Properties

Label 1104.6.a.r.1.4
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.40352\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +436.763 q^{11} -258.865 q^{13} -634.683 q^{15} +591.819 q^{17} -1663.69 q^{19} -807.419 q^{21} +529.000 q^{23} +1848.12 q^{25} -729.000 q^{27} -5121.48 q^{29} -1871.29 q^{31} -3930.87 q^{33} +6326.60 q^{35} +884.534 q^{37} +2329.78 q^{39} -18410.5 q^{41} -12334.9 q^{43} +5712.15 q^{45} -23871.6 q^{47} -8758.55 q^{49} -5326.37 q^{51} -14594.8 q^{53} +30800.7 q^{55} +14973.2 q^{57} -47739.1 q^{59} -15351.5 q^{61} +7266.77 q^{63} -18255.2 q^{65} +41174.4 q^{67} -4761.00 q^{69} +45608.3 q^{71} -7898.24 q^{73} -16633.1 q^{75} +39183.4 q^{77} -71435.2 q^{79} +6561.00 q^{81} -112580. q^{83} +41735.3 q^{85} +46093.3 q^{87} +137845. q^{89} -23223.6 q^{91} +16841.6 q^{93} -117324. q^{95} +41679.8 q^{97} +35377.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 70.5203 1.26151 0.630753 0.775984i \(-0.282746\pi\)
0.630753 + 0.775984i \(0.282746\pi\)
\(6\) 0 0
\(7\) 89.7132 0.692008 0.346004 0.938233i \(-0.387538\pi\)
0.346004 + 0.938233i \(0.387538\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 436.763 1.08834 0.544170 0.838975i \(-0.316845\pi\)
0.544170 + 0.838975i \(0.316845\pi\)
\(12\) 0 0
\(13\) −258.865 −0.424830 −0.212415 0.977180i \(-0.568133\pi\)
−0.212415 + 0.977180i \(0.568133\pi\)
\(14\) 0 0
\(15\) −634.683 −0.728331
\(16\) 0 0
\(17\) 591.819 0.496669 0.248334 0.968674i \(-0.420117\pi\)
0.248334 + 0.968674i \(0.420117\pi\)
\(18\) 0 0
\(19\) −1663.69 −1.05727 −0.528637 0.848848i \(-0.677297\pi\)
−0.528637 + 0.848848i \(0.677297\pi\)
\(20\) 0 0
\(21\) −807.419 −0.399531
\(22\) 0 0
\(23\) 529.000 0.208514
\(24\) 0 0
\(25\) 1848.12 0.591398
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −5121.48 −1.13084 −0.565419 0.824804i \(-0.691286\pi\)
−0.565419 + 0.824804i \(0.691286\pi\)
\(30\) 0 0
\(31\) −1871.29 −0.349733 −0.174867 0.984592i \(-0.555949\pi\)
−0.174867 + 0.984592i \(0.555949\pi\)
\(32\) 0 0
\(33\) −3930.87 −0.628353
\(34\) 0 0
\(35\) 6326.60 0.872972
\(36\) 0 0
\(37\) 884.534 0.106221 0.0531105 0.998589i \(-0.483086\pi\)
0.0531105 + 0.998589i \(0.483086\pi\)
\(38\) 0 0
\(39\) 2329.78 0.245275
\(40\) 0 0
\(41\) −18410.5 −1.71043 −0.855216 0.518271i \(-0.826576\pi\)
−0.855216 + 0.518271i \(0.826576\pi\)
\(42\) 0 0
\(43\) −12334.9 −1.01734 −0.508669 0.860962i \(-0.669862\pi\)
−0.508669 + 0.860962i \(0.669862\pi\)
\(44\) 0 0
\(45\) 5712.15 0.420502
\(46\) 0 0
\(47\) −23871.6 −1.57629 −0.788146 0.615489i \(-0.788959\pi\)
−0.788146 + 0.615489i \(0.788959\pi\)
\(48\) 0 0
\(49\) −8758.55 −0.521125
\(50\) 0 0
\(51\) −5326.37 −0.286752
\(52\) 0 0
\(53\) −14594.8 −0.713690 −0.356845 0.934164i \(-0.616147\pi\)
−0.356845 + 0.934164i \(0.616147\pi\)
\(54\) 0 0
\(55\) 30800.7 1.37295
\(56\) 0 0
\(57\) 14973.2 0.610418
\(58\) 0 0
\(59\) −47739.1 −1.78544 −0.892718 0.450616i \(-0.851205\pi\)
−0.892718 + 0.450616i \(0.851205\pi\)
\(60\) 0 0
\(61\) −15351.5 −0.528233 −0.264116 0.964491i \(-0.585080\pi\)
−0.264116 + 0.964491i \(0.585080\pi\)
\(62\) 0 0
\(63\) 7266.77 0.230669
\(64\) 0 0
\(65\) −18255.2 −0.535925
\(66\) 0 0
\(67\) 41174.4 1.12057 0.560287 0.828299i \(-0.310691\pi\)
0.560287 + 0.828299i \(0.310691\pi\)
\(68\) 0 0
\(69\) −4761.00 −0.120386
\(70\) 0 0
\(71\) 45608.3 1.07374 0.536869 0.843666i \(-0.319607\pi\)
0.536869 + 0.843666i \(0.319607\pi\)
\(72\) 0 0
\(73\) −7898.24 −0.173470 −0.0867348 0.996231i \(-0.527643\pi\)
−0.0867348 + 0.996231i \(0.527643\pi\)
\(74\) 0 0
\(75\) −16633.1 −0.341444
\(76\) 0 0
\(77\) 39183.4 0.753140
\(78\) 0 0
\(79\) −71435.2 −1.28779 −0.643894 0.765115i \(-0.722682\pi\)
−0.643894 + 0.765115i \(0.722682\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −112580. −1.79377 −0.896885 0.442264i \(-0.854175\pi\)
−0.896885 + 0.442264i \(0.854175\pi\)
\(84\) 0 0
\(85\) 41735.3 0.626551
\(86\) 0 0
\(87\) 46093.3 0.652890
\(88\) 0 0
\(89\) 137845. 1.84466 0.922330 0.386404i \(-0.126283\pi\)
0.922330 + 0.386404i \(0.126283\pi\)
\(90\) 0 0
\(91\) −23223.6 −0.293985
\(92\) 0 0
\(93\) 16841.6 0.201918
\(94\) 0 0
\(95\) −117324. −1.33376
\(96\) 0 0
\(97\) 41679.8 0.449776 0.224888 0.974385i \(-0.427798\pi\)
0.224888 + 0.974385i \(0.427798\pi\)
\(98\) 0 0
\(99\) 35377.8 0.362780
\(100\) 0 0
\(101\) 37100.3 0.361888 0.180944 0.983493i \(-0.442085\pi\)
0.180944 + 0.983493i \(0.442085\pi\)
\(102\) 0 0
\(103\) 123323. 1.14538 0.572692 0.819771i \(-0.305899\pi\)
0.572692 + 0.819771i \(0.305899\pi\)
\(104\) 0 0
\(105\) −56939.4 −0.504011
\(106\) 0 0
\(107\) −107455. −0.907338 −0.453669 0.891170i \(-0.649885\pi\)
−0.453669 + 0.891170i \(0.649885\pi\)
\(108\) 0 0
\(109\) 96652.7 0.779198 0.389599 0.920985i \(-0.372614\pi\)
0.389599 + 0.920985i \(0.372614\pi\)
\(110\) 0 0
\(111\) −7960.80 −0.0613267
\(112\) 0 0
\(113\) 55788.7 0.411008 0.205504 0.978656i \(-0.434117\pi\)
0.205504 + 0.978656i \(0.434117\pi\)
\(114\) 0 0
\(115\) 37305.3 0.263042
\(116\) 0 0
\(117\) −20968.1 −0.141610
\(118\) 0 0
\(119\) 53094.0 0.343699
\(120\) 0 0
\(121\) 29711.1 0.184483
\(122\) 0 0
\(123\) 165695. 0.987519
\(124\) 0 0
\(125\) −90046.2 −0.515454
\(126\) 0 0
\(127\) −167839. −0.923388 −0.461694 0.887039i \(-0.652758\pi\)
−0.461694 + 0.887039i \(0.652758\pi\)
\(128\) 0 0
\(129\) 111014. 0.587361
\(130\) 0 0
\(131\) −129729. −0.660476 −0.330238 0.943898i \(-0.607129\pi\)
−0.330238 + 0.943898i \(0.607129\pi\)
\(132\) 0 0
\(133\) −149255. −0.731643
\(134\) 0 0
\(135\) −51409.3 −0.242777
\(136\) 0 0
\(137\) −226702. −1.03194 −0.515970 0.856607i \(-0.672568\pi\)
−0.515970 + 0.856607i \(0.672568\pi\)
\(138\) 0 0
\(139\) 419175. 1.84017 0.920086 0.391716i \(-0.128118\pi\)
0.920086 + 0.391716i \(0.128118\pi\)
\(140\) 0 0
\(141\) 214844. 0.910072
\(142\) 0 0
\(143\) −113063. −0.462359
\(144\) 0 0
\(145\) −361169. −1.42656
\(146\) 0 0
\(147\) 78826.9 0.300872
\(148\) 0 0
\(149\) 503270. 1.85710 0.928551 0.371206i \(-0.121055\pi\)
0.928551 + 0.371206i \(0.121055\pi\)
\(150\) 0 0
\(151\) −517943. −1.84858 −0.924292 0.381685i \(-0.875344\pi\)
−0.924292 + 0.381685i \(0.875344\pi\)
\(152\) 0 0
\(153\) 47937.4 0.165556
\(154\) 0 0
\(155\) −131964. −0.441190
\(156\) 0 0
\(157\) −39747.9 −0.128696 −0.0643480 0.997928i \(-0.520497\pi\)
−0.0643480 + 0.997928i \(0.520497\pi\)
\(158\) 0 0
\(159\) 131353. 0.412049
\(160\) 0 0
\(161\) 47458.3 0.144294
\(162\) 0 0
\(163\) 157828. 0.465279 0.232640 0.972563i \(-0.425264\pi\)
0.232640 + 0.972563i \(0.425264\pi\)
\(164\) 0 0
\(165\) −277206. −0.792671
\(166\) 0 0
\(167\) 101000. 0.280241 0.140120 0.990134i \(-0.455251\pi\)
0.140120 + 0.990134i \(0.455251\pi\)
\(168\) 0 0
\(169\) −304282. −0.819520
\(170\) 0 0
\(171\) −134759. −0.352425
\(172\) 0 0
\(173\) 61568.5 0.156402 0.0782012 0.996938i \(-0.475082\pi\)
0.0782012 + 0.996938i \(0.475082\pi\)
\(174\) 0 0
\(175\) 165801. 0.409252
\(176\) 0 0
\(177\) 429652. 1.03082
\(178\) 0 0
\(179\) 610536. 1.42423 0.712113 0.702065i \(-0.247738\pi\)
0.712113 + 0.702065i \(0.247738\pi\)
\(180\) 0 0
\(181\) 121378. 0.275387 0.137694 0.990475i \(-0.456031\pi\)
0.137694 + 0.990475i \(0.456031\pi\)
\(182\) 0 0
\(183\) 138163. 0.304975
\(184\) 0 0
\(185\) 62377.6 0.133998
\(186\) 0 0
\(187\) 258485. 0.540544
\(188\) 0 0
\(189\) −65400.9 −0.133177
\(190\) 0 0
\(191\) 294581. 0.584280 0.292140 0.956376i \(-0.405633\pi\)
0.292140 + 0.956376i \(0.405633\pi\)
\(192\) 0 0
\(193\) −8895.53 −0.0171901 −0.00859505 0.999963i \(-0.502736\pi\)
−0.00859505 + 0.999963i \(0.502736\pi\)
\(194\) 0 0
\(195\) 164297. 0.309417
\(196\) 0 0
\(197\) −698113. −1.28162 −0.640811 0.767699i \(-0.721402\pi\)
−0.640811 + 0.767699i \(0.721402\pi\)
\(198\) 0 0
\(199\) 533271. 0.954586 0.477293 0.878744i \(-0.341618\pi\)
0.477293 + 0.878744i \(0.341618\pi\)
\(200\) 0 0
\(201\) −370570. −0.646963
\(202\) 0 0
\(203\) −459464. −0.782550
\(204\) 0 0
\(205\) −1.29831e6 −2.15772
\(206\) 0 0
\(207\) 42849.0 0.0695048
\(208\) 0 0
\(209\) −726638. −1.15067
\(210\) 0 0
\(211\) 658407. 1.01809 0.509047 0.860739i \(-0.329998\pi\)
0.509047 + 0.860739i \(0.329998\pi\)
\(212\) 0 0
\(213\) −410475. −0.619923
\(214\) 0 0
\(215\) −869864. −1.28338
\(216\) 0 0
\(217\) −167879. −0.242018
\(218\) 0 0
\(219\) 71084.2 0.100153
\(220\) 0 0
\(221\) −153201. −0.211000
\(222\) 0 0
\(223\) −79895.6 −0.107587 −0.0537936 0.998552i \(-0.517131\pi\)
−0.0537936 + 0.998552i \(0.517131\pi\)
\(224\) 0 0
\(225\) 149698. 0.197133
\(226\) 0 0
\(227\) −716111. −0.922393 −0.461196 0.887298i \(-0.652580\pi\)
−0.461196 + 0.887298i \(0.652580\pi\)
\(228\) 0 0
\(229\) −581299. −0.732506 −0.366253 0.930515i \(-0.619360\pi\)
−0.366253 + 0.930515i \(0.619360\pi\)
\(230\) 0 0
\(231\) −352651. −0.434825
\(232\) 0 0
\(233\) 122734. 0.148107 0.0740534 0.997254i \(-0.476406\pi\)
0.0740534 + 0.997254i \(0.476406\pi\)
\(234\) 0 0
\(235\) −1.68343e6 −1.98850
\(236\) 0 0
\(237\) 642917. 0.743505
\(238\) 0 0
\(239\) −210076. −0.237893 −0.118946 0.992901i \(-0.537952\pi\)
−0.118946 + 0.992901i \(0.537952\pi\)
\(240\) 0 0
\(241\) 1.23224e6 1.36664 0.683319 0.730120i \(-0.260536\pi\)
0.683319 + 0.730120i \(0.260536\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) −617656. −0.657402
\(246\) 0 0
\(247\) 430670. 0.449162
\(248\) 0 0
\(249\) 1.01322e6 1.03563
\(250\) 0 0
\(251\) 545986. 0.547012 0.273506 0.961870i \(-0.411817\pi\)
0.273506 + 0.961870i \(0.411817\pi\)
\(252\) 0 0
\(253\) 231048. 0.226934
\(254\) 0 0
\(255\) −375618. −0.361739
\(256\) 0 0
\(257\) 556705. 0.525765 0.262883 0.964828i \(-0.415327\pi\)
0.262883 + 0.964828i \(0.415327\pi\)
\(258\) 0 0
\(259\) 79354.3 0.0735057
\(260\) 0 0
\(261\) −414840. −0.376946
\(262\) 0 0
\(263\) −1.65392e6 −1.47443 −0.737217 0.675656i \(-0.763860\pi\)
−0.737217 + 0.675656i \(0.763860\pi\)
\(264\) 0 0
\(265\) −1.02923e6 −0.900324
\(266\) 0 0
\(267\) −1.24061e6 −1.06501
\(268\) 0 0
\(269\) −2.06545e6 −1.74034 −0.870168 0.492755i \(-0.835990\pi\)
−0.870168 + 0.492755i \(0.835990\pi\)
\(270\) 0 0
\(271\) −708326. −0.585881 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(272\) 0 0
\(273\) 209012. 0.169733
\(274\) 0 0
\(275\) 807190. 0.643641
\(276\) 0 0
\(277\) −17729.4 −0.0138833 −0.00694167 0.999976i \(-0.502210\pi\)
−0.00694167 + 0.999976i \(0.502210\pi\)
\(278\) 0 0
\(279\) −151574. −0.116578
\(280\) 0 0
\(281\) 754660. 0.570145 0.285073 0.958506i \(-0.407982\pi\)
0.285073 + 0.958506i \(0.407982\pi\)
\(282\) 0 0
\(283\) 417633. 0.309976 0.154988 0.987916i \(-0.450466\pi\)
0.154988 + 0.987916i \(0.450466\pi\)
\(284\) 0 0
\(285\) 1.05591e6 0.770046
\(286\) 0 0
\(287\) −1.65166e6 −1.18363
\(288\) 0 0
\(289\) −1.06961e6 −0.753320
\(290\) 0 0
\(291\) −375118. −0.259678
\(292\) 0 0
\(293\) −2.68010e6 −1.82382 −0.911908 0.410394i \(-0.865391\pi\)
−0.911908 + 0.410394i \(0.865391\pi\)
\(294\) 0 0
\(295\) −3.36658e6 −2.25234
\(296\) 0 0
\(297\) −318400. −0.209451
\(298\) 0 0
\(299\) −136940. −0.0885831
\(300\) 0 0
\(301\) −1.10661e6 −0.704007
\(302\) 0 0
\(303\) −333903. −0.208936
\(304\) 0 0
\(305\) −1.08259e6 −0.666369
\(306\) 0 0
\(307\) 2.42927e6 1.47106 0.735528 0.677494i \(-0.236934\pi\)
0.735528 + 0.677494i \(0.236934\pi\)
\(308\) 0 0
\(309\) −1.10991e6 −0.661288
\(310\) 0 0
\(311\) −1.71783e6 −1.00712 −0.503559 0.863961i \(-0.667976\pi\)
−0.503559 + 0.863961i \(0.667976\pi\)
\(312\) 0 0
\(313\) 1.56583e6 0.903406 0.451703 0.892168i \(-0.350817\pi\)
0.451703 + 0.892168i \(0.350817\pi\)
\(314\) 0 0
\(315\) 512455. 0.290991
\(316\) 0 0
\(317\) 673750. 0.376574 0.188287 0.982114i \(-0.439706\pi\)
0.188287 + 0.982114i \(0.439706\pi\)
\(318\) 0 0
\(319\) −2.23688e6 −1.23074
\(320\) 0 0
\(321\) 967099. 0.523852
\(322\) 0 0
\(323\) −984603. −0.525115
\(324\) 0 0
\(325\) −478413. −0.251243
\(326\) 0 0
\(327\) −869874. −0.449870
\(328\) 0 0
\(329\) −2.14159e6 −1.09081
\(330\) 0 0
\(331\) −1.69768e6 −0.851699 −0.425850 0.904794i \(-0.640025\pi\)
−0.425850 + 0.904794i \(0.640025\pi\)
\(332\) 0 0
\(333\) 71647.2 0.0354070
\(334\) 0 0
\(335\) 2.90363e6 1.41361
\(336\) 0 0
\(337\) −2.37281e6 −1.13812 −0.569061 0.822295i \(-0.692693\pi\)
−0.569061 + 0.822295i \(0.692693\pi\)
\(338\) 0 0
\(339\) −502099. −0.237296
\(340\) 0 0
\(341\) −817310. −0.380628
\(342\) 0 0
\(343\) −2.29357e6 −1.05263
\(344\) 0 0
\(345\) −335747. −0.151867
\(346\) 0 0
\(347\) −3.98316e6 −1.77584 −0.887921 0.459996i \(-0.847851\pi\)
−0.887921 + 0.459996i \(0.847851\pi\)
\(348\) 0 0
\(349\) −2.10463e6 −0.924935 −0.462468 0.886636i \(-0.653036\pi\)
−0.462468 + 0.886636i \(0.653036\pi\)
\(350\) 0 0
\(351\) 188712. 0.0817585
\(352\) 0 0
\(353\) 3.41107e6 1.45698 0.728490 0.685056i \(-0.240222\pi\)
0.728490 + 0.685056i \(0.240222\pi\)
\(354\) 0 0
\(355\) 3.21631e6 1.35453
\(356\) 0 0
\(357\) −477846. −0.198435
\(358\) 0 0
\(359\) 1.86760e6 0.764800 0.382400 0.923997i \(-0.375098\pi\)
0.382400 + 0.923997i \(0.375098\pi\)
\(360\) 0 0
\(361\) 291759. 0.117830
\(362\) 0 0
\(363\) −267400. −0.106511
\(364\) 0 0
\(365\) −556987. −0.218833
\(366\) 0 0
\(367\) 437210. 0.169444 0.0847218 0.996405i \(-0.473000\pi\)
0.0847218 + 0.996405i \(0.473000\pi\)
\(368\) 0 0
\(369\) −1.49125e6 −0.570144
\(370\) 0 0
\(371\) −1.30935e6 −0.493879
\(372\) 0 0
\(373\) −921734. −0.343031 −0.171516 0.985181i \(-0.554866\pi\)
−0.171516 + 0.985181i \(0.554866\pi\)
\(374\) 0 0
\(375\) 810415. 0.297598
\(376\) 0 0
\(377\) 1.32577e6 0.480414
\(378\) 0 0
\(379\) 3.80141e6 1.35940 0.679699 0.733491i \(-0.262111\pi\)
0.679699 + 0.733491i \(0.262111\pi\)
\(380\) 0 0
\(381\) 1.51055e6 0.533118
\(382\) 0 0
\(383\) 4.52311e6 1.57558 0.787790 0.615944i \(-0.211225\pi\)
0.787790 + 0.615944i \(0.211225\pi\)
\(384\) 0 0
\(385\) 2.76323e6 0.950090
\(386\) 0 0
\(387\) −999130. −0.339113
\(388\) 0 0
\(389\) 4.48168e6 1.50164 0.750822 0.660505i \(-0.229658\pi\)
0.750822 + 0.660505i \(0.229658\pi\)
\(390\) 0 0
\(391\) 313072. 0.103563
\(392\) 0 0
\(393\) 1.16756e6 0.381326
\(394\) 0 0
\(395\) −5.03763e6 −1.62455
\(396\) 0 0
\(397\) 1.76656e6 0.562539 0.281269 0.959629i \(-0.409245\pi\)
0.281269 + 0.959629i \(0.409245\pi\)
\(398\) 0 0
\(399\) 1.34329e6 0.422414
\(400\) 0 0
\(401\) −522373. −0.162226 −0.0811128 0.996705i \(-0.525847\pi\)
−0.0811128 + 0.996705i \(0.525847\pi\)
\(402\) 0 0
\(403\) 484411. 0.148577
\(404\) 0 0
\(405\) 462684. 0.140167
\(406\) 0 0
\(407\) 386332. 0.115604
\(408\) 0 0
\(409\) −1.60642e6 −0.474845 −0.237422 0.971407i \(-0.576303\pi\)
−0.237422 + 0.971407i \(0.576303\pi\)
\(410\) 0 0
\(411\) 2.04032e6 0.595790
\(412\) 0 0
\(413\) −4.28283e6 −1.23554
\(414\) 0 0
\(415\) −7.93919e6 −2.26285
\(416\) 0 0
\(417\) −3.77258e6 −1.06242
\(418\) 0 0
\(419\) 5.30658e6 1.47666 0.738329 0.674441i \(-0.235615\pi\)
0.738329 + 0.674441i \(0.235615\pi\)
\(420\) 0 0
\(421\) −2.30575e6 −0.634025 −0.317012 0.948421i \(-0.602680\pi\)
−0.317012 + 0.948421i \(0.602680\pi\)
\(422\) 0 0
\(423\) −1.93360e6 −0.525430
\(424\) 0 0
\(425\) 1.09375e6 0.293729
\(426\) 0 0
\(427\) −1.37723e6 −0.365541
\(428\) 0 0
\(429\) 1.01756e6 0.266943
\(430\) 0 0
\(431\) 1.27091e6 0.329549 0.164775 0.986331i \(-0.447310\pi\)
0.164775 + 0.986331i \(0.447310\pi\)
\(432\) 0 0
\(433\) −3.71880e6 −0.953199 −0.476599 0.879121i \(-0.658131\pi\)
−0.476599 + 0.879121i \(0.658131\pi\)
\(434\) 0 0
\(435\) 3.25052e6 0.823625
\(436\) 0 0
\(437\) −880091. −0.220457
\(438\) 0 0
\(439\) 3.51069e6 0.869423 0.434711 0.900570i \(-0.356850\pi\)
0.434711 + 0.900570i \(0.356850\pi\)
\(440\) 0 0
\(441\) −709442. −0.173708
\(442\) 0 0
\(443\) −2.67040e6 −0.646497 −0.323248 0.946314i \(-0.604775\pi\)
−0.323248 + 0.946314i \(0.604775\pi\)
\(444\) 0 0
\(445\) 9.72088e6 2.32705
\(446\) 0 0
\(447\) −4.52943e6 −1.07220
\(448\) 0 0
\(449\) 808771. 0.189326 0.0946628 0.995509i \(-0.469823\pi\)
0.0946628 + 0.995509i \(0.469823\pi\)
\(450\) 0 0
\(451\) −8.04103e6 −1.86153
\(452\) 0 0
\(453\) 4.66148e6 1.06728
\(454\) 0 0
\(455\) −1.63774e6 −0.370864
\(456\) 0 0
\(457\) −6.03865e6 −1.35254 −0.676269 0.736655i \(-0.736404\pi\)
−0.676269 + 0.736655i \(0.736404\pi\)
\(458\) 0 0
\(459\) −431436. −0.0955839
\(460\) 0 0
\(461\) −6.06334e6 −1.32880 −0.664399 0.747378i \(-0.731313\pi\)
−0.664399 + 0.747378i \(0.731313\pi\)
\(462\) 0 0
\(463\) 987185. 0.214016 0.107008 0.994258i \(-0.465873\pi\)
0.107008 + 0.994258i \(0.465873\pi\)
\(464\) 0 0
\(465\) 1.18768e6 0.254721
\(466\) 0 0
\(467\) 2.75869e6 0.585344 0.292672 0.956213i \(-0.405456\pi\)
0.292672 + 0.956213i \(0.405456\pi\)
\(468\) 0 0
\(469\) 3.69389e6 0.775446
\(470\) 0 0
\(471\) 357731. 0.0743027
\(472\) 0 0
\(473\) −5.38745e6 −1.10721
\(474\) 0 0
\(475\) −3.07469e6 −0.625270
\(476\) 0 0
\(477\) −1.18218e6 −0.237897
\(478\) 0 0
\(479\) 1.43871e6 0.286507 0.143254 0.989686i \(-0.454244\pi\)
0.143254 + 0.989686i \(0.454244\pi\)
\(480\) 0 0
\(481\) −228975. −0.0451258
\(482\) 0 0
\(483\) −427124. −0.0833080
\(484\) 0 0
\(485\) 2.93927e6 0.567395
\(486\) 0 0
\(487\) 9.54793e6 1.82426 0.912130 0.409901i \(-0.134437\pi\)
0.912130 + 0.409901i \(0.134437\pi\)
\(488\) 0 0
\(489\) −1.42045e6 −0.268629
\(490\) 0 0
\(491\) −8.94131e6 −1.67378 −0.836889 0.547373i \(-0.815628\pi\)
−0.836889 + 0.547373i \(0.815628\pi\)
\(492\) 0 0
\(493\) −3.03099e6 −0.561652
\(494\) 0 0
\(495\) 2.49486e6 0.457649
\(496\) 0 0
\(497\) 4.09167e6 0.743035
\(498\) 0 0
\(499\) −2.34135e6 −0.420934 −0.210467 0.977601i \(-0.567499\pi\)
−0.210467 + 0.977601i \(0.567499\pi\)
\(500\) 0 0
\(501\) −909002. −0.161797
\(502\) 0 0
\(503\) −41307.7 −0.00727966 −0.00363983 0.999993i \(-0.501159\pi\)
−0.00363983 + 0.999993i \(0.501159\pi\)
\(504\) 0 0
\(505\) 2.61633e6 0.456524
\(506\) 0 0
\(507\) 2.73854e6 0.473150
\(508\) 0 0
\(509\) 3.09166e6 0.528929 0.264464 0.964395i \(-0.414805\pi\)
0.264464 + 0.964395i \(0.414805\pi\)
\(510\) 0 0
\(511\) −708576. −0.120042
\(512\) 0 0
\(513\) 1.21283e6 0.203473
\(514\) 0 0
\(515\) 8.69678e6 1.44491
\(516\) 0 0
\(517\) −1.04262e7 −1.71554
\(518\) 0 0
\(519\) −554117. −0.0902990
\(520\) 0 0
\(521\) 6.02002e6 0.971635 0.485818 0.874060i \(-0.338522\pi\)
0.485818 + 0.874060i \(0.338522\pi\)
\(522\) 0 0
\(523\) 6.39458e6 1.02225 0.511125 0.859506i \(-0.329229\pi\)
0.511125 + 0.859506i \(0.329229\pi\)
\(524\) 0 0
\(525\) −1.49220e6 −0.236282
\(526\) 0 0
\(527\) −1.10746e6 −0.173701
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) −3.86687e6 −0.595145
\(532\) 0 0
\(533\) 4.76583e6 0.726642
\(534\) 0 0
\(535\) −7.57779e6 −1.14461
\(536\) 0 0
\(537\) −5.49483e6 −0.822278
\(538\) 0 0
\(539\) −3.82541e6 −0.567161
\(540\) 0 0
\(541\) −3.24248e6 −0.476304 −0.238152 0.971228i \(-0.576542\pi\)
−0.238152 + 0.971228i \(0.576542\pi\)
\(542\) 0 0
\(543\) −1.09240e6 −0.158995
\(544\) 0 0
\(545\) 6.81598e6 0.982963
\(546\) 0 0
\(547\) −9.39508e6 −1.34255 −0.671277 0.741206i \(-0.734254\pi\)
−0.671277 + 0.741206i \(0.734254\pi\)
\(548\) 0 0
\(549\) −1.24347e6 −0.176078
\(550\) 0 0
\(551\) 8.52055e6 1.19561
\(552\) 0 0
\(553\) −6.40868e6 −0.891160
\(554\) 0 0
\(555\) −561398. −0.0773640
\(556\) 0 0
\(557\) 2.95696e6 0.403839 0.201919 0.979402i \(-0.435282\pi\)
0.201919 + 0.979402i \(0.435282\pi\)
\(558\) 0 0
\(559\) 3.19308e6 0.432196
\(560\) 0 0
\(561\) −2.32636e6 −0.312083
\(562\) 0 0
\(563\) 1.35314e7 1.79917 0.899584 0.436747i \(-0.143870\pi\)
0.899584 + 0.436747i \(0.143870\pi\)
\(564\) 0 0
\(565\) 3.93424e6 0.518489
\(566\) 0 0
\(567\) 588608. 0.0768898
\(568\) 0 0
\(569\) 1.32776e7 1.71924 0.859622 0.510931i \(-0.170699\pi\)
0.859622 + 0.510931i \(0.170699\pi\)
\(570\) 0 0
\(571\) −7.74960e6 −0.994693 −0.497346 0.867552i \(-0.665692\pi\)
−0.497346 + 0.867552i \(0.665692\pi\)
\(572\) 0 0
\(573\) −2.65123e6 −0.337334
\(574\) 0 0
\(575\) 977654. 0.123315
\(576\) 0 0
\(577\) −4.83479e6 −0.604558 −0.302279 0.953220i \(-0.597747\pi\)
−0.302279 + 0.953220i \(0.597747\pi\)
\(578\) 0 0
\(579\) 80059.7 0.00992471
\(580\) 0 0
\(581\) −1.00999e7 −1.24130
\(582\) 0 0
\(583\) −6.37448e6 −0.776736
\(584\) 0 0
\(585\) −1.47867e6 −0.178642
\(586\) 0 0
\(587\) −5.13380e6 −0.614955 −0.307478 0.951555i \(-0.599485\pi\)
−0.307478 + 0.951555i \(0.599485\pi\)
\(588\) 0 0
\(589\) 3.11324e6 0.369764
\(590\) 0 0
\(591\) 6.28301e6 0.739945
\(592\) 0 0
\(593\) 1.29134e7 1.50801 0.754003 0.656871i \(-0.228121\pi\)
0.754003 + 0.656871i \(0.228121\pi\)
\(594\) 0 0
\(595\) 3.74421e6 0.433578
\(596\) 0 0
\(597\) −4.79944e6 −0.551130
\(598\) 0 0
\(599\) 5.79980e6 0.660460 0.330230 0.943901i \(-0.392874\pi\)
0.330230 + 0.943901i \(0.392874\pi\)
\(600\) 0 0
\(601\) 1.06993e7 1.20829 0.604144 0.796875i \(-0.293515\pi\)
0.604144 + 0.796875i \(0.293515\pi\)
\(602\) 0 0
\(603\) 3.33513e6 0.373525
\(604\) 0 0
\(605\) 2.09524e6 0.232726
\(606\) 0 0
\(607\) 1.48509e7 1.63600 0.817998 0.575221i \(-0.195084\pi\)
0.817998 + 0.575221i \(0.195084\pi\)
\(608\) 0 0
\(609\) 4.13518e6 0.451805
\(610\) 0 0
\(611\) 6.17951e6 0.669655
\(612\) 0 0
\(613\) −7.64415e6 −0.821633 −0.410816 0.911718i \(-0.634756\pi\)
−0.410816 + 0.911718i \(0.634756\pi\)
\(614\) 0 0
\(615\) 1.16848e7 1.24576
\(616\) 0 0
\(617\) −8.02025e6 −0.848155 −0.424077 0.905626i \(-0.639402\pi\)
−0.424077 + 0.905626i \(0.639402\pi\)
\(618\) 0 0
\(619\) −9.46459e6 −0.992831 −0.496416 0.868085i \(-0.665351\pi\)
−0.496416 + 0.868085i \(0.665351\pi\)
\(620\) 0 0
\(621\) −385641. −0.0401286
\(622\) 0 0
\(623\) 1.23665e7 1.27652
\(624\) 0 0
\(625\) −1.21255e7 −1.24165
\(626\) 0 0
\(627\) 6.53974e6 0.664342
\(628\) 0 0
\(629\) 523484. 0.0527566
\(630\) 0 0
\(631\) −8.76525e6 −0.876377 −0.438189 0.898883i \(-0.644380\pi\)
−0.438189 + 0.898883i \(0.644380\pi\)
\(632\) 0 0
\(633\) −5.92566e6 −0.587797
\(634\) 0 0
\(635\) −1.18361e7 −1.16486
\(636\) 0 0
\(637\) 2.26728e6 0.221389
\(638\) 0 0
\(639\) 3.69427e6 0.357913
\(640\) 0 0
\(641\) −8.04677e6 −0.773529 −0.386764 0.922179i \(-0.626407\pi\)
−0.386764 + 0.922179i \(0.626407\pi\)
\(642\) 0 0
\(643\) −1.81157e7 −1.72794 −0.863970 0.503543i \(-0.832030\pi\)
−0.863970 + 0.503543i \(0.832030\pi\)
\(644\) 0 0
\(645\) 7.82877e6 0.740959
\(646\) 0 0
\(647\) −6.09047e6 −0.571992 −0.285996 0.958231i \(-0.592324\pi\)
−0.285996 + 0.958231i \(0.592324\pi\)
\(648\) 0 0
\(649\) −2.08507e7 −1.94316
\(650\) 0 0
\(651\) 1.51091e6 0.139729
\(652\) 0 0
\(653\) 1.73408e7 1.59142 0.795712 0.605675i \(-0.207097\pi\)
0.795712 + 0.605675i \(0.207097\pi\)
\(654\) 0 0
\(655\) −9.14850e6 −0.833195
\(656\) 0 0
\(657\) −639757. −0.0578232
\(658\) 0 0
\(659\) −1.27618e7 −1.14471 −0.572357 0.820004i \(-0.693971\pi\)
−0.572357 + 0.820004i \(0.693971\pi\)
\(660\) 0 0
\(661\) −6.44195e6 −0.573474 −0.286737 0.958009i \(-0.592571\pi\)
−0.286737 + 0.958009i \(0.592571\pi\)
\(662\) 0 0
\(663\) 1.37881e6 0.121821
\(664\) 0 0
\(665\) −1.05255e7 −0.922972
\(666\) 0 0
\(667\) −2.70926e6 −0.235796
\(668\) 0 0
\(669\) 719060. 0.0621155
\(670\) 0 0
\(671\) −6.70496e6 −0.574897
\(672\) 0 0
\(673\) 8.64864e6 0.736055 0.368028 0.929815i \(-0.380033\pi\)
0.368028 + 0.929815i \(0.380033\pi\)
\(674\) 0 0
\(675\) −1.34728e6 −0.113815
\(676\) 0 0
\(677\) −1.22020e7 −1.02320 −0.511600 0.859223i \(-0.670947\pi\)
−0.511600 + 0.859223i \(0.670947\pi\)
\(678\) 0 0
\(679\) 3.73923e6 0.311249
\(680\) 0 0
\(681\) 6.44500e6 0.532544
\(682\) 0 0
\(683\) 1.10784e7 0.908708 0.454354 0.890821i \(-0.349870\pi\)
0.454354 + 0.890821i \(0.349870\pi\)
\(684\) 0 0
\(685\) −1.59871e7 −1.30180
\(686\) 0 0
\(687\) 5.23170e6 0.422913
\(688\) 0 0
\(689\) 3.77809e6 0.303196
\(690\) 0 0
\(691\) 1.24814e7 0.994415 0.497207 0.867632i \(-0.334359\pi\)
0.497207 + 0.867632i \(0.334359\pi\)
\(692\) 0 0
\(693\) 3.17386e6 0.251047
\(694\) 0 0
\(695\) 2.95604e7 2.32139
\(696\) 0 0
\(697\) −1.08957e7 −0.849518
\(698\) 0 0
\(699\) −1.10461e6 −0.0855094
\(700\) 0 0
\(701\) −4.71570e6 −0.362452 −0.181226 0.983441i \(-0.558007\pi\)
−0.181226 + 0.983441i \(0.558007\pi\)
\(702\) 0 0
\(703\) −1.47159e6 −0.112305
\(704\) 0 0
\(705\) 1.51509e7 1.14806
\(706\) 0 0
\(707\) 3.32839e6 0.250430
\(708\) 0 0
\(709\) 1.05958e7 0.791620 0.395810 0.918332i \(-0.370464\pi\)
0.395810 + 0.918332i \(0.370464\pi\)
\(710\) 0 0
\(711\) −5.78625e6 −0.429263
\(712\) 0 0
\(713\) −989912. −0.0729244
\(714\) 0 0
\(715\) −7.97322e6 −0.583268
\(716\) 0 0
\(717\) 1.89068e6 0.137347
\(718\) 0 0
\(719\) −2.95783e6 −0.213379 −0.106689 0.994292i \(-0.534025\pi\)
−0.106689 + 0.994292i \(0.534025\pi\)
\(720\) 0 0
\(721\) 1.10637e7 0.792615
\(722\) 0 0
\(723\) −1.10902e7 −0.789029
\(724\) 0 0
\(725\) −9.46510e6 −0.668776
\(726\) 0 0
\(727\) 1.30768e7 0.917625 0.458813 0.888533i \(-0.348275\pi\)
0.458813 + 0.888533i \(0.348275\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −7.30005e6 −0.505280
\(732\) 0 0
\(733\) 5.06819e6 0.348412 0.174206 0.984709i \(-0.444264\pi\)
0.174206 + 0.984709i \(0.444264\pi\)
\(734\) 0 0
\(735\) 5.55890e6 0.379551
\(736\) 0 0
\(737\) 1.79835e7 1.21956
\(738\) 0 0
\(739\) −5.39832e6 −0.363620 −0.181810 0.983334i \(-0.558196\pi\)
−0.181810 + 0.983334i \(0.558196\pi\)
\(740\) 0 0
\(741\) −3.87603e6 −0.259324
\(742\) 0 0
\(743\) −3.71690e6 −0.247007 −0.123503 0.992344i \(-0.539413\pi\)
−0.123503 + 0.992344i \(0.539413\pi\)
\(744\) 0 0
\(745\) 3.54908e7 2.34274
\(746\) 0 0
\(747\) −9.11900e6 −0.597923
\(748\) 0 0
\(749\) −9.64017e6 −0.627885
\(750\) 0 0
\(751\) −1.00711e7 −0.651594 −0.325797 0.945440i \(-0.605633\pi\)
−0.325797 + 0.945440i \(0.605633\pi\)
\(752\) 0 0
\(753\) −4.91387e6 −0.315818
\(754\) 0 0
\(755\) −3.65255e7 −2.33200
\(756\) 0 0
\(757\) −8.49996e6 −0.539110 −0.269555 0.962985i \(-0.586877\pi\)
−0.269555 + 0.962985i \(0.586877\pi\)
\(758\) 0 0
\(759\) −2.07943e6 −0.131021
\(760\) 0 0
\(761\) 2.62780e7 1.64487 0.822434 0.568861i \(-0.192616\pi\)
0.822434 + 0.568861i \(0.192616\pi\)
\(762\) 0 0
\(763\) 8.67102e6 0.539211
\(764\) 0 0
\(765\) 3.38056e6 0.208850
\(766\) 0 0
\(767\) 1.23580e7 0.758506
\(768\) 0 0
\(769\) 899889. 0.0548748 0.0274374 0.999624i \(-0.491265\pi\)
0.0274374 + 0.999624i \(0.491265\pi\)
\(770\) 0 0
\(771\) −5.01034e6 −0.303551
\(772\) 0 0
\(773\) −1.33758e7 −0.805140 −0.402570 0.915389i \(-0.631883\pi\)
−0.402570 + 0.915389i \(0.631883\pi\)
\(774\) 0 0
\(775\) −3.45836e6 −0.206831
\(776\) 0 0
\(777\) −714189. −0.0424385
\(778\) 0 0
\(779\) 3.06293e7 1.80840
\(780\) 0 0
\(781\) 1.99200e7 1.16859
\(782\) 0 0
\(783\) 3.73356e6 0.217630
\(784\) 0 0
\(785\) −2.80304e6 −0.162351
\(786\) 0 0
\(787\) −1.67690e7 −0.965098 −0.482549 0.875869i \(-0.660289\pi\)
−0.482549 + 0.875869i \(0.660289\pi\)
\(788\) 0 0
\(789\) 1.48853e7 0.851264
\(790\) 0 0
\(791\) 5.00498e6 0.284421
\(792\) 0 0
\(793\) 3.97396e6 0.224409
\(794\) 0 0
\(795\) 9.26309e6 0.519802
\(796\) 0 0
\(797\) 2.78280e7 1.55180 0.775902 0.630854i \(-0.217295\pi\)
0.775902 + 0.630854i \(0.217295\pi\)
\(798\) 0 0
\(799\) −1.41277e7 −0.782894
\(800\) 0 0
\(801\) 1.11654e7 0.614886
\(802\) 0 0
\(803\) −3.44966e6 −0.188794
\(804\) 0 0
\(805\) 3.34677e6 0.182027
\(806\) 0 0
\(807\) 1.85890e7 1.00478
\(808\) 0 0
\(809\) −1.98606e7 −1.06689 −0.533447 0.845834i \(-0.679103\pi\)
−0.533447 + 0.845834i \(0.679103\pi\)
\(810\) 0 0
\(811\) −1.95245e7 −1.04238 −0.521192 0.853439i \(-0.674512\pi\)
−0.521192 + 0.853439i \(0.674512\pi\)
\(812\) 0 0
\(813\) 6.37493e6 0.338259
\(814\) 0 0
\(815\) 1.11301e7 0.586953
\(816\) 0 0
\(817\) 2.05215e7 1.07561
\(818\) 0 0
\(819\) −1.88111e6 −0.0979952
\(820\) 0 0
\(821\) −2.39444e7 −1.23978 −0.619891 0.784688i \(-0.712823\pi\)
−0.619891 + 0.784688i \(0.712823\pi\)
\(822\) 0 0
\(823\) 2.78310e7 1.43229 0.716143 0.697953i \(-0.245906\pi\)
0.716143 + 0.697953i \(0.245906\pi\)
\(824\) 0 0
\(825\) −7.26471e6 −0.371607
\(826\) 0 0
\(827\) 1.94204e7 0.987405 0.493703 0.869631i \(-0.335643\pi\)
0.493703 + 0.869631i \(0.335643\pi\)
\(828\) 0 0
\(829\) 2.66039e7 1.34449 0.672247 0.740327i \(-0.265329\pi\)
0.672247 + 0.740327i \(0.265329\pi\)
\(830\) 0 0
\(831\) 159564. 0.00801555
\(832\) 0 0
\(833\) −5.18348e6 −0.258826
\(834\) 0 0
\(835\) 7.12257e6 0.353525
\(836\) 0 0
\(837\) 1.36417e6 0.0673062
\(838\) 0 0
\(839\) 1.41085e7 0.691954 0.345977 0.938243i \(-0.387547\pi\)
0.345977 + 0.938243i \(0.387547\pi\)
\(840\) 0 0
\(841\) 5.71843e6 0.278796
\(842\) 0 0
\(843\) −6.79194e6 −0.329174
\(844\) 0 0
\(845\) −2.14581e7 −1.03383
\(846\) 0 0
\(847\) 2.66548e6 0.127664
\(848\) 0 0
\(849\) −3.75869e6 −0.178965
\(850\) 0 0
\(851\) 467918. 0.0221486
\(852\) 0 0
\(853\) −1.87882e7 −0.884122 −0.442061 0.896985i \(-0.645752\pi\)
−0.442061 + 0.896985i \(0.645752\pi\)
\(854\) 0 0
\(855\) −9.50323e6 −0.444586
\(856\) 0 0
\(857\) 3.67369e7 1.70864 0.854320 0.519748i \(-0.173974\pi\)
0.854320 + 0.519748i \(0.173974\pi\)
\(858\) 0 0
\(859\) −2.03228e6 −0.0939723 −0.0469861 0.998896i \(-0.514962\pi\)
−0.0469861 + 0.998896i \(0.514962\pi\)
\(860\) 0 0
\(861\) 1.48650e7 0.683371
\(862\) 0 0
\(863\) 1.21735e7 0.556400 0.278200 0.960523i \(-0.410262\pi\)
0.278200 + 0.960523i \(0.410262\pi\)
\(864\) 0 0
\(865\) 4.34183e6 0.197303
\(866\) 0 0
\(867\) 9.62646e6 0.434930
\(868\) 0 0
\(869\) −3.12003e7 −1.40155
\(870\) 0 0
\(871\) −1.06586e7 −0.476053
\(872\) 0 0
\(873\) 3.37606e6 0.149925
\(874\) 0 0
\(875\) −8.07833e6 −0.356698
\(876\) 0 0
\(877\) 4.13453e7 1.81521 0.907605 0.419825i \(-0.137909\pi\)
0.907605 + 0.419825i \(0.137909\pi\)
\(878\) 0 0
\(879\) 2.41209e7 1.05298
\(880\) 0 0
\(881\) 1.92417e7 0.835227 0.417613 0.908625i \(-0.362867\pi\)
0.417613 + 0.908625i \(0.362867\pi\)
\(882\) 0 0
\(883\) −4.59010e6 −0.198117 −0.0990583 0.995082i \(-0.531583\pi\)
−0.0990583 + 0.995082i \(0.531583\pi\)
\(884\) 0 0
\(885\) 3.02992e7 1.30039
\(886\) 0 0
\(887\) 1.34822e7 0.575374 0.287687 0.957724i \(-0.407114\pi\)
0.287687 + 0.957724i \(0.407114\pi\)
\(888\) 0 0
\(889\) −1.50574e7 −0.638992
\(890\) 0 0
\(891\) 2.86560e6 0.120927
\(892\) 0 0
\(893\) 3.97149e7 1.66657
\(894\) 0 0
\(895\) 4.30552e7 1.79667
\(896\) 0 0
\(897\) 1.23246e6 0.0511435
\(898\) 0 0
\(899\) 9.58377e6 0.395492
\(900\) 0 0
\(901\) −8.63750e6 −0.354467
\(902\) 0 0
\(903\) 9.95945e6 0.406458
\(904\) 0 0
\(905\) 8.55962e6 0.347403
\(906\) 0 0
\(907\) −5.13039e6 −0.207077 −0.103539 0.994625i \(-0.533017\pi\)
−0.103539 + 0.994625i \(0.533017\pi\)
\(908\) 0 0
\(909\) 3.00513e6 0.120629
\(910\) 0 0
\(911\) −3.88471e7 −1.55082 −0.775411 0.631456i \(-0.782457\pi\)
−0.775411 + 0.631456i \(0.782457\pi\)
\(912\) 0 0
\(913\) −4.91709e7 −1.95223
\(914\) 0 0
\(915\) 9.74332e6 0.384728
\(916\) 0 0
\(917\) −1.16384e7 −0.457055
\(918\) 0 0
\(919\) −2.43482e7 −0.950993 −0.475496 0.879718i \(-0.657732\pi\)
−0.475496 + 0.879718i \(0.657732\pi\)
\(920\) 0 0
\(921\) −2.18634e7 −0.849315
\(922\) 0 0
\(923\) −1.18064e7 −0.456156
\(924\) 0 0
\(925\) 1.63472e6 0.0628188
\(926\) 0 0
\(927\) 9.98917e6 0.381795
\(928\) 0 0
\(929\) 2.17027e7 0.825040 0.412520 0.910948i \(-0.364649\pi\)
0.412520 + 0.910948i \(0.364649\pi\)
\(930\) 0 0
\(931\) 1.45715e7 0.550972
\(932\) 0 0
\(933\) 1.54605e7 0.581460
\(934\) 0 0
\(935\) 1.82284e7 0.681900
\(936\) 0 0
\(937\) −2.01218e6 −0.0748718 −0.0374359 0.999299i \(-0.511919\pi\)
−0.0374359 + 0.999299i \(0.511919\pi\)
\(938\) 0 0
\(939\) −1.40925e7 −0.521582
\(940\) 0 0
\(941\) −1.17136e7 −0.431237 −0.215618 0.976478i \(-0.569177\pi\)
−0.215618 + 0.976478i \(0.569177\pi\)
\(942\) 0 0
\(943\) −9.73916e6 −0.356650
\(944\) 0 0
\(945\) −4.61209e6 −0.168004
\(946\) 0 0
\(947\) −571455. −0.0207065 −0.0103533 0.999946i \(-0.503296\pi\)
−0.0103533 + 0.999946i \(0.503296\pi\)
\(948\) 0 0
\(949\) 2.04458e6 0.0736950
\(950\) 0 0
\(951\) −6.06375e6 −0.217415
\(952\) 0 0
\(953\) 2.03678e7 0.726461 0.363231 0.931699i \(-0.381674\pi\)
0.363231 + 0.931699i \(0.381674\pi\)
\(954\) 0 0
\(955\) 2.07739e7 0.737073
\(956\) 0 0
\(957\) 2.01319e7 0.710566
\(958\) 0 0
\(959\) −2.03382e7 −0.714110
\(960\) 0 0
\(961\) −2.51274e7 −0.877687
\(962\) 0 0
\(963\) −8.70389e6 −0.302446
\(964\) 0 0
\(965\) −627316. −0.0216854
\(966\) 0 0
\(967\) 3.93183e7 1.35216 0.676081 0.736827i \(-0.263677\pi\)
0.676081 + 0.736827i \(0.263677\pi\)
\(968\) 0 0
\(969\) 8.86142e6 0.303175
\(970\) 0 0
\(971\) −4.86930e7 −1.65736 −0.828682 0.559719i \(-0.810909\pi\)
−0.828682 + 0.559719i \(0.810909\pi\)
\(972\) 0 0
\(973\) 3.76055e7 1.27341
\(974\) 0 0
\(975\) 4.30572e6 0.145055
\(976\) 0 0
\(977\) −1.11758e7 −0.374577 −0.187288 0.982305i \(-0.559970\pi\)
−0.187288 + 0.982305i \(0.559970\pi\)
\(978\) 0 0
\(979\) 6.02056e7 2.00762
\(980\) 0 0
\(981\) 7.82887e6 0.259733
\(982\) 0 0
\(983\) −2.09439e7 −0.691310 −0.345655 0.938362i \(-0.612343\pi\)
−0.345655 + 0.938362i \(0.612343\pi\)
\(984\) 0 0
\(985\) −4.92311e7 −1.61677
\(986\) 0 0
\(987\) 1.92744e7 0.629777
\(988\) 0 0
\(989\) −6.52518e6 −0.212130
\(990\) 0 0
\(991\) 1.95125e7 0.631144 0.315572 0.948902i \(-0.397804\pi\)
0.315572 + 0.948902i \(0.397804\pi\)
\(992\) 0 0
\(993\) 1.52791e7 0.491729
\(994\) 0 0
\(995\) 3.76064e7 1.20422
\(996\) 0 0
\(997\) 1.32696e7 0.422786 0.211393 0.977401i \(-0.432200\pi\)
0.211393 + 0.977401i \(0.432200\pi\)
\(998\) 0 0
\(999\) −644825. −0.0204422
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 1104.6.a.r.1.4 5
4.3 odd 2 69.6.a.e.1.5 5
12.11 even 2 207.6.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.5 5 4.3 odd 2
207.6.a.f.1.1 5 12.11 even 2
1104.6.a.r.1.4 5 1.1 even 1 trivial