Properties

Label 531.6.a.d.1.12
Level $531$
Weight $6$
Character 531.1
Self dual yes
Analytic conductor $85.164$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(85.1638083207\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(10.9029\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.9029 q^{2} +86.8728 q^{4} -88.1143 q^{5} +61.7114 q^{7} +598.272 q^{8} +O(q^{10})\) \(q+10.9029 q^{2} +86.8728 q^{4} -88.1143 q^{5} +61.7114 q^{7} +598.272 q^{8} -960.699 q^{10} -423.027 q^{11} -1047.72 q^{13} +672.832 q^{14} +3742.96 q^{16} +2034.81 q^{17} -1532.11 q^{19} -7654.74 q^{20} -4612.21 q^{22} -2986.67 q^{23} +4639.12 q^{25} -11423.2 q^{26} +5361.05 q^{28} -115.477 q^{29} -3626.28 q^{31} +21664.3 q^{32} +22185.3 q^{34} -5437.66 q^{35} -15040.3 q^{37} -16704.4 q^{38} -52716.3 q^{40} +829.256 q^{41} -12294.1 q^{43} -36749.5 q^{44} -32563.3 q^{46} +16104.8 q^{47} -12998.7 q^{49} +50579.8 q^{50} -91018.7 q^{52} -34428.4 q^{53} +37274.7 q^{55} +36920.2 q^{56} -1259.04 q^{58} +3481.00 q^{59} +16418.9 q^{61} -39536.9 q^{62} +116429. q^{64} +92319.3 q^{65} +9818.25 q^{67} +176770. q^{68} -59286.1 q^{70} -3381.64 q^{71} +27012.0 q^{73} -163982. q^{74} -133098. q^{76} -26105.6 q^{77} +20426.3 q^{79} -329808. q^{80} +9041.29 q^{82} +54511.3 q^{83} -179296. q^{85} -134041. q^{86} -253085. q^{88} +69700.2 q^{89} -64656.5 q^{91} -259460. q^{92} +175589. q^{94} +135000. q^{95} +48526.4 q^{97} -141723. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{2} + 198q^{4} - 36q^{5} - 411q^{7} + 69q^{8} + O(q^{10}) \) \( 12q + 4q^{2} + 198q^{4} - 36q^{5} - 411q^{7} + 69q^{8} - 863q^{10} - 492q^{11} - 974q^{13} + 967q^{14} + 6370q^{16} + 1463q^{17} - 3189q^{19} + 835q^{20} - 2726q^{22} + 2617q^{23} + 8642q^{25} - 2414q^{26} - 20458q^{28} + 1963q^{29} - 11929q^{31} + 14382q^{32} - 20744q^{34} - 1829q^{35} - 28105q^{37} + 23475q^{38} - 100576q^{40} + 7585q^{41} - 33146q^{43} - 26014q^{44} - 142851q^{46} + 79215q^{47} - 32569q^{49} + 136019q^{50} - 248218q^{52} + 12220q^{53} - 117770q^{55} + 186728q^{56} - 188072q^{58} + 41772q^{59} - 54195q^{61} - 36230q^{62} + 45197q^{64} - 42368q^{65} + 24224q^{67} + 209639q^{68} - 35684q^{70} - 60254q^{71} - 15385q^{73} - 214638q^{74} - 167504q^{76} + 17169q^{77} - 27054q^{79} - 216899q^{80} + 37917q^{82} + 117595q^{83} - 121585q^{85} - 306756q^{86} - 105799q^{88} + 36033q^{89} - 32217q^{91} + 30906q^{92} + 128392q^{94} + 50721q^{95} - 196914q^{97} - 574100q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.9029 1.92738 0.963688 0.267032i \(-0.0860429\pi\)
0.963688 + 0.267032i \(0.0860429\pi\)
\(3\) 0 0
\(4\) 86.8728 2.71478
\(5\) −88.1143 −1.57624 −0.788118 0.615524i \(-0.788944\pi\)
−0.788118 + 0.615524i \(0.788944\pi\)
\(6\) 0 0
\(7\) 61.7114 0.476015 0.238007 0.971263i \(-0.423506\pi\)
0.238007 + 0.971263i \(0.423506\pi\)
\(8\) 598.272 3.30502
\(9\) 0 0
\(10\) −960.699 −3.03800
\(11\) −423.027 −1.05411 −0.527055 0.849831i \(-0.676704\pi\)
−0.527055 + 0.849831i \(0.676704\pi\)
\(12\) 0 0
\(13\) −1047.72 −1.71944 −0.859722 0.510762i \(-0.829363\pi\)
−0.859722 + 0.510762i \(0.829363\pi\)
\(14\) 672.832 0.917459
\(15\) 0 0
\(16\) 3742.96 3.65523
\(17\) 2034.81 1.70766 0.853831 0.520550i \(-0.174273\pi\)
0.853831 + 0.520550i \(0.174273\pi\)
\(18\) 0 0
\(19\) −1532.11 −0.973654 −0.486827 0.873498i \(-0.661846\pi\)
−0.486827 + 0.873498i \(0.661846\pi\)
\(20\) −7654.74 −4.27913
\(21\) 0 0
\(22\) −4612.21 −2.03167
\(23\) −2986.67 −1.17725 −0.588623 0.808408i \(-0.700330\pi\)
−0.588623 + 0.808408i \(0.700330\pi\)
\(24\) 0 0
\(25\) 4639.12 1.48452
\(26\) −11423.2 −3.31402
\(27\) 0 0
\(28\) 5361.05 1.29227
\(29\) −115.477 −0.0254978 −0.0127489 0.999919i \(-0.504058\pi\)
−0.0127489 + 0.999919i \(0.504058\pi\)
\(30\) 0 0
\(31\) −3626.28 −0.677730 −0.338865 0.940835i \(-0.610043\pi\)
−0.338865 + 0.940835i \(0.610043\pi\)
\(32\) 21664.3 3.73999
\(33\) 0 0
\(34\) 22185.3 3.29131
\(35\) −5437.66 −0.750312
\(36\) 0 0
\(37\) −15040.3 −1.80614 −0.903070 0.429493i \(-0.858692\pi\)
−0.903070 + 0.429493i \(0.858692\pi\)
\(38\) −16704.4 −1.87660
\(39\) 0 0
\(40\) −52716.3 −5.20949
\(41\) 829.256 0.0770423 0.0385211 0.999258i \(-0.487735\pi\)
0.0385211 + 0.999258i \(0.487735\pi\)
\(42\) 0 0
\(43\) −12294.1 −1.01397 −0.506985 0.861955i \(-0.669240\pi\)
−0.506985 + 0.861955i \(0.669240\pi\)
\(44\) −36749.5 −2.86167
\(45\) 0 0
\(46\) −32563.3 −2.26900
\(47\) 16104.8 1.06343 0.531717 0.846922i \(-0.321547\pi\)
0.531717 + 0.846922i \(0.321547\pi\)
\(48\) 0 0
\(49\) −12998.7 −0.773410
\(50\) 50579.8 2.86123
\(51\) 0 0
\(52\) −91018.7 −4.66791
\(53\) −34428.4 −1.68356 −0.841778 0.539824i \(-0.818491\pi\)
−0.841778 + 0.539824i \(0.818491\pi\)
\(54\) 0 0
\(55\) 37274.7 1.66153
\(56\) 36920.2 1.57324
\(57\) 0 0
\(58\) −1259.04 −0.0491438
\(59\) 3481.00 0.130189
\(60\) 0 0
\(61\) 16418.9 0.564961 0.282481 0.959273i \(-0.408843\pi\)
0.282481 + 0.959273i \(0.408843\pi\)
\(62\) −39536.9 −1.30624
\(63\) 0 0
\(64\) 116429. 3.55313
\(65\) 92319.3 2.71025
\(66\) 0 0
\(67\) 9818.25 0.267206 0.133603 0.991035i \(-0.457345\pi\)
0.133603 + 0.991035i \(0.457345\pi\)
\(68\) 176770. 4.63592
\(69\) 0 0
\(70\) −59286.1 −1.44613
\(71\) −3381.64 −0.0796125 −0.0398062 0.999207i \(-0.512674\pi\)
−0.0398062 + 0.999207i \(0.512674\pi\)
\(72\) 0 0
\(73\) 27012.0 0.593267 0.296634 0.954991i \(-0.404136\pi\)
0.296634 + 0.954991i \(0.404136\pi\)
\(74\) −163982. −3.48111
\(75\) 0 0
\(76\) −133098. −2.64325
\(77\) −26105.6 −0.501772
\(78\) 0 0
\(79\) 20426.3 0.368233 0.184117 0.982904i \(-0.441058\pi\)
0.184117 + 0.982904i \(0.441058\pi\)
\(80\) −329808. −5.76151
\(81\) 0 0
\(82\) 9041.29 0.148489
\(83\) 54511.3 0.868543 0.434272 0.900782i \(-0.357006\pi\)
0.434272 + 0.900782i \(0.357006\pi\)
\(84\) 0 0
\(85\) −179296. −2.69168
\(86\) −134041. −1.95430
\(87\) 0 0
\(88\) −253085. −3.48385
\(89\) 69700.2 0.932736 0.466368 0.884591i \(-0.345562\pi\)
0.466368 + 0.884591i \(0.345562\pi\)
\(90\) 0 0
\(91\) −64656.5 −0.818481
\(92\) −259460. −3.19596
\(93\) 0 0
\(94\) 175589. 2.04964
\(95\) 135000. 1.53471
\(96\) 0 0
\(97\) 48526.4 0.523660 0.261830 0.965114i \(-0.415674\pi\)
0.261830 + 0.965114i \(0.415674\pi\)
\(98\) −141723. −1.49065
\(99\) 0 0
\(100\) 403014. 4.03014
\(101\) −107128. −1.04496 −0.522478 0.852653i \(-0.674992\pi\)
−0.522478 + 0.852653i \(0.674992\pi\)
\(102\) 0 0
\(103\) −93523.6 −0.868617 −0.434308 0.900764i \(-0.643007\pi\)
−0.434308 + 0.900764i \(0.643007\pi\)
\(104\) −626823. −5.68279
\(105\) 0 0
\(106\) −375369. −3.24484
\(107\) −51795.7 −0.437356 −0.218678 0.975797i \(-0.570174\pi\)
−0.218678 + 0.975797i \(0.570174\pi\)
\(108\) 0 0
\(109\) −147885. −1.19223 −0.596113 0.802901i \(-0.703289\pi\)
−0.596113 + 0.802901i \(0.703289\pi\)
\(110\) 406401. 3.20238
\(111\) 0 0
\(112\) 230983. 1.73995
\(113\) 83662.7 0.616362 0.308181 0.951328i \(-0.400280\pi\)
0.308181 + 0.951328i \(0.400280\pi\)
\(114\) 0 0
\(115\) 263168. 1.85562
\(116\) −10031.9 −0.0692208
\(117\) 0 0
\(118\) 37952.9 0.250923
\(119\) 125571. 0.812873
\(120\) 0 0
\(121\) 17900.5 0.111148
\(122\) 179013. 1.08889
\(123\) 0 0
\(124\) −315025. −1.83988
\(125\) −133416. −0.763716
\(126\) 0 0
\(127\) −155048. −0.853016 −0.426508 0.904484i \(-0.640256\pi\)
−0.426508 + 0.904484i \(0.640256\pi\)
\(128\) 576153. 3.10823
\(129\) 0 0
\(130\) 1.00655e6 5.22367
\(131\) 370672. 1.88717 0.943587 0.331124i \(-0.107428\pi\)
0.943587 + 0.331124i \(0.107428\pi\)
\(132\) 0 0
\(133\) −94548.4 −0.463474
\(134\) 107047. 0.515007
\(135\) 0 0
\(136\) 1.21737e6 5.64385
\(137\) −194058. −0.883345 −0.441673 0.897176i \(-0.645615\pi\)
−0.441673 + 0.897176i \(0.645615\pi\)
\(138\) 0 0
\(139\) −80499.1 −0.353390 −0.176695 0.984266i \(-0.556541\pi\)
−0.176695 + 0.984266i \(0.556541\pi\)
\(140\) −472385. −2.03693
\(141\) 0 0
\(142\) −36869.6 −0.153443
\(143\) 443215. 1.81248
\(144\) 0 0
\(145\) 10175.2 0.0401905
\(146\) 294509. 1.14345
\(147\) 0 0
\(148\) −1.30659e6 −4.90327
\(149\) −229972. −0.848611 −0.424305 0.905519i \(-0.639482\pi\)
−0.424305 + 0.905519i \(0.639482\pi\)
\(150\) 0 0
\(151\) −103481. −0.369333 −0.184667 0.982801i \(-0.559121\pi\)
−0.184667 + 0.982801i \(0.559121\pi\)
\(152\) −916616. −3.21794
\(153\) 0 0
\(154\) −284626. −0.967103
\(155\) 319527. 1.06826
\(156\) 0 0
\(157\) 222025. 0.718875 0.359438 0.933169i \(-0.382969\pi\)
0.359438 + 0.933169i \(0.382969\pi\)
\(158\) 222706. 0.709723
\(159\) 0 0
\(160\) −1.90894e6 −5.89510
\(161\) −184312. −0.560387
\(162\) 0 0
\(163\) −1674.56 −0.00493664 −0.00246832 0.999997i \(-0.500786\pi\)
−0.00246832 + 0.999997i \(0.500786\pi\)
\(164\) 72039.9 0.209153
\(165\) 0 0
\(166\) 594331. 1.67401
\(167\) 633831. 1.75866 0.879330 0.476212i \(-0.157991\pi\)
0.879330 + 0.476212i \(0.157991\pi\)
\(168\) 0 0
\(169\) 726431. 1.95649
\(170\) −1.95484e6 −5.18788
\(171\) 0 0
\(172\) −1.06802e6 −2.75270
\(173\) −304757. −0.774175 −0.387087 0.922043i \(-0.626519\pi\)
−0.387087 + 0.922043i \(0.626519\pi\)
\(174\) 0 0
\(175\) 286287. 0.706653
\(176\) −1.58337e6 −3.85302
\(177\) 0 0
\(178\) 759933. 1.79773
\(179\) −67132.3 −0.156603 −0.0783013 0.996930i \(-0.524950\pi\)
−0.0783013 + 0.996930i \(0.524950\pi\)
\(180\) 0 0
\(181\) 111781. 0.253613 0.126807 0.991927i \(-0.459527\pi\)
0.126807 + 0.991927i \(0.459527\pi\)
\(182\) −704942. −1.57752
\(183\) 0 0
\(184\) −1.78684e6 −3.89082
\(185\) 1.32526e6 2.84690
\(186\) 0 0
\(187\) −860780. −1.80006
\(188\) 1.39907e6 2.88698
\(189\) 0 0
\(190\) 1.47189e6 2.95796
\(191\) 731353. 1.45059 0.725294 0.688440i \(-0.241704\pi\)
0.725294 + 0.688440i \(0.241704\pi\)
\(192\) 0 0
\(193\) 685155. 1.32402 0.662011 0.749494i \(-0.269703\pi\)
0.662011 + 0.749494i \(0.269703\pi\)
\(194\) 529078. 1.00929
\(195\) 0 0
\(196\) −1.12923e6 −2.09963
\(197\) 723750. 1.32869 0.664344 0.747427i \(-0.268711\pi\)
0.664344 + 0.747427i \(0.268711\pi\)
\(198\) 0 0
\(199\) −186876. −0.334519 −0.167260 0.985913i \(-0.553492\pi\)
−0.167260 + 0.985913i \(0.553492\pi\)
\(200\) 2.77546e6 4.90636
\(201\) 0 0
\(202\) −1.16800e6 −2.01402
\(203\) −7126.28 −0.0121373
\(204\) 0 0
\(205\) −73069.3 −0.121437
\(206\) −1.01968e6 −1.67415
\(207\) 0 0
\(208\) −3.92158e6 −6.28497
\(209\) 648121. 1.02634
\(210\) 0 0
\(211\) −143708. −0.222216 −0.111108 0.993808i \(-0.535440\pi\)
−0.111108 + 0.993808i \(0.535440\pi\)
\(212\) −2.99089e6 −4.57048
\(213\) 0 0
\(214\) −564723. −0.842948
\(215\) 1.08329e6 1.59826
\(216\) 0 0
\(217\) −223783. −0.322609
\(218\) −1.61237e6 −2.29787
\(219\) 0 0
\(220\) 3.23816e6 4.51067
\(221\) −2.13192e6 −2.93623
\(222\) 0 0
\(223\) 299009. 0.402645 0.201322 0.979525i \(-0.435476\pi\)
0.201322 + 0.979525i \(0.435476\pi\)
\(224\) 1.33694e6 1.78029
\(225\) 0 0
\(226\) 912164. 1.18796
\(227\) −106091. −0.136651 −0.0683255 0.997663i \(-0.521766\pi\)
−0.0683255 + 0.997663i \(0.521766\pi\)
\(228\) 0 0
\(229\) −222769. −0.280715 −0.140358 0.990101i \(-0.544825\pi\)
−0.140358 + 0.990101i \(0.544825\pi\)
\(230\) 2.86929e6 3.57647
\(231\) 0 0
\(232\) −69086.9 −0.0842706
\(233\) 877575. 1.05900 0.529498 0.848311i \(-0.322380\pi\)
0.529498 + 0.848311i \(0.322380\pi\)
\(234\) 0 0
\(235\) −1.41906e6 −1.67622
\(236\) 302404. 0.353434
\(237\) 0 0
\(238\) 1.36909e6 1.56671
\(239\) −1.55403e6 −1.75980 −0.879902 0.475156i \(-0.842392\pi\)
−0.879902 + 0.475156i \(0.842392\pi\)
\(240\) 0 0
\(241\) −65896.5 −0.0730836 −0.0365418 0.999332i \(-0.511634\pi\)
−0.0365418 + 0.999332i \(0.511634\pi\)
\(242\) 195167. 0.214223
\(243\) 0 0
\(244\) 1.42635e6 1.53374
\(245\) 1.14537e6 1.21908
\(246\) 0 0
\(247\) 1.60522e6 1.67414
\(248\) −2.16950e6 −2.23991
\(249\) 0 0
\(250\) −1.45462e6 −1.47197
\(251\) 24842.7 0.0248894 0.0124447 0.999923i \(-0.496039\pi\)
0.0124447 + 0.999923i \(0.496039\pi\)
\(252\) 0 0
\(253\) 1.26344e6 1.24095
\(254\) −1.69047e6 −1.64408
\(255\) 0 0
\(256\) 2.55600e6 2.43759
\(257\) −501967. −0.474070 −0.237035 0.971501i \(-0.576176\pi\)
−0.237035 + 0.971501i \(0.576176\pi\)
\(258\) 0 0
\(259\) −928157. −0.859750
\(260\) 8.02004e6 7.35772
\(261\) 0 0
\(262\) 4.04140e6 3.63729
\(263\) −1.26902e6 −1.13130 −0.565652 0.824644i \(-0.691376\pi\)
−0.565652 + 0.824644i \(0.691376\pi\)
\(264\) 0 0
\(265\) 3.03363e6 2.65368
\(266\) −1.03085e6 −0.893288
\(267\) 0 0
\(268\) 852939. 0.725406
\(269\) 1.04097e6 0.877121 0.438560 0.898702i \(-0.355489\pi\)
0.438560 + 0.898702i \(0.355489\pi\)
\(270\) 0 0
\(271\) −2.07017e6 −1.71231 −0.856155 0.516718i \(-0.827153\pi\)
−0.856155 + 0.516718i \(0.827153\pi\)
\(272\) 7.61622e6 6.24191
\(273\) 0 0
\(274\) −2.11579e6 −1.70254
\(275\) −1.96247e6 −1.56485
\(276\) 0 0
\(277\) 1.93149e6 1.51249 0.756247 0.654287i \(-0.227031\pi\)
0.756247 + 0.654287i \(0.227031\pi\)
\(278\) −877672. −0.681115
\(279\) 0 0
\(280\) −3.25320e6 −2.47979
\(281\) −859707. −0.649508 −0.324754 0.945799i \(-0.605282\pi\)
−0.324754 + 0.945799i \(0.605282\pi\)
\(282\) 0 0
\(283\) −789427. −0.585930 −0.292965 0.956123i \(-0.594642\pi\)
−0.292965 + 0.956123i \(0.594642\pi\)
\(284\) −293772. −0.216130
\(285\) 0 0
\(286\) 4.83232e6 3.49334
\(287\) 51174.6 0.0366733
\(288\) 0 0
\(289\) 2.72060e6 1.91611
\(290\) 110939. 0.0774622
\(291\) 0 0
\(292\) 2.34661e6 1.61059
\(293\) 1.83732e6 1.25030 0.625152 0.780503i \(-0.285037\pi\)
0.625152 + 0.780503i \(0.285037\pi\)
\(294\) 0 0
\(295\) −306726. −0.205208
\(296\) −8.99818e6 −5.96933
\(297\) 0 0
\(298\) −2.50735e6 −1.63559
\(299\) 3.12920e6 2.02421
\(300\) 0 0
\(301\) −758686. −0.482665
\(302\) −1.12824e6 −0.711843
\(303\) 0 0
\(304\) −5.73461e6 −3.55893
\(305\) −1.44674e6 −0.890512
\(306\) 0 0
\(307\) −443750. −0.268715 −0.134358 0.990933i \(-0.542897\pi\)
−0.134358 + 0.990933i \(0.542897\pi\)
\(308\) −2.26787e6 −1.36220
\(309\) 0 0
\(310\) 3.48376e6 2.05894
\(311\) 651682. 0.382063 0.191031 0.981584i \(-0.438817\pi\)
0.191031 + 0.981584i \(0.438817\pi\)
\(312\) 0 0
\(313\) −1.28063e6 −0.738859 −0.369430 0.929259i \(-0.620447\pi\)
−0.369430 + 0.929259i \(0.620447\pi\)
\(314\) 2.42072e6 1.38554
\(315\) 0 0
\(316\) 1.77449e6 0.999670
\(317\) −1.85275e6 −1.03554 −0.517772 0.855518i \(-0.673239\pi\)
−0.517772 + 0.855518i \(0.673239\pi\)
\(318\) 0 0
\(319\) 48850.0 0.0268775
\(320\) −1.02591e7 −5.60057
\(321\) 0 0
\(322\) −2.00953e6 −1.08008
\(323\) −3.11755e6 −1.66267
\(324\) 0 0
\(325\) −4.86052e6 −2.55255
\(326\) −18257.5 −0.00951475
\(327\) 0 0
\(328\) 496121. 0.254626
\(329\) 993849. 0.506210
\(330\) 0 0
\(331\) −646915. −0.324547 −0.162273 0.986746i \(-0.551883\pi\)
−0.162273 + 0.986746i \(0.551883\pi\)
\(332\) 4.73555e6 2.35790
\(333\) 0 0
\(334\) 6.91058e6 3.38960
\(335\) −865128. −0.421180
\(336\) 0 0
\(337\) −1.93315e6 −0.927235 −0.463618 0.886035i \(-0.653449\pi\)
−0.463618 + 0.886035i \(0.653449\pi\)
\(338\) 7.92019e6 3.77089
\(339\) 0 0
\(340\) −1.55759e7 −7.30731
\(341\) 1.53401e6 0.714402
\(342\) 0 0
\(343\) −1.83935e6 −0.844169
\(344\) −7.35521e6 −3.35119
\(345\) 0 0
\(346\) −3.32273e6 −1.49213
\(347\) 2.55520e6 1.13920 0.569601 0.821921i \(-0.307098\pi\)
0.569601 + 0.821921i \(0.307098\pi\)
\(348\) 0 0
\(349\) −106267. −0.0467019 −0.0233509 0.999727i \(-0.507434\pi\)
−0.0233509 + 0.999727i \(0.507434\pi\)
\(350\) 3.12135e6 1.36199
\(351\) 0 0
\(352\) −9.16459e6 −3.94236
\(353\) −421809. −0.180169 −0.0900844 0.995934i \(-0.528714\pi\)
−0.0900844 + 0.995934i \(0.528714\pi\)
\(354\) 0 0
\(355\) 297970. 0.125488
\(356\) 6.05505e6 2.53217
\(357\) 0 0
\(358\) −731936. −0.301832
\(359\) −2.33843e6 −0.957607 −0.478804 0.877922i \(-0.658929\pi\)
−0.478804 + 0.877922i \(0.658929\pi\)
\(360\) 0 0
\(361\) −128752. −0.0519980
\(362\) 1.21874e6 0.488808
\(363\) 0 0
\(364\) −5.61689e6 −2.22199
\(365\) −2.38015e6 −0.935129
\(366\) 0 0
\(367\) 1.76169e6 0.682755 0.341377 0.939926i \(-0.389107\pi\)
0.341377 + 0.939926i \(0.389107\pi\)
\(368\) −1.11790e7 −4.30311
\(369\) 0 0
\(370\) 1.44492e7 5.48705
\(371\) −2.12463e6 −0.801397
\(372\) 0 0
\(373\) 555814. 0.206851 0.103425 0.994637i \(-0.467020\pi\)
0.103425 + 0.994637i \(0.467020\pi\)
\(374\) −9.38498e6 −3.46940
\(375\) 0 0
\(376\) 9.63504e6 3.51467
\(377\) 120988. 0.0438420
\(378\) 0 0
\(379\) 1.23190e6 0.440532 0.220266 0.975440i \(-0.429307\pi\)
0.220266 + 0.975440i \(0.429307\pi\)
\(380\) 1.17279e7 4.16639
\(381\) 0 0
\(382\) 7.97386e6 2.79583
\(383\) −2.07159e6 −0.721617 −0.360808 0.932640i \(-0.617499\pi\)
−0.360808 + 0.932640i \(0.617499\pi\)
\(384\) 0 0
\(385\) 2.30027e6 0.790911
\(386\) 7.47016e6 2.55189
\(387\) 0 0
\(388\) 4.21563e6 1.42162
\(389\) 4.06653e6 1.36254 0.681271 0.732031i \(-0.261427\pi\)
0.681271 + 0.732031i \(0.261427\pi\)
\(390\) 0 0
\(391\) −6.07731e6 −2.01034
\(392\) −7.77676e6 −2.55613
\(393\) 0 0
\(394\) 7.89096e6 2.56088
\(395\) −1.79985e6 −0.580422
\(396\) 0 0
\(397\) −4.63853e6 −1.47708 −0.738541 0.674209i \(-0.764485\pi\)
−0.738541 + 0.674209i \(0.764485\pi\)
\(398\) −2.03749e6 −0.644744
\(399\) 0 0
\(400\) 1.73640e7 5.42626
\(401\) −2.63194e6 −0.817363 −0.408681 0.912677i \(-0.634011\pi\)
−0.408681 + 0.912677i \(0.634011\pi\)
\(402\) 0 0
\(403\) 3.79933e6 1.16532
\(404\) −9.30647e6 −2.83682
\(405\) 0 0
\(406\) −77697.0 −0.0233932
\(407\) 6.36244e6 1.90387
\(408\) 0 0
\(409\) 3.04976e6 0.901482 0.450741 0.892655i \(-0.351160\pi\)
0.450741 + 0.892655i \(0.351160\pi\)
\(410\) −796666. −0.234054
\(411\) 0 0
\(412\) −8.12466e6 −2.35810
\(413\) 214817. 0.0619719
\(414\) 0 0
\(415\) −4.80322e6 −1.36903
\(416\) −2.26982e7 −6.43070
\(417\) 0 0
\(418\) 7.06639e6 1.97814
\(419\) −1.34161e6 −0.373329 −0.186664 0.982424i \(-0.559768\pi\)
−0.186664 + 0.982424i \(0.559768\pi\)
\(420\) 0 0
\(421\) 2.54019e6 0.698492 0.349246 0.937031i \(-0.386438\pi\)
0.349246 + 0.937031i \(0.386438\pi\)
\(422\) −1.56683e6 −0.428293
\(423\) 0 0
\(424\) −2.05976e7 −5.56418
\(425\) 9.43974e6 2.53506
\(426\) 0 0
\(427\) 1.01323e6 0.268930
\(428\) −4.49964e6 −1.18732
\(429\) 0 0
\(430\) 1.18109e7 3.08044
\(431\) −4.17832e6 −1.08345 −0.541725 0.840556i \(-0.682228\pi\)
−0.541725 + 0.840556i \(0.682228\pi\)
\(432\) 0 0
\(433\) −74580.5 −0.0191164 −0.00955819 0.999954i \(-0.503043\pi\)
−0.00955819 + 0.999954i \(0.503043\pi\)
\(434\) −2.43988e6 −0.621790
\(435\) 0 0
\(436\) −1.28472e7 −3.23663
\(437\) 4.57589e6 1.14623
\(438\) 0 0
\(439\) 7.12201e6 1.76377 0.881884 0.471466i \(-0.156275\pi\)
0.881884 + 0.471466i \(0.156275\pi\)
\(440\) 2.23004e7 5.49137
\(441\) 0 0
\(442\) −2.32441e7 −5.65922
\(443\) 7.36242e6 1.78243 0.891213 0.453585i \(-0.149855\pi\)
0.891213 + 0.453585i \(0.149855\pi\)
\(444\) 0 0
\(445\) −6.14158e6 −1.47021
\(446\) 3.26006e6 0.776047
\(447\) 0 0
\(448\) 7.18500e6 1.69134
\(449\) 5.60487e6 1.31205 0.656023 0.754741i \(-0.272237\pi\)
0.656023 + 0.754741i \(0.272237\pi\)
\(450\) 0 0
\(451\) −350798. −0.0812111
\(452\) 7.26801e6 1.67328
\(453\) 0 0
\(454\) −1.15669e6 −0.263378
\(455\) 5.69716e6 1.29012
\(456\) 0 0
\(457\) −2.27985e6 −0.510641 −0.255320 0.966856i \(-0.582181\pi\)
−0.255320 + 0.966856i \(0.582181\pi\)
\(458\) −2.42883e6 −0.541044
\(459\) 0 0
\(460\) 2.28621e7 5.03759
\(461\) −5.70375e6 −1.24999 −0.624997 0.780627i \(-0.714900\pi\)
−0.624997 + 0.780627i \(0.714900\pi\)
\(462\) 0 0
\(463\) 3.15681e6 0.684378 0.342189 0.939631i \(-0.388832\pi\)
0.342189 + 0.939631i \(0.388832\pi\)
\(464\) −432227. −0.0932003
\(465\) 0 0
\(466\) 9.56810e6 2.04108
\(467\) −4.10668e6 −0.871362 −0.435681 0.900101i \(-0.643492\pi\)
−0.435681 + 0.900101i \(0.643492\pi\)
\(468\) 0 0
\(469\) 605898. 0.127194
\(470\) −1.54719e7 −3.23071
\(471\) 0 0
\(472\) 2.08259e6 0.430277
\(473\) 5.20073e6 1.06884
\(474\) 0 0
\(475\) −7.10762e6 −1.44541
\(476\) 1.09087e7 2.20677
\(477\) 0 0
\(478\) −1.69434e7 −3.39180
\(479\) 3.96139e6 0.788876 0.394438 0.918923i \(-0.370939\pi\)
0.394438 + 0.918923i \(0.370939\pi\)
\(480\) 0 0
\(481\) 1.57580e7 3.10556
\(482\) −718462. −0.140860
\(483\) 0 0
\(484\) 1.55506e6 0.301741
\(485\) −4.27587e6 −0.825411
\(486\) 0 0
\(487\) −9.18635e6 −1.75517 −0.877587 0.479417i \(-0.840848\pi\)
−0.877587 + 0.479417i \(0.840848\pi\)
\(488\) 9.82295e6 1.86721
\(489\) 0 0
\(490\) 1.24878e7 2.34962
\(491\) −3.85969e6 −0.722518 −0.361259 0.932465i \(-0.617653\pi\)
−0.361259 + 0.932465i \(0.617653\pi\)
\(492\) 0 0
\(493\) −234975. −0.0435416
\(494\) 1.75015e7 3.22670
\(495\) 0 0
\(496\) −1.35730e7 −2.47726
\(497\) −208686. −0.0378967
\(498\) 0 0
\(499\) −3.44774e6 −0.619845 −0.309922 0.950762i \(-0.600303\pi\)
−0.309922 + 0.950762i \(0.600303\pi\)
\(500\) −1.15902e7 −2.07332
\(501\) 0 0
\(502\) 270857. 0.0479712
\(503\) −5.17527e6 −0.912038 −0.456019 0.889970i \(-0.650725\pi\)
−0.456019 + 0.889970i \(0.650725\pi\)
\(504\) 0 0
\(505\) 9.43946e6 1.64710
\(506\) 1.37751e7 2.39177
\(507\) 0 0
\(508\) −1.34695e7 −2.31575
\(509\) 86953.7 0.0148763 0.00743813 0.999972i \(-0.497632\pi\)
0.00743813 + 0.999972i \(0.497632\pi\)
\(510\) 0 0
\(511\) 1.66695e6 0.282404
\(512\) 9.43086e6 1.58992
\(513\) 0 0
\(514\) −5.47289e6 −0.913710
\(515\) 8.24077e6 1.36915
\(516\) 0 0
\(517\) −6.81275e6 −1.12098
\(518\) −1.01196e7 −1.65706
\(519\) 0 0
\(520\) 5.52321e7 8.95742
\(521\) 6.15153e6 0.992861 0.496431 0.868076i \(-0.334644\pi\)
0.496431 + 0.868076i \(0.334644\pi\)
\(522\) 0 0
\(523\) 1.72407e6 0.275613 0.137807 0.990459i \(-0.455995\pi\)
0.137807 + 0.990459i \(0.455995\pi\)
\(524\) 3.22014e7 5.12326
\(525\) 0 0
\(526\) −1.38360e7 −2.18045
\(527\) −7.37879e6 −1.15733
\(528\) 0 0
\(529\) 2.48384e6 0.385909
\(530\) 3.30754e7 5.11464
\(531\) 0 0
\(532\) −8.21369e6 −1.25823
\(533\) −868831. −0.132470
\(534\) 0 0
\(535\) 4.56394e6 0.689375
\(536\) 5.87398e6 0.883122
\(537\) 0 0
\(538\) 1.13496e7 1.69054
\(539\) 5.49879e6 0.815259
\(540\) 0 0
\(541\) −8.39290e6 −1.23287 −0.616437 0.787404i \(-0.711425\pi\)
−0.616437 + 0.787404i \(0.711425\pi\)
\(542\) −2.25708e7 −3.30027
\(543\) 0 0
\(544\) 4.40829e7 6.38664
\(545\) 1.30308e7 1.87923
\(546\) 0 0
\(547\) −1.01323e7 −1.44790 −0.723952 0.689850i \(-0.757676\pi\)
−0.723952 + 0.689850i \(0.757676\pi\)
\(548\) −1.68584e7 −2.39808
\(549\) 0 0
\(550\) −2.13966e7 −3.01605
\(551\) 176924. 0.0248260
\(552\) 0 0
\(553\) 1.26054e6 0.175284
\(554\) 2.10588e7 2.91514
\(555\) 0 0
\(556\) −6.99318e6 −0.959374
\(557\) −9.82722e6 −1.34212 −0.671062 0.741401i \(-0.734162\pi\)
−0.671062 + 0.741401i \(0.734162\pi\)
\(558\) 0 0
\(559\) 1.28808e7 1.74347
\(560\) −2.03529e7 −2.74256
\(561\) 0 0
\(562\) −9.37328e6 −1.25185
\(563\) −9.96382e6 −1.32481 −0.662407 0.749144i \(-0.730465\pi\)
−0.662407 + 0.749144i \(0.730465\pi\)
\(564\) 0 0
\(565\) −7.37187e6 −0.971531
\(566\) −8.60703e6 −1.12931
\(567\) 0 0
\(568\) −2.02314e6 −0.263121
\(569\) −1.30624e7 −1.69138 −0.845692 0.533672i \(-0.820812\pi\)
−0.845692 + 0.533672i \(0.820812\pi\)
\(570\) 0 0
\(571\) 1.30362e7 1.67325 0.836625 0.547776i \(-0.184525\pi\)
0.836625 + 0.547776i \(0.184525\pi\)
\(572\) 3.85033e7 4.92049
\(573\) 0 0
\(574\) 557951. 0.0706832
\(575\) −1.38555e7 −1.74764
\(576\) 0 0
\(577\) −1.37814e7 −1.72327 −0.861637 0.507524i \(-0.830561\pi\)
−0.861637 + 0.507524i \(0.830561\pi\)
\(578\) 2.96624e7 3.69307
\(579\) 0 0
\(580\) 883949. 0.109108
\(581\) 3.36397e6 0.413440
\(582\) 0 0
\(583\) 1.45641e7 1.77465
\(584\) 1.61606e7 1.96076
\(585\) 0 0
\(586\) 2.00321e7 2.40981
\(587\) 4.70405e6 0.563478 0.281739 0.959491i \(-0.409089\pi\)
0.281739 + 0.959491i \(0.409089\pi\)
\(588\) 0 0
\(589\) 5.55584e6 0.659874
\(590\) −3.34419e6 −0.395514
\(591\) 0 0
\(592\) −5.62951e7 −6.60187
\(593\) −4.77416e6 −0.557519 −0.278760 0.960361i \(-0.589923\pi\)
−0.278760 + 0.960361i \(0.589923\pi\)
\(594\) 0 0
\(595\) −1.10646e7 −1.28128
\(596\) −1.99783e7 −2.30379
\(597\) 0 0
\(598\) 3.41173e7 3.90141
\(599\) 1.34470e6 0.153129 0.0765645 0.997065i \(-0.475605\pi\)
0.0765645 + 0.997065i \(0.475605\pi\)
\(600\) 0 0
\(601\) −1.25261e7 −1.41458 −0.707292 0.706921i \(-0.750084\pi\)
−0.707292 + 0.706921i \(0.750084\pi\)
\(602\) −8.27187e6 −0.930277
\(603\) 0 0
\(604\) −8.98969e6 −1.00266
\(605\) −1.57728e6 −0.175195
\(606\) 0 0
\(607\) −1.01433e7 −1.11740 −0.558698 0.829371i \(-0.688699\pi\)
−0.558698 + 0.829371i \(0.688699\pi\)
\(608\) −3.31920e7 −3.64146
\(609\) 0 0
\(610\) −1.57736e7 −1.71635
\(611\) −1.68734e7 −1.82851
\(612\) 0 0
\(613\) 5.39122e6 0.579476 0.289738 0.957106i \(-0.406432\pi\)
0.289738 + 0.957106i \(0.406432\pi\)
\(614\) −4.83815e6 −0.517915
\(615\) 0 0
\(616\) −1.56182e7 −1.65837
\(617\) −1.83062e6 −0.193591 −0.0967957 0.995304i \(-0.530859\pi\)
−0.0967957 + 0.995304i \(0.530859\pi\)
\(618\) 0 0
\(619\) 1.04088e7 1.09188 0.545939 0.837825i \(-0.316173\pi\)
0.545939 + 0.837825i \(0.316173\pi\)
\(620\) 2.77582e7 2.90009
\(621\) 0 0
\(622\) 7.10521e6 0.736378
\(623\) 4.30130e6 0.443996
\(624\) 0 0
\(625\) −2.74143e6 −0.280722
\(626\) −1.39625e7 −1.42406
\(627\) 0 0
\(628\) 1.92880e7 1.95159
\(629\) −3.06041e7 −3.08428
\(630\) 0 0
\(631\) 4.99395e6 0.499310 0.249655 0.968335i \(-0.419683\pi\)
0.249655 + 0.968335i \(0.419683\pi\)
\(632\) 1.22205e7 1.21702
\(633\) 0 0
\(634\) −2.02003e7 −1.99588
\(635\) 1.36620e7 1.34455
\(636\) 0 0
\(637\) 1.36190e7 1.32984
\(638\) 532606. 0.0518030
\(639\) 0 0
\(640\) −5.07673e7 −4.89930
\(641\) −6.06889e6 −0.583397 −0.291699 0.956510i \(-0.594220\pi\)
−0.291699 + 0.956510i \(0.594220\pi\)
\(642\) 0 0
\(643\) −1.56458e7 −1.49235 −0.746176 0.665748i \(-0.768112\pi\)
−0.746176 + 0.665748i \(0.768112\pi\)
\(644\) −1.60117e7 −1.52132
\(645\) 0 0
\(646\) −3.39902e7 −3.20459
\(647\) −1.46905e7 −1.37967 −0.689837 0.723964i \(-0.742318\pi\)
−0.689837 + 0.723964i \(0.742318\pi\)
\(648\) 0 0
\(649\) −1.47256e6 −0.137233
\(650\) −5.29936e7 −4.91972
\(651\) 0 0
\(652\) −145474. −0.0134019
\(653\) −1.07967e7 −0.990852 −0.495426 0.868650i \(-0.664988\pi\)
−0.495426 + 0.868650i \(0.664988\pi\)
\(654\) 0 0
\(655\) −3.26615e7 −2.97463
\(656\) 3.10387e6 0.281608
\(657\) 0 0
\(658\) 1.08358e7 0.975657
\(659\) 1.33499e7 1.19747 0.598734 0.800948i \(-0.295671\pi\)
0.598734 + 0.800948i \(0.295671\pi\)
\(660\) 0 0
\(661\) −1.52627e7 −1.35871 −0.679354 0.733810i \(-0.737740\pi\)
−0.679354 + 0.733810i \(0.737740\pi\)
\(662\) −7.05324e6 −0.625524
\(663\) 0 0
\(664\) 3.26126e7 2.87055
\(665\) 8.33106e6 0.730544
\(666\) 0 0
\(667\) 344893. 0.0300172
\(668\) 5.50627e7 4.77437
\(669\) 0 0
\(670\) −9.43238e6 −0.811773
\(671\) −6.94562e6 −0.595531
\(672\) 0 0
\(673\) 700630. 0.0596281 0.0298141 0.999555i \(-0.490508\pi\)
0.0298141 + 0.999555i \(0.490508\pi\)
\(674\) −2.10769e7 −1.78713
\(675\) 0 0
\(676\) 6.31071e7 5.31143
\(677\) −1.76900e7 −1.48339 −0.741697 0.670735i \(-0.765979\pi\)
−0.741697 + 0.670735i \(0.765979\pi\)
\(678\) 0 0
\(679\) 2.99464e6 0.249270
\(680\) −1.07268e8 −8.89605
\(681\) 0 0
\(682\) 1.67251e7 1.37692
\(683\) 1.72148e7 1.41205 0.706024 0.708187i \(-0.250487\pi\)
0.706024 + 0.708187i \(0.250487\pi\)
\(684\) 0 0
\(685\) 1.70993e7 1.39236
\(686\) −2.00542e7 −1.62703
\(687\) 0 0
\(688\) −4.60163e7 −3.70630
\(689\) 3.60715e7 2.89478
\(690\) 0 0
\(691\) −1.54571e7 −1.23149 −0.615746 0.787944i \(-0.711145\pi\)
−0.615746 + 0.787944i \(0.711145\pi\)
\(692\) −2.64751e7 −2.10171
\(693\) 0 0
\(694\) 2.78590e7 2.19567
\(695\) 7.09312e6 0.557025
\(696\) 0 0
\(697\) 1.68738e6 0.131562
\(698\) −1.15862e6 −0.0900120
\(699\) 0 0
\(700\) 2.48706e7 1.91841
\(701\) −90599.9 −0.00696358 −0.00348179 0.999994i \(-0.501108\pi\)
−0.00348179 + 0.999994i \(0.501108\pi\)
\(702\) 0 0
\(703\) 2.30433e7 1.75856
\(704\) −4.92526e7 −3.74539
\(705\) 0 0
\(706\) −4.59894e6 −0.347253
\(707\) −6.61099e6 −0.497414
\(708\) 0 0
\(709\) 7.81917e6 0.584178 0.292089 0.956391i \(-0.405650\pi\)
0.292089 + 0.956391i \(0.405650\pi\)
\(710\) 3.24874e6 0.241863
\(711\) 0 0
\(712\) 4.16997e7 3.08271
\(713\) 1.08305e7 0.797855
\(714\) 0 0
\(715\) −3.90535e7 −2.85690
\(716\) −5.83197e6 −0.425141
\(717\) 0 0
\(718\) −2.54956e7 −1.84567
\(719\) −2.06917e7 −1.49270 −0.746352 0.665552i \(-0.768196\pi\)
−0.746352 + 0.665552i \(0.768196\pi\)
\(720\) 0 0
\(721\) −5.77148e6 −0.413475
\(722\) −1.40377e6 −0.100220
\(723\) 0 0
\(724\) 9.71073e6 0.688503
\(725\) −535714. −0.0378519
\(726\) 0 0
\(727\) −1.54163e7 −1.08179 −0.540896 0.841089i \(-0.681915\pi\)
−0.540896 + 0.841089i \(0.681915\pi\)
\(728\) −3.86822e7 −2.70509
\(729\) 0 0
\(730\) −2.59505e7 −1.80234
\(731\) −2.50162e7 −1.73152
\(732\) 0 0
\(733\) 1.32653e7 0.911924 0.455962 0.889999i \(-0.349295\pi\)
0.455962 + 0.889999i \(0.349295\pi\)
\(734\) 1.92075e7 1.31592
\(735\) 0 0
\(736\) −6.47042e7 −4.40289
\(737\) −4.15338e6 −0.281665
\(738\) 0 0
\(739\) 1.43430e7 0.966114 0.483057 0.875589i \(-0.339526\pi\)
0.483057 + 0.875589i \(0.339526\pi\)
\(740\) 1.15129e8 7.72871
\(741\) 0 0
\(742\) −2.31646e7 −1.54459
\(743\) −6.61514e6 −0.439609 −0.219805 0.975544i \(-0.570542\pi\)
−0.219805 + 0.975544i \(0.570542\pi\)
\(744\) 0 0
\(745\) 2.02638e7 1.33761
\(746\) 6.05998e6 0.398679
\(747\) 0 0
\(748\) −7.47784e7 −4.88677
\(749\) −3.19639e6 −0.208188
\(750\) 0 0
\(751\) 5.45848e6 0.353160 0.176580 0.984286i \(-0.443497\pi\)
0.176580 + 0.984286i \(0.443497\pi\)
\(752\) 6.02795e7 3.88710
\(753\) 0 0
\(754\) 1.31912e6 0.0845000
\(755\) 9.11815e6 0.582156
\(756\) 0 0
\(757\) 1.40547e7 0.891416 0.445708 0.895178i \(-0.352952\pi\)
0.445708 + 0.895178i \(0.352952\pi\)
\(758\) 1.34313e7 0.849070
\(759\) 0 0
\(760\) 8.07669e7 5.07224
\(761\) 7.51836e6 0.470610 0.235305 0.971922i \(-0.424391\pi\)
0.235305 + 0.971922i \(0.424391\pi\)
\(762\) 0 0
\(763\) −9.12621e6 −0.567517
\(764\) 6.35347e7 3.93802
\(765\) 0 0
\(766\) −2.25863e7 −1.39083
\(767\) −3.64712e6 −0.223853
\(768\) 0 0
\(769\) 1.72875e7 1.05418 0.527091 0.849809i \(-0.323283\pi\)
0.527091 + 0.849809i \(0.323283\pi\)
\(770\) 2.50796e7 1.52438
\(771\) 0 0
\(772\) 5.95213e7 3.59442
\(773\) −7.13686e6 −0.429594 −0.214797 0.976659i \(-0.568909\pi\)
−0.214797 + 0.976659i \(0.568909\pi\)
\(774\) 0 0
\(775\) −1.68227e7 −1.00610
\(776\) 2.90320e7 1.73070
\(777\) 0 0
\(778\) 4.43369e7 2.62613
\(779\) −1.27051e6 −0.0750125
\(780\) 0 0
\(781\) 1.43052e6 0.0839203
\(782\) −6.62602e7 −3.87468
\(783\) 0 0
\(784\) −4.86536e7 −2.82699
\(785\) −1.95636e7 −1.13312
\(786\) 0 0
\(787\) −5.86391e6 −0.337482 −0.168741 0.985660i \(-0.553970\pi\)
−0.168741 + 0.985660i \(0.553970\pi\)
\(788\) 6.28742e7 3.60709
\(789\) 0 0
\(790\) −1.96236e7 −1.11869
\(791\) 5.16294e6 0.293397
\(792\) 0 0
\(793\) −1.72024e7 −0.971419
\(794\) −5.05734e7 −2.84689
\(795\) 0 0
\(796\) −1.62344e7 −0.908144
\(797\) −1.96169e7 −1.09392 −0.546958 0.837160i \(-0.684214\pi\)
−0.546958 + 0.837160i \(0.684214\pi\)
\(798\) 0 0
\(799\) 3.27702e7 1.81599
\(800\) 1.00503e8 5.55209
\(801\) 0 0
\(802\) −2.86957e7 −1.57536
\(803\) −1.14268e7 −0.625369
\(804\) 0 0
\(805\) 1.62405e7 0.883302
\(806\) 4.14237e7 2.24601
\(807\) 0 0
\(808\) −6.40914e7 −3.45360
\(809\) 2.49811e7 1.34196 0.670982 0.741474i \(-0.265873\pi\)
0.670982 + 0.741474i \(0.265873\pi\)
\(810\) 0 0
\(811\) −1.44411e7 −0.770987 −0.385493 0.922711i \(-0.625969\pi\)
−0.385493 + 0.922711i \(0.625969\pi\)
\(812\) −619080. −0.0329501
\(813\) 0 0
\(814\) 6.93689e7 3.66947
\(815\) 147552. 0.00778130
\(816\) 0 0
\(817\) 1.88359e7 0.987257
\(818\) 3.32512e7 1.73749
\(819\) 0 0
\(820\) −6.34774e6 −0.329674
\(821\) 625911. 0.0324082 0.0162041 0.999869i \(-0.494842\pi\)
0.0162041 + 0.999869i \(0.494842\pi\)
\(822\) 0 0
\(823\) 2.15073e7 1.10684 0.553422 0.832901i \(-0.313322\pi\)
0.553422 + 0.832901i \(0.313322\pi\)
\(824\) −5.59526e7 −2.87079
\(825\) 0 0
\(826\) 2.34213e6 0.119443
\(827\) −2.06626e7 −1.05056 −0.525280 0.850929i \(-0.676040\pi\)
−0.525280 + 0.850929i \(0.676040\pi\)
\(828\) 0 0
\(829\) −5.93386e6 −0.299882 −0.149941 0.988695i \(-0.547908\pi\)
−0.149941 + 0.988695i \(0.547908\pi\)
\(830\) −5.23690e7 −2.63863
\(831\) 0 0
\(832\) −1.21985e8 −6.10941
\(833\) −2.64499e7 −1.32072
\(834\) 0 0
\(835\) −5.58495e7 −2.77206
\(836\) 5.63041e7 2.78628
\(837\) 0 0
\(838\) −1.46274e7 −0.719544
\(839\) 925271. 0.0453800 0.0226900 0.999743i \(-0.492777\pi\)
0.0226900 + 0.999743i \(0.492777\pi\)
\(840\) 0 0
\(841\) −2.04978e7 −0.999350
\(842\) 2.76954e7 1.34626
\(843\) 0 0
\(844\) −1.24843e7 −0.603265
\(845\) −6.40089e7 −3.08389
\(846\) 0 0
\(847\) 1.10466e6 0.0529080
\(848\) −1.28864e8 −6.15379
\(849\) 0 0
\(850\) 1.02920e8 4.88601
\(851\) 4.49203e7 2.12627
\(852\) 0 0
\(853\) 1.48698e7 0.699732 0.349866 0.936800i \(-0.386227\pi\)
0.349866 + 0.936800i \(0.386227\pi\)
\(854\) 1.10471e7 0.518329
\(855\) 0 0
\(856\) −3.09879e7 −1.44547
\(857\) 4.11203e7 1.91251 0.956256 0.292531i \(-0.0944975\pi\)
0.956256 + 0.292531i \(0.0944975\pi\)
\(858\) 0 0
\(859\) 1.38706e7 0.641378 0.320689 0.947185i \(-0.396086\pi\)
0.320689 + 0.947185i \(0.396086\pi\)
\(860\) 9.41081e7 4.33891
\(861\) 0 0
\(862\) −4.55557e7 −2.08821
\(863\) −1.87918e7 −0.858898 −0.429449 0.903091i \(-0.641292\pi\)
−0.429449 + 0.903091i \(0.641292\pi\)
\(864\) 0 0
\(865\) 2.68535e7 1.22028
\(866\) −813143. −0.0368445
\(867\) 0 0
\(868\) −1.94406e7 −0.875812
\(869\) −8.64088e6 −0.388158
\(870\) 0 0
\(871\) −1.02868e7 −0.459447
\(872\) −8.84756e7 −3.94033
\(873\) 0 0
\(874\) 4.98904e7 2.20922
\(875\) −8.23327e6 −0.363540
\(876\) 0 0
\(877\) 4.12113e7 1.80933 0.904665 0.426124i \(-0.140121\pi\)
0.904665 + 0.426124i \(0.140121\pi\)
\(878\) 7.76505e7 3.39944
\(879\) 0 0
\(880\) 1.39518e8 6.07326
\(881\) −3.73000e7 −1.61908 −0.809542 0.587062i \(-0.800284\pi\)
−0.809542 + 0.587062i \(0.800284\pi\)
\(882\) 0 0
\(883\) −3.73701e7 −1.61295 −0.806477 0.591265i \(-0.798629\pi\)
−0.806477 + 0.591265i \(0.798629\pi\)
\(884\) −1.85206e8 −7.97121
\(885\) 0 0
\(886\) 8.02716e7 3.43540
\(887\) 2.39606e6 0.102256 0.0511280 0.998692i \(-0.483718\pi\)
0.0511280 + 0.998692i \(0.483718\pi\)
\(888\) 0 0
\(889\) −9.56824e6 −0.406048
\(890\) −6.69609e7 −2.83365
\(891\) 0 0
\(892\) 2.59758e7 1.09309
\(893\) −2.46742e7 −1.03542
\(894\) 0 0
\(895\) 5.91531e6 0.246843
\(896\) 3.55552e7 1.47956
\(897\) 0 0
\(898\) 6.11092e7 2.52881
\(899\) 418753. 0.0172806
\(900\) 0 0
\(901\) −7.00554e7 −2.87494
\(902\) −3.82470e6 −0.156524
\(903\) 0 0
\(904\) 5.00530e7 2.03709
\(905\) −9.84950e6 −0.399754
\(906\) 0 0
\(907\) −6.32877e6 −0.255447 −0.127724 0.991810i \(-0.540767\pi\)
−0.127724 + 0.991810i \(0.540767\pi\)
\(908\) −9.21640e6 −0.370977
\(909\) 0 0
\(910\) 6.21155e7 2.48654
\(911\) 1.22155e6 0.0487659 0.0243830 0.999703i \(-0.492238\pi\)
0.0243830 + 0.999703i \(0.492238\pi\)
\(912\) 0 0
\(913\) −2.30597e7 −0.915540
\(914\) −2.48569e7 −0.984197
\(915\) 0 0
\(916\) −1.93526e7 −0.762080
\(917\) 2.28747e7 0.898323
\(918\) 0 0
\(919\) −2.44744e7 −0.955924 −0.477962 0.878381i \(-0.658624\pi\)
−0.477962 + 0.878381i \(0.658624\pi\)
\(920\) 1.57446e8 6.13285
\(921\) 0 0
\(922\) −6.21873e7 −2.40921
\(923\) 3.54302e6 0.136889
\(924\) 0 0
\(925\) −6.97737e7 −2.68125
\(926\) 3.44184e7 1.31905
\(927\) 0 0
\(928\) −2.50174e6 −0.0953614
\(929\) −471008. −0.0179056 −0.00895280 0.999960i \(-0.502850\pi\)
−0.00895280 + 0.999960i \(0.502850\pi\)
\(930\) 0 0
\(931\) 1.99154e7 0.753034
\(932\) 7.62374e7 2.87494
\(933\) 0 0
\(934\) −4.47746e7 −1.67944
\(935\) 7.58470e7 2.83733
\(936\) 0 0
\(937\) 9.92896e6 0.369449 0.184725 0.982790i \(-0.440861\pi\)
0.184725 + 0.982790i \(0.440861\pi\)
\(938\) 6.60604e6 0.245151
\(939\) 0 0
\(940\) −1.23278e8 −4.55057
\(941\) 2.51107e7 0.924452 0.462226 0.886762i \(-0.347051\pi\)
0.462226 + 0.886762i \(0.347051\pi\)
\(942\) 0 0
\(943\) −2.47671e6 −0.0906978
\(944\) 1.30292e7 0.475871
\(945\) 0 0
\(946\) 5.67029e7 2.06005
\(947\) −1.98803e7 −0.720356 −0.360178 0.932884i \(-0.617284\pi\)
−0.360178 + 0.932884i \(0.617284\pi\)
\(948\) 0 0
\(949\) −2.83011e7 −1.02009
\(950\) −7.74936e7 −2.78584
\(951\) 0 0
\(952\) 7.51257e7 2.68656
\(953\) −3.07521e7 −1.09684 −0.548418 0.836204i \(-0.684770\pi\)
−0.548418 + 0.836204i \(0.684770\pi\)
\(954\) 0 0
\(955\) −6.44427e7 −2.28647
\(956\) −1.35003e8 −4.77747
\(957\) 0 0
\(958\) 4.31905e7 1.52046
\(959\) −1.19756e7 −0.420486
\(960\) 0 0
\(961\) −1.54793e7 −0.540682
\(962\) 1.71808e8 5.98558
\(963\) 0 0
\(964\) −5.72462e6 −0.198406
\(965\) −6.03719e7 −2.08697
\(966\) 0 0
\(967\) 3.47128e7 1.19378 0.596888 0.802324i \(-0.296404\pi\)
0.596888 + 0.802324i \(0.296404\pi\)
\(968\) 1.07093e7 0.367345
\(969\) 0 0
\(970\) −4.66193e7 −1.59088
\(971\) −6.63760e6 −0.225924 −0.112962 0.993599i \(-0.536034\pi\)
−0.112962 + 0.993599i \(0.536034\pi\)
\(972\) 0 0
\(973\) −4.96771e6 −0.168219
\(974\) −1.00158e8 −3.38288
\(975\) 0 0
\(976\) 6.14551e7 2.06506
\(977\) −2.24812e7 −0.753500 −0.376750 0.926315i \(-0.622958\pi\)
−0.376750 + 0.926315i \(0.622958\pi\)
\(978\) 0 0
\(979\) −2.94850e7 −0.983207
\(980\) 9.95016e7 3.30952
\(981\) 0 0
\(982\) −4.20818e7 −1.39256
\(983\) 4.47702e7 1.47776 0.738882 0.673835i \(-0.235354\pi\)
0.738882 + 0.673835i \(0.235354\pi\)
\(984\) 0 0
\(985\) −6.37727e7 −2.09432
\(986\) −2.56190e6 −0.0839210
\(987\) 0 0
\(988\) 1.39450e8 4.54493
\(989\) 3.67184e7 1.19369
\(990\) 0 0
\(991\) −173229. −0.00560320 −0.00280160 0.999996i \(-0.500892\pi\)
−0.00280160 + 0.999996i \(0.500892\pi\)
\(992\) −7.85608e7 −2.53470
\(993\) 0 0
\(994\) −2.27528e6 −0.0730412
\(995\) 1.64664e7 0.527281
\(996\) 0 0
\(997\) 1.72883e7 0.550825 0.275413 0.961326i \(-0.411186\pi\)
0.275413 + 0.961326i \(0.411186\pi\)
\(998\) −3.75903e7 −1.19467
\(999\) 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 531.6.a.d.1.12 12
3.2 odd 2 177.6.a.b.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.1 12 3.2 odd 2
531.6.a.d.1.12 12 1.1 even 1 trivial