6 −0.955960
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.45768e6 −1.67246 −0.836230 0.548378i \(-0.815245\pi\)
−0.836230 + 0.548378i \(0.815245\pi\)
\(294\) 0 0
\(295\) −284538. −0.190364
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.25540e6 2.10585
\(300\) 0 0
\(301\) 445205. 0.283233
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −31302.8 −0.0192679
\(306\) 0 0
\(307\) 1.77336e6 1.07387 0.536934 0.843624i \(-0.319583\pi\)
0.536934 + 0.843624i \(0.319583\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.50571e6 −0.882756 −0.441378 0.897321i \(-0.645510\pi\)
−0.441378 + 0.897321i \(0.645510\pi\)
\(312\) 0 0
\(313\) −812705. −0.468891 −0.234446 0.972129i \(-0.575327\pi\)
−0.234446 + 0.972129i \(0.575327\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.04760e6 −0.585525 −0.292763 0.956185i \(-0.594575\pi\)
−0.292763 + 0.956185i \(0.594575\pi\)
\(318\) 0 0
\(319\) −774329. −0.426038
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −275668. −0.147021
\(324\) 0 0
\(325\) 1.79568e6 0.943020
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −943717. −0.480675
\(330\) 0 0
\(331\) −68005.3 −0.0341172 −0.0170586 0.999854i \(-0.505430\pi\)
−0.0170586 + 0.999854i \(0.505430\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.55766e6 0.758334
\(336\) 0 0
\(337\) −416472. −0.199761 −0.0998806 0.994999i \(-0.531846\pi\)
−0.0998806 + 0.994999i \(0.531846\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 453007. 0.210969
\(342\) 0 0
\(343\) −1.87321e6 −0.859709
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.80740e6 1.25164 0.625822 0.779966i \(-0.284764\pi\)
0.625822 + 0.779966i \(0.284764\pi\)
\(348\) 0 0
\(349\) −481973. −0.211816 −0.105908 0.994376i \(-0.533775\pi\)
−0.105908 + 0.994376i \(0.533775\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4.00037e6 −1.70869 −0.854346 0.519705i \(-0.826042\pi\)
−0.854346 + 0.519705i \(0.826042\pi\)
\(354\) 0 0
\(355\) −377082. −0.158805
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 800858. 0.327959 0.163979 0.986464i \(-0.447567\pi\)
0.163979 + 0.986464i \(0.447567\pi\)
\(360\) 0 0
\(361\) −1.26080e6 −0.509189
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.39757e6 −0.549087
\(366\) 0 0
\(367\) 3.00972e6 1.16644 0.583218 0.812316i \(-0.301793\pi\)
0.583218 + 0.812316i \(0.301793\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.42024e6 −0.535707
\(372\) 0 0
\(373\) 1.59331e6 0.592962 0.296481 0.955039i \(-0.404187\pi\)
0.296481 + 0.955039i \(0.404187\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.81925e6 −2.10869
\(378\) 0 0
\(379\) −297930. −0.106541 −0.0532703 0.998580i \(-0.516965\pi\)
−0.0532703 + 0.998580i \(0.516965\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.29030e6 0.449464 0.224732 0.974421i \(-0.427849\pi\)
0.224732 + 0.974421i \(0.427849\pi\)
\(384\) 0 0
\(385\) −163564. −0.0562389
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.99873e6 −1.67489 −0.837444 0.546523i \(-0.815951\pi\)
−0.837444 + 0.546523i \(0.815951\pi\)
\(390\) 0 0
\(391\) 1.10262e6 0.364740
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −984153. −0.317373
\(396\) 0 0
\(397\) −558980. −0.178000 −0.0890000 0.996032i \(-0.528367\pi\)
−0.0890000 + 0.996032i \(0.528367\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −94207.2 −0.0292565 −0.0146283 0.999893i \(-0.504656\pi\)
−0.0146283 + 0.999893i \(0.504656\pi\)
\(402\) 0 0
\(403\) 3.40445e6 1.04420
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.16868e6 −0.349711
\(408\) 0 0
\(409\) 5.46230e6 1.61461 0.807304 0.590135i \(-0.200926\pi\)
0.807304 + 0.590135i \(0.200926\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −683845. −0.197280
\(414\) 0 0
\(415\) −3.18159e6 −0.906827
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 836972. 0.232903 0.116452 0.993196i \(-0.462848\pi\)
0.116452 + 0.993196i \(0.462848\pi\)
\(420\) 0 0
\(421\) −6.42809e6 −1.76757 −0.883785 0.467894i \(-0.845013\pi\)
−0.883785 + 0.467894i \(0.845013\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 608204. 0.163334
\(426\) 0 0
\(427\) −75231.7 −0.0199678
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.11509e6 1.32636 0.663178 0.748462i \(-0.269207\pi\)
0.663178 + 0.748462i \(0.269207\pi\)
\(432\) 0 0
\(433\) −1.39089e6 −0.356512 −0.178256 0.983984i \(-0.557045\pi\)
−0.178256 + 0.983984i \(0.557045\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.86095e6 −1.21764
\(438\) 0 0
\(439\) −3.52532e6 −0.873045 −0.436523 0.899693i \(-0.643790\pi\)
−0.436523 + 0.899693i \(0.643790\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.04432e6 −0.494924 −0.247462 0.968898i \(-0.579597\pi\)
−0.247462 + 0.968898i \(0.579597\pi\)
\(444\) 0 0
\(445\) −2.57607e6 −0.616677
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.70508e6 −1.80369 −0.901844 0.432062i \(-0.857786\pi\)
−0.901844 + 0.432062i \(0.857786\pi\)
\(450\) 0 0
\(451\) 990361. 0.229273
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.22922e6 −0.278357
\(456\) 0 0
\(457\) 8.57692e6 1.92106 0.960529 0.278179i \(-0.0897308\pi\)
0.960529 + 0.278179i \(0.0897308\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.43263e6 −0.313966 −0.156983 0.987601i \(-0.550177\pi\)
−0.156983 + 0.987601i \(0.550177\pi\)
\(462\) 0 0
\(463\) 2.94447e6 0.638343 0.319172 0.947697i \(-0.396595\pi\)
0.319172 + 0.947697i \(0.396595\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.96160e6 −1.05276 −0.526380 0.850249i \(-0.676451\pi\)
−0.526380 + 0.850249i \(0.676451\pi\)
\(468\) 0 0
\(469\) 3.74361e6 0.785884
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −691402. −0.142095
\(474\) 0 0
\(475\) −2.68130e6 −0.545271
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 241513. 0.0480953 0.0240476 0.999711i \(-0.492345\pi\)
0.0240476 + 0.999711i \(0.492345\pi\)
\(480\) 0 0
\(481\) −8.78289e6 −1.73091
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.80875e6 0.542199
\(486\) 0 0
\(487\) 325055. 0.0621061 0.0310530 0.999518i \(-0.490114\pi\)
0.0310530 + 0.999518i \(0.490114\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.98047e6 −0.932323 −0.466162 0.884700i \(-0.654363\pi\)
−0.466162 + 0.884700i \(0.654363\pi\)
\(492\) 0 0
\(493\) −1.97100e6 −0.365233
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −906262. −0.164575
\(498\) 0 0
\(499\) −5.86495e6 −1.05442 −0.527209 0.849736i \(-0.676761\pi\)
−0.527209 + 0.849736i \(0.676761\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.36396e6 0.240370 0.120185 0.992751i \(-0.461651\pi\)
0.120185 + 0.992751i \(0.461651\pi\)
\(504\) 0 0
\(505\) −2.53919e6 −0.443065
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.70047e6 −0.633085 −0.316542 0.948578i \(-0.602522\pi\)
−0.316542 + 0.948578i \(0.602522\pi\)
\(510\) 0 0
\(511\) −3.35885e6 −0.569035
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 788153. 0.130946
\(516\) 0 0
\(517\) 1.46559e6 0.241149
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 285260. 0.0460412 0.0230206 0.999735i \(-0.492672\pi\)
0.0230206 + 0.999735i \(0.492672\pi\)
\(522\) 0 0
\(523\) 8.28809e6 1.32495 0.662476 0.749083i \(-0.269506\pi\)
0.662476 + 0.749083i \(0.269506\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.15310e6 0.180859
\(528\) 0 0
\(529\) 1.30065e7 2.02080
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.44278e6 1.13479
\(534\) 0 0
\(535\) −599027. −0.0904819
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.25800e6 0.186512
\(540\) 0 0
\(541\) 1.29750e7 1.90596 0.952982 0.303028i \(-0.0979975\pi\)
0.952982 + 0.303028i \(0.0979975\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −70326.5 −0.0101421
\(546\) 0 0
\(547\) −9.31601e6 −1.33126 −0.665628 0.746284i \(-0.731836\pi\)
−0.665628 + 0.746284i \(0.731836\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.68929e6 1.21928
\(552\) 0 0
\(553\) −2.36527e6 −0.328903
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.18234e6 0.571192 0.285596 0.958350i \(-0.407809\pi\)
0.285596 + 0.958350i \(0.407809\pi\)
\(558\) 0 0
\(559\) −5.19604e6 −0.703304
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.31319e6 0.440531 0.220265 0.975440i \(-0.429308\pi\)
0.220265 + 0.975440i \(0.429308\pi\)
\(564\) 0 0
\(565\) −3.45692e6 −0.455583
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.85391e6 0.757993 0.378996 0.925398i \(-0.376269\pi\)
0.378996 + 0.925398i \(0.376269\pi\)
\(570\) 0 0
\(571\) 1.35086e7 1.73388 0.866941 0.498411i \(-0.166083\pi\)
0.866941 + 0.498411i \(0.166083\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.07247e7 1.35274
\(576\) 0 0
\(577\) −1.12898e7 −1.41171 −0.705855 0.708356i \(-0.749437\pi\)
−0.705855 + 0.708356i \(0.749437\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.64649e6 −0.939771
\(582\) 0 0
\(583\) 2.20563e6 0.268758
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.98280e6 −0.237510 −0.118755 0.992924i \(-0.537890\pi\)
−0.118755 + 0.992924i \(0.537890\pi\)
\(588\) 0 0
\(589\) −5.08351e6 −0.603776
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.22629e7 −1.43204 −0.716021 0.698078i \(-0.754039\pi\)
−0.716021 + 0.698078i \(0.754039\pi\)
\(594\) 0 0
\(595\) −416342. −0.0482123
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.56995e7 1.78780 0.893902 0.448263i \(-0.147957\pi\)
0.893902 + 0.448263i \(0.147957\pi\)
\(600\) 0 0
\(601\) −5.31944e6 −0.600731 −0.300366 0.953824i \(-0.597109\pi\)
−0.300366 + 0.953824i \(0.597109\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.98493e6 −0.442621
\(606\) 0 0
\(607\) 4.36964e6 0.481365 0.240682 0.970604i \(-0.422629\pi\)
0.240682 + 0.970604i \(0.422629\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.10142e7 1.19358
\(612\) 0 0
\(613\) −7.62875e6 −0.819978 −0.409989 0.912091i \(-0.634467\pi\)
−0.409989 + 0.912091i \(0.634467\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.43300e7 1.51542 0.757708 0.652594i \(-0.226319\pi\)
0.757708 + 0.652594i \(0.226319\pi\)
\(618\) 0 0
\(619\) 1.08181e7 1.13482 0.567408 0.823437i \(-0.307946\pi\)
0.567408 + 0.823437i \(0.307946\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.19121e6 −0.639080
\(624\) 0 0
\(625\) 3.75085e6 0.384087
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.97480e6 −0.299800
\(630\) 0 0
\(631\) 7.96579e6 0.796444 0.398222 0.917289i \(-0.369627\pi\)
0.398222 + 0.917289i \(0.369627\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.21194e6 0.611354
\(636\) 0 0
\(637\) 9.45411e6 0.923149
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.80448e6 −0.654108 −0.327054 0.945006i \(-0.606056\pi\)
−0.327054 + 0.945006i \(0.606056\pi\)
\(642\) 0 0
\(643\) −5.32929e6 −0.508325 −0.254163 0.967161i \(-0.581800\pi\)
−0.254163 + 0.967161i \(0.581800\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.03021e6 0.472417 0.236208 0.971702i \(-0.424095\pi\)
0.236208 + 0.971702i \(0.424095\pi\)
\(648\) 0 0
\(649\) 1.06201e6 0.0989731
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.13341e7 1.04017 0.520084 0.854115i \(-0.325901\pi\)
0.520084 + 0.854115i \(0.325901\pi\)
\(654\) 0 0
\(655\) −9.48502e6 −0.863843
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.82007e6 −0.342655 −0.171328 0.985214i \(-0.554806\pi\)
−0.171328 + 0.985214i \(0.554806\pi\)
\(660\) 0 0
\(661\) 926573. 0.0824852 0.0412426 0.999149i \(-0.486868\pi\)
0.0412426 + 0.999149i \(0.486868\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.83547e6 0.160951
\(666\) 0 0
\(667\) −3.47555e7 −3.02488
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 116835. 0.0100176
\(672\) 0 0
\(673\) 4.90291e6 0.417269 0.208635 0.977994i \(-0.433098\pi\)
0.208635 + 0.977994i \(0.433098\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.47440e7 1.23636 0.618179 0.786037i \(-0.287871\pi\)
0.618179 + 0.786037i \(0.287871\pi\)
\(678\) 0 0
\(679\) 6.75041e6 0.561896
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.53893e6 0.536358 0.268179 0.963369i \(-0.413578\pi\)
0.268179 + 0.963369i \(0.413578\pi\)
\(684\) 0 0
\(685\) 6.25548e6 0.509371
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.65758e7 1.33023
\(690\) 0 0
\(691\) −1.09050e7 −0.868818 −0.434409 0.900716i \(-0.643043\pi\)
−0.434409 + 0.900716i \(0.643043\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.29937e6 0.337631
\(696\) 0 0
\(697\) 2.52090e6 0.196550
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.89162e7 1.45391 0.726957 0.686683i \(-0.240934\pi\)
0.726957 + 0.686683i \(0.240934\pi\)
\(702\) 0 0
\(703\) 1.31146e7 1.00084
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.10258e6 −0.459161
\(708\) 0 0
\(709\) −1.92732e7 −1.43992 −0.719960 0.694015i \(-0.755840\pi\)
−0.719960 + 0.694015i \(0.755840\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.03331e7 1.49789
\(714\) 0 0
\(715\) 1.90898e6 0.139648
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.42859e7 −1.03059 −0.515293 0.857014i \(-0.672317\pi\)
−0.515293 + 0.857014i \(0.672317\pi\)
\(720\) 0 0
\(721\) 1.89421e6 0.135703
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.91711e7 −1.35457
\(726\) 0 0
\(727\) 2.82771e7 1.98426 0.992132 0.125198i \(-0.0399567\pi\)
0.992132 + 0.125198i \(0.0399567\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.75992e6 −0.121815
\(732\) 0 0
\(733\) 9.87699e6 0.678992 0.339496 0.940607i \(-0.389743\pi\)
0.339496 + 0.940607i \(0.389743\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.81381e6 −0.394269
\(738\) 0 0
\(739\) 6.31081e6 0.425083 0.212542 0.977152i \(-0.431826\pi\)
0.212542 + 0.977152i \(0.431826\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 63059.8 0.00419064 0.00209532 0.999998i \(-0.499333\pi\)
0.00209532 + 0.999998i \(0.499333\pi\)
\(744\) 0 0
\(745\) −8.83333e6 −0.583088
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.43967e6 −0.0937690
\(750\) 0 0
\(751\) −1.73837e7 −1.12471 −0.562356 0.826895i \(-0.690105\pi\)
−0.562356 + 0.826895i \(0.690105\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.56845e6 −0.227831
\(756\) 0 0
\(757\) 9.43592e6 0.598473 0.299237 0.954179i \(-0.403268\pi\)
0.299237 + 0.954179i \(0.403268\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.61282e7 −1.00954 −0.504771 0.863254i \(-0.668423\pi\)
−0.504771 + 0.863254i \(0.668423\pi\)
\(762\) 0 0
\(763\) −169020. −0.0105106
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.98124e6 0.489871
\(768\) 0 0
\(769\) −2.56863e7 −1.56634 −0.783169 0.621809i \(-0.786398\pi\)
−0.783169 + 0.621809i \(0.786398\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.01235e7 1.21131 0.605654 0.795728i \(-0.292912\pi\)
0.605654 + 0.795728i \(0.292912\pi\)
\(774\) 0 0
\(775\) 1.12157e7 0.670769
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.11135e7 −0.656158
\(780\) 0 0
\(781\) 1.40742e6 0.0825652
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.11976e7 −0.648559
\(786\) 0 0
\(787\) 5.75281e6 0.331088 0.165544 0.986202i \(-0.447062\pi\)
0.165544 + 0.986202i \(0.447062\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.30819e6 −0.472134
\(792\) 0 0
\(793\) 878038. 0.0495827
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.01964e7 0.568595 0.284298 0.958736i \(-0.408240\pi\)
0.284298 + 0.958736i \(0.408240\pi\)
\(798\) 0 0
\(799\) 3.73056e6 0.206732
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.21629e6 0.285478
\(804\) 0 0
\(805\) −7.34152e6 −0.399297
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00420e6 −0.322540 −0.161270 0.986910i \(-0.551559\pi\)
−0.161270 + 0.986910i \(0.551559\pi\)
\(810\) 0 0
\(811\) 3.23880e7 1.72915 0.864574 0.502506i \(-0.167589\pi\)
0.864574 + 0.502506i \(0.167589\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.17651e6 −0.220252
\(816\) 0 0
\(817\) 7.75871e6 0.406663
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.56747e7 −0.811600 −0.405800 0.913962i \(-0.633007\pi\)
−0.405800 + 0.913962i \(0.633007\pi\)
\(822\) 0 0
\(823\) −2.71466e7 −1.39706 −0.698532 0.715579i \(-0.746163\pi\)
−0.698532 + 0.715579i \(0.746163\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.28543e7 −1.16199 −0.580997 0.813906i \(-0.697337\pi\)
−0.580997 + 0.813906i \(0.697337\pi\)
\(828\) 0 0
\(829\) −2.05418e7 −1.03813 −0.519066 0.854734i \(-0.673720\pi\)
−0.519066 + 0.854734i \(0.673720\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.20214e6 0.159893
\(834\) 0 0
\(835\) 1.36458e7 0.677301
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.47887e7 0.725313 0.362657 0.931923i \(-0.381870\pi\)
0.362657 + 0.931923i \(0.381870\pi\)
\(840\) 0 0
\(841\) 4.16166e7 2.02897
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.57378e6 0.220360
\(846\) 0 0
\(847\) −9.57719e6 −0.458701
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.24558e7 −2.48296
\(852\) 0 0
\(853\) −1.05245e7 −0.495253 −0.247627 0.968856i \(-0.579651\pi\)
−0.247627 + 0.968856i \(0.579651\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.48671e7 −0.691473 −0.345736 0.938332i \(-0.612371\pi\)
−0.345736 + 0.938332i \(0.612371\pi\)
\(858\) 0 0
\(859\) −8.64679e6 −0.399827 −0.199913 0.979814i \(-0.564066\pi\)
−0.199913 + 0.979814i \(0.564066\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.38401e7 −1.08964 −0.544818 0.838555i \(-0.683401\pi\)
−0.544818 + 0.838555i \(0.683401\pi\)
\(864\) 0 0
\(865\) 4.03172e6 0.183210
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.67325e6 0.165007
\(870\) 0 0
\(871\) −4.36921e7 −1.95145
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.25261e6 −0.408549
\(876\) 0 0
\(877\) −38315.7 −0.00168220 −0.000841100 1.00000i \(-0.500268\pi\)
−0.000841100 1.00000i \(0.500268\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.55072e7 1.10719 0.553596 0.832785i \(-0.313255\pi\)
0.553596 + 0.832785i \(0.313255\pi\)
\(882\) 0 0
\(883\) −3.37096e7 −1.45496 −0.727482 0.686127i \(-0.759309\pi\)
−0.727482 + 0.686127i \(0.759309\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.64222e6 0.112761 0.0563806 0.998409i \(-0.482044\pi\)
0.0563806 + 0.998409i \(0.482044\pi\)
\(888\) 0 0
\(889\) 1.49295e7 0.633564
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.64464e7 −0.690148
\(894\) 0 0
\(895\) 1.93941e7 0.809306
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.63467e7 −1.49991
\(900\) 0 0
\(901\) 5.61429e6 0.230400
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 741812. 0.0301073
\(906\) 0 0
\(907\) 7.13018e6 0.287795 0.143897 0.989593i \(-0.454037\pi\)
0.143897 + 0.989593i \(0.454037\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.13154e7 −1.25015 −0.625075 0.780564i \(-0.714932\pi\)
−0.625075 + 0.780564i \(0.714932\pi\)
\(912\) 0 0
\(913\) 1.18750e7 0.471472
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.27959e7 −0.895226
\(918\) 0 0
\(919\) 2.28042e6 0.0890687 0.0445344 0.999008i \(-0.485820\pi\)
0.0445344 + 0.999008i \(0.485820\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.05771e7 0.408660
\(924\) 0 0
\(925\) −2.89346e7 −1.11189
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.05725e7 −0.401919 −0.200959 0.979600i \(-0.564406\pi\)
−0.200959 + 0.979600i \(0.564406\pi\)
\(930\) 0 0
\(931\) −1.41168e7 −0.533781
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 646579. 0.0241876
\(936\) 0 0
\(937\) −3.99327e7 −1.48587 −0.742934 0.669365i \(-0.766566\pi\)
−0.742934 + 0.669365i \(0.766566\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.32632e6 0.306534 0.153267 0.988185i \(-0.451021\pi\)
0.153267 + 0.988185i \(0.451021\pi\)
\(942\) 0 0
\(943\) 4.44520e7 1.62784
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.75667e7 −0.998873 −0.499437 0.866350i \(-0.666460\pi\)
−0.499437 + 0.866350i \(0.666460\pi\)
\(948\) 0 0
\(949\) 3.92016e7 1.41299
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.81280e7 1.35992 0.679958 0.733251i \(-0.261998\pi\)
0.679958 + 0.733251i \(0.261998\pi\)
\(954\) 0 0
\(955\) −2.61418e6 −0.0927528
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.50341e7 0.527876
\(960\) 0 0
\(961\) −7.36516e6 −0.257261
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.09828e7 0.379660
\(966\) 0 0
\(967\) −2.72711e6 −0.0937857 −0.0468928 0.998900i \(-0.514932\pi\)
−0.0468928 + 0.998900i \(0.514932\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.26521e7 −0.430641 −0.215320 0.976543i \(-0.569080\pi\)
−0.215320 + 0.976543i \(0.569080\pi\)
\(972\) 0 0
\(973\) 1.03329e7 0.349897
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.37252e7 1.13036 0.565182 0.824967i \(-0.308806\pi\)
0.565182 + 0.824967i \(0.308806\pi\)
\(978\) 0 0
\(979\) 9.61493e6 0.320619
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.18743e7 −0.391944 −0.195972 0.980610i \(-0.562786\pi\)
−0.195972 + 0.980610i \(0.562786\pi\)
\(984\) 0 0
\(985\) −6.70027e6 −0.220040
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.10333e7 −1.00888
\(990\) 0 0
\(991\) −2.31910e7 −0.750128 −0.375064 0.926999i \(-0.622379\pi\)
−0.375064 + 0.926999i \(0.622379\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.57845e7 −0.505443
\(996\) 0 0
\(997\) 3.86264e7 1.23068 0.615342 0.788260i \(-0.289018\pi\)
0.615342 + 0.788260i \(0.289018\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 324.6.a.d.1.4 5
3.2 odd 2 324.6.a.e.1.2 5
9.2 odd 6 36.6.e.a.13.4 10
9.4 even 3 108.6.e.a.73.2 10
9.5 odd 6 36.6.e.a.25.4 yes 10
9.7 even 3 108.6.e.a.37.2 10
36.7 odd 6 432.6.i.d.145.2 10
36.11 even 6 144.6.i.d.49.2 10
36.23 even 6 144.6.i.d.97.2 10
36.31 odd 6 432.6.i.d.289.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.6.e.a.13.4 10 9.2 odd 6
36.6.e.a.25.4 yes 10 9.5 odd 6
108.6.e.a.37.2 10 9.7 even 3
108.6.e.a.73.2 10 9.4 even 3
144.6.i.d.49.2 10 36.11 even 6
144.6.i.d.97.2 10 36.23 even 6
324.6.a.d.1.4 5 1.1 even 1 trivial
324.6.a.e.1.2 5 3.2 odd 2
432.6.i.d.145.2 10 36.7 odd 6
432.6.i.d.289.2 10 36.31 odd 6