Properties

Label 400.6.c.i.49.2
Level $400$
Weight $6$
Character 400.49
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.6.c.i.49.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.00000i q^{3} +118.000i q^{7} +207.000 q^{9} +O(q^{10})\) \(q+6.00000i q^{3} +118.000i q^{7} +207.000 q^{9} -192.000 q^{11} -1106.00i q^{13} +762.000i q^{17} -2740.00 q^{19} -708.000 q^{21} +1566.00i q^{23} +2700.00i q^{27} -5910.00 q^{29} +6868.00 q^{31} -1152.00i q^{33} -5518.00i q^{37} +6636.00 q^{39} -378.000 q^{41} -2434.00i q^{43} -13122.0i q^{47} +2883.00 q^{49} -4572.00 q^{51} +9174.00i q^{53} -16440.0i q^{57} -34980.0 q^{59} -9838.00 q^{61} +24426.0i q^{63} -33722.0i q^{67} -9396.00 q^{69} -70212.0 q^{71} -21986.0i q^{73} -22656.0i q^{77} +4520.00 q^{79} +34101.0 q^{81} -109074. i q^{83} -35460.0i q^{87} -38490.0 q^{89} +130508. q^{91} +41208.0i q^{93} -1918.00i q^{97} -39744.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 414q^{9} + O(q^{10}) \) \( 2q + 414q^{9} - 384q^{11} - 5480q^{19} - 1416q^{21} - 11820q^{29} + 13736q^{31} + 13272q^{39} - 756q^{41} + 5766q^{49} - 9144q^{51} - 69960q^{59} - 19676q^{61} - 18792q^{69} - 140424q^{71} + 9040q^{79} + 68202q^{81} - 76980q^{89} + 261016q^{91} - 79488q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 6.00000i 0.384900i 0.981307 + 0.192450i \(0.0616434\pi\)
−0.981307 + 0.192450i \(0.938357\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 118.000i 0.910200i 0.890440 + 0.455100i \(0.150397\pi\)
−0.890440 + 0.455100i \(0.849603\pi\)
\(8\) 0 0
\(9\) 207.000 0.851852
\(10\) 0 0
\(11\) −192.000 −0.478431 −0.239216 0.970966i \(-0.576890\pi\)
−0.239216 + 0.970966i \(0.576890\pi\)
\(12\) 0 0
\(13\) − 1106.00i − 1.81508i −0.419961 0.907542i \(-0.637956\pi\)
0.419961 0.907542i \(-0.362044\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 762.000i 0.639488i 0.947504 + 0.319744i \(0.103597\pi\)
−0.947504 + 0.319744i \(0.896403\pi\)
\(18\) 0 0
\(19\) −2740.00 −1.74127 −0.870636 0.491928i \(-0.836292\pi\)
−0.870636 + 0.491928i \(0.836292\pi\)
\(20\) 0 0
\(21\) −708.000 −0.350336
\(22\) 0 0
\(23\) 1566.00i 0.617266i 0.951181 + 0.308633i \(0.0998714\pi\)
−0.951181 + 0.308633i \(0.900129\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2700.00i 0.712778i
\(28\) 0 0
\(29\) −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 \(-0.278190\pi\)
0.641795 + 0.766877i \(0.278190\pi\)
\(32\) 0 0
\(33\) − 1152.00i − 0.184148i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5518.00i − 0.662640i −0.943519 0.331320i \(-0.892506\pi\)
0.943519 0.331320i \(-0.107494\pi\)
\(38\) 0 0
\(39\) 6636.00 0.698626
\(40\) 0 0
\(41\) −378.000 −0.0351182 −0.0175591 0.999846i \(-0.505590\pi\)
−0.0175591 + 0.999846i \(0.505590\pi\)
\(42\) 0 0
\(43\) − 2434.00i − 0.200747i −0.994950 0.100374i \(-0.967996\pi\)
0.994950 0.100374i \(-0.0320038\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13122.0i − 0.866474i −0.901280 0.433237i \(-0.857371\pi\)
0.901280 0.433237i \(-0.142629\pi\)
\(48\) 0 0
\(49\) 2883.00 0.171536
\(50\) 0 0
\(51\) −4572.00 −0.246139
\(52\) 0 0
\(53\) 9174.00i 0.448610i 0.974519 + 0.224305i \(0.0720112\pi\)
−0.974519 + 0.224305i \(0.927989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 16440.0i − 0.670216i
\(58\) 0 0
\(59\) −34980.0 −1.30825 −0.654124 0.756388i \(-0.726962\pi\)
−0.654124 + 0.756388i \(0.726962\pi\)
\(60\) 0 0
\(61\) −9838.00 −0.338518 −0.169259 0.985572i \(-0.554137\pi\)
−0.169259 + 0.985572i \(0.554137\pi\)
\(62\) 0 0
\(63\) 24426.0i 0.775356i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 33722.0i − 0.917754i −0.888500 0.458877i \(-0.848252\pi\)
0.888500 0.458877i \(-0.151748\pi\)
\(68\) 0 0
\(69\) −9396.00 −0.237586
\(70\) 0 0
\(71\) −70212.0 −1.65297 −0.826486 0.562957i \(-0.809664\pi\)
−0.826486 + 0.562957i \(0.809664\pi\)
\(72\) 0 0
\(73\) − 21986.0i − 0.482880i −0.970416 0.241440i \(-0.922380\pi\)
0.970416 0.241440i \(-0.0776197\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 22656.0i − 0.435468i
\(78\) 0 0
\(79\) 4520.00 0.0814837 0.0407418 0.999170i \(-0.487028\pi\)
0.0407418 + 0.999170i \(0.487028\pi\)
\(80\) 0 0
\(81\) 34101.0 0.577503
\(82\) 0 0
\(83\) − 109074.i − 1.73790i −0.494896 0.868952i \(-0.664794\pi\)
0.494896 0.868952i \(-0.335206\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 35460.0i − 0.502274i
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) 41208.0i 0.494054i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1918.00i − 0.0206976i −0.999946 0.0103488i \(-0.996706\pi\)
0.999946 0.0103488i \(-0.00329418\pi\)
\(98\) 0 0
\(99\) −39744.0 −0.407553
\(100\) 0 0
\(101\) 77622.0 0.757149 0.378575 0.925571i \(-0.376414\pi\)
0.378575 + 0.925571i \(0.376414\pi\)
\(102\) 0 0
\(103\) − 46714.0i − 0.433864i −0.976187 0.216932i \(-0.930395\pi\)
0.976187 0.216932i \(-0.0696051\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1038.00i 0.00876472i 0.999990 + 0.00438236i \(0.00139495\pi\)
−0.999990 + 0.00438236i \(0.998605\pi\)
\(108\) 0 0
\(109\) −206930. −1.66823 −0.834117 0.551587i \(-0.814023\pi\)
−0.834117 + 0.551587i \(0.814023\pi\)
\(110\) 0 0
\(111\) 33108.0 0.255050
\(112\) 0 0
\(113\) − 139386.i − 1.02689i −0.858123 0.513444i \(-0.828369\pi\)
0.858123 0.513444i \(-0.171631\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 228942.i − 1.54618i
\(118\) 0 0
\(119\) −89916.0 −0.582062
\(120\) 0 0
\(121\) −124187. −0.771104
\(122\) 0 0
\(123\) − 2268.00i − 0.0135170i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 299882.i − 1.64984i −0.565252 0.824919i \(-0.691221\pi\)
0.565252 0.824919i \(-0.308779\pi\)
\(128\) 0 0
\(129\) 14604.0 0.0772676
\(130\) 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.i − 1.58491i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 164238.i − 0.747605i −0.927508 0.373803i \(-0.878054\pi\)
0.927508 0.373803i \(-0.121946\pi\)
\(138\) 0 0
\(139\) −282100. −1.23841 −0.619207 0.785228i \(-0.712546\pi\)
−0.619207 + 0.785228i \(0.712546\pi\)
\(140\) 0 0
\(141\) 78732.0 0.333506
\(142\) 0 0
\(143\) 212352.i 0.868393i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 17298.0i 0.0660241i
\(148\) 0 0
\(149\) 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 \(-0.444074\pi\)
0.174793 + 0.984605i \(0.444074\pi\)
\(152\) 0 0
\(153\) 157734.i 0.544749i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 3718.00i − 0.0120382i −0.999982 0.00601908i \(-0.998084\pi\)
0.999982 0.00601908i \(-0.00191594\pi\)
\(158\) 0 0
\(159\) −55044.0 −0.172670
\(160\) 0 0
\(161\) −184788. −0.561835
\(162\) 0 0
\(163\) − 43234.0i − 0.127455i −0.997967 0.0637274i \(-0.979701\pi\)
0.997967 0.0637274i \(-0.0202988\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 186522.i − 0.517534i −0.965940 0.258767i \(-0.916684\pi\)
0.965940 0.258767i \(-0.0833162\pi\)
\(168\) 0 0
\(169\) −851943. −2.29453
\(170\) 0 0
\(171\) −567180. −1.48331
\(172\) 0 0
\(173\) 374454.i 0.951225i 0.879655 + 0.475612i \(0.157774\pi\)
−0.879655 + 0.475612i \(0.842226\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 209880.i − 0.503545i
\(178\) 0 0
\(179\) 272100. 0.634740 0.317370 0.948302i \(-0.397200\pi\)
0.317370 + 0.948302i \(0.397200\pi\)
\(180\) 0 0
\(181\) −75418.0 −0.171111 −0.0855556 0.996333i \(-0.527267\pi\)
−0.0855556 + 0.996333i \(0.527267\pi\)
\(182\) 0 0
\(183\) − 59028.0i − 0.130296i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 146304.i − 0.305951i
\(188\) 0 0
\(189\) −318600. −0.648771
\(190\) 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.i 0.847751i 0.905720 + 0.423876i \(0.139331\pi\)
−0.905720 + 0.423876i \(0.860669\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 156798.i − 0.287856i −0.989588 0.143928i \(-0.954027\pi\)
0.989588 0.143928i \(-0.0459733\pi\)
\(198\) 0 0
\(199\) −162520. −0.290920 −0.145460 0.989364i \(-0.546466\pi\)
−0.145460 + 0.989364i \(0.546466\pi\)
\(200\) 0 0
\(201\) 202332. 0.353244
\(202\) 0 0
\(203\) − 697380.i − 1.18776i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 324162.i 0.525819i
\(208\) 0 0
\(209\) 526080. 0.833079
\(210\) 0 0
\(211\) 181648. 0.280882 0.140441 0.990089i \(-0.455148\pi\)
0.140441 + 0.990089i \(0.455148\pi\)
\(212\) 0 0
\(213\) − 421272.i − 0.636229i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 810424.i 1.16832i
\(218\) 0 0
\(219\) 131916. 0.185861
\(220\) 0 0
\(221\) 842772. 1.16073
\(222\) 0 0
\(223\) − 288274.i − 0.388189i −0.980983 0.194095i \(-0.937823\pi\)
0.980983 0.194095i \(-0.0621769\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.12552e6i − 1.44974i −0.688887 0.724869i \(-0.741900\pi\)
0.688887 0.724869i \(-0.258100\pi\)
\(228\) 0 0
\(229\) 415810. 0.523970 0.261985 0.965072i \(-0.415623\pi\)
0.261985 + 0.965072i \(0.415623\pi\)
\(230\) 0 0
\(231\) 135936. 0.167612
\(232\) 0 0
\(233\) − 770586.i − 0.929889i −0.885340 0.464945i \(-0.846074\pi\)
0.885340 0.464945i \(-0.153926\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 27120.0i 0.0313631i
\(238\) 0 0
\(239\) −595320. −0.674149 −0.337074 0.941478i \(-0.609437\pi\)
−0.337074 + 0.941478i \(0.609437\pi\)
\(240\) 0 0
\(241\) 273902. 0.303775 0.151888 0.988398i \(-0.451465\pi\)
0.151888 + 0.988398i \(0.451465\pi\)
\(242\) 0 0
\(243\) 860706.i 0.935059i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.03044e6i 3.16055i
\(248\) 0 0
\(249\) 654444. 0.668920
\(250\) 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.i − 0.295319i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 825402.i 0.779530i 0.920914 + 0.389765i \(0.127444\pi\)
−0.920914 + 0.389765i \(0.872556\pi\)
\(258\) 0 0
\(259\) 651124. 0.603135
\(260\) 0 0
\(261\) −1.22337e6 −1.11162
\(262\) 0 0
\(263\) 1.36465e6i 1.21655i 0.793726 + 0.608276i \(0.208139\pi\)
−0.793726 + 0.608276i \(0.791861\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 230940.i − 0.198254i
\(268\) 0 0
\(269\) 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 \(-0.385713\pi\)
0.351379 + 0.936233i \(0.385713\pi\)
\(272\) 0 0
\(273\) 783048.i 0.635890i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 438602.i 0.343456i 0.985144 + 0.171728i \(0.0549350\pi\)
−0.985144 + 0.171728i \(0.945065\pi\)
\(278\) 0 0
\(279\) 1.42168e6 1.09343
\(280\) 0 0
\(281\) −1.45670e6 −1.10053 −0.550267 0.834989i \(-0.685474\pi\)
−0.550267 + 0.834989i \(0.685474\pi\)
\(282\) 0 0
\(283\) − 120394.i − 0.0893591i −0.999001 0.0446795i \(-0.985773\pi\)
0.999001 0.0446795i \(-0.0142267\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 44604.0i − 0.0319646i
\(288\) 0 0
\(289\) 839213. 0.591055
\(290\) 0 0
\(291\) 11508.0 0.00796650
\(292\) 0 0
\(293\) 2.64209e6i 1.79796i 0.437993 + 0.898978i \(0.355689\pi\)
−0.437993 + 0.898978i \(0.644311\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 518400.i − 0.341015i
\(298\) 0 0
\(299\) 1.73200e6 1.12039
\(300\) 0 0
\(301\) 287212. 0.182720
\(302\) 0 0
\(303\) 465732.i 0.291427i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.44756e6i 0.876577i 0.898834 + 0.438288i \(0.144415\pi\)
−0.898834 + 0.438288i \(0.855585\pi\)
\(308\) 0 0
\(309\) 280284. 0.166994
\(310\) 0 0
\(311\) 928068. 0.544100 0.272050 0.962283i \(-0.412298\pi\)
0.272050 + 0.962283i \(0.412298\pi\)
\(312\) 0 0
\(313\) − 2.29563e6i − 1.32446i −0.749299 0.662232i \(-0.769609\pi\)
0.749299 0.662232i \(-0.230391\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.73652e6i 1.52950i 0.644324 + 0.764752i \(0.277139\pi\)
−0.644324 + 0.764752i \(0.722861\pi\)
\(318\) 0 0
\(319\) 1.13472e6 0.624327
\(320\) 0 0
\(321\) −6228.00 −0.00337354
\(322\) 0 0
\(323\) − 2.08788e6i − 1.11352i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 1.24158e6i − 0.642104i
\(328\) 0 0
\(329\) 1.54840e6 0.788665
\(330\) 0 0
\(331\) −3.81879e6 −1.91583 −0.957913 0.287059i \(-0.907322\pi\)
−0.957913 + 0.287059i \(0.907322\pi\)
\(332\) 0 0
\(333\) − 1.14223e6i − 0.564471i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.21088e6i − 1.06045i −0.847857 0.530225i \(-0.822108\pi\)
0.847857 0.530225i \(-0.177892\pi\)
\(338\) 0 0
\(339\) 836316. 0.395249
\(340\) 0 0
\(341\) −1.31866e6 −0.614109
\(342\) 0 0
\(343\) 2.32342e6i 1.06633i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.32724e6i 1.03757i 0.854905 + 0.518785i \(0.173615\pi\)
−0.854905 + 0.518785i \(0.826385\pi\)
\(348\) 0 0
\(349\) 311290. 0.136805 0.0684024 0.997658i \(-0.478210\pi\)
0.0684024 + 0.997658i \(0.478210\pi\)
\(350\) 0 0
\(351\) 2.98620e6 1.29375
\(352\) 0 0
\(353\) 3.08657e6i 1.31838i 0.751977 + 0.659189i \(0.229100\pi\)
−0.751977 + 0.659189i \(0.770900\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 539496.i − 0.224036i
\(358\) 0 0
\(359\) −3.53076e6 −1.44588 −0.722940 0.690911i \(-0.757210\pi\)
−0.722940 + 0.690911i \(0.757210\pi\)
\(360\) 0 0
\(361\) 5.03150e6 2.03203
\(362\) 0 0
\(363\) − 745122.i − 0.296798i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 35762.0i − 0.0138598i −0.999976 0.00692989i \(-0.997794\pi\)
0.999976 0.00692989i \(-0.00220587\pi\)
\(368\) 0 0
\(369\) −78246.0 −0.0299155
\(370\) 0 0
\(371\) −1.08253e6 −0.408325
\(372\) 0 0
\(373\) 1.71525e6i 0.638346i 0.947696 + 0.319173i \(0.103405\pi\)
−0.947696 + 0.319173i \(0.896595\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.53646e6i 2.36859i
\(378\) 0 0
\(379\) −3.10174e6 −1.10919 −0.554597 0.832119i \(-0.687127\pi\)
−0.554597 + 0.832119i \(0.687127\pi\)
\(380\) 0 0
\(381\) 1.79929e6 0.635023
\(382\) 0 0
\(383\) 5.31949e6i 1.85299i 0.376309 + 0.926494i \(0.377193\pi\)
−0.376309 + 0.926494i \(0.622807\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 503838.i − 0.171007i
\(388\) 0 0
\(389\) −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
\(392\) 0 0
\(393\) − 47232.0i − 0.0154261i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 628562.i 0.200157i 0.994980 + 0.100079i \(0.0319095\pi\)
−0.994980 + 0.100079i \(0.968091\pi\)
\(398\) 0 0
\(399\) 1.93992e6 0.610031
\(400\) 0 0
\(401\) −2.72432e6 −0.846052 −0.423026 0.906118i \(-0.639032\pi\)
−0.423026 + 0.906118i \(0.639032\pi\)
\(402\) 0 0
\(403\) − 7.59601e6i − 2.32982i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.05946e6i 0.317027i
\(408\) 0 0
\(409\) −1.78019e6 −0.526209 −0.263104 0.964767i \(-0.584746\pi\)
−0.263104 + 0.964767i \(0.584746\pi\)
\(410\) 0 0
\(411\) 985428. 0.287753
\(412\) 0 0
\(413\) − 4.12764e6i − 1.19077i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 1.69260e6i − 0.476666i
\(418\) 0 0
\(419\) 650580. 0.181036 0.0905181 0.995895i \(-0.471148\pi\)
0.0905181 + 0.995895i \(0.471148\pi\)
\(420\) 0 0
\(421\) −3.54060e6 −0.973579 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(422\) 0 0
\(423\) − 2.71625e6i − 0.738107i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.16088e6i − 0.308119i
\(428\) 0 0
\(429\) −1.27411e6 −0.334245
\(430\) 0 0
\(431\) 548748. 0.142292 0.0711459 0.997466i \(-0.477334\pi\)
0.0711459 + 0.997466i \(0.477334\pi\)
\(432\) 0 0
\(433\) 1.49241e6i 0.382534i 0.981538 + 0.191267i \(0.0612596\pi\)
−0.981538 + 0.191267i \(0.938740\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.29084e6i − 1.07483i
\(438\) 0 0
\(439\) 4.86212e6 1.20411 0.602053 0.798456i \(-0.294350\pi\)
0.602053 + 0.798456i \(0.294350\pi\)
\(440\) 0 0
\(441\) 596781. 0.146123
\(442\) 0 0
\(443\) − 1.86155e6i − 0.450678i −0.974280 0.225339i \(-0.927651\pi\)
0.974280 0.225339i \(-0.0723490\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.33370e6i 0.552429i
\(448\) 0 0
\(449\) −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
\(452\) 0 0
\(453\) 587688.i 0.134555i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.48276e6i − 1.45201i −0.687690 0.726005i \(-0.741375\pi\)
0.687690 0.726005i \(-0.258625\pi\)
\(458\) 0 0
\(459\) −2.05740e6 −0.455813
\(460\) 0 0
\(461\) 1.50910e6 0.330724 0.165362 0.986233i \(-0.447121\pi\)
0.165362 + 0.986233i \(0.447121\pi\)
\(462\) 0 0
\(463\) 8.68401e6i 1.88264i 0.337513 + 0.941321i \(0.390414\pi\)
−0.337513 + 0.941321i \(0.609586\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.96412e6i − 1.47766i −0.673893 0.738829i \(-0.735379\pi\)
0.673893 0.738829i \(-0.264621\pi\)
\(468\) 0 0
\(469\) 3.97920e6 0.835340
\(470\) 0 0
\(471\) 22308.0 0.00463349
\(472\) 0 0
\(473\) 467328.i 0.0960437i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.89902e6i 0.382149i
\(478\) 0 0
\(479\) −5.51052e6 −1.09737 −0.548686 0.836029i \(-0.684872\pi\)
−0.548686 + 0.836029i \(0.684872\pi\)
\(480\) 0 0
\(481\) −6.10291e6 −1.20275
\(482\) 0 0
\(483\) − 1.10873e6i − 0.216251i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.51808e6i − 1.05430i −0.849771 0.527152i \(-0.823260\pi\)
0.849771 0.527152i \(-0.176740\pi\)
\(488\) 0 0
\(489\) 259404. 0.0490574
\(490\) 0 0
\(491\) 1.51277e6 0.283184 0.141592 0.989925i \(-0.454778\pi\)
0.141592 + 0.989925i \(0.454778\pi\)
\(492\) 0 0
\(493\) − 4.50342e6i − 0.834498i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.28502e6i − 1.50454i
\(498\) 0 0
\(499\) −1.93042e6 −0.347057 −0.173528 0.984829i \(-0.555517\pi\)
−0.173528 + 0.984829i \(0.555517\pi\)
\(500\) 0 0
\(501\) 1.11913e6 0.199199
\(502\) 0 0
\(503\) 6.73105e6i 1.18621i 0.805124 + 0.593106i \(0.202099\pi\)
−0.805124 + 0.593106i \(0.797901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 5.11166e6i − 0.883165i
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) − 7.39800e6i − 1.24114i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.51942e6i 0.414548i
\(518\) 0 0
\(519\) −2.24672e6 −0.366127
\(520\) 0 0
\(521\) 1.01110e7 1.63192 0.815962 0.578106i \(-0.196208\pi\)
0.815962 + 0.578106i \(0.196208\pi\)
\(522\) 0 0
\(523\) − 7.03719e6i − 1.12498i −0.826804 0.562491i \(-0.809843\pi\)
0.826804 0.562491i \(-0.190157\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.23342e6i 0.820840i
\(528\) 0 0
\(529\) 3.98399e6 0.618983
\(530\) 0 0
\(531\) −7.24086e6 −1.11443
\(532\) 0 0
\(533\) 418068.i 0.0637425i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.63260e6i 0.244312i
\(538\) 0 0
\(539\) −553536. −0.0820680
\(540\) 0 0
\(541\) −4.23114e6 −0.621533 −0.310766 0.950486i \(-0.600586\pi\)
−0.310766 + 0.950486i \(0.600586\pi\)
\(542\) 0 0
\(543\) − 452508.i − 0.0658608i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.44024e6i − 0.634510i −0.948340 0.317255i \(-0.897239\pi\)
0.948340 0.317255i \(-0.102761\pi\)
\(548\) 0 0
\(549\) −2.03647e6 −0.288367
\(550\) 0 0
\(551\) 1.61934e7 2.27227
\(552\) 0 0
\(553\) 533360.i 0.0741665i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 9.01448e6i − 1.23113i −0.788088 0.615563i \(-0.788929\pi\)
0.788088 0.615563i \(-0.211071\pi\)
\(558\) 0 0
\(559\) −2.69200e6 −0.364373
\(560\) 0 0
\(561\) 877824. 0.117761
\(562\) 0 0
\(563\) − 9.81287e6i − 1.30474i −0.757899 0.652372i \(-0.773774\pi\)
0.757899 0.652372i \(-0.226226\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.02392e6i 0.525644i
\(568\) 0 0
\(569\) −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 \(-0.720708\pi\)
−0.639136 + 0.769094i \(0.720708\pi\)
\(572\) 0 0
\(573\) 2.14193e6i 0.272533i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.50372e6i 0.563160i 0.959538 + 0.281580i \(0.0908585\pi\)
−0.959538 + 0.281580i \(0.909141\pi\)
\(578\) 0 0
\(579\) −2.63216e6 −0.326300
\(580\) 0 0
\(581\) 1.28707e7 1.58184
\(582\) 0 0
\(583\) − 1.76141e6i − 0.214629i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 625842.i − 0.0749669i −0.999297 0.0374834i \(-0.988066\pi\)
0.999297 0.0374834i \(-0.0119341\pi\)
\(588\) 0 0
\(589\) −1.88183e7 −2.23508
\(590\) 0 0
\(591\) 940788. 0.110796
\(592\) 0 0
\(593\) 2.50385e6i 0.292397i 0.989255 + 0.146198i \(0.0467038\pi\)
−0.989255 + 0.146198i \(0.953296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 975120.i − 0.111975i
\(598\) 0 0
\(599\) −756480. −0.0861451 −0.0430725 0.999072i \(-0.513715\pi\)
−0.0430725 + 0.999072i \(0.513715\pi\)
\(600\) 0 0
\(601\) −1.38565e7 −1.56483 −0.782413 0.622760i \(-0.786011\pi\)
−0.782413 + 0.622760i \(0.786011\pi\)
\(602\) 0 0
\(603\) − 6.98045e6i − 0.781791i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 1.13772e7i − 1.25333i −0.779291 0.626663i \(-0.784420\pi\)
0.779291 0.626663i \(-0.215580\pi\)
\(608\) 0 0
\(609\) 4.18428e6 0.457170
\(610\) 0 0
\(611\) −1.45129e7 −1.57272
\(612\) 0 0
\(613\) 7.00161e6i 0.752570i 0.926504 + 0.376285i \(0.122799\pi\)
−0.926504 + 0.376285i \(0.877201\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.90300e6i 0.835755i 0.908503 + 0.417878i \(0.137226\pi\)
−0.908503 + 0.417878i \(0.862774\pi\)
\(618\) 0 0
\(619\) 4.02362e6 0.422076 0.211038 0.977478i \(-0.432316\pi\)
0.211038 + 0.977478i \(0.432316\pi\)
\(620\) 0 0
\(621\) −4.22820e6 −0.439974
\(622\) 0 0
\(623\) − 4.54182e6i − 0.468824i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.15648e6i 0.320652i
\(628\) 0 0
\(629\) 4.20472e6 0.423750
\(630\) 0 0
\(631\) 1.00227e7 1.00210 0.501049 0.865419i \(-0.332948\pi\)
0.501049 + 0.865419i \(0.332948\pi\)
\(632\) 0 0
\(633\) 1.08989e6i 0.108112i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.18860e6i − 0.311352i
\(638\) 0 0
\(639\) −1.45339e7 −1.40809
\(640\) 0 0
\(641\) 6.37390e6 0.612718 0.306359 0.951916i \(-0.400889\pi\)
0.306359 + 0.951916i \(0.400889\pi\)
\(642\) 0 0
\(643\) 5.00457e6i 0.477352i 0.971099 + 0.238676i \(0.0767134\pi\)
−0.971099 + 0.238676i \(0.923287\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.71928e6i 0.818879i 0.912337 + 0.409440i \(0.134276\pi\)
−0.912337 + 0.409440i \(0.865724\pi\)
\(648\) 0 0
\(649\) 6.71616e6 0.625906
\(650\) 0 0
\(651\) −4.86254e6 −0.449688
\(652\) 0 0
\(653\) 1.58477e6i 0.145440i 0.997352 + 0.0727201i \(0.0231680\pi\)
−0.997352 + 0.0727201i \(0.976832\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 4.55110e6i − 0.411342i
\(658\) 0 0
\(659\) 1.26410e7 1.13388 0.566940 0.823759i \(-0.308127\pi\)
0.566940 + 0.823759i \(0.308127\pi\)
\(660\) 0 0
\(661\) −3.61572e6 −0.321878 −0.160939 0.986964i \(-0.551452\pi\)
−0.160939 + 0.986964i \(0.551452\pi\)
\(662\) 0 0
\(663\) 5.05663e6i 0.446763i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.25506e6i − 0.805498i
\(668\) 0 0
\(669\) 1.72964e6 0.149414
\(670\) 0 0
\(671\) 1.88890e6 0.161958
\(672\) 0 0
\(673\) − 1.11313e7i − 0.947349i −0.880700 0.473675i \(-0.842927\pi\)
0.880700 0.473675i \(-0.157073\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 235518.i − 0.0197493i −0.999951 0.00987467i \(-0.996857\pi\)
0.999951 0.00987467i \(-0.00314326\pi\)
\(678\) 0 0
\(679\) 226324. 0.0188389
\(680\) 0 0
\(681\) 6.75313e6 0.558004
\(682\) 0 0
\(683\) 2.05830e7i 1.68833i 0.536084 + 0.844164i \(0.319903\pi\)
−0.536084 + 0.844164i \(0.680097\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.49486e6i 0.201676i
\(688\) 0 0
\(689\) 1.01464e7 0.814265
\(690\) 0 0
\(691\) 9.54825e6 0.760727 0.380363 0.924837i \(-0.375799\pi\)
0.380363 + 0.924837i \(0.375799\pi\)
\(692\) 0 0
\(693\) − 4.68979e6i − 0.370954i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 288036.i − 0.0224577i
\(698\) 0 0
\(699\) 4.62352e6 0.357915
\(700\) 0 0
\(701\) 1.29304e6 0.0993843 0.0496921 0.998765i \(-0.484176\pi\)
0.0496921 + 0.998765i \(0.484176\pi\)
\(702\) 0 0
\(703\) 1.51193e7i 1.15384i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.15940e6i 0.689157i
\(708\) 0 0
\(709\) 2.12720e7 1.58926 0.794628 0.607097i \(-0.207666\pi\)
0.794628 + 0.607097i \(0.207666\pi\)
\(710\) 0 0
\(711\) 935640. 0.0694120
\(712\) 0 0
\(713\) 1.07553e7i 0.792316i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.57192e6i − 0.259480i
\(718\) 0 0
\(719\) 8.31732e6 0.600014 0.300007 0.953937i \(-0.403011\pi\)
0.300007 + 0.953937i \(0.403011\pi\)
\(720\) 0 0
\(721\) 5.51225e6 0.394903
\(722\) 0 0
\(723\) 1.64341e6i 0.116923i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.36740e6i 0.306469i 0.988190 + 0.153235i \(0.0489690\pi\)
−0.988190 + 0.153235i \(0.951031\pi\)
\(728\) 0 0
\(729\) 3.12231e6 0.217599
\(730\) 0 0
\(731\) 1.85471e6 0.128375
\(732\) 0 0
\(733\) 4.05645e6i 0.278860i 0.990232 + 0.139430i \(0.0445271\pi\)
−0.990232 + 0.139430i \(0.955473\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.47462e6i 0.439082i
\(738\) 0 0
\(739\) 768260. 0.0517484 0.0258742 0.999665i \(-0.491763\pi\)
0.0258742 + 0.999665i \(0.491763\pi\)
\(740\) 0 0
\(741\) −1.81826e7 −1.21650
\(742\) 0 0
\(743\) 6.18781e6i 0.411211i 0.978635 + 0.205605i \(0.0659164\pi\)
−0.978635 + 0.205605i \(0.934084\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2.25783e7i − 1.48044i
\(748\) 0 0
\(749\) −122484. −0.00797765
\(750\) 0 0
\(751\) −1.81698e7 −1.17557 −0.587787 0.809016i \(-0.700001\pi\)
−0.587787 + 0.809016i \(0.700001\pi\)
\(752\) 0 0
\(753\) − 5.10451e6i − 0.328070i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.93494e7i 1.22724i 0.789603 + 0.613618i \(0.210286\pi\)
−0.789603 + 0.613618i \(0.789714\pi\)
\(758\) 0 0
\(759\) 1.80403e6 0.113668
\(760\) 0 0
\(761\) −3.01992e7 −1.89031 −0.945155 0.326621i \(-0.894090\pi\)
−0.945155 + 0.326621i \(0.894090\pi\)
\(762\) 0 0
\(763\) − 2.44177e7i − 1.51843i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.86879e7i 2.37458i
\(768\) 0 0
\(769\) −2.15854e7 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(770\) 0 0
\(771\) −4.95241e6 −0.300041
\(772\) 0 0
\(773\) − 3.90895e6i − 0.235294i −0.993055 0.117647i \(-0.962465\pi\)
0.993055 0.117647i \(-0.0375351\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.90674e6i 0.232147i
\(778\) 0 0
\(779\) 1.03572e6 0.0611503
\(780\) 0 0
\(781\) 1.34807e7 0.790833
\(782\) 0 0
\(783\) − 1.59570e7i − 0.930137i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.65082e7i 1.52561i 0.646628 + 0.762806i \(0.276179\pi\)
−0.646628 + 0.762806i \(0.723821\pi\)
\(788\) 0 0
\(789\) −8.18788e6 −0.468251
\(790\) 0 0
\(791\) 1.64475e7 0.934674
\(792\) 0 0
\(793\) 1.08808e7i 0.614439i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.07940e7i 0.601919i 0.953637 + 0.300960i \(0.0973070\pi\)
−0.953637 + 0.300960i \(0.902693\pi\)
\(798\) 0 0
\(799\) 9.99896e6 0.554100
\(800\) 0 0
\(801\) −7.96743e6 −0.438770
\(802\) 0 0
\(803\) 4.22131e6i 0.231025i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 679860.i 0.0367482i
\(808\) 0 0
\(809\) 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 \(-0.400800\pi\)
0.306626 + 0.951830i \(0.400800\pi\)
\(812\) 0 0
\(813\) 5.09777e6i 0.270492i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.66916e6i 0.349555i
\(818\) 0 0
\(819\) 2.70152e7 1.40734
\(820\) 0 0
\(821\) 3.04347e7 1.57584 0.787918 0.615781i \(-0.211159\pi\)
0.787918 + 0.615781i \(0.211159\pi\)
\(822\) 0 0
\(823\) 4.09773e6i 0.210884i 0.994425 + 0.105442i \(0.0336257\pi\)
−0.994425 + 0.105442i \(0.966374\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.70652e7i 0.867654i 0.900996 + 0.433827i \(0.142837\pi\)
−0.900996 + 0.433827i \(0.857163\pi\)
\(828\) 0 0
\(829\) 2.47617e7 1.25139 0.625697 0.780066i \(-0.284815\pi\)
0.625697 + 0.780066i \(0.284815\pi\)
\(830\) 0 0
\(831\) −2.63161e6 −0.132196
\(832\) 0 0
\(833\) 2.19685e6i 0.109695i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.85436e7i 0.914914i
\(838\) 0 0
\(839\) 3.16529e7 1.55242 0.776208 0.630476i \(-0.217140\pi\)
0.776208 + 0.630476i \(0.217140\pi\)
\(840\) 0 0
\(841\) 1.44170e7 0.702884
\(842\) 0 0
\(843\) − 8.74019e6i − 0.423596i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.46541e7i − 0.701859i
\(848\) 0 0
\(849\) 722364. 0.0343943
\(850\) 0 0
\(851\) 8.64119e6 0.409025
\(852\) 0 0
\(853\) − 2.82671e7i − 1.33017i −0.746765 0.665087i \(-0.768394\pi\)
0.746765 0.665087i \(-0.231606\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.60870e7i 1.21331i 0.794966 + 0.606655i \(0.207489\pi\)
−0.794966 + 0.606655i \(0.792511\pi\)
\(858\) 0 0
\(859\) −3.38111e7 −1.56342 −0.781710 0.623642i \(-0.785652\pi\)
−0.781710 + 0.623642i \(0.785652\pi\)
\(860\) 0 0
\(861\) 267624. 0.0123032
\(862\) 0 0
\(863\) 2.22817e7i 1.01841i 0.860646 + 0.509204i \(0.170060\pi\)
−0.860646 + 0.509204i \(0.829940\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5.03528e6i 0.227497i
\(868\) 0 0
\(869\) −867840. −0.0389843
\(870\) 0 0
\(871\) −3.72965e7 −1.66580
\(872\) 0 0
\(873\) − 397026.i − 0.0176313i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 3.46748e7i − 1.52235i −0.648545 0.761177i \(-0.724622\pi\)
0.648545 0.761177i \(-0.275378\pi\)
\(878\) 0 0
\(879\) −1.58526e7 −0.692034
\(880\) 0 0
\(881\) 1.42603e7 0.618998 0.309499 0.950900i \(-0.399839\pi\)
0.309499 + 0.950900i \(0.399839\pi\)
\(882\) 0 0
\(883\) − 3.75177e7i − 1.61933i −0.586895 0.809663i \(-0.699650\pi\)
0.586895 0.809663i \(-0.300350\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 4.07657e7i − 1.73975i −0.493275 0.869873i \(-0.664200\pi\)
0.493275 0.869873i \(-0.335800\pi\)
\(888\) 0 0
\(889\) 3.53861e7 1.50168
\(890\) 0 0
\(891\) −6.54739e6 −0.276296
\(892\) 0 0
\(893\) 3.59543e7i 1.50877i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.03920e7i 0.431238i
\(898\) 0 0
\(899\) −4.05899e7 −1.67501
\(900\) 0 0
\(901\) −6.99059e6 −0.286881
\(902\) 0 0
\(903\) 1.72327e6i 0.0703290i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.57116e7i 1.44142i 0.693235 + 0.720712i \(0.256185\pi\)
−0.693235 + 0.720712i \(0.743815\pi\)
\(908\) 0 0
\(909\) 1.60678e7 0.644979
\(910\) 0 0
\(911\) 2.11389e7 0.843893 0.421947 0.906621i \(-0.361347\pi\)
0.421947 + 0.906621i \(0.361347\pi\)
\(912\) 0 0
\(913\) 2.09422e7i 0.831468i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 928896.i − 0.0364791i
\(918\) 0 0
\(919\) 1.85996e7 0.726465 0.363233 0.931698i \(-0.381673\pi\)
0.363233 + 0.931698i \(0.381673\pi\)
\(920\) 0 0
\(921\) −8.68535e6 −0.337395
\(922\) 0 0
\(923\) 7.76545e7i 3.00028i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 9.66980e6i − 0.369588i
\(928\) 0 0
\(929\) −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
\(932\) 0 0
\(933\) 5.56841e6i 0.209424i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.19419e7i − 0.816441i −0.912883 0.408221i \(-0.866149\pi\)
0.912883 0.408221i \(-0.133851\pi\)
\(938\) 0 0
\(939\) 1.37738e7 0.509787
\(940\) 0 0
\(941\) −7.77722e6 −0.286319 −0.143160 0.989700i \(-0.545726\pi\)
−0.143160 + 0.989700i \(0.545726\pi\)
\(942\) 0 0
\(943\) − 591948.i − 0.0216773i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.17199e7i − 1.14936i −0.818378 0.574681i \(-0.805126\pi\)
0.818378 0.574681i \(-0.194874\pi\)
\(948\) 0 0
\(949\) −2.43165e7 −0.876468
\(950\) 0 0
\(951\) −1.64191e7 −0.588707
\(952\) 0 0
\(953\) 5.60285e6i 0.199838i 0.994996 + 0.0999188i \(0.0318583\pi\)
−0.994996 + 0.0999188i \(0.968142\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.80832e6i 0.240304i
\(958\) 0 0
\(959\) 1.93801e7 0.680470
\(960\) 0 0
\(961\) 1.85403e7 0.647601
\(962\) 0 0
\(963\) 214866.i 0.00746624i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.03532e7i 0.699949i 0.936759 + 0.349975i \(0.113810\pi\)
−0.936759 + 0.349975i \(0.886190\pi\)
\(968\) 0 0
\(969\) 1.25273e7 0.428595
\(970\) 0 0
\(971\) 2.34306e7 0.797510 0.398755 0.917057i \(-0.369442\pi\)
0.398755 + 0.917057i \(0.369442\pi\)
\(972\) 0 0
\(973\) − 3.32878e7i − 1.12721i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 4.30412e7i − 1.44261i −0.692619 0.721303i \(-0.743543\pi\)
0.692619 0.721303i \(-0.256457\pi\)
\(978\) 0 0
\(979\) 7.39008e6 0.246429
\(980\) 0 0
\(981\) −4.28345e7 −1.42109
\(982\) 0 0
\(983\) − 4.75003e7i − 1.56788i −0.620837 0.783940i \(-0.713207\pi\)
0.620837 0.783940i \(-0.286793\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.29038e6i 0.303557i
\(988\) 0 0
\(989\) 3.81164e6 0.123914
\(990\) 0 0
\(991\) −2.09231e7 −0.676770 −0.338385 0.941008i \(-0.609881\pi\)
−0.338385 + 0.941008i \(0.609881\pi\)
\(992\) 0 0
\(993\) − 2.29128e7i − 0.737402i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.96332e7i 0.944148i 0.881559 + 0.472074i \(0.156495\pi\)
−0.881559 + 0.472074i \(0.843505\pi\)
\(998\) 0 0
\(999\) 1.48986e7 0.472315
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 400.6.c.i.49.2 2
4.3 odd 2 50.6.b.b.49.2 2
5.2 odd 4 400.6.a.i.1.1 1
5.3 odd 4 80.6.a.c.1.1 1
5.4 even 2 inner 400.6.c.i.49.1 2
12.11 even 2 450.6.c.f.199.1 2
15.8 even 4 720.6.a.v.1.1 1
20.3 even 4 10.6.a.c.1.1 1
20.7 even 4 50.6.a.b.1.1 1
20.19 odd 2 50.6.b.b.49.1 2
40.3 even 4 320.6.a.f.1.1 1
40.13 odd 4 320.6.a.k.1.1 1
60.23 odd 4 90.6.a.b.1.1 1
60.47 odd 4 450.6.a.u.1.1 1
60.59 even 2 450.6.c.f.199.2 2
140.83 odd 4 490.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 20.3 even 4
50.6.a.b.1.1 1 20.7 even 4
50.6.b.b.49.1 2 20.19 odd 2
50.6.b.b.49.2 2 4.3 odd 2
80.6.a.c.1.1 1 5.3 odd 4
90.6.a.b.1.1 1 60.23 odd 4
320.6.a.f.1.1 1 40.3 even 4
320.6.a.k.1.1 1 40.13 odd 4
400.6.a.i.1.1 1 5.2 odd 4
400.6.c.i.49.1 2 5.4 even 2 inner
400.6.c.i.49.2 2 1.1 even 1 trivial
450.6.a.u.1.1 1 60.47 odd 4
450.6.c.f.199.1 2 12.11 even 2
450.6.c.f.199.2 2 60.59 even 2
490.6.a.k.1.1 1 140.83 odd 4
720.6.a.v.1.1 1 15.8 even 4