Properties

Label 450.6.a.u.1.1
Level $450$
Weight $6$
Character 450.1
Self dual yes
Analytic conductor $72.173$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +118.000 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} +118.000 q^{7} +64.0000 q^{8} -192.000 q^{11} -1106.00 q^{13} +472.000 q^{14} +256.000 q^{16} +762.000 q^{17} -2740.00 q^{19} -768.000 q^{22} +1566.00 q^{23} -4424.00 q^{26} +1888.00 q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00 q^{32} +3048.00 q^{34} +5518.00 q^{37} -10960.0 q^{38} +378.000 q^{41} +2434.00 q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0 q^{47} -2883.00 q^{49} -17696.0 q^{52} -9174.00 q^{53} +7552.00 q^{56} -23640.0 q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0 q^{62} +4096.00 q^{64} -33722.0 q^{67} +12192.0 q^{68} -70212.0 q^{71} -21986.0 q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0 q^{77} +4520.00 q^{79} +1512.00 q^{82} -109074. q^{83} +9736.00 q^{86} -12288.0 q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0 q^{92} +52488.0 q^{94} +1918.00 q^{97} -11532.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 118.000 0.910200 0.455100 0.890440i \(-0.349603\pi\)
0.455100 + 0.890440i \(0.349603\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −192.000 −0.478431 −0.239216 0.970966i \(-0.576890\pi\)
−0.239216 + 0.970966i \(0.576890\pi\)
\(12\) 0 0
\(13\) −1106.00 −1.81508 −0.907542 0.419961i \(-0.862044\pi\)
−0.907542 + 0.419961i \(0.862044\pi\)
\(14\) 472.000 0.643609
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 762.000 0.639488 0.319744 0.947504i \(-0.396403\pi\)
0.319744 + 0.947504i \(0.396403\pi\)
\(18\) 0 0
\(19\) −2740.00 −1.74127 −0.870636 0.491928i \(-0.836292\pi\)
−0.870636 + 0.491928i \(0.836292\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −768.000 −0.338302
\(23\) 1566.00 0.617266 0.308633 0.951181i \(-0.400129\pi\)
0.308633 + 0.951181i \(0.400129\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4424.00 −1.28346
\(27\) 0 0
\(28\) 1888.00 0.455100
\(29\) −5910.00 −1.30495 −0.652473 0.757812i \(-0.726268\pi\)
−0.652473 + 0.757812i \(0.726268\pi\)
\(30\) 0 0
\(31\) −6868.00 −1.28359 −0.641795 0.766877i \(-0.721810\pi\)
−0.641795 + 0.766877i \(0.721810\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 3048.00 0.452187
\(35\) 0 0
\(36\) 0 0
\(37\) 5518.00 0.662640 0.331320 0.943519i \(-0.392506\pi\)
0.331320 + 0.943519i \(0.392506\pi\)
\(38\) −10960.0 −1.23127
\(39\) 0 0
\(40\) 0 0
\(41\) 378.000 0.0351182 0.0175591 0.999846i \(-0.494410\pi\)
0.0175591 + 0.999846i \(0.494410\pi\)
\(42\) 0 0
\(43\) 2434.00 0.200747 0.100374 0.994950i \(-0.467996\pi\)
0.100374 + 0.994950i \(0.467996\pi\)
\(44\) −3072.00 −0.239216
\(45\) 0 0
\(46\) 6264.00 0.436473
\(47\) 13122.0 0.866474 0.433237 0.901280i \(-0.357371\pi\)
0.433237 + 0.901280i \(0.357371\pi\)
\(48\) 0 0
\(49\) −2883.00 −0.171536
\(50\) 0 0
\(51\) 0 0
\(52\) −17696.0 −0.907542
\(53\) −9174.00 −0.448610 −0.224305 0.974519i \(-0.572011\pi\)
−0.224305 + 0.974519i \(0.572011\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7552.00 0.321804
\(57\) 0 0
\(58\) −23640.0 −0.922736
\(59\) 34980.0 1.30825 0.654124 0.756388i \(-0.273038\pi\)
0.654124 + 0.756388i \(0.273038\pi\)
\(60\) 0 0
\(61\) −9838.00 −0.338518 −0.169259 0.985572i \(-0.554137\pi\)
−0.169259 + 0.985572i \(0.554137\pi\)
\(62\) −27472.0 −0.907635
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −33722.0 −0.917754 −0.458877 0.888500i \(-0.651748\pi\)
−0.458877 + 0.888500i \(0.651748\pi\)
\(68\) 12192.0 0.319744
\(69\) 0 0
\(70\) 0 0
\(71\) −70212.0 −1.65297 −0.826486 0.562957i \(-0.809664\pi\)
−0.826486 + 0.562957i \(0.809664\pi\)
\(72\) 0 0
\(73\) −21986.0 −0.482880 −0.241440 0.970416i \(-0.577620\pi\)
−0.241440 + 0.970416i \(0.577620\pi\)
\(74\) 22072.0 0.468557
\(75\) 0 0
\(76\) −43840.0 −0.870636
\(77\) −22656.0 −0.435468
\(78\) 0 0
\(79\) 4520.00 0.0814837 0.0407418 0.999170i \(-0.487028\pi\)
0.0407418 + 0.999170i \(0.487028\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1512.00 0.0248323
\(83\) −109074. −1.73790 −0.868952 0.494896i \(-0.835206\pi\)
−0.868952 + 0.494896i \(0.835206\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9736.00 0.141950
\(87\) 0 0
\(88\) −12288.0 −0.169151
\(89\) −38490.0 −0.515078 −0.257539 0.966268i \(-0.582912\pi\)
−0.257539 + 0.966268i \(0.582912\pi\)
\(90\) 0 0
\(91\) −130508. −1.65209
\(92\) 25056.0 0.308633
\(93\) 0 0
\(94\) 52488.0 0.612689
\(95\) 0 0
\(96\) 0 0
\(97\) 1918.00 0.0206976 0.0103488 0.999946i \(-0.496706\pi\)
0.0103488 + 0.999946i \(0.496706\pi\)
\(98\) −11532.0 −0.121294
\(99\) 0 0
\(100\) 0 0
\(101\) −77622.0 −0.757149 −0.378575 0.925571i \(-0.623586\pi\)
−0.378575 + 0.925571i \(0.623586\pi\)
\(102\) 0 0
\(103\) 46714.0 0.433864 0.216932 0.976187i \(-0.430395\pi\)
0.216932 + 0.976187i \(0.430395\pi\)
\(104\) −70784.0 −0.641729
\(105\) 0 0
\(106\) −36696.0 −0.317215
\(107\) −1038.00 −0.00876472 −0.00438236 0.999990i \(-0.501395\pi\)
−0.00438236 + 0.999990i \(0.501395\pi\)
\(108\) 0 0
\(109\) 206930. 1.66823 0.834117 0.551587i \(-0.185977\pi\)
0.834117 + 0.551587i \(0.185977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 30208.0 0.227550
\(113\) 139386. 1.02689 0.513444 0.858123i \(-0.328369\pi\)
0.513444 + 0.858123i \(0.328369\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −94560.0 −0.652473
\(117\) 0 0
\(118\) 139920. 0.925070
\(119\) 89916.0 0.582062
\(120\) 0 0
\(121\) −124187. −0.771104
\(122\) −39352.0 −0.239369
\(123\) 0 0
\(124\) −109888. −0.641795
\(125\) 0 0
\(126\) 0 0
\(127\) −299882. −1.64984 −0.824919 0.565252i \(-0.808779\pi\)
−0.824919 + 0.565252i \(0.808779\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −7872.00 −0.0400781 −0.0200390 0.999799i \(-0.506379\pi\)
−0.0200390 + 0.999799i \(0.506379\pi\)
\(132\) 0 0
\(133\) −323320. −1.58491
\(134\) −134888. −0.648950
\(135\) 0 0
\(136\) 48768.0 0.226093
\(137\) −164238. −0.747605 −0.373803 0.927508i \(-0.621946\pi\)
−0.373803 + 0.927508i \(0.621946\pi\)
\(138\) 0 0
\(139\) −282100. −1.23841 −0.619207 0.785228i \(-0.712546\pi\)
−0.619207 + 0.785228i \(0.712546\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −280848. −1.16883
\(143\) 212352. 0.868393
\(144\) 0 0
\(145\) 0 0
\(146\) −87944.0 −0.341448
\(147\) 0 0
\(148\) 88288.0 0.331320
\(149\) 388950. 1.43525 0.717626 0.696429i \(-0.245229\pi\)
0.717626 + 0.696429i \(0.245229\pi\)
\(150\) 0 0
\(151\) −97948.0 −0.349585 −0.174793 0.984605i \(-0.555926\pi\)
−0.174793 + 0.984605i \(0.555926\pi\)
\(152\) −175360. −0.615633
\(153\) 0 0
\(154\) −90624.0 −0.307923
\(155\) 0 0
\(156\) 0 0
\(157\) 3718.00 0.0120382 0.00601908 0.999982i \(-0.498084\pi\)
0.00601908 + 0.999982i \(0.498084\pi\)
\(158\) 18080.0 0.0576177
\(159\) 0 0
\(160\) 0 0
\(161\) 184788. 0.561835
\(162\) 0 0
\(163\) 43234.0 0.127455 0.0637274 0.997967i \(-0.479701\pi\)
0.0637274 + 0.997967i \(0.479701\pi\)
\(164\) 6048.00 0.0175591
\(165\) 0 0
\(166\) −436296. −1.22888
\(167\) 186522. 0.517534 0.258767 0.965940i \(-0.416684\pi\)
0.258767 + 0.965940i \(0.416684\pi\)
\(168\) 0 0
\(169\) 851943. 2.29453
\(170\) 0 0
\(171\) 0 0
\(172\) 38944.0 0.100374
\(173\) −374454. −0.951225 −0.475612 0.879655i \(-0.657774\pi\)
−0.475612 + 0.879655i \(0.657774\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −49152.0 −0.119608
\(177\) 0 0
\(178\) −153960. −0.364215
\(179\) −272100. −0.634740 −0.317370 0.948302i \(-0.602800\pi\)
−0.317370 + 0.948302i \(0.602800\pi\)
\(180\) 0 0
\(181\) −75418.0 −0.171111 −0.0855556 0.996333i \(-0.527267\pi\)
−0.0855556 + 0.996333i \(0.527267\pi\)
\(182\) −522032. −1.16820
\(183\) 0 0
\(184\) 100224. 0.218236
\(185\) 0 0
\(186\) 0 0
\(187\) −146304. −0.305951
\(188\) 209952. 0.433237
\(189\) 0 0
\(190\) 0 0
\(191\) 356988. 0.708060 0.354030 0.935234i \(-0.384811\pi\)
0.354030 + 0.935234i \(0.384811\pi\)
\(192\) 0 0
\(193\) 438694. 0.847751 0.423876 0.905720i \(-0.360669\pi\)
0.423876 + 0.905720i \(0.360669\pi\)
\(194\) 7672.00 0.0146354
\(195\) 0 0
\(196\) −46128.0 −0.0857678
\(197\) −156798. −0.287856 −0.143928 0.989588i \(-0.545973\pi\)
−0.143928 + 0.989588i \(0.545973\pi\)
\(198\) 0 0
\(199\) −162520. −0.290920 −0.145460 0.989364i \(-0.546466\pi\)
−0.145460 + 0.989364i \(0.546466\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −310488. −0.535385
\(203\) −697380. −1.18776
\(204\) 0 0
\(205\) 0 0
\(206\) 186856. 0.306788
\(207\) 0 0
\(208\) −283136. −0.453771
\(209\) 526080. 0.833079
\(210\) 0 0
\(211\) −181648. −0.280882 −0.140441 0.990089i \(-0.544852\pi\)
−0.140441 + 0.990089i \(0.544852\pi\)
\(212\) −146784. −0.224305
\(213\) 0 0
\(214\) −4152.00 −0.00619759
\(215\) 0 0
\(216\) 0 0
\(217\) −810424. −1.16832
\(218\) 827720. 1.17962
\(219\) 0 0
\(220\) 0 0
\(221\) −842772. −1.16073
\(222\) 0 0
\(223\) 288274. 0.388189 0.194095 0.980983i \(-0.437823\pi\)
0.194095 + 0.980983i \(0.437823\pi\)
\(224\) 120832. 0.160902
\(225\) 0 0
\(226\) 557544. 0.726119
\(227\) 1.12552e6 1.44974 0.724869 0.688887i \(-0.241900\pi\)
0.724869 + 0.688887i \(0.241900\pi\)
\(228\) 0 0
\(229\) −415810. −0.523970 −0.261985 0.965072i \(-0.584377\pi\)
−0.261985 + 0.965072i \(0.584377\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −378240. −0.461368
\(233\) 770586. 0.929889 0.464945 0.885340i \(-0.346074\pi\)
0.464945 + 0.885340i \(0.346074\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 559680. 0.654124
\(237\) 0 0
\(238\) 359664. 0.411580
\(239\) 595320. 0.674149 0.337074 0.941478i \(-0.390563\pi\)
0.337074 + 0.941478i \(0.390563\pi\)
\(240\) 0 0
\(241\) 273902. 0.303775 0.151888 0.988398i \(-0.451465\pi\)
0.151888 + 0.988398i \(0.451465\pi\)
\(242\) −496748. −0.545253
\(243\) 0 0
\(244\) −157408. −0.169259
\(245\) 0 0
\(246\) 0 0
\(247\) 3.03044e6 3.16055
\(248\) −439552. −0.453817
\(249\) 0 0
\(250\) 0 0
\(251\) −850752. −0.852351 −0.426176 0.904640i \(-0.640139\pi\)
−0.426176 + 0.904640i \(0.640139\pi\)
\(252\) 0 0
\(253\) −300672. −0.295319
\(254\) −1.19953e6 −1.16661
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 825402. 0.779530 0.389765 0.920914i \(-0.372556\pi\)
0.389765 + 0.920914i \(0.372556\pi\)
\(258\) 0 0
\(259\) 651124. 0.603135
\(260\) 0 0
\(261\) 0 0
\(262\) −31488.0 −0.0283395
\(263\) 1.36465e6 1.21655 0.608276 0.793726i \(-0.291861\pi\)
0.608276 + 0.793726i \(0.291861\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.29328e6 −1.12070
\(267\) 0 0
\(268\) −539552. −0.458877
\(269\) 113310. 0.0954745 0.0477373 0.998860i \(-0.484799\pi\)
0.0477373 + 0.998860i \(0.484799\pi\)
\(270\) 0 0
\(271\) −849628. −0.702758 −0.351379 0.936233i \(-0.614287\pi\)
−0.351379 + 0.936233i \(0.614287\pi\)
\(272\) 195072. 0.159872
\(273\) 0 0
\(274\) −656952. −0.528637
\(275\) 0 0
\(276\) 0 0
\(277\) −438602. −0.343456 −0.171728 0.985144i \(-0.554935\pi\)
−0.171728 + 0.985144i \(0.554935\pi\)
\(278\) −1.12840e6 −0.875691
\(279\) 0 0
\(280\) 0 0
\(281\) 1.45670e6 1.10053 0.550267 0.834989i \(-0.314526\pi\)
0.550267 + 0.834989i \(0.314526\pi\)
\(282\) 0 0
\(283\) 120394. 0.0893591 0.0446795 0.999001i \(-0.485773\pi\)
0.0446795 + 0.999001i \(0.485773\pi\)
\(284\) −1.12339e6 −0.826486
\(285\) 0 0
\(286\) 849408. 0.614047
\(287\) 44604.0 0.0319646
\(288\) 0 0
\(289\) −839213. −0.591055
\(290\) 0 0
\(291\) 0 0
\(292\) −351776. −0.241440
\(293\) −2.64209e6 −1.79796 −0.898978 0.437993i \(-0.855689\pi\)
−0.898978 + 0.437993i \(0.855689\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 353152. 0.234278
\(297\) 0 0
\(298\) 1.55580e6 1.01488
\(299\) −1.73200e6 −1.12039
\(300\) 0 0
\(301\) 287212. 0.182720
\(302\) −391792. −0.247194
\(303\) 0 0
\(304\) −701440. −0.435318
\(305\) 0 0
\(306\) 0 0
\(307\) 1.44756e6 0.876577 0.438288 0.898834i \(-0.355585\pi\)
0.438288 + 0.898834i \(0.355585\pi\)
\(308\) −362496. −0.217734
\(309\) 0 0
\(310\) 0 0
\(311\) 928068. 0.544100 0.272050 0.962283i \(-0.412298\pi\)
0.272050 + 0.962283i \(0.412298\pi\)
\(312\) 0 0
\(313\) −2.29563e6 −1.32446 −0.662232 0.749299i \(-0.730391\pi\)
−0.662232 + 0.749299i \(0.730391\pi\)
\(314\) 14872.0 0.00851227
\(315\) 0 0
\(316\) 72320.0 0.0407418
\(317\) 2.73652e6 1.52950 0.764752 0.644324i \(-0.222861\pi\)
0.764752 + 0.644324i \(0.222861\pi\)
\(318\) 0 0
\(319\) 1.13472e6 0.624327
\(320\) 0 0
\(321\) 0 0
\(322\) 739152. 0.397278
\(323\) −2.08788e6 −1.11352
\(324\) 0 0
\(325\) 0 0
\(326\) 172936. 0.0901242
\(327\) 0 0
\(328\) 24192.0 0.0124162
\(329\) 1.54840e6 0.788665
\(330\) 0 0
\(331\) 3.81879e6 1.91583 0.957913 0.287059i \(-0.0926776\pi\)
0.957913 + 0.287059i \(0.0926776\pi\)
\(332\) −1.74518e6 −0.868952
\(333\) 0 0
\(334\) 746088. 0.365952
\(335\) 0 0
\(336\) 0 0
\(337\) 2.21088e6 1.06045 0.530225 0.847857i \(-0.322108\pi\)
0.530225 + 0.847857i \(0.322108\pi\)
\(338\) 3.40777e6 1.62248
\(339\) 0 0
\(340\) 0 0
\(341\) 1.31866e6 0.614109
\(342\) 0 0
\(343\) −2.32342e6 −1.06633
\(344\) 155776. 0.0709748
\(345\) 0 0
\(346\) −1.49782e6 −0.672618
\(347\) −2.32724e6 −1.03757 −0.518785 0.854905i \(-0.673615\pi\)
−0.518785 + 0.854905i \(0.673615\pi\)
\(348\) 0 0
\(349\) −311290. −0.136805 −0.0684024 0.997658i \(-0.521790\pi\)
−0.0684024 + 0.997658i \(0.521790\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −196608. −0.0845755
\(353\) −3.08657e6 −1.31838 −0.659189 0.751977i \(-0.729100\pi\)
−0.659189 + 0.751977i \(0.729100\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −615840. −0.257539
\(357\) 0 0
\(358\) −1.08840e6 −0.448829
\(359\) 3.53076e6 1.44588 0.722940 0.690911i \(-0.242790\pi\)
0.722940 + 0.690911i \(0.242790\pi\)
\(360\) 0 0
\(361\) 5.03150e6 2.03203
\(362\) −301672. −0.120994
\(363\) 0 0
\(364\) −2.08813e6 −0.826045
\(365\) 0 0
\(366\) 0 0
\(367\) −35762.0 −0.0138598 −0.00692989 0.999976i \(-0.502206\pi\)
−0.00692989 + 0.999976i \(0.502206\pi\)
\(368\) 400896. 0.154316
\(369\) 0 0
\(370\) 0 0
\(371\) −1.08253e6 −0.408325
\(372\) 0 0
\(373\) 1.71525e6 0.638346 0.319173 0.947696i \(-0.396595\pi\)
0.319173 + 0.947696i \(0.396595\pi\)
\(374\) −585216. −0.216340
\(375\) 0 0
\(376\) 839808. 0.306345
\(377\) 6.53646e6 2.36859
\(378\) 0 0
\(379\) −3.10174e6 −1.10919 −0.554597 0.832119i \(-0.687127\pi\)
−0.554597 + 0.832119i \(0.687127\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.42795e6 0.500674
\(383\) 5.31949e6 1.85299 0.926494 0.376309i \(-0.122807\pi\)
0.926494 + 0.376309i \(0.122807\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.75478e6 0.599451
\(387\) 0 0
\(388\) 30688.0 0.0103488
\(389\) −1.16145e6 −0.389158 −0.194579 0.980887i \(-0.562334\pi\)
−0.194579 + 0.980887i \(0.562334\pi\)
\(390\) 0 0
\(391\) 1.19329e6 0.394734
\(392\) −184512. −0.0606470
\(393\) 0 0
\(394\) −627192. −0.203545
\(395\) 0 0
\(396\) 0 0
\(397\) −628562. −0.200157 −0.100079 0.994980i \(-0.531909\pi\)
−0.100079 + 0.994980i \(0.531909\pi\)
\(398\) −650080. −0.205712
\(399\) 0 0
\(400\) 0 0
\(401\) 2.72432e6 0.846052 0.423026 0.906118i \(-0.360968\pi\)
0.423026 + 0.906118i \(0.360968\pi\)
\(402\) 0 0
\(403\) 7.59601e6 2.32982
\(404\) −1.24195e6 −0.378575
\(405\) 0 0
\(406\) −2.78952e6 −0.839875
\(407\) −1.05946e6 −0.317027
\(408\) 0 0
\(409\) 1.78019e6 0.526209 0.263104 0.964767i \(-0.415254\pi\)
0.263104 + 0.964767i \(0.415254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 747424. 0.216932
\(413\) 4.12764e6 1.19077
\(414\) 0 0
\(415\) 0 0
\(416\) −1.13254e6 −0.320865
\(417\) 0 0
\(418\) 2.10432e6 0.589076
\(419\) −650580. −0.181036 −0.0905181 0.995895i \(-0.528852\pi\)
−0.0905181 + 0.995895i \(0.528852\pi\)
\(420\) 0 0
\(421\) −3.54060e6 −0.973579 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(422\) −726592. −0.198614
\(423\) 0 0
\(424\) −587136. −0.158608
\(425\) 0 0
\(426\) 0 0
\(427\) −1.16088e6 −0.308119
\(428\) −16608.0 −0.00438236
\(429\) 0 0
\(430\) 0 0
\(431\) 548748. 0.142292 0.0711459 0.997466i \(-0.477334\pi\)
0.0711459 + 0.997466i \(0.477334\pi\)
\(432\) 0 0
\(433\) 1.49241e6 0.382534 0.191267 0.981538i \(-0.438740\pi\)
0.191267 + 0.981538i \(0.438740\pi\)
\(434\) −3.24170e6 −0.826129
\(435\) 0 0
\(436\) 3.31088e6 0.834117
\(437\) −4.29084e6 −1.07483
\(438\) 0 0
\(439\) 4.86212e6 1.20411 0.602053 0.798456i \(-0.294350\pi\)
0.602053 + 0.798456i \(0.294350\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.37109e6 −0.820757
\(443\) −1.86155e6 −0.450678 −0.225339 0.974280i \(-0.572349\pi\)
−0.225339 + 0.974280i \(0.572349\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.15310e6 0.274491
\(447\) 0 0
\(448\) 483328. 0.113775
\(449\) −3.73719e6 −0.874841 −0.437421 0.899257i \(-0.644108\pi\)
−0.437421 + 0.899257i \(0.644108\pi\)
\(450\) 0 0
\(451\) −72576.0 −0.0168016
\(452\) 2.23018e6 0.513444
\(453\) 0 0
\(454\) 4.50209e6 1.02512
\(455\) 0 0
\(456\) 0 0
\(457\) 6.48276e6 1.45201 0.726005 0.687690i \(-0.241375\pi\)
0.726005 + 0.687690i \(0.241375\pi\)
\(458\) −1.66324e6 −0.370503
\(459\) 0 0
\(460\) 0 0
\(461\) −1.50910e6 −0.330724 −0.165362 0.986233i \(-0.552879\pi\)
−0.165362 + 0.986233i \(0.552879\pi\)
\(462\) 0 0
\(463\) −8.68401e6 −1.88264 −0.941321 0.337513i \(-0.890414\pi\)
−0.941321 + 0.337513i \(0.890414\pi\)
\(464\) −1.51296e6 −0.326236
\(465\) 0 0
\(466\) 3.08234e6 0.657531
\(467\) 6.96412e6 1.47766 0.738829 0.673893i \(-0.235379\pi\)
0.738829 + 0.673893i \(0.235379\pi\)
\(468\) 0 0
\(469\) −3.97920e6 −0.835340
\(470\) 0 0
\(471\) 0 0
\(472\) 2.23872e6 0.462535
\(473\) −467328. −0.0960437
\(474\) 0 0
\(475\) 0 0
\(476\) 1.43866e6 0.291031
\(477\) 0 0
\(478\) 2.38128e6 0.476695
\(479\) 5.51052e6 1.09737 0.548686 0.836029i \(-0.315128\pi\)
0.548686 + 0.836029i \(0.315128\pi\)
\(480\) 0 0
\(481\) −6.10291e6 −1.20275
\(482\) 1.09561e6 0.214802
\(483\) 0 0
\(484\) −1.98699e6 −0.385552
\(485\) 0 0
\(486\) 0 0
\(487\) −5.51808e6 −1.05430 −0.527152 0.849771i \(-0.676740\pi\)
−0.527152 + 0.849771i \(0.676740\pi\)
\(488\) −629632. −0.119684
\(489\) 0 0
\(490\) 0 0
\(491\) 1.51277e6 0.283184 0.141592 0.989925i \(-0.454778\pi\)
0.141592 + 0.989925i \(0.454778\pi\)
\(492\) 0 0
\(493\) −4.50342e6 −0.834498
\(494\) 1.21218e7 2.23485
\(495\) 0 0
\(496\) −1.75821e6 −0.320897
\(497\) −8.28502e6 −1.50454
\(498\) 0 0
\(499\) −1.93042e6 −0.347057 −0.173528 0.984829i \(-0.555517\pi\)
−0.173528 + 0.984829i \(0.555517\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.40301e6 −0.602703
\(503\) 6.73105e6 1.18621 0.593106 0.805124i \(-0.297901\pi\)
0.593106 + 0.805124i \(0.297901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.20269e6 −0.208822
\(507\) 0 0
\(508\) −4.79811e6 −0.824919
\(509\) 556650. 0.0952331 0.0476165 0.998866i \(-0.484837\pi\)
0.0476165 + 0.998866i \(0.484837\pi\)
\(510\) 0 0
\(511\) −2.59435e6 −0.439517
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) 3.30161e6 0.551211
\(515\) 0 0
\(516\) 0 0
\(517\) −2.51942e6 −0.414548
\(518\) 2.60450e6 0.426481
\(519\) 0 0
\(520\) 0 0
\(521\) −1.01110e7 −1.63192 −0.815962 0.578106i \(-0.803792\pi\)
−0.815962 + 0.578106i \(0.803792\pi\)
\(522\) 0 0
\(523\) 7.03719e6 1.12498 0.562491 0.826804i \(-0.309843\pi\)
0.562491 + 0.826804i \(0.309843\pi\)
\(524\) −125952. −0.0200390
\(525\) 0 0
\(526\) 5.45858e6 0.860232
\(527\) −5.23342e6 −0.820840
\(528\) 0 0
\(529\) −3.98399e6 −0.618983
\(530\) 0 0
\(531\) 0 0
\(532\) −5.17312e6 −0.792453
\(533\) −418068. −0.0637425
\(534\) 0 0
\(535\) 0 0
\(536\) −2.15821e6 −0.324475
\(537\) 0 0
\(538\) 453240. 0.0675107
\(539\) 553536. 0.0820680
\(540\) 0 0
\(541\) −4.23114e6 −0.621533 −0.310766 0.950486i \(-0.600586\pi\)
−0.310766 + 0.950486i \(0.600586\pi\)
\(542\) −3.39851e6 −0.496925
\(543\) 0 0
\(544\) 780288. 0.113047
\(545\) 0 0
\(546\) 0 0
\(547\) −4.44024e6 −0.634510 −0.317255 0.948340i \(-0.602761\pi\)
−0.317255 + 0.948340i \(0.602761\pi\)
\(548\) −2.62781e6 −0.373803
\(549\) 0 0
\(550\) 0 0
\(551\) 1.61934e7 2.27227
\(552\) 0 0
\(553\) 533360. 0.0741665
\(554\) −1.75441e6 −0.242860
\(555\) 0 0
\(556\) −4.51360e6 −0.619207
\(557\) −9.01448e6 −1.23113 −0.615563 0.788088i \(-0.711071\pi\)
−0.615563 + 0.788088i \(0.711071\pi\)
\(558\) 0 0
\(559\) −2.69200e6 −0.364373
\(560\) 0 0
\(561\) 0 0
\(562\) 5.82679e6 0.778196
\(563\) −9.81287e6 −1.30474 −0.652372 0.757899i \(-0.726226\pi\)
−0.652372 + 0.757899i \(0.726226\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 481576. 0.0631864
\(567\) 0 0
\(568\) −4.49357e6 −0.584414
\(569\) −1.33152e7 −1.72412 −0.862061 0.506804i \(-0.830827\pi\)
−0.862061 + 0.506804i \(0.830827\pi\)
\(570\) 0 0
\(571\) 9.95895e6 1.27827 0.639136 0.769094i \(-0.279292\pi\)
0.639136 + 0.769094i \(0.279292\pi\)
\(572\) 3.39763e6 0.434196
\(573\) 0 0
\(574\) 178416. 0.0226024
\(575\) 0 0
\(576\) 0 0
\(577\) −4.50372e6 −0.563160 −0.281580 0.959538i \(-0.590859\pi\)
−0.281580 + 0.959538i \(0.590859\pi\)
\(578\) −3.35685e6 −0.417939
\(579\) 0 0
\(580\) 0 0
\(581\) −1.28707e7 −1.58184
\(582\) 0 0
\(583\) 1.76141e6 0.214629
\(584\) −1.40710e6 −0.170724
\(585\) 0 0
\(586\) −1.05684e7 −1.27135
\(587\) 625842. 0.0749669 0.0374834 0.999297i \(-0.488066\pi\)
0.0374834 + 0.999297i \(0.488066\pi\)
\(588\) 0 0
\(589\) 1.88183e7 2.23508
\(590\) 0 0
\(591\) 0 0
\(592\) 1.41261e6 0.165660
\(593\) −2.50385e6 −0.292397 −0.146198 0.989255i \(-0.546704\pi\)
−0.146198 + 0.989255i \(0.546704\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.22320e6 0.717626
\(597\) 0 0
\(598\) −6.92798e6 −0.792235
\(599\) 756480. 0.0861451 0.0430725 0.999072i \(-0.486285\pi\)
0.0430725 + 0.999072i \(0.486285\pi\)
\(600\) 0 0
\(601\) −1.38565e7 −1.56483 −0.782413 0.622760i \(-0.786011\pi\)
−0.782413 + 0.622760i \(0.786011\pi\)
\(602\) 1.14885e6 0.129203
\(603\) 0 0
\(604\) −1.56717e6 −0.174793
\(605\) 0 0
\(606\) 0 0
\(607\) −1.13772e7 −1.25333 −0.626663 0.779291i \(-0.715580\pi\)
−0.626663 + 0.779291i \(0.715580\pi\)
\(608\) −2.80576e6 −0.307816
\(609\) 0 0
\(610\) 0 0
\(611\) −1.45129e7 −1.57272
\(612\) 0 0
\(613\) 7.00161e6 0.752570 0.376285 0.926504i \(-0.377201\pi\)
0.376285 + 0.926504i \(0.377201\pi\)
\(614\) 5.79023e6 0.619833
\(615\) 0 0
\(616\) −1.44998e6 −0.153961
\(617\) 7.90300e6 0.835755 0.417878 0.908503i \(-0.362774\pi\)
0.417878 + 0.908503i \(0.362774\pi\)
\(618\) 0 0
\(619\) 4.02362e6 0.422076 0.211038 0.977478i \(-0.432316\pi\)
0.211038 + 0.977478i \(0.432316\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.71227e6 0.384737
\(623\) −4.54182e6 −0.468824
\(624\) 0 0
\(625\) 0 0
\(626\) −9.18250e6 −0.936538
\(627\) 0 0
\(628\) 59488.0 0.00601908
\(629\) 4.20472e6 0.423750
\(630\) 0 0
\(631\) −1.00227e7 −1.00210 −0.501049 0.865419i \(-0.667052\pi\)
−0.501049 + 0.865419i \(0.667052\pi\)
\(632\) 289280. 0.0288088
\(633\) 0 0
\(634\) 1.09461e7 1.08152
\(635\) 0 0
\(636\) 0 0
\(637\) 3.18860e6 0.311352
\(638\) 4.53888e6 0.441466
\(639\) 0 0
\(640\) 0 0
\(641\) −6.37390e6 −0.612718 −0.306359 0.951916i \(-0.599111\pi\)
−0.306359 + 0.951916i \(0.599111\pi\)
\(642\) 0 0
\(643\) −5.00457e6 −0.477352 −0.238676 0.971099i \(-0.576713\pi\)
−0.238676 + 0.971099i \(0.576713\pi\)
\(644\) 2.95661e6 0.280918
\(645\) 0 0
\(646\) −8.35152e6 −0.787380
\(647\) −8.71928e6 −0.818879 −0.409440 0.912337i \(-0.634276\pi\)
−0.409440 + 0.912337i \(0.634276\pi\)
\(648\) 0 0
\(649\) −6.71616e6 −0.625906
\(650\) 0 0
\(651\) 0 0
\(652\) 691744. 0.0637274
\(653\) −1.58477e6 −0.145440 −0.0727201 0.997352i \(-0.523168\pi\)
−0.0727201 + 0.997352i \(0.523168\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 96768.0 0.00877955
\(657\) 0 0
\(658\) 6.19358e6 0.557670
\(659\) −1.26410e7 −1.13388 −0.566940 0.823759i \(-0.691873\pi\)
−0.566940 + 0.823759i \(0.691873\pi\)
\(660\) 0 0
\(661\) −3.61572e6 −0.321878 −0.160939 0.986964i \(-0.551452\pi\)
−0.160939 + 0.986964i \(0.551452\pi\)
\(662\) 1.52752e7 1.35469
\(663\) 0 0
\(664\) −6.98074e6 −0.614442
\(665\) 0 0
\(666\) 0 0
\(667\) −9.25506e6 −0.805498
\(668\) 2.98435e6 0.258767
\(669\) 0 0
\(670\) 0 0
\(671\) 1.88890e6 0.161958
\(672\) 0 0
\(673\) −1.11313e7 −0.947349 −0.473675 0.880700i \(-0.657073\pi\)
−0.473675 + 0.880700i \(0.657073\pi\)
\(674\) 8.84351e6 0.749851
\(675\) 0 0
\(676\) 1.36311e7 1.14727
\(677\) −235518. −0.0197493 −0.00987467 0.999951i \(-0.503143\pi\)
−0.00987467 + 0.999951i \(0.503143\pi\)
\(678\) 0 0
\(679\) 226324. 0.0188389
\(680\) 0 0
\(681\) 0 0
\(682\) 5.27462e6 0.434241
\(683\) 2.05830e7 1.68833 0.844164 0.536084i \(-0.180097\pi\)
0.844164 + 0.536084i \(0.180097\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.29368e6 −0.754011
\(687\) 0 0
\(688\) 623104. 0.0501868
\(689\) 1.01464e7 0.814265
\(690\) 0 0
\(691\) −9.54825e6 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(692\) −5.99126e6 −0.475612
\(693\) 0 0
\(694\) −9.30895e6 −0.733672
\(695\) 0 0
\(696\) 0 0
\(697\) 288036. 0.0224577
\(698\) −1.24516e6 −0.0967357
\(699\) 0 0
\(700\) 0 0
\(701\) −1.29304e6 −0.0993843 −0.0496921 0.998765i \(-0.515824\pi\)
−0.0496921 + 0.998765i \(0.515824\pi\)
\(702\) 0 0
\(703\) −1.51193e7 −1.15384
\(704\) −786432. −0.0598039
\(705\) 0 0
\(706\) −1.23463e7 −0.932234
\(707\) −9.15940e6 −0.689157
\(708\) 0 0
\(709\) −2.12720e7 −1.58926 −0.794628 0.607097i \(-0.792334\pi\)
−0.794628 + 0.607097i \(0.792334\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.46336e6 −0.182108
\(713\) −1.07553e7 −0.792316
\(714\) 0 0
\(715\) 0 0
\(716\) −4.35360e6 −0.317370
\(717\) 0 0
\(718\) 1.41230e7 1.02239
\(719\) −8.31732e6 −0.600014 −0.300007 0.953937i \(-0.596989\pi\)
−0.300007 + 0.953937i \(0.596989\pi\)
\(720\) 0 0
\(721\) 5.51225e6 0.394903
\(722\) 2.01260e7 1.43686
\(723\) 0 0
\(724\) −1.20669e6 −0.0855556
\(725\) 0 0
\(726\) 0 0
\(727\) 4.36740e6 0.306469 0.153235 0.988190i \(-0.451031\pi\)
0.153235 + 0.988190i \(0.451031\pi\)
\(728\) −8.35251e6 −0.584102
\(729\) 0 0
\(730\) 0 0
\(731\) 1.85471e6 0.128375
\(732\) 0 0
\(733\) 4.05645e6 0.278860 0.139430 0.990232i \(-0.455473\pi\)
0.139430 + 0.990232i \(0.455473\pi\)
\(734\) −143048. −0.00980035
\(735\) 0 0
\(736\) 1.60358e6 0.109118
\(737\) 6.47462e6 0.439082
\(738\) 0 0
\(739\) 768260. 0.0517484 0.0258742 0.999665i \(-0.491763\pi\)
0.0258742 + 0.999665i \(0.491763\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.33013e6 −0.288729
\(743\) 6.18781e6 0.411211 0.205605 0.978635i \(-0.434084\pi\)
0.205605 + 0.978635i \(0.434084\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.86102e6 0.451379
\(747\) 0 0
\(748\) −2.34086e6 −0.152976
\(749\) −122484. −0.00797765
\(750\) 0 0
\(751\) 1.81698e7 1.17557 0.587787 0.809016i \(-0.299999\pi\)
0.587787 + 0.809016i \(0.299999\pi\)
\(752\) 3.35923e6 0.216618
\(753\) 0 0
\(754\) 2.61458e7 1.67484
\(755\) 0 0
\(756\) 0 0
\(757\) −1.93494e7 −1.22724 −0.613618 0.789603i \(-0.710286\pi\)
−0.613618 + 0.789603i \(0.710286\pi\)
\(758\) −1.24070e7 −0.784318
\(759\) 0 0
\(760\) 0 0
\(761\) 3.01992e7 1.89031 0.945155 0.326621i \(-0.105910\pi\)
0.945155 + 0.326621i \(0.105910\pi\)
\(762\) 0 0
\(763\) 2.44177e7 1.51843
\(764\) 5.71181e6 0.354030
\(765\) 0 0
\(766\) 2.12779e7 1.31026
\(767\) −3.86879e7 −2.37458
\(768\) 0 0
\(769\) 2.15854e7 1.31627 0.658134 0.752901i \(-0.271346\pi\)
0.658134 + 0.752901i \(0.271346\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.01910e6 0.423876
\(773\) 3.90895e6 0.235294 0.117647 0.993055i \(-0.462465\pi\)
0.117647 + 0.993055i \(0.462465\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 122752. 0.00731769
\(777\) 0 0
\(778\) −4.64580e6 −0.275177
\(779\) −1.03572e6 −0.0611503
\(780\) 0 0
\(781\) 1.34807e7 0.790833
\(782\) 4.77317e6 0.279119
\(783\) 0 0
\(784\) −738048. −0.0428839
\(785\) 0 0
\(786\) 0 0
\(787\) 2.65082e7 1.52561 0.762806 0.646628i \(-0.223821\pi\)
0.762806 + 0.646628i \(0.223821\pi\)
\(788\) −2.50877e6 −0.143928
\(789\) 0 0
\(790\) 0 0
\(791\) 1.64475e7 0.934674
\(792\) 0 0
\(793\) 1.08808e7 0.614439
\(794\) −2.51425e6 −0.141533
\(795\) 0 0
\(796\) −2.60032e6 −0.145460
\(797\) 1.07940e7 0.601919 0.300960 0.953637i \(-0.402693\pi\)
0.300960 + 0.953637i \(0.402693\pi\)
\(798\) 0 0
\(799\) 9.99896e6 0.554100
\(800\) 0 0
\(801\) 0 0
\(802\) 1.08973e7 0.598249
\(803\) 4.22131e6 0.231025
\(804\) 0 0
\(805\) 0 0
\(806\) 3.03840e7 1.64743
\(807\) 0 0
\(808\) −4.96781e6 −0.267693
\(809\) 1.11446e7 0.598675 0.299338 0.954147i \(-0.403234\pi\)
0.299338 + 0.954147i \(0.403234\pi\)
\(810\) 0 0
\(811\) −1.14866e7 −0.613253 −0.306626 0.951830i \(-0.599200\pi\)
−0.306626 + 0.951830i \(0.599200\pi\)
\(812\) −1.11581e7 −0.593881
\(813\) 0 0
\(814\) −4.23782e6 −0.224172
\(815\) 0 0
\(816\) 0 0
\(817\) −6.66916e6 −0.349555
\(818\) 7.12076e6 0.372086
\(819\) 0 0
\(820\) 0 0
\(821\) −3.04347e7 −1.57584 −0.787918 0.615781i \(-0.788841\pi\)
−0.787918 + 0.615781i \(0.788841\pi\)
\(822\) 0 0
\(823\) −4.09773e6 −0.210884 −0.105442 0.994425i \(-0.533626\pi\)
−0.105442 + 0.994425i \(0.533626\pi\)
\(824\) 2.98970e6 0.153394
\(825\) 0 0
\(826\) 1.65106e7 0.841999
\(827\) −1.70652e7 −0.867654 −0.433827 0.900996i \(-0.642837\pi\)
−0.433827 + 0.900996i \(0.642837\pi\)
\(828\) 0 0
\(829\) −2.47617e7 −1.25139 −0.625697 0.780066i \(-0.715185\pi\)
−0.625697 + 0.780066i \(0.715185\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.53018e6 −0.226886
\(833\) −2.19685e6 −0.109695
\(834\) 0 0
\(835\) 0 0
\(836\) 8.41728e6 0.416539
\(837\) 0 0
\(838\) −2.60232e6 −0.128012
\(839\) −3.16529e7 −1.55242 −0.776208 0.630476i \(-0.782860\pi\)
−0.776208 + 0.630476i \(0.782860\pi\)
\(840\) 0 0
\(841\) 1.44170e7 0.702884
\(842\) −1.41624e7 −0.688425
\(843\) 0 0
\(844\) −2.90637e6 −0.140441
\(845\) 0 0
\(846\) 0 0
\(847\) −1.46541e7 −0.701859
\(848\) −2.34854e6 −0.112153
\(849\) 0 0
\(850\) 0 0
\(851\) 8.64119e6 0.409025
\(852\) 0 0
\(853\) −2.82671e7 −1.33017 −0.665087 0.746765i \(-0.731606\pi\)
−0.665087 + 0.746765i \(0.731606\pi\)
\(854\) −4.64354e6 −0.217873
\(855\) 0 0
\(856\) −66432.0 −0.00309880
\(857\) 2.60870e7 1.21331 0.606655 0.794966i \(-0.292511\pi\)
0.606655 + 0.794966i \(0.292511\pi\)
\(858\) 0 0
\(859\) −3.38111e7 −1.56342 −0.781710 0.623642i \(-0.785652\pi\)
−0.781710 + 0.623642i \(0.785652\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.19499e6 0.100615
\(863\) 2.22817e7 1.01841 0.509204 0.860646i \(-0.329940\pi\)
0.509204 + 0.860646i \(0.329940\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5.96966e6 0.270492
\(867\) 0 0
\(868\) −1.29668e7 −0.584162
\(869\) −867840. −0.0389843
\(870\) 0 0
\(871\) 3.72965e7 1.66580
\(872\) 1.32435e7 0.589810
\(873\) 0 0
\(874\) −1.71634e7 −0.760018
\(875\) 0 0
\(876\) 0 0
\(877\) 3.46748e7 1.52235 0.761177 0.648545i \(-0.224622\pi\)
0.761177 + 0.648545i \(0.224622\pi\)
\(878\) 1.94485e7 0.851431
\(879\) 0 0
\(880\) 0 0
\(881\) −1.42603e7 −0.618998 −0.309499 0.950900i \(-0.600161\pi\)
−0.309499 + 0.950900i \(0.600161\pi\)
\(882\) 0 0
\(883\) 3.75177e7 1.61933 0.809663 0.586895i \(-0.199650\pi\)
0.809663 + 0.586895i \(0.199650\pi\)
\(884\) −1.34844e7 −0.580363
\(885\) 0 0
\(886\) −7.44622e6 −0.318677
\(887\) 4.07657e7 1.73975 0.869873 0.493275i \(-0.164200\pi\)
0.869873 + 0.493275i \(0.164200\pi\)
\(888\) 0 0
\(889\) −3.53861e7 −1.50168
\(890\) 0 0
\(891\) 0 0
\(892\) 4.61238e6 0.194095
\(893\) −3.59543e7 −1.50877
\(894\) 0 0
\(895\) 0 0
\(896\) 1.93331e6 0.0804511
\(897\) 0 0
\(898\) −1.49488e7 −0.618606
\(899\) 4.05899e7 1.67501
\(900\) 0 0
\(901\) −6.99059e6 −0.286881
\(902\) −290304. −0.0118806
\(903\) 0 0
\(904\) 8.92070e6 0.363060
\(905\) 0 0
\(906\) 0 0
\(907\) 3.57116e7 1.44142 0.720712 0.693235i \(-0.243815\pi\)
0.720712 + 0.693235i \(0.243815\pi\)
\(908\) 1.80084e7 0.724869
\(909\) 0 0
\(910\) 0 0
\(911\) 2.11389e7 0.843893 0.421947 0.906621i \(-0.361347\pi\)
0.421947 + 0.906621i \(0.361347\pi\)
\(912\) 0 0
\(913\) 2.09422e7 0.831468
\(914\) 2.59310e7 1.02673
\(915\) 0 0
\(916\) −6.65296e6 −0.261985
\(917\) −928896. −0.0364791
\(918\) 0 0
\(919\) 1.85996e7 0.726465 0.363233 0.931698i \(-0.381673\pi\)
0.363233 + 0.931698i \(0.381673\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.03641e6 −0.233857
\(923\) 7.76545e7 3.00028
\(924\) 0 0
\(925\) 0 0
\(926\) −3.47360e7 −1.33123
\(927\) 0 0
\(928\) −6.05184e6 −0.230684
\(929\) −4.45110e7 −1.69211 −0.846055 0.533096i \(-0.821028\pi\)
−0.846055 + 0.533096i \(0.821028\pi\)
\(930\) 0 0
\(931\) 7.89942e6 0.298690
\(932\) 1.23294e7 0.464945
\(933\) 0 0
\(934\) 2.78565e7 1.04486
\(935\) 0 0
\(936\) 0 0
\(937\) 2.19419e7 0.816441 0.408221 0.912883i \(-0.366149\pi\)
0.408221 + 0.912883i \(0.366149\pi\)
\(938\) −1.59168e7 −0.590675
\(939\) 0 0
\(940\) 0 0
\(941\) 7.77722e6 0.286319 0.143160 0.989700i \(-0.454274\pi\)
0.143160 + 0.989700i \(0.454274\pi\)
\(942\) 0 0
\(943\) 591948. 0.0216773
\(944\) 8.95488e6 0.327062
\(945\) 0 0
\(946\) −1.86931e6 −0.0679132
\(947\) 3.17199e7 1.14936 0.574681 0.818378i \(-0.305126\pi\)
0.574681 + 0.818378i \(0.305126\pi\)
\(948\) 0 0
\(949\) 2.43165e7 0.876468
\(950\) 0 0
\(951\) 0 0
\(952\) 5.75462e6 0.205790
\(953\) −5.60285e6 −0.199838 −0.0999188 0.994996i \(-0.531858\pi\)
−0.0999188 + 0.994996i \(0.531858\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.52512e6 0.337074
\(957\) 0 0
\(958\) 2.20421e7 0.775959
\(959\) −1.93801e7 −0.680470
\(960\) 0 0
\(961\) 1.85403e7 0.647601
\(962\) −2.44116e7 −0.850470
\(963\) 0 0
\(964\) 4.38243e6 0.151888
\(965\) 0 0
\(966\) 0 0
\(967\) 2.03532e7 0.699949 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(968\) −7.94797e6 −0.272626
\(969\) 0 0
\(970\) 0 0
\(971\) 2.34306e7 0.797510 0.398755 0.917057i \(-0.369442\pi\)
0.398755 + 0.917057i \(0.369442\pi\)
\(972\) 0 0
\(973\) −3.32878e7 −1.12721
\(974\) −2.20723e7 −0.745505
\(975\) 0 0
\(976\) −2.51853e6 −0.0846296
\(977\) −4.30412e7 −1.44261 −0.721303 0.692619i \(-0.756457\pi\)
−0.721303 + 0.692619i \(0.756457\pi\)
\(978\) 0 0
\(979\) 7.39008e6 0.246429
\(980\) 0 0
\(981\) 0 0
\(982\) 6.05107e6 0.200241
\(983\) −4.75003e7 −1.56788 −0.783940 0.620837i \(-0.786793\pi\)
−0.783940 + 0.620837i \(0.786793\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.80137e7 −0.590079
\(987\) 0 0
\(988\) 4.84870e7 1.58028
\(989\) 3.81164e6 0.123914
\(990\) 0 0
\(991\) 2.09231e7 0.676770 0.338385 0.941008i \(-0.390119\pi\)
0.338385 + 0.941008i \(0.390119\pi\)
\(992\) −7.03283e6 −0.226909
\(993\) 0 0
\(994\) −3.31401e7 −1.06387
\(995\) 0 0
\(996\) 0 0
\(997\) −2.96332e7 −0.944148 −0.472074 0.881559i \(-0.656495\pi\)
−0.472074 + 0.881559i \(0.656495\pi\)
\(998\) −7.72168e6 −0.245406
\(999\) 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 450.6.a.u.1.1 1
3.2 odd 2 50.6.a.b.1.1 1
5.2 odd 4 450.6.c.f.199.2 2
5.3 odd 4 450.6.c.f.199.1 2
5.4 even 2 90.6.a.b.1.1 1
12.11 even 2 400.6.a.i.1.1 1
15.2 even 4 50.6.b.b.49.1 2
15.8 even 4 50.6.b.b.49.2 2
15.14 odd 2 10.6.a.c.1.1 1
20.19 odd 2 720.6.a.v.1.1 1
60.23 odd 4 400.6.c.i.49.2 2
60.47 odd 4 400.6.c.i.49.1 2
60.59 even 2 80.6.a.c.1.1 1
105.104 even 2 490.6.a.k.1.1 1
120.29 odd 2 320.6.a.f.1.1 1
120.59 even 2 320.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 15.14 odd 2
50.6.a.b.1.1 1 3.2 odd 2
50.6.b.b.49.1 2 15.2 even 4
50.6.b.b.49.2 2 15.8 even 4
80.6.a.c.1.1 1 60.59 even 2
90.6.a.b.1.1 1 5.4 even 2
320.6.a.f.1.1 1 120.29 odd 2
320.6.a.k.1.1 1 120.59 even 2
400.6.a.i.1.1 1 12.11 even 2
400.6.c.i.49.1 2 60.47 odd 4
400.6.c.i.49.2 2 60.23 odd 4
450.6.a.u.1.1 1 1.1 even 1 trivial
450.6.c.f.199.1 2 5.3 odd 4
450.6.c.f.199.2 2 5.2 odd 4
490.6.a.k.1.1 1 105.104 even 2
720.6.a.v.1.1 1 20.19 odd 2