Properties

Label 50.6.a.b.1.1
Level $50$
Weight $6$
Character 50.1
Self dual yes
Analytic conductor $8.019$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.01919099065\)
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 \(\chi\) \(=\) 50.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} -6.00000 q^{3} +16.0000 q^{4} +24.0000 q^{6} +118.000 q^{7} -64.0000 q^{8} -207.000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -6.00000 q^{3} +16.0000 q^{4} +24.0000 q^{6} +118.000 q^{7} -64.0000 q^{8} -207.000 q^{9} +192.000 q^{11} -96.0000 q^{12} -1106.00 q^{13} -472.000 q^{14} +256.000 q^{16} -762.000 q^{17} +828.000 q^{18} -2740.00 q^{19} -708.000 q^{21} -768.000 q^{22} -1566.00 q^{23} +384.000 q^{24} +4424.00 q^{26} +2700.00 q^{27} +1888.00 q^{28} +5910.00 q^{29} -6868.00 q^{31} -1024.00 q^{32} -1152.00 q^{33} +3048.00 q^{34} -3312.00 q^{36} +5518.00 q^{37} +10960.0 q^{38} +6636.00 q^{39} -378.000 q^{41} +2832.00 q^{42} +2434.00 q^{43} +3072.00 q^{44} +6264.00 q^{46} -13122.0 q^{47} -1536.00 q^{48} -2883.00 q^{49} +4572.00 q^{51} -17696.0 q^{52} +9174.00 q^{53} -10800.0 q^{54} -7552.00 q^{56} +16440.0 q^{57} -23640.0 q^{58} -34980.0 q^{59} -9838.00 q^{61} +27472.0 q^{62} -24426.0 q^{63} +4096.00 q^{64} +4608.00 q^{66} -33722.0 q^{67} -12192.0 q^{68} +9396.00 q^{69} +70212.0 q^{71} +13248.0 q^{72} -21986.0 q^{73} -22072.0 q^{74} -43840.0 q^{76} +22656.0 q^{77} -26544.0 q^{78} +4520.00 q^{79} +34101.0 q^{81} +1512.00 q^{82} +109074. q^{83} -11328.0 q^{84} -9736.00 q^{86} -35460.0 q^{87} -12288.0 q^{88} +38490.0 q^{89} -130508. q^{91} -25056.0 q^{92} +41208.0 q^{93} +52488.0 q^{94} +6144.00 q^{96} +1918.00 q^{97} +11532.0 q^{98} -39744.0 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −6.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 24.0000 0.272166
\(7\) 118.000 0.910200 0.455100 0.890440i \(-0.349603\pi\)
0.455100 + 0.890440i \(0.349603\pi\)
\(8\) −64.0000 −0.353553
\(9\) −207.000 −0.851852
\(10\) 0 0
\(11\) 192.000 0.478431 0.239216 0.970966i \(-0.423110\pi\)
0.239216 + 0.970966i \(0.423110\pi\)
\(12\) −96.0000 −0.192450
\(13\) −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 \(-0.603597\pi\)
−0.319744 + 0.947504i \(0.603597\pi\)
\(18\) 828.000 0.602350
\(19\) −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
\(22\) −768.000 −0.338302
\(23\) −1566.00 −0.617266 −0.308633 0.951181i \(-0.599871\pi\)
−0.308633 + 0.951181i \(0.599871\pi\)
\(24\) 384.000 0.136083
\(25\) 0 0
\(26\) 4424.00 1.28346
\(27\) 2700.00 0.712778
\(28\) 1888.00 0.455100
\(29\) 5910.00 1.30495 0.652473 0.757812i \(-0.273732\pi\)
0.652473 + 0.757812i \(0.273732\pi\)
\(30\) 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
\(33\) −1152.00 −0.184148
\(34\) 3048.00 0.452187
\(35\) 0 0
\(36\) −3312.00 −0.425926
\(37\) 5518.00 0.662640 0.331320 0.943519i \(-0.392506\pi\)
0.331320 + 0.943519i \(0.392506\pi\)
\(38\) 10960.0 1.23127
\(39\) 6636.00 0.698626
\(40\) 0 0
\(41\) −378.000 −0.0351182 −0.0175591 0.999846i \(-0.505590\pi\)
−0.0175591 + 0.999846i \(0.505590\pi\)
\(42\) 2832.00 0.247725
\(43\) 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 \(-0.642629\pi\)
−0.433237 + 0.901280i \(0.642629\pi\)
\(48\) −1536.00 −0.0962250
\(49\) −2883.00 −0.171536
\(50\) 0 0
\(51\) 4572.00 0.246139
\(52\) −17696.0 −0.907542
\(53\) 9174.00 0.448610 0.224305 0.974519i \(-0.427989\pi\)
0.224305 + 0.974519i \(0.427989\pi\)
\(54\) −10800.0 −0.504010
\(55\) 0 0
\(56\) −7552.00 −0.321804
\(57\) 16440.0 0.670216
\(58\) −23640.0 −0.922736
\(59\) −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 \(0.554137\pi\)
\(62\) 27472.0 0.907635
\(63\) −24426.0 −0.775356
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 4608.00 0.130212
\(67\) −33722.0 −0.917754 −0.458877 0.888500i \(-0.651748\pi\)
−0.458877 + 0.888500i \(0.651748\pi\)
\(68\) −12192.0 −0.319744
\(69\) 9396.00 0.237586
\(70\) 0 0
\(71\) 70212.0 1.65297 0.826486 0.562957i \(-0.190336\pi\)
0.826486 + 0.562957i \(0.190336\pi\)
\(72\) 13248.0 0.301175
\(73\) −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
\(78\) −26544.0 −0.494003
\(79\) 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
\(82\) 1512.00 0.0248323
\(83\) 109074. 1.73790 0.868952 0.494896i \(-0.164794\pi\)
0.868952 + 0.494896i \(0.164794\pi\)
\(84\) −11328.0 −0.175168
\(85\) 0 0
\(86\) −9736.00 −0.141950
\(87\) −35460.0 −0.502274
\(88\) −12288.0 −0.169151
\(89\) 38490.0 0.515078 0.257539 0.966268i \(-0.417088\pi\)
0.257539 + 0.966268i \(0.417088\pi\)
\(90\) 0 0
\(91\) −130508. −1.65209
\(92\) −25056.0 −0.308633
\(93\) 41208.0 0.494054
\(94\) 52488.0 0.612689
\(95\) 0 0
\(96\) 6144.00 0.0680414
\(97\) 1918.00 0.0206976 0.0103488 0.999946i \(-0.496706\pi\)
0.0103488 + 0.999946i \(0.496706\pi\)
\(98\) 11532.0 0.121294
\(99\) −39744.0 −0.407553
\(100\) 0 0
\(101\) 77622.0 0.757149 0.378575 0.925571i \(-0.376414\pi\)
0.378575 + 0.925571i \(0.376414\pi\)
\(102\) −18288.0 −0.174047
\(103\) 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 \(-0.498605\pi\)
0.00438236 + 0.999990i \(0.498605\pi\)
\(108\) 43200.0 0.356389
\(109\) 206930. 1.66823 0.834117 0.551587i \(-0.185977\pi\)
0.834117 + 0.551587i \(0.185977\pi\)
\(110\) 0 0
\(111\) −33108.0 −0.255050
\(112\) 30208.0 0.227550
\(113\) −139386. −1.02689 −0.513444 0.858123i \(-0.671631\pi\)
−0.513444 + 0.858123i \(0.671631\pi\)
\(114\) −65760.0 −0.473914
\(115\) 0 0
\(116\) 94560.0 0.652473
\(117\) 228942. 1.54618
\(118\) 139920. 0.925070
\(119\) −89916.0 −0.582062
\(120\) 0 0
\(121\) −124187. −0.771104
\(122\) 39352.0 0.239369
\(123\) 2268.00 0.0135170
\(124\) −109888. −0.641795
\(125\) 0 0
\(126\) 97704.0 0.548259
\(127\) −299882. −1.64984 −0.824919 0.565252i \(-0.808779\pi\)
−0.824919 + 0.565252i \(0.808779\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −14604.0 −0.0772676
\(130\) 0 0
\(131\) 7872.00 0.0400781 0.0200390 0.999799i \(-0.493621\pi\)
0.0200390 + 0.999799i \(0.493621\pi\)
\(132\) −18432.0 −0.0920741
\(133\) −323320. −1.58491
\(134\) 134888. 0.648950
\(135\) 0 0
\(136\) 48768.0 0.226093
\(137\) 164238. 0.747605 0.373803 0.927508i \(-0.378054\pi\)
0.373803 + 0.927508i \(0.378054\pi\)
\(138\) −37584.0 −0.167998
\(139\) −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
\(142\) −280848. −1.16883
\(143\) −212352. −0.868393
\(144\) −52992.0 −0.212963
\(145\) 0 0
\(146\) 87944.0 0.341448
\(147\) 17298.0 0.0660241
\(148\) 88288.0 0.331320
\(149\) −388950. −1.43525 −0.717626 0.696429i \(-0.754771\pi\)
−0.717626 + 0.696429i \(0.754771\pi\)
\(150\) 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
\(153\) 157734. 0.544749
\(154\) −90624.0 −0.307923
\(155\) 0 0
\(156\) 106176. 0.349313
\(157\) 3718.00 0.0120382 0.00601908 0.999982i \(-0.498084\pi\)
0.00601908 + 0.999982i \(0.498084\pi\)
\(158\) −18080.0 −0.0576177
\(159\) −55044.0 −0.172670
\(160\) 0 0
\(161\) −184788. −0.561835
\(162\) −136404. −0.408357
\(163\) 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 \(-0.583316\pi\)
−0.258767 + 0.965940i \(0.583316\pi\)
\(168\) 45312.0 0.123863
\(169\) 851943. 2.29453
\(170\) 0 0
\(171\) 567180. 1.48331
\(172\) 38944.0 0.100374
\(173\) 374454. 0.951225 0.475612 0.879655i \(-0.342226\pi\)
0.475612 + 0.879655i \(0.342226\pi\)
\(174\) 141840. 0.355161
\(175\) 0 0
\(176\) 49152.0 0.119608
\(177\) 209880. 0.503545
\(178\) −153960. −0.364215
\(179\) 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 \(0.527267\pi\)
\(182\) 522032. 1.16820
\(183\) 59028.0 0.130296
\(184\) 100224. 0.218236
\(185\) 0 0
\(186\) −164832. −0.349349
\(187\) −146304. −0.305951
\(188\) −209952. −0.433237
\(189\) 318600. 0.648771
\(190\) 0 0
\(191\) −356988. −0.708060 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(192\) −24576.0 −0.0481125
\(193\) 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 \(-0.454027\pi\)
0.143928 + 0.989588i \(0.454027\pi\)
\(198\) 158976. 0.288183
\(199\) −162520. −0.290920 −0.145460 0.989364i \(-0.546466\pi\)
−0.145460 + 0.989364i \(0.546466\pi\)
\(200\) 0 0
\(201\) 202332. 0.353244
\(202\) −310488. −0.535385
\(203\) 697380. 1.18776
\(204\) 73152.0 0.123070
\(205\) 0 0
\(206\) −186856. −0.306788
\(207\) 324162. 0.525819
\(208\) −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
\(213\) −421272. −0.636229
\(214\) −4152.00 −0.00619759
\(215\) 0 0
\(216\) −172800. −0.252005
\(217\) −810424. −1.16832
\(218\) −827720. −1.17962
\(219\) 131916. 0.185861
\(220\) 0 0
\(221\) 842772. 1.16073
\(222\) 132432. 0.180348
\(223\) 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 \(-0.758100\pi\)
−0.724869 + 0.688887i \(0.758100\pi\)
\(228\) 263040. 0.335108
\(229\) −415810. −0.523970 −0.261985 0.965072i \(-0.584377\pi\)
−0.261985 + 0.965072i \(0.584377\pi\)
\(230\) 0 0
\(231\) −135936. −0.167612
\(232\) −378240. −0.461368
\(233\) −770586. −0.929889 −0.464945 0.885340i \(-0.653926\pi\)
−0.464945 + 0.885340i \(0.653926\pi\)
\(234\) −915768. −1.09332
\(235\) 0 0
\(236\) −559680. −0.654124
\(237\) −27120.0 −0.0313631
\(238\) 359664. 0.411580
\(239\) −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 \(0.451465\pi\)
\(242\) 496748. 0.545253
\(243\) −860706. −0.935059
\(244\) −157408. −0.169259
\(245\) 0 0
\(246\) −9072.00 −0.00955796
\(247\) 3.03044e6 3.16055
\(248\) 439552. 0.453817
\(249\) −654444. −0.668920
\(250\) 0 0
\(251\) 850752. 0.852351 0.426176 0.904640i \(-0.359861\pi\)
0.426176 + 0.904640i \(0.359861\pi\)
\(252\) −390816. −0.387678
\(253\) −300672. −0.295319
\(254\) 1.19953e6 1.16661
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −825402. −0.779530 −0.389765 0.920914i \(-0.627444\pi\)
−0.389765 + 0.920914i \(0.627444\pi\)
\(258\) 58416.0 0.0546365
\(259\) 651124. 0.603135
\(260\) 0 0
\(261\) −1.22337e6 −1.11162
\(262\) −31488.0 −0.0283395
\(263\) −1.36465e6 −1.21655 −0.608276 0.793726i \(-0.708139\pi\)
−0.608276 + 0.793726i \(0.708139\pi\)
\(264\) 73728.0 0.0651062
\(265\) 0 0
\(266\) 1.29328e6 1.12070
\(267\) −230940. −0.198254
\(268\) −539552. −0.458877
\(269\) −113310. −0.0954745 −0.0477373 0.998860i \(-0.515201\pi\)
−0.0477373 + 0.998860i \(0.515201\pi\)
\(270\) 0 0
\(271\) −849628. −0.702758 −0.351379 0.936233i \(-0.614287\pi\)
−0.351379 + 0.936233i \(0.614287\pi\)
\(272\) −195072. −0.159872
\(273\) 783048. 0.635890
\(274\) −656952. −0.528637
\(275\) 0 0
\(276\) 150336. 0.118793
\(277\) −438602. −0.343456 −0.171728 0.985144i \(-0.554935\pi\)
−0.171728 + 0.985144i \(0.554935\pi\)
\(278\) 1.12840e6 0.875691
\(279\) 1.42168e6 1.09343
\(280\) 0 0
\(281\) −1.45670e6 −1.10053 −0.550267 0.834989i \(-0.685474\pi\)
−0.550267 + 0.834989i \(0.685474\pi\)
\(282\) −314928. −0.235824
\(283\) 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
\(288\) 211968. 0.150588
\(289\) −839213. −0.591055
\(290\) 0 0
\(291\) −11508.0 −0.00796650
\(292\) −351776. −0.241440
\(293\) 2.64209e6 1.79796 0.898978 0.437993i \(-0.144311\pi\)
0.898978 + 0.437993i \(0.144311\pi\)
\(294\) −69192.0 −0.0466861
\(295\) 0 0
\(296\) −353152. −0.234278
\(297\) 518400. 0.341015
\(298\) 1.55580e6 1.01488
\(299\) 1.73200e6 1.12039
\(300\) 0 0
\(301\) 287212. 0.182720
\(302\) 391792. 0.247194
\(303\) −465732. −0.291427
\(304\) −701440. −0.435318
\(305\) 0 0
\(306\) −630936. −0.385196
\(307\) 1.44756e6 0.876577 0.438288 0.898834i \(-0.355585\pi\)
0.438288 + 0.898834i \(0.355585\pi\)
\(308\) 362496. 0.217734
\(309\) −280284. −0.166994
\(310\) 0 0
\(311\) −928068. −0.544100 −0.272050 0.962283i \(-0.587702\pi\)
−0.272050 + 0.962283i \(0.587702\pi\)
\(312\) −424704. −0.247002
\(313\) −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 \(-0.777139\pi\)
−0.764752 + 0.644324i \(0.777139\pi\)
\(318\) 220176. 0.122096
\(319\) 1.13472e6 0.624327
\(320\) 0 0
\(321\) −6228.00 −0.00337354
\(322\) 739152. 0.397278
\(323\) 2.08788e6 1.11352
\(324\) 545616. 0.288752
\(325\) 0 0
\(326\) −172936. −0.0901242
\(327\) −1.24158e6 −0.642104
\(328\) 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
\(333\) −1.14223e6 −0.564471
\(334\) 746088. 0.365952
\(335\) 0 0
\(336\) −181248. −0.0875841
\(337\) 2.21088e6 1.06045 0.530225 0.847857i \(-0.322108\pi\)
0.530225 + 0.847857i \(0.322108\pi\)
\(338\) −3.40777e6 −1.62248
\(339\) 836316. 0.395249
\(340\) 0 0
\(341\) −1.31866e6 −0.614109
\(342\) −2.26872e6 −1.04886
\(343\) −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 \(-0.326385\pi\)
0.518785 + 0.854905i \(0.326385\pi\)
\(348\) −567360. −0.251137
\(349\) −311290. −0.136805 −0.0684024 0.997658i \(-0.521790\pi\)
−0.0684024 + 0.997658i \(0.521790\pi\)
\(350\) 0 0
\(351\) −2.98620e6 −1.29375
\(352\) −196608. −0.0845755
\(353\) 3.08657e6 1.31838 0.659189 0.751977i \(-0.270900\pi\)
0.659189 + 0.751977i \(0.270900\pi\)
\(354\) −839520. −0.356060
\(355\) 0 0
\(356\) 615840. 0.257539
\(357\) 539496. 0.224036
\(358\) −1.08840e6 −0.448829
\(359\) −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
\(362\) 301672. 0.120994
\(363\) 745122. 0.296798
\(364\) −2.08813e6 −0.826045
\(365\) 0 0
\(366\) −236112. −0.0921330
\(367\) −35762.0 −0.0138598 −0.00692989 0.999976i \(-0.502206\pi\)
−0.00692989 + 0.999976i \(0.502206\pi\)
\(368\) −400896. −0.154316
\(369\) 78246.0 0.0299155
\(370\) 0 0
\(371\) 1.08253e6 0.408325
\(372\) 659328. 0.247027
\(373\) 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
\(378\) −1.27440e6 −0.458750
\(379\) −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
\(382\) 1.42795e6 0.500674
\(383\) −5.31949e6 −1.85299 −0.926494 0.376309i \(-0.877193\pi\)
−0.926494 + 0.376309i \(0.877193\pi\)
\(384\) 98304.0 0.0340207
\(385\) 0 0
\(386\) −1.75478e6 −0.599451
\(387\) −503838. −0.171007
\(388\) 30688.0 0.0103488
\(389\) 1.16145e6 0.389158 0.194579 0.980887i \(-0.437666\pi\)
0.194579 + 0.980887i \(0.437666\pi\)
\(390\) 0 0
\(391\) 1.19329e6 0.394734
\(392\) 184512. 0.0606470
\(393\) −47232.0 −0.0154261
\(394\) −627192. −0.203545
\(395\) 0 0
\(396\) −635904. −0.203776
\(397\) −628562. −0.200157 −0.100079 0.994980i \(-0.531909\pi\)
−0.100079 + 0.994980i \(0.531909\pi\)
\(398\) 650080. 0.205712
\(399\) 1.93992e6 0.610031
\(400\) 0 0
\(401\) −2.72432e6 −0.846052 −0.423026 0.906118i \(-0.639032\pi\)
−0.423026 + 0.906118i \(0.639032\pi\)
\(402\) −809328. −0.249781
\(403\) 7.59601e6 2.32982
\(404\) 1.24195e6 0.378575
\(405\) 0 0
\(406\) −2.78952e6 −0.839875
\(407\) 1.05946e6 0.317027
\(408\) −292608. −0.0870233
\(409\) 1.78019e6 0.526209 0.263104 0.964767i \(-0.415254\pi\)
0.263104 + 0.964767i \(0.415254\pi\)
\(410\) 0 0
\(411\) −985428. −0.287753
\(412\) 747424. 0.216932
\(413\) −4.12764e6 −1.19077
\(414\) −1.29665e6 −0.371810
\(415\) 0 0
\(416\) 1.13254e6 0.320865
\(417\) 1.69260e6 0.476666
\(418\) 2.10432e6 0.589076
\(419\) 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 \(0.661832\pi\)
\(422\) 726592. 0.198614
\(423\) 2.71625e6 0.738107
\(424\) −587136. −0.158608
\(425\) 0 0
\(426\) 1.68509e6 0.449882
\(427\) −1.16088e6 −0.308119
\(428\) 16608.0 0.00438236
\(429\) 1.27411e6 0.334245
\(430\) 0 0
\(431\) −548748. −0.142292 −0.0711459 0.997466i \(-0.522666\pi\)
−0.0711459 + 0.997466i \(0.522666\pi\)
\(432\) 691200. 0.178195
\(433\) 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
\(438\) −527664. −0.131423
\(439\) 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
\(442\) −3.37109e6 −0.820757
\(443\) 1.86155e6 0.450678 0.225339 0.974280i \(-0.427651\pi\)
0.225339 + 0.974280i \(0.427651\pi\)
\(444\) −529728. −0.127525
\(445\) 0 0
\(446\) −1.15310e6 −0.274491
\(447\) 2.33370e6 0.552429
\(448\) 483328. 0.113775
\(449\) 3.73719e6 0.874841 0.437421 0.899257i \(-0.355892\pi\)
0.437421 + 0.899257i \(0.355892\pi\)
\(450\) 0 0
\(451\) −72576.0 −0.0168016
\(452\) −2.23018e6 −0.513444
\(453\) 587688. 0.134555
\(454\) 4.50209e6 1.02512
\(455\) 0 0
\(456\) −1.05216e6 −0.236957
\(457\) 6.48276e6 1.45201 0.726005 0.687690i \(-0.241375\pi\)
0.726005 + 0.687690i \(0.241375\pi\)
\(458\) 1.66324e6 0.370503
\(459\) −2.05740e6 −0.455813
\(460\) 0 0
\(461\) 1.50910e6 0.330724 0.165362 0.986233i \(-0.447121\pi\)
0.165362 + 0.986233i \(0.447121\pi\)
\(462\) 543744. 0.118519
\(463\) −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 \(-0.764621\pi\)
−0.738829 + 0.673893i \(0.764621\pi\)
\(468\) 3.66307e6 0.773091
\(469\) −3.97920e6 −0.835340
\(470\) 0 0
\(471\) −22308.0 −0.00463349
\(472\) 2.23872e6 0.462535
\(473\) 467328. 0.0960437
\(474\) 108480. 0.0221771
\(475\) 0 0
\(476\) −1.43866e6 −0.291031
\(477\) −1.89902e6 −0.382149
\(478\) 2.38128e6 0.476695
\(479\) −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
\(482\) −1.09561e6 −0.214802
\(483\) 1.10873e6 0.216251
\(484\) −1.98699e6 −0.385552
\(485\) 0 0
\(486\) 3.44282e6 0.661187
\(487\) −5.51808e6 −1.05430 −0.527152 0.849771i \(-0.676740\pi\)
−0.527152 + 0.849771i \(0.676740\pi\)
\(488\) 629632. 0.119684
\(489\) −259404. −0.0490574
\(490\) 0 0
\(491\) −1.51277e6 −0.283184 −0.141592 0.989925i \(-0.545222\pi\)
−0.141592 + 0.989925i \(0.545222\pi\)
\(492\) 36288.0 0.00675850
\(493\) −4.50342e6 −0.834498
\(494\) −1.21218e7 −2.23485
\(495\) 0 0
\(496\) −1.75821e6 −0.320897
\(497\) 8.28502e6 1.50454
\(498\) 2.61778e6 0.472998
\(499\) −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
\(502\) −3.40301e6 −0.602703
\(503\) −6.73105e6 −1.18621 −0.593106 0.805124i \(-0.702099\pi\)
−0.593106 + 0.805124i \(0.702099\pi\)
\(504\) 1.56326e6 0.274130
\(505\) 0 0
\(506\) 1.20269e6 0.208822
\(507\) −5.11166e6 −0.883165
\(508\) −4.79811e6 −0.824919
\(509\) −556650. −0.0952331 −0.0476165 0.998866i \(-0.515163\pi\)
−0.0476165 + 0.998866i \(0.515163\pi\)
\(510\) 0 0
\(511\) −2.59435e6 −0.439517
\(512\) −262144. −0.0441942
\(513\) −7.39800e6 −1.24114
\(514\) 3.30161e6 0.551211
\(515\) 0 0
\(516\) −233664. −0.0386338
\(517\) −2.51942e6 −0.414548
\(518\) −2.60450e6 −0.426481
\(519\) −2.24672e6 −0.366127
\(520\) 0 0
\(521\) 1.01110e7 1.63192 0.815962 0.578106i \(-0.196208\pi\)
0.815962 + 0.578106i \(0.196208\pi\)
\(522\) 4.89348e6 0.786034
\(523\) 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
\(528\) −294912. −0.0460371
\(529\) −3.98399e6 −0.618983
\(530\) 0 0
\(531\) 7.24086e6 1.11443
\(532\) −5.17312e6 −0.792453
\(533\) 418068. 0.0637425
\(534\) 923760. 0.140186
\(535\) 0 0
\(536\) 2.15821e6 0.324475
\(537\) −1.63260e6 −0.244312
\(538\) 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
\(543\) 452508. 0.0658608
\(544\) 780288. 0.113047
\(545\) 0 0
\(546\) −3.13219e6 −0.449642
\(547\) −4.44024e6 −0.634510 −0.317255 0.948340i \(-0.602761\pi\)
−0.317255 + 0.948340i \(0.602761\pi\)
\(548\) 2.62781e6 0.373803
\(549\) 2.03647e6 0.288367
\(550\) 0 0
\(551\) −1.61934e7 −2.27227
\(552\) −601344. −0.0839992
\(553\) 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 \(-0.288929\pi\)
0.615563 + 0.788088i \(0.288929\pi\)
\(558\) −5.68670e6 −0.773170
\(559\) −2.69200e6 −0.364373
\(560\) 0 0
\(561\) 877824. 0.117761
\(562\) 5.82679e6 0.778196
\(563\) 9.81287e6 1.30474 0.652372 0.757899i \(-0.273774\pi\)
0.652372 + 0.757899i \(0.273774\pi\)
\(564\) 1.25971e6 0.166753
\(565\) 0 0
\(566\) −481576. −0.0631864
\(567\) 4.02392e6 0.525644
\(568\) −4.49357e6 −0.584414
\(569\) 1.33152e7 1.72412 0.862061 0.506804i \(-0.169173\pi\)
0.862061 + 0.506804i \(0.169173\pi\)
\(570\) 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
\(573\) 2.14193e6 0.272533
\(574\) 178416. 0.0226024
\(575\) 0 0
\(576\) −847872. −0.106481
\(577\) −4.50372e6 −0.563160 −0.281580 0.959538i \(-0.590859\pi\)
−0.281580 + 0.959538i \(0.590859\pi\)
\(578\) 3.35685e6 0.417939
\(579\) −2.63216e6 −0.326300
\(580\) 0 0
\(581\) 1.28707e7 1.58184
\(582\) 46032.0 0.00563316
\(583\) 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 \(-0.511934\pi\)
−0.0374834 + 0.999297i \(0.511934\pi\)
\(588\) 276768. 0.0330121
\(589\) 1.88183e7 2.23508
\(590\) 0 0
\(591\) −940788. −0.110796
\(592\) 1.41261e6 0.165660
\(593\) 2.50385e6 0.292397 0.146198 0.989255i \(-0.453296\pi\)
0.146198 + 0.989255i \(0.453296\pi\)
\(594\) −2.07360e6 −0.241134
\(595\) 0 0
\(596\) −6.22320e6 −0.717626
\(597\) 975120. 0.111975
\(598\) −6.92798e6 −0.792235
\(599\) −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 \(0.786011\pi\)
\(602\) −1.14885e6 −0.129203
\(603\) 6.98045e6 0.781791
\(604\) −1.56717e6 −0.174793
\(605\) 0 0
\(606\) 1.86293e6 0.206070
\(607\) −1.13772e7 −1.25333 −0.626663 0.779291i \(-0.715580\pi\)
−0.626663 + 0.779291i \(0.715580\pi\)
\(608\) 2.80576e6 0.307816
\(609\) −4.18428e6 −0.457170
\(610\) 0 0
\(611\) 1.45129e7 1.57272
\(612\) 2.52374e6 0.272375
\(613\) 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 \(-0.637226\pi\)
−0.417878 + 0.908503i \(0.637226\pi\)
\(618\) 1.12114e6 0.118083
\(619\) 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
\(622\) 3.71227e6 0.384737
\(623\) 4.54182e6 0.468824
\(624\) 1.69882e6 0.174657
\(625\) 0 0
\(626\) 9.18250e6 0.936538
\(627\) 3.15648e6 0.320652
\(628\) 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
\(633\) 1.08989e6 0.108112
\(634\) 1.09461e7 1.08152
\(635\) 0 0
\(636\) −880704. −0.0863351
\(637\) 3.18860e6 0.311352
\(638\) −4.53888e6 −0.441466
\(639\) −1.45339e7 −1.40809
\(640\) 0 0
\(641\) 6.37390e6 0.612718 0.306359 0.951916i \(-0.400889\pi\)
0.306359 + 0.951916i \(0.400889\pi\)
\(642\) 24912.0 0.00238545
\(643\) −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 \(-0.365724\pi\)
0.409440 + 0.912337i \(0.365724\pi\)
\(648\) −2.18246e6 −0.204178
\(649\) −6.71616e6 −0.625906
\(650\) 0 0
\(651\) 4.86254e6 0.449688
\(652\) 691744. 0.0637274
\(653\) 1.58477e6 0.145440 0.0727201 0.997352i \(-0.476832\pi\)
0.0727201 + 0.997352i \(0.476832\pi\)
\(654\) 4.96632e6 0.454036
\(655\) 0 0
\(656\) −96768.0 −0.00877955
\(657\) 4.55110e6 0.411342
\(658\) 6.19358e6 0.557670
\(659\) 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 \(0.551452\pi\)
\(662\) −1.52752e7 −1.35469
\(663\) −5.05663e6 −0.446763
\(664\) −6.98074e6 −0.614442
\(665\) 0 0
\(666\) 4.56890e6 0.399141
\(667\) −9.25506e6 −0.805498
\(668\) −2.98435e6 −0.258767
\(669\) −1.72964e6 −0.149414
\(670\) 0 0
\(671\) −1.88890e6 −0.161958
\(672\) 724992. 0.0619313
\(673\) −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 \(-0.496857\pi\)
0.00987467 + 0.999951i \(0.496857\pi\)
\(678\) −3.34526e6 −0.279483
\(679\) 226324. 0.0188389
\(680\) 0 0
\(681\) 6.75313e6 0.558004
\(682\) 5.27462e6 0.434241
\(683\) −2.05830e7 −1.68833 −0.844164 0.536084i \(-0.819903\pi\)
−0.844164 + 0.536084i \(0.819903\pi\)
\(684\) 9.07488e6 0.741653
\(685\) 0 0
\(686\) 9.29368e6 0.754011
\(687\) 2.49486e6 0.201676
\(688\) 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
\(693\) −4.68979e6 −0.370954
\(694\) −9.30895e6 −0.733672
\(695\) 0 0
\(696\) 2.26944e6 0.177581
\(697\) 288036. 0.0224577
\(698\) 1.24516e6 0.0967357
\(699\) 4.62352e6 0.357915
\(700\) 0 0
\(701\) 1.29304e6 0.0993843 0.0496921 0.998765i \(-0.484176\pi\)
0.0496921 + 0.998765i \(0.484176\pi\)
\(702\) 1.19448e7 0.914821
\(703\) −1.51193e7 −1.15384
\(704\) 786432. 0.0598039
\(705\) 0 0
\(706\) −1.23463e7 −0.932234
\(707\) 9.15940e6 0.689157
\(708\) 3.35808e6 0.251772
\(709\) −2.12720e7 −1.58926 −0.794628 0.607097i \(-0.792334\pi\)
−0.794628 + 0.607097i \(0.792334\pi\)
\(710\) 0 0
\(711\) −935640. −0.0694120
\(712\) −2.46336e6 −0.182108
\(713\) 1.07553e7 0.792316
\(714\) −2.15798e6 −0.158417
\(715\) 0 0
\(716\) 4.35360e6 0.317370
\(717\) 3.57192e6 0.259480
\(718\) 1.41230e7 1.02239
\(719\) 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
\(722\) −2.01260e7 −1.43686
\(723\) −1.64341e6 −0.116923
\(724\) −1.20669e6 −0.0855556
\(725\) 0 0
\(726\) −2.98049e6 −0.209868
\(727\) 4.36740e6 0.306469 0.153235 0.988190i \(-0.451031\pi\)
0.153235 + 0.988190i \(0.451031\pi\)
\(728\) 8.35251e6 0.584102
\(729\) −3.12231e6 −0.217599
\(730\) 0 0
\(731\) −1.85471e6 −0.128375
\(732\) 944448. 0.0651479
\(733\) 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
\(738\) −312984. −0.0211535
\(739\) 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
\(742\) −4.33013e6 −0.288729
\(743\) −6.18781e6 −0.411211 −0.205605 0.978635i \(-0.565916\pi\)
−0.205605 + 0.978635i \(0.565916\pi\)
\(744\) −2.63731e6 −0.174674
\(745\) 0 0
\(746\) −6.86102e6 −0.451379
\(747\) −2.25783e7 −1.48044
\(748\) −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
\(753\) −5.10451e6 −0.328070
\(754\) 2.61458e7 1.67484
\(755\) 0 0
\(756\) 5.09760e6 0.324385
\(757\) −1.93494e7 −1.22724 −0.613618 0.789603i \(-0.710286\pi\)
−0.613618 + 0.789603i \(0.710286\pi\)
\(758\) 1.24070e7 0.784318
\(759\) 1.80403e6 0.113668
\(760\) 0 0
\(761\) −3.01992e7 −1.89031 −0.945155 0.326621i \(-0.894090\pi\)
−0.945155 + 0.326621i \(0.894090\pi\)
\(762\) −7.19717e6 −0.449029
\(763\) 2.44177e7 1.51843
\(764\) −5.71181e6 −0.354030
\(765\) 0 0
\(766\) 2.12779e7 1.31026
\(767\) 3.86879e7 2.37458
\(768\) −393216. −0.0240563
\(769\) 2.15854e7 1.31627 0.658134 0.752901i \(-0.271346\pi\)
0.658134 + 0.752901i \(0.271346\pi\)
\(770\) 0 0
\(771\) 4.95241e6 0.300041
\(772\) 7.01910e6 0.423876
\(773\) −3.90895e6 −0.235294 −0.117647 0.993055i \(-0.537535\pi\)
−0.117647 + 0.993055i \(0.537535\pi\)
\(774\) 2.01535e6 0.120920
\(775\) 0 0
\(776\) −122752. −0.00731769
\(777\) −3.90674e6 −0.232147
\(778\) −4.64580e6 −0.275177
\(779\) 1.03572e6 0.0611503
\(780\) 0 0
\(781\) 1.34807e7 0.790833
\(782\) −4.77317e6 −0.279119
\(783\) 1.59570e7 0.930137
\(784\) −738048. −0.0428839
\(785\) 0 0
\(786\) 188928. 0.0109079
\(787\) 2.65082e7 1.52561 0.762806 0.646628i \(-0.223821\pi\)
0.762806 + 0.646628i \(0.223821\pi\)
\(788\) 2.50877e6 0.143928
\(789\) 8.18788e6 0.468251
\(790\) 0 0
\(791\) −1.64475e7 −0.934674
\(792\) 2.54362e6 0.144092
\(793\) 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 \(-0.597307\pi\)
−0.300960 + 0.953637i \(0.597307\pi\)
\(798\) −7.75968e6 −0.431357
\(799\) 9.99896e6 0.554100
\(800\) 0 0
\(801\) −7.96743e6 −0.438770
\(802\) 1.08973e7 0.598249
\(803\) −4.22131e6 −0.231025
\(804\) 3.23731e6 0.176622
\(805\) 0 0
\(806\) −3.03840e7 −1.64743
\(807\) 679860. 0.0367482
\(808\) −4.96781e6 −0.267693
\(809\) −1.11446e7 −0.598675 −0.299338 0.954147i \(-0.596766\pi\)
−0.299338 + 0.954147i \(0.596766\pi\)
\(810\) 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
\(813\) 5.09777e6 0.270492
\(814\) −4.23782e6 −0.224172
\(815\) 0 0
\(816\) 1.17043e6 0.0615348
\(817\) −6.66916e6 −0.349555
\(818\) −7.12076e6 −0.372086
\(819\) 2.70152e7 1.40734
\(820\) 0 0
\(821\) 3.04347e7 1.57584 0.787918 0.615781i \(-0.211159\pi\)
0.787918 + 0.615781i \(0.211159\pi\)
\(822\) 3.94171e6 0.203472
\(823\) −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 \(-0.357163\pi\)
0.433827 + 0.900996i \(0.357163\pi\)
\(828\) 5.18659e6 0.262909
\(829\) −2.47617e7 −1.25139 −0.625697 0.780066i \(-0.715185\pi\)
−0.625697 + 0.780066i \(0.715185\pi\)
\(830\) 0 0
\(831\) 2.63161e6 0.132196
\(832\) −4.53018e6 −0.226886
\(833\) 2.19685e6 0.109695
\(834\) −6.77040e6 −0.337054
\(835\) 0 0
\(836\) −8.41728e6 −0.416539
\(837\) −1.85436e7 −0.914914
\(838\) −2.60232e6 −0.128012
\(839\) 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
\(842\) 1.41624e7 0.688425
\(843\) 8.74019e6 0.423596
\(844\) −2.90637e6 −0.140441
\(845\) 0 0
\(846\) −1.08650e7 −0.521921
\(847\) −1.46541e7 −0.701859
\(848\) 2.34854e6 0.112153
\(849\) −722364. −0.0343943
\(850\) 0 0
\(851\) −8.64119e6 −0.409025
\(852\) −6.74035e6 −0.318115
\(853\) −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 \(-0.707489\pi\)
−0.606655 + 0.794966i \(0.707489\pi\)
\(858\) −5.09645e6 −0.236347
\(859\) −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
\(862\) 2.19499e6 0.100615
\(863\) −2.22817e7 −1.01841 −0.509204 0.860646i \(-0.670060\pi\)
−0.509204 + 0.860646i \(0.670060\pi\)
\(864\) −2.76480e6 −0.126003
\(865\) 0 0
\(866\) −5.96966e6 −0.270492
\(867\) 5.03528e6 0.227497
\(868\) −1.29668e7 −0.584162
\(869\) 867840. 0.0389843
\(870\) 0 0
\(871\) 3.72965e7 1.66580
\(872\) −1.32435e7 −0.589810
\(873\) −397026. −0.0176313
\(874\) −1.71634e7 −0.760018
\(875\) 0 0
\(876\) 2.11066e6 0.0929303
\(877\) 3.46748e7 1.52235 0.761177 0.648545i \(-0.224622\pi\)
0.761177 + 0.648545i \(0.224622\pi\)
\(878\) −1.94485e7 −0.851431
\(879\) −1.58526e7 −0.692034
\(880\) 0 0
\(881\) 1.42603e7 0.618998 0.309499 0.950900i \(-0.399839\pi\)
0.309499 + 0.950900i \(0.399839\pi\)
\(882\) −2.38712e6 −0.103325
\(883\) 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 \(-0.835800\pi\)
−0.869873 + 0.493275i \(0.835800\pi\)
\(888\) 2.11891e6 0.0901738
\(889\) −3.53861e7 −1.50168
\(890\) 0 0
\(891\) 6.54739e6 0.276296
\(892\) 4.61238e6 0.194095
\(893\) 3.59543e7 1.50877
\(894\) −9.33480e6 −0.390626
\(895\) 0 0
\(896\) −1.93331e6 −0.0804511
\(897\) −1.03920e7 −0.431238
\(898\) −1.49488e7 −0.618606
\(899\) −4.05899e7 −1.67501
\(900\) 0 0
\(901\) −6.99059e6 −0.286881
\(902\) 290304. 0.0118806
\(903\) −1.72327e6 −0.0703290
\(904\) 8.92070e6 0.363060
\(905\) 0 0
\(906\) −2.35075e6 −0.0951451
\(907\) 3.57116e7 1.44142 0.720712 0.693235i \(-0.243815\pi\)
0.720712 + 0.693235i \(0.243815\pi\)
\(908\) −1.80084e7 −0.724869
\(909\) −1.60678e7 −0.644979
\(910\) 0 0
\(911\) −2.11389e7 −0.843893 −0.421947 0.906621i \(-0.638653\pi\)
−0.421947 + 0.906621i \(0.638653\pi\)
\(912\) 4.20864e6 0.167554
\(913\) 2.09422e7 0.831468
\(914\) −2.59310e7 −1.02673
\(915\) 0 0
\(916\) −6.65296e6 −0.261985
\(917\) 928896. 0.0364791
\(918\) 8.22960e6 0.322309
\(919\) 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
\(922\) −6.03641e6 −0.233857
\(923\) −7.76545e7 −3.00028
\(924\) −2.17498e6 −0.0838059
\(925\) 0 0
\(926\) 3.47360e7 1.33123
\(927\) −9.66980e6 −0.369588
\(928\) −6.05184e6 −0.230684
\(929\) 4.45110e7 1.69211 0.846055 0.533096i \(-0.178972\pi\)
0.846055 + 0.533096i \(0.178972\pi\)
\(930\) 0 0
\(931\) 7.89942e6 0.298690
\(932\) −1.23294e7 −0.464945
\(933\) 5.56841e6 0.209424
\(934\) 2.78565e7 1.04486
\(935\) 0 0
\(936\) −1.46523e7 −0.546658
\(937\) 2.19419e7 0.816441 0.408221 0.912883i \(-0.366149\pi\)
0.408221 + 0.912883i \(0.366149\pi\)
\(938\) 1.59168e7 0.590675
\(939\) 1.37738e7 0.509787
\(940\) 0 0
\(941\) −7.77722e6 −0.286319 −0.143160 0.989700i \(-0.545726\pi\)
−0.143160 + 0.989700i \(0.545726\pi\)
\(942\) 89232.0 0.00327637
\(943\) 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 \(-0.694874\pi\)
−0.574681 + 0.818378i \(0.694874\pi\)
\(948\) −433920. −0.0156815
\(949\) 2.43165e7 0.876468
\(950\) 0 0
\(951\) 1.64191e7 0.588707
\(952\) 5.75462e6 0.205790
\(953\) 5.60285e6 0.199838 0.0999188 0.994996i \(-0.468142\pi\)
0.0999188 + 0.994996i \(0.468142\pi\)
\(954\) 7.59607e6 0.270220
\(955\) 0 0
\(956\) −9.52512e6 −0.337074
\(957\) −6.80832e6 −0.240304
\(958\) 2.20421e7 0.775959
\(959\) 1.93801e7 0.680470
\(960\) 0 0
\(961\) 1.85403e7 0.647601
\(962\) 2.44116e7 0.850470
\(963\) −214866. −0.00746624
\(964\) 4.38243e6 0.151888
\(965\) 0 0
\(966\) −4.43491e6 −0.152912
\(967\) 2.03532e7 0.699949 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(968\) 7.94797e6 0.272626
\(969\) −1.25273e7 −0.428595
\(970\) 0 0
\(971\) −2.34306e7 −0.797510 −0.398755 0.917057i \(-0.630558\pi\)
−0.398755 + 0.917057i \(0.630558\pi\)
\(972\) −1.37713e7 −0.467530
\(973\) −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 \(-0.243543\pi\)
0.721303 + 0.692619i \(0.243543\pi\)
\(978\) 1.03762e6 0.0346888
\(979\) 7.39008e6 0.246429
\(980\) 0 0
\(981\) −4.28345e7 −1.42109
\(982\) 6.05107e6 0.200241
\(983\) 4.75003e7 1.56788 0.783940 0.620837i \(-0.213207\pi\)
0.783940 + 0.620837i \(0.213207\pi\)
\(984\) −145152. −0.00477898
\(985\) 0 0
\(986\) 1.80137e7 0.590079
\(987\) 9.29038e6 0.303557
\(988\) 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
\(993\) −2.29128e7 −0.737402
\(994\) −3.31401e7 −1.06387
\(995\) 0 0
\(996\) −1.04711e7 −0.334460
\(997\) −2.96332e7 −0.944148 −0.472074 0.881559i \(-0.656495\pi\)
−0.472074 + 0.881559i \(0.656495\pi\)
\(998\) 7.72168e6 0.245406
\(999\) 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 50.6.a.b.1.1 1
3.2 odd 2 450.6.a.u.1.1 1
4.3 odd 2 400.6.a.i.1.1 1
5.2 odd 4 50.6.b.b.49.1 2
5.3 odd 4 50.6.b.b.49.2 2
5.4 even 2 10.6.a.c.1.1 1
15.2 even 4 450.6.c.f.199.2 2
15.8 even 4 450.6.c.f.199.1 2
15.14 odd 2 90.6.a.b.1.1 1
20.3 even 4 400.6.c.i.49.2 2
20.7 even 4 400.6.c.i.49.1 2
20.19 odd 2 80.6.a.c.1.1 1
35.34 odd 2 490.6.a.k.1.1 1
40.19 odd 2 320.6.a.k.1.1 1
40.29 even 2 320.6.a.f.1.1 1
60.59 even 2 720.6.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 5.4 even 2
50.6.a.b.1.1 1 1.1 even 1 trivial
50.6.b.b.49.1 2 5.2 odd 4
50.6.b.b.49.2 2 5.3 odd 4
80.6.a.c.1.1 1 20.19 odd 2
90.6.a.b.1.1 1 15.14 odd 2
320.6.a.f.1.1 1 40.29 even 2
320.6.a.k.1.1 1 40.19 odd 2
400.6.a.i.1.1 1 4.3 odd 2
400.6.c.i.49.1 2 20.7 even 4
400.6.c.i.49.2 2 20.3 even 4
450.6.a.u.1.1 1 3.2 odd 2
450.6.c.f.199.1 2 15.8 even 4
450.6.c.f.199.2 2 15.2 even 4
490.6.a.k.1.1 1 35.34 odd 2
720.6.a.v.1.1 1 60.59 even 2