Properties

Label 912.6.a.y.1.4
Level $912$
Weight $6$
Character 912.1
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(49.1493\) of defining polynomial
Character \(\chi\) \(=\) 912.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{3} +28.9245 q^{5} -210.915 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} +28.9245 q^{5} -210.915 q^{7} +81.0000 q^{9} -169.342 q^{11} -1019.58 q^{13} +260.320 q^{15} -1848.05 q^{17} -361.000 q^{19} -1898.23 q^{21} +1415.83 q^{23} -2288.37 q^{25} +729.000 q^{27} +2799.23 q^{29} -61.3714 q^{31} -1524.08 q^{33} -6100.61 q^{35} +2247.44 q^{37} -9176.19 q^{39} +997.106 q^{41} +3731.90 q^{43} +2342.88 q^{45} +16075.6 q^{47} +27678.1 q^{49} -16632.5 q^{51} +39562.9 q^{53} -4898.14 q^{55} -3249.00 q^{57} +35454.3 q^{59} -56219.8 q^{61} -17084.1 q^{63} -29490.7 q^{65} +6653.47 q^{67} +12742.5 q^{69} +8888.19 q^{71} -12509.4 q^{73} -20595.4 q^{75} +35716.8 q^{77} +63220.1 q^{79} +6561.00 q^{81} +37207.9 q^{83} -53454.0 q^{85} +25193.1 q^{87} +46963.0 q^{89} +215044. q^{91} -552.343 q^{93} -10441.7 q^{95} +46146.9 q^{97} -13716.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + O(q^{10}) \) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + 203 q^{11} - 298 q^{13} - 585 q^{15} + 1319 q^{17} - 2166 q^{19} + 1341 q^{21} - 1234 q^{23} + 4589 q^{25} + 4374 q^{27} + 7356 q^{29} - 1632 q^{31} + 1827 q^{33} - 4383 q^{35} + 14204 q^{37} - 2682 q^{39} + 14734 q^{41} + 4693 q^{43} - 5265 q^{45} + 10955 q^{47} + 38561 q^{49} + 11871 q^{51} + 47500 q^{53} - 769 q^{55} - 19494 q^{57} + 61744 q^{59} - 81581 q^{61} + 12069 q^{63} - 59686 q^{65} + 45756 q^{67} - 11106 q^{69} + 10416 q^{71} - 54615 q^{73} + 41301 q^{75} - 29515 q^{77} + 145594 q^{79} + 39366 q^{81} + 160548 q^{83} - 53947 q^{85} + 66204 q^{87} - 97728 q^{89} + 418294 q^{91} - 14688 q^{93} + 23465 q^{95} - 760 q^{97} + 16443 q^{99} + O(q^{100}) \)

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
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) 28.9245 0.517417 0.258708 0.965955i \(-0.416703\pi\)
0.258708 + 0.965955i \(0.416703\pi\)
\(6\) 0 0
\(7\) −210.915 −1.62691 −0.813453 0.581631i \(-0.802415\pi\)
−0.813453 + 0.581631i \(0.802415\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −169.342 −0.421972 −0.210986 0.977489i \(-0.567667\pi\)
−0.210986 + 0.977489i \(0.567667\pi\)
\(12\) 0 0
\(13\) −1019.58 −1.67325 −0.836626 0.547774i \(-0.815475\pi\)
−0.836626 + 0.547774i \(0.815475\pi\)
\(14\) 0 0
\(15\) 260.320 0.298731
\(16\) 0 0
\(17\) −1848.05 −1.55093 −0.775465 0.631390i \(-0.782485\pi\)
−0.775465 + 0.631390i \(0.782485\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) −1898.23 −0.939294
\(22\) 0 0
\(23\) 1415.83 0.558074 0.279037 0.960280i \(-0.409985\pi\)
0.279037 + 0.960280i \(0.409985\pi\)
\(24\) 0 0
\(25\) −2288.37 −0.732280
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 2799.23 0.618079 0.309039 0.951049i \(-0.399993\pi\)
0.309039 + 0.951049i \(0.399993\pi\)
\(30\) 0 0
\(31\) −61.3714 −0.0114700 −0.00573498 0.999984i \(-0.501826\pi\)
−0.00573498 + 0.999984i \(0.501826\pi\)
\(32\) 0 0
\(33\) −1524.08 −0.243626
\(34\) 0 0
\(35\) −6100.61 −0.841788
\(36\) 0 0
\(37\) 2247.44 0.269888 0.134944 0.990853i \(-0.456914\pi\)
0.134944 + 0.990853i \(0.456914\pi\)
\(38\) 0 0
\(39\) −9176.19 −0.966053
\(40\) 0 0
\(41\) 997.106 0.0926364 0.0463182 0.998927i \(-0.485251\pi\)
0.0463182 + 0.998927i \(0.485251\pi\)
\(42\) 0 0
\(43\) 3731.90 0.307793 0.153897 0.988087i \(-0.450818\pi\)
0.153897 + 0.988087i \(0.450818\pi\)
\(44\) 0 0
\(45\) 2342.88 0.172472
\(46\) 0 0
\(47\) 16075.6 1.06151 0.530754 0.847526i \(-0.321909\pi\)
0.530754 + 0.847526i \(0.321909\pi\)
\(48\) 0 0
\(49\) 27678.1 1.64682
\(50\) 0 0
\(51\) −16632.5 −0.895430
\(52\) 0 0
\(53\) 39562.9 1.93463 0.967317 0.253570i \(-0.0816047\pi\)
0.967317 + 0.253570i \(0.0816047\pi\)
\(54\) 0 0
\(55\) −4898.14 −0.218335
\(56\) 0 0
\(57\) −3249.00 −0.132453
\(58\) 0 0
\(59\) 35454.3 1.32599 0.662993 0.748626i \(-0.269286\pi\)
0.662993 + 0.748626i \(0.269286\pi\)
\(60\) 0 0
\(61\) −56219.8 −1.93448 −0.967241 0.253859i \(-0.918300\pi\)
−0.967241 + 0.253859i \(0.918300\pi\)
\(62\) 0 0
\(63\) −17084.1 −0.542302
\(64\) 0 0
\(65\) −29490.7 −0.865769
\(66\) 0 0
\(67\) 6653.47 0.181076 0.0905381 0.995893i \(-0.471141\pi\)
0.0905381 + 0.995893i \(0.471141\pi\)
\(68\) 0 0
\(69\) 12742.5 0.322204
\(70\) 0 0
\(71\) 8888.19 0.209251 0.104626 0.994512i \(-0.466636\pi\)
0.104626 + 0.994512i \(0.466636\pi\)
\(72\) 0 0
\(73\) −12509.4 −0.274746 −0.137373 0.990519i \(-0.543866\pi\)
−0.137373 + 0.990519i \(0.543866\pi\)
\(74\) 0 0
\(75\) −20595.4 −0.422782
\(76\) 0 0
\(77\) 35716.8 0.686508
\(78\) 0 0
\(79\) 63220.1 1.13969 0.569846 0.821752i \(-0.307003\pi\)
0.569846 + 0.821752i \(0.307003\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 37207.9 0.592843 0.296422 0.955057i \(-0.404207\pi\)
0.296422 + 0.955057i \(0.404207\pi\)
\(84\) 0 0
\(85\) −53454.0 −0.802478
\(86\) 0 0
\(87\) 25193.1 0.356848
\(88\) 0 0
\(89\) 46963.0 0.628464 0.314232 0.949346i \(-0.398253\pi\)
0.314232 + 0.949346i \(0.398253\pi\)
\(90\) 0 0
\(91\) 215044. 2.72222
\(92\) 0 0
\(93\) −552.343 −0.00662219
\(94\) 0 0
\(95\) −10441.7 −0.118704
\(96\) 0 0
\(97\) 46146.9 0.497981 0.248991 0.968506i \(-0.419901\pi\)
0.248991 + 0.968506i \(0.419901\pi\)
\(98\) 0 0
\(99\) −13716.7 −0.140657
\(100\) 0 0
\(101\) 2530.94 0.0246875 0.0123438 0.999924i \(-0.496071\pi\)
0.0123438 + 0.999924i \(0.496071\pi\)
\(102\) 0 0
\(103\) −93712.2 −0.870369 −0.435184 0.900341i \(-0.643317\pi\)
−0.435184 + 0.900341i \(0.643317\pi\)
\(104\) 0 0
\(105\) −54905.5 −0.486007
\(106\) 0 0
\(107\) 153860. 1.29917 0.649585 0.760289i \(-0.274943\pi\)
0.649585 + 0.760289i \(0.274943\pi\)
\(108\) 0 0
\(109\) −93199.1 −0.751355 −0.375678 0.926750i \(-0.622590\pi\)
−0.375678 + 0.926750i \(0.622590\pi\)
\(110\) 0 0
\(111\) 20227.0 0.155820
\(112\) 0 0
\(113\) 88014.1 0.648419 0.324210 0.945985i \(-0.394902\pi\)
0.324210 + 0.945985i \(0.394902\pi\)
\(114\) 0 0
\(115\) 40952.2 0.288757
\(116\) 0 0
\(117\) −82585.7 −0.557751
\(118\) 0 0
\(119\) 389782. 2.52322
\(120\) 0 0
\(121\) −132374. −0.821940
\(122\) 0 0
\(123\) 8973.95 0.0534836
\(124\) 0 0
\(125\) −156579. −0.896311
\(126\) 0 0
\(127\) −137253. −0.755114 −0.377557 0.925986i \(-0.623236\pi\)
−0.377557 + 0.925986i \(0.623236\pi\)
\(128\) 0 0
\(129\) 33587.1 0.177705
\(130\) 0 0
\(131\) 97239.0 0.495065 0.247532 0.968880i \(-0.420380\pi\)
0.247532 + 0.968880i \(0.420380\pi\)
\(132\) 0 0
\(133\) 76140.3 0.373238
\(134\) 0 0
\(135\) 21086.0 0.0995769
\(136\) 0 0
\(137\) 244529. 1.11309 0.556543 0.830819i \(-0.312128\pi\)
0.556543 + 0.830819i \(0.312128\pi\)
\(138\) 0 0
\(139\) 4472.71 0.0196351 0.00981756 0.999952i \(-0.496875\pi\)
0.00981756 + 0.999952i \(0.496875\pi\)
\(140\) 0 0
\(141\) 144681. 0.612862
\(142\) 0 0
\(143\) 172657. 0.706066
\(144\) 0 0
\(145\) 80966.3 0.319804
\(146\) 0 0
\(147\) 249103. 0.950792
\(148\) 0 0
\(149\) −261625. −0.965415 −0.482707 0.875782i \(-0.660347\pi\)
−0.482707 + 0.875782i \(0.660347\pi\)
\(150\) 0 0
\(151\) −245463. −0.876080 −0.438040 0.898956i \(-0.644327\pi\)
−0.438040 + 0.898956i \(0.644327\pi\)
\(152\) 0 0
\(153\) −149692. −0.516977
\(154\) 0 0
\(155\) −1775.14 −0.00593475
\(156\) 0 0
\(157\) −448479. −1.45209 −0.726045 0.687647i \(-0.758644\pi\)
−0.726045 + 0.687647i \(0.758644\pi\)
\(158\) 0 0
\(159\) 356066. 1.11696
\(160\) 0 0
\(161\) −298620. −0.907933
\(162\) 0 0
\(163\) −439435. −1.29547 −0.647733 0.761867i \(-0.724283\pi\)
−0.647733 + 0.761867i \(0.724283\pi\)
\(164\) 0 0
\(165\) −44083.2 −0.126056
\(166\) 0 0
\(167\) 171577. 0.476067 0.238033 0.971257i \(-0.423497\pi\)
0.238033 + 0.971257i \(0.423497\pi\)
\(168\) 0 0
\(169\) 668244. 1.79977
\(170\) 0 0
\(171\) −29241.0 −0.0764719
\(172\) 0 0
\(173\) 20210.0 0.0513395 0.0256698 0.999670i \(-0.491828\pi\)
0.0256698 + 0.999670i \(0.491828\pi\)
\(174\) 0 0
\(175\) 482652. 1.19135
\(176\) 0 0
\(177\) 319089. 0.765558
\(178\) 0 0
\(179\) 207989. 0.485185 0.242593 0.970128i \(-0.422002\pi\)
0.242593 + 0.970128i \(0.422002\pi\)
\(180\) 0 0
\(181\) −799903. −1.81485 −0.907425 0.420214i \(-0.861955\pi\)
−0.907425 + 0.420214i \(0.861955\pi\)
\(182\) 0 0
\(183\) −505978. −1.11687
\(184\) 0 0
\(185\) 65006.1 0.139645
\(186\) 0 0
\(187\) 312954. 0.654449
\(188\) 0 0
\(189\) −153757. −0.313098
\(190\) 0 0
\(191\) −580340. −1.15106 −0.575531 0.817780i \(-0.695205\pi\)
−0.575531 + 0.817780i \(0.695205\pi\)
\(192\) 0 0
\(193\) −99920.5 −0.193091 −0.0965453 0.995329i \(-0.530779\pi\)
−0.0965453 + 0.995329i \(0.530779\pi\)
\(194\) 0 0
\(195\) −265417. −0.499852
\(196\) 0 0
\(197\) −333306. −0.611897 −0.305948 0.952048i \(-0.598973\pi\)
−0.305948 + 0.952048i \(0.598973\pi\)
\(198\) 0 0
\(199\) 840198. 1.50400 0.752002 0.659161i \(-0.229088\pi\)
0.752002 + 0.659161i \(0.229088\pi\)
\(200\) 0 0
\(201\) 59881.2 0.104544
\(202\) 0 0
\(203\) −590399. −1.00556
\(204\) 0 0
\(205\) 28840.8 0.0479316
\(206\) 0 0
\(207\) 114682. 0.186025
\(208\) 0 0
\(209\) 61132.5 0.0968070
\(210\) 0 0
\(211\) 819075. 1.26654 0.633268 0.773933i \(-0.281713\pi\)
0.633268 + 0.773933i \(0.281713\pi\)
\(212\) 0 0
\(213\) 79993.8 0.120811
\(214\) 0 0
\(215\) 107943. 0.159257
\(216\) 0 0
\(217\) 12944.1 0.0186605
\(218\) 0 0
\(219\) −112585. −0.158624
\(220\) 0 0
\(221\) 1.88423e6 2.59510
\(222\) 0 0
\(223\) 471476. 0.634888 0.317444 0.948277i \(-0.397175\pi\)
0.317444 + 0.948277i \(0.397175\pi\)
\(224\) 0 0
\(225\) −185358. −0.244093
\(226\) 0 0
\(227\) −669991. −0.862987 −0.431494 0.902116i \(-0.642013\pi\)
−0.431494 + 0.902116i \(0.642013\pi\)
\(228\) 0 0
\(229\) −1.04824e6 −1.32091 −0.660453 0.750867i \(-0.729636\pi\)
−0.660453 + 0.750867i \(0.729636\pi\)
\(230\) 0 0
\(231\) 321451. 0.396356
\(232\) 0 0
\(233\) −456245. −0.550564 −0.275282 0.961363i \(-0.588771\pi\)
−0.275282 + 0.961363i \(0.588771\pi\)
\(234\) 0 0
\(235\) 464979. 0.549242
\(236\) 0 0
\(237\) 568981. 0.658001
\(238\) 0 0
\(239\) 87877.9 0.0995142 0.0497571 0.998761i \(-0.484155\pi\)
0.0497571 + 0.998761i \(0.484155\pi\)
\(240\) 0 0
\(241\) 996858. 1.10558 0.552791 0.833320i \(-0.313563\pi\)
0.552791 + 0.833320i \(0.313563\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) 800575. 0.852093
\(246\) 0 0
\(247\) 368067. 0.383870
\(248\) 0 0
\(249\) 334871. 0.342278
\(250\) 0 0
\(251\) −651472. −0.652696 −0.326348 0.945250i \(-0.605818\pi\)
−0.326348 + 0.945250i \(0.605818\pi\)
\(252\) 0 0
\(253\) −239760. −0.235491
\(254\) 0 0
\(255\) −481086. −0.463311
\(256\) 0 0
\(257\) 1.96985e6 1.86037 0.930185 0.367090i \(-0.119646\pi\)
0.930185 + 0.367090i \(0.119646\pi\)
\(258\) 0 0
\(259\) −474019. −0.439083
\(260\) 0 0
\(261\) 226738. 0.206026
\(262\) 0 0
\(263\) 1.13782e6 1.01434 0.507171 0.861846i \(-0.330691\pi\)
0.507171 + 0.861846i \(0.330691\pi\)
\(264\) 0 0
\(265\) 1.14434e6 1.00101
\(266\) 0 0
\(267\) 422667. 0.362844
\(268\) 0 0
\(269\) −641291. −0.540349 −0.270174 0.962811i \(-0.587081\pi\)
−0.270174 + 0.962811i \(0.587081\pi\)
\(270\) 0 0
\(271\) 938680. 0.776416 0.388208 0.921572i \(-0.373094\pi\)
0.388208 + 0.921572i \(0.373094\pi\)
\(272\) 0 0
\(273\) 1.93540e6 1.57168
\(274\) 0 0
\(275\) 387518. 0.309001
\(276\) 0 0
\(277\) −1.02519e6 −0.802795 −0.401398 0.915904i \(-0.631475\pi\)
−0.401398 + 0.915904i \(0.631475\pi\)
\(278\) 0 0
\(279\) −4971.09 −0.00382332
\(280\) 0 0
\(281\) −841492. −0.635747 −0.317873 0.948133i \(-0.602969\pi\)
−0.317873 + 0.948133i \(0.602969\pi\)
\(282\) 0 0
\(283\) 1.57993e6 1.17266 0.586330 0.810073i \(-0.300572\pi\)
0.586330 + 0.810073i \(0.300572\pi\)
\(284\) 0 0
\(285\) −93975.7 −0.0685335
\(286\) 0 0
\(287\) −210305. −0.150711
\(288\) 0 0
\(289\) 1.99545e6 1.40539
\(290\) 0 0
\(291\) 415322. 0.287510
\(292\) 0 0
\(293\) 2.28841e6 1.55727 0.778635 0.627477i \(-0.215912\pi\)
0.778635 + 0.627477i \(0.215912\pi\)
\(294\) 0 0
\(295\) 1.02550e6 0.686088
\(296\) 0 0
\(297\) −123450. −0.0812085
\(298\) 0 0
\(299\) −1.44355e6 −0.933799
\(300\) 0 0
\(301\) −787114. −0.500751
\(302\) 0 0
\(303\) 22778.4 0.0142534
\(304\) 0 0
\(305\) −1.62613e6 −1.00093
\(306\) 0 0
\(307\) 1.93279e6 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(308\) 0 0
\(309\) −843410. −0.502507
\(310\) 0 0
\(311\) 2.50607e6 1.46924 0.734619 0.678479i \(-0.237361\pi\)
0.734619 + 0.678479i \(0.237361\pi\)
\(312\) 0 0
\(313\) −744655. −0.429629 −0.214815 0.976655i \(-0.568915\pi\)
−0.214815 + 0.976655i \(0.568915\pi\)
\(314\) 0 0
\(315\) −494149. −0.280596
\(316\) 0 0
\(317\) 1.95216e6 1.09111 0.545554 0.838076i \(-0.316319\pi\)
0.545554 + 0.838076i \(0.316319\pi\)
\(318\) 0 0
\(319\) −474028. −0.260812
\(320\) 0 0
\(321\) 1.38474e6 0.750076
\(322\) 0 0
\(323\) 667147. 0.355808
\(324\) 0 0
\(325\) 2.33317e6 1.22529
\(326\) 0 0
\(327\) −838792. −0.433795
\(328\) 0 0
\(329\) −3.39059e6 −1.72697
\(330\) 0 0
\(331\) −2.09353e6 −1.05029 −0.525145 0.851013i \(-0.675989\pi\)
−0.525145 + 0.851013i \(0.675989\pi\)
\(332\) 0 0
\(333\) 182043. 0.0899628
\(334\) 0 0
\(335\) 192448. 0.0936919
\(336\) 0 0
\(337\) −939240. −0.450507 −0.225254 0.974300i \(-0.572321\pi\)
−0.225254 + 0.974300i \(0.572321\pi\)
\(338\) 0 0
\(339\) 792127. 0.374365
\(340\) 0 0
\(341\) 10392.8 0.00484000
\(342\) 0 0
\(343\) −2.29288e6 −1.05231
\(344\) 0 0
\(345\) 368570. 0.166714
\(346\) 0 0
\(347\) −1.82627e6 −0.814219 −0.407109 0.913379i \(-0.633463\pi\)
−0.407109 + 0.913379i \(0.633463\pi\)
\(348\) 0 0
\(349\) −2.03096e6 −0.892561 −0.446280 0.894893i \(-0.647252\pi\)
−0.446280 + 0.894893i \(0.647252\pi\)
\(350\) 0 0
\(351\) −743271. −0.322018
\(352\) 0 0
\(353\) 3.63876e6 1.55424 0.777118 0.629355i \(-0.216681\pi\)
0.777118 + 0.629355i \(0.216681\pi\)
\(354\) 0 0
\(355\) 257086. 0.108270
\(356\) 0 0
\(357\) 3.50804e6 1.45678
\(358\) 0 0
\(359\) −737834. −0.302150 −0.151075 0.988522i \(-0.548274\pi\)
−0.151075 + 0.988522i \(0.548274\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) −1.19137e6 −0.474547
\(364\) 0 0
\(365\) −361829. −0.142158
\(366\) 0 0
\(367\) −834883. −0.323564 −0.161782 0.986826i \(-0.551724\pi\)
−0.161782 + 0.986826i \(0.551724\pi\)
\(368\) 0 0
\(369\) 80765.6 0.0308788
\(370\) 0 0
\(371\) −8.34441e6 −3.14747
\(372\) 0 0
\(373\) 4.71333e6 1.75410 0.877052 0.480395i \(-0.159507\pi\)
0.877052 + 0.480395i \(0.159507\pi\)
\(374\) 0 0
\(375\) −1.40921e6 −0.517485
\(376\) 0 0
\(377\) −2.85403e6 −1.03420
\(378\) 0 0
\(379\) 5.30886e6 1.89847 0.949234 0.314572i \(-0.101861\pi\)
0.949234 + 0.314572i \(0.101861\pi\)
\(380\) 0 0
\(381\) −1.23528e6 −0.435965
\(382\) 0 0
\(383\) −5.06430e6 −1.76410 −0.882048 0.471159i \(-0.843836\pi\)
−0.882048 + 0.471159i \(0.843836\pi\)
\(384\) 0 0
\(385\) 1.03309e6 0.355211
\(386\) 0 0
\(387\) 302284. 0.102598
\(388\) 0 0
\(389\) 1.65007e6 0.552877 0.276438 0.961032i \(-0.410846\pi\)
0.276438 + 0.961032i \(0.410846\pi\)
\(390\) 0 0
\(391\) −2.61653e6 −0.865534
\(392\) 0 0
\(393\) 875151. 0.285826
\(394\) 0 0
\(395\) 1.82861e6 0.589696
\(396\) 0 0
\(397\) 3.64454e6 1.16056 0.580279 0.814418i \(-0.302944\pi\)
0.580279 + 0.814418i \(0.302944\pi\)
\(398\) 0 0
\(399\) 685263. 0.215489
\(400\) 0 0
\(401\) 3.54899e6 1.10216 0.551079 0.834453i \(-0.314216\pi\)
0.551079 + 0.834453i \(0.314216\pi\)
\(402\) 0 0
\(403\) 62572.9 0.0191921
\(404\) 0 0
\(405\) 189774. 0.0574908
\(406\) 0 0
\(407\) −380587. −0.113885
\(408\) 0 0
\(409\) −4.24114e6 −1.25364 −0.626822 0.779163i \(-0.715645\pi\)
−0.626822 + 0.779163i \(0.715645\pi\)
\(410\) 0 0
\(411\) 2.20076e6 0.642640
\(412\) 0 0
\(413\) −7.47784e6 −2.15725
\(414\) 0 0
\(415\) 1.07622e6 0.306747
\(416\) 0 0
\(417\) 40254.4 0.0113363
\(418\) 0 0
\(419\) −513024. −0.142759 −0.0713793 0.997449i \(-0.522740\pi\)
−0.0713793 + 0.997449i \(0.522740\pi\)
\(420\) 0 0
\(421\) 1.30360e6 0.358460 0.179230 0.983807i \(-0.442639\pi\)
0.179230 + 0.983807i \(0.442639\pi\)
\(422\) 0 0
\(423\) 1.30213e6 0.353836
\(424\) 0 0
\(425\) 4.22904e6 1.13571
\(426\) 0 0
\(427\) 1.18576e7 3.14722
\(428\) 0 0
\(429\) 1.55392e6 0.407647
\(430\) 0 0
\(431\) −6.04300e6 −1.56697 −0.783483 0.621413i \(-0.786559\pi\)
−0.783483 + 0.621413i \(0.786559\pi\)
\(432\) 0 0
\(433\) 4.01535e6 1.02921 0.514605 0.857427i \(-0.327939\pi\)
0.514605 + 0.857427i \(0.327939\pi\)
\(434\) 0 0
\(435\) 728697. 0.184639
\(436\) 0 0
\(437\) −511115. −0.128031
\(438\) 0 0
\(439\) 5.54934e6 1.37429 0.687147 0.726518i \(-0.258863\pi\)
0.687147 + 0.726518i \(0.258863\pi\)
\(440\) 0 0
\(441\) 2.24193e6 0.548940
\(442\) 0 0
\(443\) 6.39785e6 1.54890 0.774452 0.632632i \(-0.218026\pi\)
0.774452 + 0.632632i \(0.218026\pi\)
\(444\) 0 0
\(445\) 1.35838e6 0.325178
\(446\) 0 0
\(447\) −2.35463e6 −0.557383
\(448\) 0 0
\(449\) −273835. −0.0641023 −0.0320511 0.999486i \(-0.510204\pi\)
−0.0320511 + 0.999486i \(0.510204\pi\)
\(450\) 0 0
\(451\) −168852. −0.0390900
\(452\) 0 0
\(453\) −2.20917e6 −0.505805
\(454\) 0 0
\(455\) 6.22004e6 1.40852
\(456\) 0 0
\(457\) 606862. 0.135925 0.0679626 0.997688i \(-0.478350\pi\)
0.0679626 + 0.997688i \(0.478350\pi\)
\(458\) 0 0
\(459\) −1.34723e6 −0.298477
\(460\) 0 0
\(461\) 6.65261e6 1.45794 0.728970 0.684545i \(-0.239999\pi\)
0.728970 + 0.684545i \(0.239999\pi\)
\(462\) 0 0
\(463\) −6.17901e6 −1.33957 −0.669786 0.742554i \(-0.733614\pi\)
−0.669786 + 0.742554i \(0.733614\pi\)
\(464\) 0 0
\(465\) −15976.2 −0.00342643
\(466\) 0 0
\(467\) 4.11852e6 0.873874 0.436937 0.899492i \(-0.356063\pi\)
0.436937 + 0.899492i \(0.356063\pi\)
\(468\) 0 0
\(469\) −1.40332e6 −0.294594
\(470\) 0 0
\(471\) −4.03632e6 −0.838364
\(472\) 0 0
\(473\) −631969. −0.129880
\(474\) 0 0
\(475\) 826103. 0.167996
\(476\) 0 0
\(477\) 3.20460e6 0.644878
\(478\) 0 0
\(479\) −2.35240e6 −0.468459 −0.234230 0.972181i \(-0.575257\pi\)
−0.234230 + 0.972181i \(0.575257\pi\)
\(480\) 0 0
\(481\) −2.29144e6 −0.451591
\(482\) 0 0
\(483\) −2.68758e6 −0.524195
\(484\) 0 0
\(485\) 1.33477e6 0.257664
\(486\) 0 0
\(487\) 2.77278e6 0.529776 0.264888 0.964279i \(-0.414665\pi\)
0.264888 + 0.964279i \(0.414665\pi\)
\(488\) 0 0
\(489\) −3.95492e6 −0.747938
\(490\) 0 0
\(491\) −7.43178e6 −1.39120 −0.695600 0.718430i \(-0.744861\pi\)
−0.695600 + 0.718430i \(0.744861\pi\)
\(492\) 0 0
\(493\) −5.17313e6 −0.958597
\(494\) 0 0
\(495\) −396749. −0.0727785
\(496\) 0 0
\(497\) −1.87465e6 −0.340432
\(498\) 0 0
\(499\) 1.43910e6 0.258726 0.129363 0.991597i \(-0.458707\pi\)
0.129363 + 0.991597i \(0.458707\pi\)
\(500\) 0 0
\(501\) 1.54419e6 0.274857
\(502\) 0 0
\(503\) 7.91258e6 1.39444 0.697218 0.716860i \(-0.254421\pi\)
0.697218 + 0.716860i \(0.254421\pi\)
\(504\) 0 0
\(505\) 73206.0 0.0127737
\(506\) 0 0
\(507\) 6.01419e6 1.03910
\(508\) 0 0
\(509\) 4.37164e6 0.747910 0.373955 0.927447i \(-0.378001\pi\)
0.373955 + 0.927447i \(0.378001\pi\)
\(510\) 0 0
\(511\) 2.63843e6 0.446985
\(512\) 0 0
\(513\) −263169. −0.0441511
\(514\) 0 0
\(515\) −2.71058e6 −0.450343
\(516\) 0 0
\(517\) −2.72228e6 −0.447927
\(518\) 0 0
\(519\) 181890. 0.0296409
\(520\) 0 0
\(521\) −4.31485e6 −0.696420 −0.348210 0.937416i \(-0.613210\pi\)
−0.348210 + 0.937416i \(0.613210\pi\)
\(522\) 0 0
\(523\) 5.04725e6 0.806864 0.403432 0.915010i \(-0.367817\pi\)
0.403432 + 0.915010i \(0.367817\pi\)
\(524\) 0 0
\(525\) 4.34387e6 0.687826
\(526\) 0 0
\(527\) 113418. 0.0177891
\(528\) 0 0
\(529\) −4.43177e6 −0.688554
\(530\) 0 0
\(531\) 2.87180e6 0.441995
\(532\) 0 0
\(533\) −1.01663e6 −0.155004
\(534\) 0 0
\(535\) 4.45032e6 0.672213
\(536\) 0 0
\(537\) 1.87190e6 0.280122
\(538\) 0 0
\(539\) −4.68707e6 −0.694912
\(540\) 0 0
\(541\) −9.77646e6 −1.43611 −0.718056 0.695985i \(-0.754968\pi\)
−0.718056 + 0.695985i \(0.754968\pi\)
\(542\) 0 0
\(543\) −7.19913e6 −1.04780
\(544\) 0 0
\(545\) −2.69573e6 −0.388764
\(546\) 0 0
\(547\) 7.32800e6 1.04717 0.523585 0.851974i \(-0.324594\pi\)
0.523585 + 0.851974i \(0.324594\pi\)
\(548\) 0 0
\(549\) −4.55381e6 −0.644828
\(550\) 0 0
\(551\) −1.01052e6 −0.141797
\(552\) 0 0
\(553\) −1.33341e7 −1.85417
\(554\) 0 0
\(555\) 585055. 0.0806240
\(556\) 0 0
\(557\) −7.91105e6 −1.08043 −0.540214 0.841527i \(-0.681657\pi\)
−0.540214 + 0.841527i \(0.681657\pi\)
\(558\) 0 0
\(559\) −3.80496e6 −0.515016
\(560\) 0 0
\(561\) 2.81658e6 0.377846
\(562\) 0 0
\(563\) 207075. 0.0275333 0.0137666 0.999905i \(-0.495618\pi\)
0.0137666 + 0.999905i \(0.495618\pi\)
\(564\) 0 0
\(565\) 2.54576e6 0.335503
\(566\) 0 0
\(567\) −1.38381e6 −0.180767
\(568\) 0 0
\(569\) 1.15559e7 1.49631 0.748156 0.663523i \(-0.230939\pi\)
0.748156 + 0.663523i \(0.230939\pi\)
\(570\) 0 0
\(571\) −1.42478e7 −1.82876 −0.914379 0.404858i \(-0.867321\pi\)
−0.914379 + 0.404858i \(0.867321\pi\)
\(572\) 0 0
\(573\) −5.22306e6 −0.664566
\(574\) 0 0
\(575\) −3.23995e6 −0.408666
\(576\) 0 0
\(577\) −3.50790e6 −0.438639 −0.219320 0.975653i \(-0.570384\pi\)
−0.219320 + 0.975653i \(0.570384\pi\)
\(578\) 0 0
\(579\) −899285. −0.111481
\(580\) 0 0
\(581\) −7.84770e6 −0.964499
\(582\) 0 0
\(583\) −6.69968e6 −0.816361
\(584\) 0 0
\(585\) −2.38875e6 −0.288590
\(586\) 0 0
\(587\) 1.36708e6 0.163756 0.0818780 0.996642i \(-0.473908\pi\)
0.0818780 + 0.996642i \(0.473908\pi\)
\(588\) 0 0
\(589\) 22155.1 0.00263139
\(590\) 0 0
\(591\) −2.99976e6 −0.353279
\(592\) 0 0
\(593\) 9.26761e6 1.08226 0.541129 0.840939i \(-0.317997\pi\)
0.541129 + 0.840939i \(0.317997\pi\)
\(594\) 0 0
\(595\) 1.12742e7 1.30555
\(596\) 0 0
\(597\) 7.56178e6 0.868337
\(598\) 0 0
\(599\) 1.19662e6 0.136267 0.0681333 0.997676i \(-0.478296\pi\)
0.0681333 + 0.997676i \(0.478296\pi\)
\(600\) 0 0
\(601\) −5.19257e6 −0.586403 −0.293201 0.956051i \(-0.594721\pi\)
−0.293201 + 0.956051i \(0.594721\pi\)
\(602\) 0 0
\(603\) 538931. 0.0603587
\(604\) 0 0
\(605\) −3.82886e6 −0.425286
\(606\) 0 0
\(607\) 6.94465e6 0.765031 0.382515 0.923949i \(-0.375058\pi\)
0.382515 + 0.923949i \(0.375058\pi\)
\(608\) 0 0
\(609\) −5.31360e6 −0.580558
\(610\) 0 0
\(611\) −1.63903e7 −1.77617
\(612\) 0 0
\(613\) −1.78506e7 −1.91868 −0.959339 0.282257i \(-0.908917\pi\)
−0.959339 + 0.282257i \(0.908917\pi\)
\(614\) 0 0
\(615\) 259567. 0.0276733
\(616\) 0 0
\(617\) −1.92760e6 −0.203847 −0.101923 0.994792i \(-0.532500\pi\)
−0.101923 + 0.994792i \(0.532500\pi\)
\(618\) 0 0
\(619\) 8.17886e6 0.857958 0.428979 0.903315i \(-0.358873\pi\)
0.428979 + 0.903315i \(0.358873\pi\)
\(620\) 0 0
\(621\) 1.03214e6 0.107401
\(622\) 0 0
\(623\) −9.90519e6 −1.02245
\(624\) 0 0
\(625\) 2.62220e6 0.268513
\(626\) 0 0
\(627\) 550193. 0.0558915
\(628\) 0 0
\(629\) −4.15339e6 −0.418578
\(630\) 0 0
\(631\) −3.39940e6 −0.339883 −0.169941 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(632\) 0 0
\(633\) 7.37167e6 0.731234
\(634\) 0 0
\(635\) −3.96997e6 −0.390709
\(636\) 0 0
\(637\) −2.82200e7 −2.75555
\(638\) 0 0
\(639\) 719944. 0.0697504
\(640\) 0 0
\(641\) −1.56117e6 −0.150074 −0.0750371 0.997181i \(-0.523908\pi\)
−0.0750371 + 0.997181i \(0.523908\pi\)
\(642\) 0 0
\(643\) −1.07265e7 −1.02313 −0.511567 0.859243i \(-0.670935\pi\)
−0.511567 + 0.859243i \(0.670935\pi\)
\(644\) 0 0
\(645\) 971490. 0.0919474
\(646\) 0 0
\(647\) 4.28656e6 0.402576 0.201288 0.979532i \(-0.435487\pi\)
0.201288 + 0.979532i \(0.435487\pi\)
\(648\) 0 0
\(649\) −6.00391e6 −0.559529
\(650\) 0 0
\(651\) 116497. 0.0107737
\(652\) 0 0
\(653\) −2.00405e7 −1.83918 −0.919592 0.392875i \(-0.871480\pi\)
−0.919592 + 0.392875i \(0.871480\pi\)
\(654\) 0 0
\(655\) 2.81259e6 0.256155
\(656\) 0 0
\(657\) −1.01327e6 −0.0915819
\(658\) 0 0
\(659\) −4.00472e6 −0.359218 −0.179609 0.983738i \(-0.557483\pi\)
−0.179609 + 0.983738i \(0.557483\pi\)
\(660\) 0 0
\(661\) −7.40685e6 −0.659371 −0.329685 0.944091i \(-0.606943\pi\)
−0.329685 + 0.944091i \(0.606943\pi\)
\(662\) 0 0
\(663\) 1.69581e7 1.49828
\(664\) 0 0
\(665\) 2.20232e6 0.193119
\(666\) 0 0
\(667\) 3.96324e6 0.344933
\(668\) 0 0
\(669\) 4.24328e6 0.366553
\(670\) 0 0
\(671\) 9.52039e6 0.816297
\(672\) 0 0
\(673\) 1.33052e7 1.13236 0.566180 0.824282i \(-0.308421\pi\)
0.566180 + 0.824282i \(0.308421\pi\)
\(674\) 0 0
\(675\) −1.66822e6 −0.140927
\(676\) 0 0
\(677\) −1.07656e7 −0.902751 −0.451375 0.892334i \(-0.649066\pi\)
−0.451375 + 0.892334i \(0.649066\pi\)
\(678\) 0 0
\(679\) −9.73306e6 −0.810168
\(680\) 0 0
\(681\) −6.02992e6 −0.498246
\(682\) 0 0
\(683\) 1.55246e7 1.27341 0.636706 0.771107i \(-0.280297\pi\)
0.636706 + 0.771107i \(0.280297\pi\)
\(684\) 0 0
\(685\) 7.07287e6 0.575929
\(686\) 0 0
\(687\) −9.43416e6 −0.762626
\(688\) 0 0
\(689\) −4.03375e7 −3.23713
\(690\) 0 0
\(691\) 5.48083e6 0.436668 0.218334 0.975874i \(-0.429938\pi\)
0.218334 + 0.975874i \(0.429938\pi\)
\(692\) 0 0
\(693\) 2.89306e6 0.228836
\(694\) 0 0
\(695\) 129371. 0.0101595
\(696\) 0 0
\(697\) −1.84271e6 −0.143673
\(698\) 0 0
\(699\) −4.10620e6 −0.317869
\(700\) 0 0
\(701\) −1.35701e7 −1.04301 −0.521504 0.853249i \(-0.674629\pi\)
−0.521504 + 0.853249i \(0.674629\pi\)
\(702\) 0 0
\(703\) −811327. −0.0619166
\(704\) 0 0
\(705\) 4.18481e6 0.317105
\(706\) 0 0
\(707\) −533812. −0.0401643
\(708\) 0 0
\(709\) 1.08328e7 0.809332 0.404666 0.914465i \(-0.367388\pi\)
0.404666 + 0.914465i \(0.367388\pi\)
\(710\) 0 0
\(711\) 5.12083e6 0.379897
\(712\) 0 0
\(713\) −86891.5 −0.00640109
\(714\) 0 0
\(715\) 4.99403e6 0.365330
\(716\) 0 0
\(717\) 790901. 0.0574546
\(718\) 0 0
\(719\) −1.84578e7 −1.33155 −0.665774 0.746154i \(-0.731898\pi\)
−0.665774 + 0.746154i \(0.731898\pi\)
\(720\) 0 0
\(721\) 1.97653e7 1.41601
\(722\) 0 0
\(723\) 8.97172e6 0.638308
\(724\) 0 0
\(725\) −6.40569e6 −0.452606
\(726\) 0 0
\(727\) −2.36232e7 −1.65769 −0.828843 0.559482i \(-0.811000\pi\)
−0.828843 + 0.559482i \(0.811000\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −6.89676e6 −0.477366
\(732\) 0 0
\(733\) 2.32984e7 1.60165 0.800823 0.598901i \(-0.204396\pi\)
0.800823 + 0.598901i \(0.204396\pi\)
\(734\) 0 0
\(735\) 7.20517e6 0.491956
\(736\) 0 0
\(737\) −1.12671e6 −0.0764091
\(738\) 0 0
\(739\) −2.76787e7 −1.86438 −0.932190 0.361969i \(-0.882105\pi\)
−0.932190 + 0.361969i \(0.882105\pi\)
\(740\) 0 0
\(741\) 3.31260e6 0.221628
\(742\) 0 0
\(743\) −2.15463e7 −1.43186 −0.715930 0.698172i \(-0.753997\pi\)
−0.715930 + 0.698172i \(0.753997\pi\)
\(744\) 0 0
\(745\) −7.56738e6 −0.499522
\(746\) 0 0
\(747\) 3.01384e6 0.197614
\(748\) 0 0
\(749\) −3.24513e7 −2.11363
\(750\) 0 0
\(751\) −1.62799e6 −0.105330 −0.0526649 0.998612i \(-0.516772\pi\)
−0.0526649 + 0.998612i \(0.516772\pi\)
\(752\) 0 0
\(753\) −5.86324e6 −0.376834
\(754\) 0 0
\(755\) −7.09989e6 −0.453298
\(756\) 0 0
\(757\) 3.12417e7 1.98151 0.990753 0.135678i \(-0.0433214\pi\)
0.990753 + 0.135678i \(0.0433214\pi\)
\(758\) 0 0
\(759\) −2.15784e6 −0.135961
\(760\) 0 0
\(761\) 2.26372e7 1.41697 0.708485 0.705726i \(-0.249379\pi\)
0.708485 + 0.705726i \(0.249379\pi\)
\(762\) 0 0
\(763\) 1.96571e7 1.22238
\(764\) 0 0
\(765\) −4.32977e6 −0.267493
\(766\) 0 0
\(767\) −3.61484e7 −2.21871
\(768\) 0 0
\(769\) 1.46708e6 0.0894618 0.0447309 0.998999i \(-0.485757\pi\)
0.0447309 + 0.998999i \(0.485757\pi\)
\(770\) 0 0
\(771\) 1.77286e7 1.07409
\(772\) 0 0
\(773\) −3.85297e6 −0.231925 −0.115962 0.993254i \(-0.536995\pi\)
−0.115962 + 0.993254i \(0.536995\pi\)
\(774\) 0 0
\(775\) 140441. 0.00839922
\(776\) 0 0
\(777\) −4.26617e6 −0.253505
\(778\) 0 0
\(779\) −359955. −0.0212522
\(780\) 0 0
\(781\) −1.50515e6 −0.0882981
\(782\) 0 0
\(783\) 2.04064e6 0.118949
\(784\) 0 0
\(785\) −1.29720e7 −0.751336
\(786\) 0 0
\(787\) 2.50359e7 1.44088 0.720439 0.693519i \(-0.243941\pi\)
0.720439 + 0.693519i \(0.243941\pi\)
\(788\) 0 0
\(789\) 1.02404e7 0.585630
\(790\) 0 0
\(791\) −1.85635e7 −1.05492
\(792\) 0 0
\(793\) 5.73204e7 3.23688
\(794\) 0 0
\(795\) 1.02990e7 0.577935
\(796\) 0 0
\(797\) 3.36793e7 1.87809 0.939047 0.343788i \(-0.111710\pi\)
0.939047 + 0.343788i \(0.111710\pi\)
\(798\) 0 0
\(799\) −2.97086e7 −1.64633
\(800\) 0 0
\(801\) 3.80400e6 0.209488
\(802\) 0 0
\(803\) 2.11838e6 0.115935
\(804\) 0 0
\(805\) −8.63742e6 −0.469780
\(806\) 0 0
\(807\) −5.77162e6 −0.311971
\(808\) 0 0
\(809\) −1.99641e7 −1.07245 −0.536227 0.844074i \(-0.680151\pi\)
−0.536227 + 0.844074i \(0.680151\pi\)
\(810\) 0 0
\(811\) 8.00386e6 0.427314 0.213657 0.976909i \(-0.431462\pi\)
0.213657 + 0.976909i \(0.431462\pi\)
\(812\) 0 0
\(813\) 8.44812e6 0.448264
\(814\) 0 0
\(815\) −1.27104e7 −0.670296
\(816\) 0 0
\(817\) −1.34722e6 −0.0706126
\(818\) 0 0
\(819\) 1.74186e7 0.907408
\(820\) 0 0
\(821\) 1.19433e7 0.618397 0.309199 0.950998i \(-0.399939\pi\)
0.309199 + 0.950998i \(0.399939\pi\)
\(822\) 0 0
\(823\) −7.36634e6 −0.379098 −0.189549 0.981871i \(-0.560703\pi\)
−0.189549 + 0.981871i \(0.560703\pi\)
\(824\) 0 0
\(825\) 3.48767e6 0.178402
\(826\) 0 0
\(827\) 8.37344e6 0.425736 0.212868 0.977081i \(-0.431720\pi\)
0.212868 + 0.977081i \(0.431720\pi\)
\(828\) 0 0
\(829\) 3.71149e7 1.87569 0.937846 0.347052i \(-0.112817\pi\)
0.937846 + 0.347052i \(0.112817\pi\)
\(830\) 0 0
\(831\) −9.22670e6 −0.463494
\(832\) 0 0
\(833\) −5.11506e7 −2.55410
\(834\) 0 0
\(835\) 4.96278e6 0.246325
\(836\) 0 0
\(837\) −44739.8 −0.00220740
\(838\) 0 0
\(839\) 1.03736e7 0.508774 0.254387 0.967102i \(-0.418126\pi\)
0.254387 + 0.967102i \(0.418126\pi\)
\(840\) 0 0
\(841\) −1.26755e7 −0.617979
\(842\) 0 0
\(843\) −7.57343e6 −0.367049
\(844\) 0 0
\(845\) 1.93286e7 0.931234
\(846\) 0 0
\(847\) 2.79197e7 1.33722
\(848\) 0 0
\(849\) 1.42194e7 0.677035
\(850\) 0 0
\(851\) 3.18200e6 0.150618
\(852\) 0 0
\(853\) 2.88586e7 1.35801 0.679004 0.734135i \(-0.262412\pi\)
0.679004 + 0.734135i \(0.262412\pi\)
\(854\) 0 0
\(855\) −845781. −0.0395679
\(856\) 0 0
\(857\) 2.87680e7 1.33800 0.669002 0.743261i \(-0.266722\pi\)
0.669002 + 0.743261i \(0.266722\pi\)
\(858\) 0 0
\(859\) −2.81738e6 −0.130275 −0.0651376 0.997876i \(-0.520749\pi\)
−0.0651376 + 0.997876i \(0.520749\pi\)
\(860\) 0 0
\(861\) −1.89274e6 −0.0870128
\(862\) 0 0
\(863\) −2.34321e7 −1.07099 −0.535493 0.844539i \(-0.679874\pi\)
−0.535493 + 0.844539i \(0.679874\pi\)
\(864\) 0 0
\(865\) 584565. 0.0265639
\(866\) 0 0
\(867\) 1.79590e7 0.811399
\(868\) 0 0
\(869\) −1.07058e7 −0.480918
\(870\) 0 0
\(871\) −6.78372e6 −0.302986
\(872\) 0 0
\(873\) 3.73790e6 0.165994
\(874\) 0 0
\(875\) 3.30249e7 1.45821
\(876\) 0 0
\(877\) −4.12225e7 −1.80982 −0.904909 0.425604i \(-0.860062\pi\)
−0.904909 + 0.425604i \(0.860062\pi\)
\(878\) 0 0
\(879\) 2.05957e7 0.899091
\(880\) 0 0
\(881\) 1.30218e7 0.565238 0.282619 0.959232i \(-0.408797\pi\)
0.282619 + 0.959232i \(0.408797\pi\)
\(882\) 0 0
\(883\) 406852. 0.0175604 0.00878021 0.999961i \(-0.497205\pi\)
0.00878021 + 0.999961i \(0.497205\pi\)
\(884\) 0 0
\(885\) 9.22948e6 0.396113
\(886\) 0 0
\(887\) 2.48049e7 1.05859 0.529296 0.848437i \(-0.322456\pi\)
0.529296 + 0.848437i \(0.322456\pi\)
\(888\) 0 0
\(889\) 2.89487e7 1.22850
\(890\) 0 0
\(891\) −1.11105e6 −0.0468858
\(892\) 0 0
\(893\) −5.80330e6 −0.243527
\(894\) 0 0
\(895\) 6.01597e6 0.251043
\(896\) 0 0
\(897\) −1.29919e7 −0.539129
\(898\) 0 0
\(899\) −171793. −0.00708934
\(900\) 0 0
\(901\) −7.31144e7 −3.00048
\(902\) 0 0
\(903\) −7.08403e6 −0.289108
\(904\) 0 0
\(905\) −2.31368e7 −0.939034
\(906\) 0 0
\(907\) −2.35678e6 −0.0951265 −0.0475633 0.998868i \(-0.515146\pi\)
−0.0475633 + 0.998868i \(0.515146\pi\)
\(908\) 0 0
\(909\) 205006. 0.00822918
\(910\) 0 0
\(911\) −3.11062e7 −1.24180 −0.620898 0.783891i \(-0.713232\pi\)
−0.620898 + 0.783891i \(0.713232\pi\)
\(912\) 0 0
\(913\) −6.30087e6 −0.250163
\(914\) 0 0
\(915\) −1.46352e7 −0.577890
\(916\) 0 0
\(917\) −2.05092e7 −0.805424
\(918\) 0 0
\(919\) −1.17745e6 −0.0459890 −0.0229945 0.999736i \(-0.507320\pi\)
−0.0229945 + 0.999736i \(0.507320\pi\)
\(920\) 0 0
\(921\) 1.73951e7 0.675737
\(922\) 0 0
\(923\) −9.06220e6 −0.350130
\(924\) 0 0
\(925\) −5.14299e6 −0.197634
\(926\) 0 0
\(927\) −7.59069e6 −0.290123
\(928\) 0 0
\(929\) −3.01092e7 −1.14462 −0.572308 0.820039i \(-0.693952\pi\)
−0.572308 + 0.820039i \(0.693952\pi\)
\(930\) 0 0
\(931\) −9.99180e6 −0.377806
\(932\) 0 0
\(933\) 2.25546e7 0.848265
\(934\) 0 0
\(935\) 9.05202e6 0.338623
\(936\) 0 0
\(937\) 1.22792e7 0.456902 0.228451 0.973555i \(-0.426634\pi\)
0.228451 + 0.973555i \(0.426634\pi\)
\(938\) 0 0
\(939\) −6.70189e6 −0.248047
\(940\) 0 0
\(941\) 4.46062e7 1.64218 0.821090 0.570799i \(-0.193366\pi\)
0.821090 + 0.570799i \(0.193366\pi\)
\(942\) 0 0
\(943\) 1.41173e6 0.0516979
\(944\) 0 0
\(945\) −4.44734e6 −0.162002
\(946\) 0 0
\(947\) −5.16996e7 −1.87332 −0.936661 0.350238i \(-0.886101\pi\)
−0.936661 + 0.350238i \(0.886101\pi\)
\(948\) 0 0
\(949\) 1.27543e7 0.459719
\(950\) 0 0
\(951\) 1.75694e7 0.629951
\(952\) 0 0
\(953\) −3.83133e7 −1.36652 −0.683262 0.730173i \(-0.739439\pi\)
−0.683262 + 0.730173i \(0.739439\pi\)
\(954\) 0 0
\(955\) −1.67860e7 −0.595579
\(956\) 0 0
\(957\) −4.26625e6 −0.150580
\(958\) 0 0
\(959\) −5.15748e7 −1.81088
\(960\) 0 0
\(961\) −2.86254e7 −0.999868
\(962\) 0 0
\(963\) 1.24627e7 0.433057
\(964\) 0 0
\(965\) −2.89015e6 −0.0999084
\(966\) 0 0
\(967\) 1.46233e7 0.502896 0.251448 0.967871i \(-0.419093\pi\)
0.251448 + 0.967871i \(0.419093\pi\)
\(968\) 0 0
\(969\) 6.00433e6 0.205426
\(970\) 0 0
\(971\) −1.06002e7 −0.360800 −0.180400 0.983593i \(-0.557739\pi\)
−0.180400 + 0.983593i \(0.557739\pi\)
\(972\) 0 0
\(973\) −943361. −0.0319445
\(974\) 0 0
\(975\) 2.09986e7 0.707421
\(976\) 0 0
\(977\) 1.93574e7 0.648799 0.324400 0.945920i \(-0.394838\pi\)
0.324400 + 0.945920i \(0.394838\pi\)
\(978\) 0 0
\(979\) −7.95281e6 −0.265194
\(980\) 0 0
\(981\) −7.54912e6 −0.250452
\(982\) 0 0
\(983\) 3.37569e7 1.11424 0.557120 0.830432i \(-0.311906\pi\)
0.557120 + 0.830432i \(0.311906\pi\)
\(984\) 0 0
\(985\) −9.64072e6 −0.316606
\(986\) 0 0
\(987\) −3.05153e7 −0.997069
\(988\) 0 0
\(989\) 5.28374e6 0.171771
\(990\) 0 0
\(991\) −4.24463e7 −1.37295 −0.686477 0.727152i \(-0.740844\pi\)
−0.686477 + 0.727152i \(0.740844\pi\)
\(992\) 0 0
\(993\) −1.88418e7 −0.606385
\(994\) 0 0
\(995\) 2.43023e7 0.778197
\(996\) 0 0
\(997\) 5.54861e7 1.76785 0.883927 0.467624i \(-0.154890\pi\)
0.883927 + 0.467624i \(0.154890\pi\)
\(998\) 0 0
\(999\) 1.63839e6 0.0519400
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 912.6.a.y.1.4 6
4.3 odd 2 456.6.a.f.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.6.a.f.1.4 6 4.3 odd 2
912.6.a.y.1.4 6 1.1 even 1 trivial