Properties

Label 90.6.a.b.1.1
Level $90$
Weight $6$
Character 90.1
Self dual yes
Analytic conductor $14.435$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4345437832\)
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 \(\chi\) \(=\) 90.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -118.000 q^{7} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -118.000 q^{7} -64.0000 q^{8} -100.000 q^{10} -192.000 q^{11} +1106.00 q^{13} +472.000 q^{14} +256.000 q^{16} -762.000 q^{17} -2740.00 q^{19} +400.000 q^{20} +768.000 q^{22} -1566.00 q^{23} +625.000 q^{25} -4424.00 q^{26} -1888.00 q^{28} -5910.00 q^{29} -6868.00 q^{31} -1024.00 q^{32} +3048.00 q^{34} -2950.00 q^{35} -5518.00 q^{37} +10960.0 q^{38} -1600.00 q^{40} +378.000 q^{41} -2434.00 q^{43} -3072.00 q^{44} +6264.00 q^{46} -13122.0 q^{47} -2883.00 q^{49} -2500.00 q^{50} +17696.0 q^{52} +9174.00 q^{53} -4800.00 q^{55} +7552.00 q^{56} +23640.0 q^{58} +34980.0 q^{59} -9838.00 q^{61} +27472.0 q^{62} +4096.00 q^{64} +27650.0 q^{65} +33722.0 q^{67} -12192.0 q^{68} +11800.0 q^{70} -70212.0 q^{71} +21986.0 q^{73} +22072.0 q^{74} -43840.0 q^{76} +22656.0 q^{77} +4520.00 q^{79} +6400.00 q^{80} -1512.00 q^{82} +109074. q^{83} -19050.0 q^{85} +9736.00 q^{86} +12288.0 q^{88} -38490.0 q^{89} -130508. q^{91} -25056.0 q^{92} +52488.0 q^{94} -68500.0 q^{95} -1918.00 q^{97} +11532.0 q^{98} +O(q^{100})\)

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
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −118.000 −0.910200 −0.455100 0.890440i \(-0.650397\pi\)
−0.455100 + 0.890440i \(0.650397\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) −100.000 −0.316228
\(11\) −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 \(-0.137956\pi\)
0.907542 + 0.419961i \(0.137956\pi\)
\(14\) 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 \(0.603597\pi\)
\(18\) 0 0
\(19\) −2740.00 −1.74127 −0.870636 0.491928i \(-0.836292\pi\)
−0.870636 + 0.491928i \(0.836292\pi\)
\(20\) 400.000 0.223607
\(21\) 0 0
\(22\) 768.000 0.338302
\(23\) −1566.00 −0.617266 −0.308633 0.951181i \(-0.599871\pi\)
−0.308633 + 0.951181i \(0.599871\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −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
\(35\) −2950.00 −0.407054
\(36\) 0 0
\(37\) −5518.00 −0.662640 −0.331320 0.943519i \(-0.607494\pi\)
−0.331320 + 0.943519i \(0.607494\pi\)
\(38\) 10960.0 1.23127
\(39\) 0 0
\(40\) −1600.00 −0.158114
\(41\) 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 \(-0.532004\pi\)
−0.100374 + 0.994950i \(0.532004\pi\)
\(44\) −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 \(0.642629\pi\)
\(48\) 0 0
\(49\) −2883.00 −0.171536
\(50\) −2500.00 −0.141421
\(51\) 0 0
\(52\) 17696.0 0.907542
\(53\) 9174.00 0.448610 0.224305 0.974519i \(-0.427989\pi\)
0.224305 + 0.974519i \(0.427989\pi\)
\(54\) 0 0
\(55\) −4800.00 −0.213961
\(56\) 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
\(65\) 27650.0 0.811730
\(66\) 0 0
\(67\) 33722.0 0.917754 0.458877 0.888500i \(-0.348252\pi\)
0.458877 + 0.888500i \(0.348252\pi\)
\(68\) −12192.0 −0.319744
\(69\) 0 0
\(70\) 11800.0 0.287831
\(71\) −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 \(-0.422380\pi\)
0.241440 + 0.970416i \(0.422380\pi\)
\(74\) 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 \(0.487028\pi\)
\(80\) 6400.00 0.111803
\(81\) 0 0
\(82\) −1512.00 −0.0248323
\(83\) 109074. 1.73790 0.868952 0.494896i \(-0.164794\pi\)
0.868952 + 0.494896i \(0.164794\pi\)
\(84\) 0 0
\(85\) −19050.0 −0.285988
\(86\) 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
\(95\) −68500.0 −0.778720
\(96\) 0 0
\(97\) −1918.00 −0.0206976 −0.0103488 0.999946i \(-0.503294\pi\)
−0.0103488 + 0.999946i \(0.503294\pi\)
\(98\) 11532.0 0.121294
\(99\) 0 0
\(100\) 10000.0 0.100000
\(101\) −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 \(-0.569605\pi\)
−0.216932 + 0.976187i \(0.569605\pi\)
\(104\) −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 \(0.498605\pi\)
\(108\) 0 0
\(109\) 206930. 1.66823 0.834117 0.551587i \(-0.185977\pi\)
0.834117 + 0.551587i \(0.185977\pi\)
\(110\) 19200.0 0.151293
\(111\) 0 0
\(112\) −30208.0 −0.227550
\(113\) −139386. −1.02689 −0.513444 0.858123i \(-0.671631\pi\)
−0.513444 + 0.858123i \(0.671631\pi\)
\(114\) 0 0
\(115\) −39150.0 −0.276050
\(116\) −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
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 299882. 1.64984 0.824919 0.565252i \(-0.191221\pi\)
0.824919 + 0.565252i \(0.191221\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) −110600. −0.573980
\(131\) −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 \(-0.378054\pi\)
0.373803 + 0.927508i \(0.378054\pi\)
\(138\) 0 0
\(139\) −282100. −1.23841 −0.619207 0.785228i \(-0.712546\pi\)
−0.619207 + 0.785228i \(0.712546\pi\)
\(140\) −47200.0 −0.203527
\(141\) 0 0
\(142\) 280848. 1.16883
\(143\) −212352. −0.868393
\(144\) 0 0
\(145\) −147750. −0.583590
\(146\) −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
\(155\) −171700. −0.574039
\(156\) 0 0
\(157\) −3718.00 −0.0120382 −0.00601908 0.999982i \(-0.501916\pi\)
−0.00601908 + 0.999982i \(0.501916\pi\)
\(158\) −18080.0 −0.0576177
\(159\) 0 0
\(160\) −25600.0 −0.0790569
\(161\) 184788. 0.561835
\(162\) 0 0
\(163\) −43234.0 −0.127455 −0.0637274 0.997967i \(-0.520299\pi\)
−0.0637274 + 0.997967i \(0.520299\pi\)
\(164\) 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 \(0.583316\pi\)
\(168\) 0 0
\(169\) 851943. 2.29453
\(170\) 76200.0 0.202224
\(171\) 0 0
\(172\) −38944.0 −0.100374
\(173\) 374454. 0.951225 0.475612 0.879655i \(-0.342226\pi\)
0.475612 + 0.879655i \(0.342226\pi\)
\(174\) 0 0
\(175\) −73750.0 −0.182040
\(176\) −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
\(185\) −137950. −0.296341
\(186\) 0 0
\(187\) 146304. 0.305951
\(188\) −209952. −0.433237
\(189\) 0 0
\(190\) 274000. 0.550638
\(191\) 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 \(-0.639331\pi\)
−0.423876 + 0.905720i \(0.639331\pi\)
\(194\) 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 \(0.454027\pi\)
\(198\) 0 0
\(199\) −162520. −0.290920 −0.145460 0.989364i \(-0.546466\pi\)
−0.145460 + 0.989364i \(0.546466\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 0 0
\(202\) 310488. 0.535385
\(203\) 697380. 1.18776
\(204\) 0 0
\(205\) 9450.00 0.0157053
\(206\) 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
\(215\) −60850.0 −0.0897769
\(216\) 0 0
\(217\) 810424. 1.16832
\(218\) −827720. −1.17962
\(219\) 0 0
\(220\) −76800.0 −0.106980
\(221\) −842772. −1.16073
\(222\) 0 0
\(223\) −288274. −0.388189 −0.194095 0.980983i \(-0.562177\pi\)
−0.194095 + 0.980983i \(0.562177\pi\)
\(224\) 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 \(0.758100\pi\)
\(228\) 0 0
\(229\) −415810. −0.523970 −0.261985 0.965072i \(-0.584377\pi\)
−0.261985 + 0.965072i \(0.584377\pi\)
\(230\) 156600. 0.195197
\(231\) 0 0
\(232\) 378240. 0.461368
\(233\) −770586. −0.929889 −0.464945 0.885340i \(-0.653926\pi\)
−0.464945 + 0.885340i \(0.653926\pi\)
\(234\) 0 0
\(235\) −328050. −0.387499
\(236\) 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
\(245\) −72075.0 −0.0767131
\(246\) 0 0
\(247\) −3.03044e6 −3.16055
\(248\) 439552. 0.453817
\(249\) 0 0
\(250\) −62500.0 −0.0632456
\(251\) −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 \(-0.627444\pi\)
−0.389765 + 0.920914i \(0.627444\pi\)
\(258\) 0 0
\(259\) 651124. 0.603135
\(260\) 442400. 0.405865
\(261\) 0 0
\(262\) 31488.0 0.0283395
\(263\) −1.36465e6 −1.21655 −0.608276 0.793726i \(-0.708139\pi\)
−0.608276 + 0.793726i \(0.708139\pi\)
\(264\) 0 0
\(265\) 229350. 0.200625
\(266\) −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
\(275\) −120000. −0.0956862
\(276\) 0 0
\(277\) 438602. 0.343456 0.171728 0.985144i \(-0.445065\pi\)
0.171728 + 0.985144i \(0.445065\pi\)
\(278\) 1.12840e6 0.875691
\(279\) 0 0
\(280\) 188800. 0.143915
\(281\) 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 \(-0.514227\pi\)
−0.0446795 + 0.999001i \(0.514227\pi\)
\(284\) −1.12339e6 −0.826486
\(285\) 0 0
\(286\) 849408. 0.614047
\(287\) −44604.0 −0.0319646
\(288\) 0 0
\(289\) −839213. −0.591055
\(290\) 591000. 0.412660
\(291\) 0 0
\(292\) 351776. 0.241440
\(293\) 2.64209e6 1.79796 0.898978 0.437993i \(-0.144311\pi\)
0.898978 + 0.437993i \(0.144311\pi\)
\(294\) 0 0
\(295\) 874500. 0.585066
\(296\) 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
\(305\) −245950. −0.151390
\(306\) 0 0
\(307\) −1.44756e6 −0.876577 −0.438288 0.898834i \(-0.644415\pi\)
−0.438288 + 0.898834i \(0.644415\pi\)
\(308\) 362496. 0.217734
\(309\) 0 0
\(310\) 686800. 0.405907
\(311\) 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 \(-0.269609\pi\)
0.662232 + 0.749299i \(0.269609\pi\)
\(314\) 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 \(0.777139\pi\)
\(318\) 0 0
\(319\) 1.13472e6 0.624327
\(320\) 102400. 0.0559017
\(321\) 0 0
\(322\) −739152. −0.397278
\(323\) 2.08788e6 1.11352
\(324\) 0 0
\(325\) 691250. 0.363017
\(326\) 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
\(335\) 843050. 0.410432
\(336\) 0 0
\(337\) −2.21088e6 −1.06045 −0.530225 0.847857i \(-0.677892\pi\)
−0.530225 + 0.847857i \(0.677892\pi\)
\(338\) −3.40777e6 −1.62248
\(339\) 0 0
\(340\) −304800. −0.142994
\(341\) 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 \(-0.326385\pi\)
0.518785 + 0.854905i \(0.326385\pi\)
\(348\) 0 0
\(349\) −311290. −0.136805 −0.0684024 0.997658i \(-0.521790\pi\)
−0.0684024 + 0.997658i \(0.521790\pi\)
\(350\) 295000. 0.128722
\(351\) 0 0
\(352\) 196608. 0.0845755
\(353\) 3.08657e6 1.31838 0.659189 0.751977i \(-0.270900\pi\)
0.659189 + 0.751977i \(0.270900\pi\)
\(354\) 0 0
\(355\) −1.75530e6 −0.739232
\(356\) −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
\(365\) 549650. 0.215950
\(366\) 0 0
\(367\) 35762.0 0.0138598 0.00692989 0.999976i \(-0.497794\pi\)
0.00692989 + 0.999976i \(0.497794\pi\)
\(368\) −400896. −0.154316
\(369\) 0 0
\(370\) 551800. 0.209545
\(371\) −1.08253e6 −0.408325
\(372\) 0 0
\(373\) −1.71525e6 −0.638346 −0.319173 0.947696i \(-0.603405\pi\)
−0.319173 + 0.947696i \(0.603405\pi\)
\(374\) −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 \(0.687127\pi\)
\(380\) −1.09600e6 −0.389360
\(381\) 0 0
\(382\) −1.42795e6 −0.500674
\(383\) −5.31949e6 −1.85299 −0.926494 0.376309i \(-0.877193\pi\)
−0.926494 + 0.376309i \(0.877193\pi\)
\(384\) 0 0
\(385\) 566400. 0.194747
\(386\) 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
\(395\) 113000. 0.0364406
\(396\) 0 0
\(397\) 628562. 0.200157 0.100079 0.994980i \(-0.468091\pi\)
0.100079 + 0.994980i \(0.468091\pi\)
\(398\) 650080. 0.205712
\(399\) 0 0
\(400\) 160000. 0.0500000
\(401\) 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 \(0.415254\pi\)
\(410\) −37800.0 −0.0111053
\(411\) 0 0
\(412\) −747424. −0.216932
\(413\) −4.12764e6 −1.19077
\(414\) 0 0
\(415\) 2.72685e6 0.777215
\(416\) −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
\(425\) −476250. −0.127898
\(426\) 0 0
\(427\) 1.16088e6 0.308119
\(428\) 16608.0 0.00438236
\(429\) 0 0
\(430\) 243400. 0.0634818
\(431\) 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 \(-0.561260\pi\)
−0.191267 + 0.981538i \(0.561260\pi\)
\(434\) −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 \(0.294350\pi\)
\(440\) 307200. 0.0756466
\(441\) 0 0
\(442\) 3.37109e6 0.820757
\(443\) 1.86155e6 0.450678 0.225339 0.974280i \(-0.427651\pi\)
0.225339 + 0.974280i \(0.427651\pi\)
\(444\) 0 0
\(445\) −962250. −0.230350
\(446\) 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
\(455\) −3.26270e6 −0.738837
\(456\) 0 0
\(457\) −6.48276e6 −1.45201 −0.726005 0.687690i \(-0.758625\pi\)
−0.726005 + 0.687690i \(0.758625\pi\)
\(458\) 1.66324e6 0.370503
\(459\) 0 0
\(460\) −626400. −0.138025
\(461\) −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 \(-0.109586\pi\)
0.941321 + 0.337513i \(0.109586\pi\)
\(464\) −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 \(0.764621\pi\)
\(468\) 0 0
\(469\) −3.97920e6 −0.835340
\(470\) 1.31220e6 0.274003
\(471\) 0 0
\(472\) −2.23872e6 −0.462535
\(473\) 467328. 0.0960437
\(474\) 0 0
\(475\) −1.71250e6 −0.348254
\(476\) 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
\(485\) −47950.0 −0.00925623
\(486\) 0 0
\(487\) 5.51808e6 1.05430 0.527152 0.849771i \(-0.323260\pi\)
0.527152 + 0.849771i \(0.323260\pi\)
\(488\) 629632. 0.119684
\(489\) 0 0
\(490\) 288300. 0.0542443
\(491\) 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 \(0.555517\pi\)
\(500\) 250000. 0.0447214
\(501\) 0 0
\(502\) 3.40301e6 0.602703
\(503\) −6.73105e6 −1.18621 −0.593106 0.805124i \(-0.702099\pi\)
−0.593106 + 0.805124i \(0.702099\pi\)
\(504\) 0 0
\(505\) −1.94055e6 −0.338607
\(506\) −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
\(515\) −1.16785e6 −0.194030
\(516\) 0 0
\(517\) 2.51942e6 0.414548
\(518\) −2.60450e6 −0.426481
\(519\) 0 0
\(520\) −1.76960e6 −0.286990
\(521\) −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 \(-0.690157\pi\)
−0.562491 + 0.826804i \(0.690157\pi\)
\(524\) −125952. −0.0200390
\(525\) 0 0
\(526\) 5.45858e6 0.860232
\(527\) 5.23342e6 0.820840
\(528\) 0 0
\(529\) −3.98399e6 −0.618983
\(530\) −917400. −0.141863
\(531\) 0 0
\(532\) 5.17312e6 0.792453
\(533\) 418068. 0.0637425
\(534\) 0 0
\(535\) 25950.0 0.00391970
\(536\) −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
\(545\) 5.17325e6 0.746057
\(546\) 0 0
\(547\) 4.44024e6 0.634510 0.317255 0.948340i \(-0.397239\pi\)
0.317255 + 0.948340i \(0.397239\pi\)
\(548\) 2.62781e6 0.373803
\(549\) 0 0
\(550\) 480000. 0.0676604
\(551\) 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 \(-0.288929\pi\)
0.615563 + 0.788088i \(0.288929\pi\)
\(558\) 0 0
\(559\) −2.69200e6 −0.364373
\(560\) −755200. −0.101763
\(561\) 0 0
\(562\) −5.82679e6 −0.778196
\(563\) 9.81287e6 1.30474 0.652372 0.757899i \(-0.273774\pi\)
0.652372 + 0.757899i \(0.273774\pi\)
\(564\) 0 0
\(565\) −3.48465e6 −0.459238
\(566\) 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
\(575\) −978750. −0.123453
\(576\) 0 0
\(577\) 4.50372e6 0.563160 0.281580 0.959538i \(-0.409141\pi\)
0.281580 + 0.959538i \(0.409141\pi\)
\(578\) 3.35685e6 0.417939
\(579\) 0 0
\(580\) −2.36400e6 −0.291795
\(581\) −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 \(-0.511934\pi\)
−0.0374834 + 0.999297i \(0.511934\pi\)
\(588\) 0 0
\(589\) 1.88183e7 2.23508
\(590\) −3.49800e6 −0.413704
\(591\) 0 0
\(592\) −1.41261e6 −0.165660
\(593\) 2.50385e6 0.292397 0.146198 0.989255i \(-0.453296\pi\)
0.146198 + 0.989255i \(0.453296\pi\)
\(594\) 0 0
\(595\) 2.24790e6 0.260306
\(596\) 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
\(605\) −3.10468e6 −0.344848
\(606\) 0 0
\(607\) 1.13772e7 1.25333 0.626663 0.779291i \(-0.284420\pi\)
0.626663 + 0.779291i \(0.284420\pi\)
\(608\) 2.80576e6 0.307816
\(609\) 0 0
\(610\) 983800. 0.107049
\(611\) −1.45129e7 −1.57272
\(612\) 0 0
\(613\) −7.00161e6 −0.752570 −0.376285 0.926504i \(-0.622799\pi\)
−0.376285 + 0.926504i \(0.622799\pi\)
\(614\) 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 \(0.637226\pi\)
\(618\) 0 0
\(619\) 4.02362e6 0.422076 0.211038 0.977478i \(-0.432316\pi\)
0.211038 + 0.977478i \(0.432316\pi\)
\(620\) −2.74720e6 −0.287019
\(621\) 0 0
\(622\) −3.71227e6 −0.384737
\(623\) 4.54182e6 0.468824
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) −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
\(635\) 7.49705e6 0.737830
\(636\) 0 0
\(637\) −3.18860e6 −0.311352
\(638\) −4.53888e6 −0.441466
\(639\) 0 0
\(640\) −409600. −0.0395285
\(641\) −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 \(-0.423287\pi\)
0.238676 + 0.971099i \(0.423287\pi\)
\(644\) 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 \(0.365724\pi\)
\(648\) 0 0
\(649\) −6.71616e6 −0.625906
\(650\) −2.76500e6 −0.256692
\(651\) 0 0
\(652\) −691744. −0.0637274
\(653\) 1.58477e6 0.145440 0.0727201 0.997352i \(-0.476832\pi\)
0.0727201 + 0.997352i \(0.476832\pi\)
\(654\) 0 0
\(655\) −196800. −0.0179235
\(656\) 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
\(665\) 8.08300e6 0.708791
\(666\) 0 0
\(667\) 9.25506e6 0.805498
\(668\) −2.98435e6 −0.258767
\(669\) 0 0
\(670\) −3.37220e6 −0.290219
\(671\) 1.88890e6 0.161958
\(672\) 0 0
\(673\) 1.11313e7 0.947349 0.473675 0.880700i \(-0.342927\pi\)
0.473675 + 0.880700i \(0.342927\pi\)
\(674\) 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 \(0.496857\pi\)
\(678\) 0 0
\(679\) 226324. 0.0188389
\(680\) 1.21920e6 0.101112
\(681\) 0 0
\(682\) −5.27462e6 −0.434241
\(683\) −2.05830e7 −1.68833 −0.844164 0.536084i \(-0.819903\pi\)
−0.844164 + 0.536084i \(0.819903\pi\)
\(684\) 0 0
\(685\) 4.10595e6 0.334339
\(686\) −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
\(695\) −7.05250e6 −0.553836
\(696\) 0 0
\(697\) −288036. −0.0224577
\(698\) 1.24516e6 0.0967357
\(699\) 0 0
\(700\) −1.18000e6 −0.0910200
\(701\) −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 \(0.792334\pi\)
\(710\) 7.02120e6 0.522716
\(711\) 0 0
\(712\) 2.46336e6 0.182108
\(713\) 1.07553e7 0.792316
\(714\) 0 0
\(715\) −5.30880e6 −0.388357
\(716\) −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
\(725\) −3.69375e6 −0.260989
\(726\) 0 0
\(727\) −4.36740e6 −0.306469 −0.153235 0.988190i \(-0.548969\pi\)
−0.153235 + 0.988190i \(0.548969\pi\)
\(728\) 8.35251e6 0.584102
\(729\) 0 0
\(730\) −2.19860e6 −0.152700
\(731\) 1.85471e6 0.128375
\(732\) 0 0
\(733\) −4.05645e6 −0.278860 −0.139430 0.990232i \(-0.544527\pi\)
−0.139430 + 0.990232i \(0.544527\pi\)
\(734\) −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 \(0.491763\pi\)
\(740\) −2.20720e6 −0.148171
\(741\) 0 0
\(742\) 4.33013e6 0.288729
\(743\) −6.18781e6 −0.411211 −0.205605 0.978635i \(-0.565916\pi\)
−0.205605 + 0.978635i \(0.565916\pi\)
\(744\) 0 0
\(745\) 9.72375e6 0.641864
\(746\) 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
\(755\) −2.44870e6 −0.156339
\(756\) 0 0
\(757\) 1.93494e7 1.22724 0.613618 0.789603i \(-0.289714\pi\)
0.613618 + 0.789603i \(0.289714\pi\)
\(758\) 1.24070e7 0.784318
\(759\) 0 0
\(760\) 4.38400e6 0.275319
\(761\) 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 \(0.271346\pi\)
\(770\) −2.26560e6 −0.137707
\(771\) 0 0
\(772\) −7.01910e6 −0.423876
\(773\) −3.90895e6 −0.235294 −0.117647 0.993055i \(-0.537535\pi\)
−0.117647 + 0.993055i \(0.537535\pi\)
\(774\) 0 0
\(775\) −4.29250e6 −0.256718
\(776\) 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
\(785\) −92950.0 −0.00538363
\(786\) 0 0
\(787\) −2.65082e7 −1.52561 −0.762806 0.646628i \(-0.776179\pi\)
−0.762806 + 0.646628i \(0.776179\pi\)
\(788\) 2.50877e6 0.143928
\(789\) 0 0
\(790\) −452000. −0.0257674
\(791\) 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 \(-0.597307\pi\)
−0.300960 + 0.953637i \(0.597307\pi\)
\(798\) 0 0
\(799\) 9.99896e6 0.554100
\(800\) −640000. −0.0353553
\(801\) 0 0
\(802\) −1.08973e7 −0.598249
\(803\) −4.22131e6 −0.231025
\(804\) 0 0
\(805\) 4.61970e6 0.251260
\(806\) 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
\(815\) −1.08085e6 −0.0569995
\(816\) 0 0
\(817\) 6.66916e6 0.349555
\(818\) −7.12076e6 −0.372086
\(819\) 0 0
\(820\) 151200. 0.00785267
\(821\) −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 \(-0.466374\pi\)
0.105442 + 0.994425i \(0.466374\pi\)
\(824\) 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 \(0.357163\pi\)
\(828\) 0 0
\(829\) −2.47617e7 −1.25139 −0.625697 0.780066i \(-0.715185\pi\)
−0.625697 + 0.780066i \(0.715185\pi\)
\(830\) −1.09074e7 −0.549574
\(831\) 0 0
\(832\) 4.53018e6 0.226886
\(833\) 2.19685e6 0.109695
\(834\) 0 0
\(835\) −4.66305e6 −0.231448
\(836\) 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
\(845\) 2.12986e7 1.02615
\(846\) 0 0
\(847\) 1.46541e7 0.701859
\(848\) 2.34854e6 0.112153
\(849\) 0 0
\(850\) 1.90500e6 0.0904373
\(851\) 8.64119e6 0.409025
\(852\) 0 0
\(853\) 2.82671e7 1.33017 0.665087 0.746765i \(-0.268394\pi\)
0.665087 + 0.746765i \(0.268394\pi\)
\(854\) −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 \(0.707489\pi\)
\(858\) 0 0
\(859\) −3.38111e7 −1.56342 −0.781710 0.623642i \(-0.785652\pi\)
−0.781710 + 0.623642i \(0.785652\pi\)
\(860\) −973600. −0.0448884
\(861\) 0 0
\(862\) −2.19499e6 −0.100615
\(863\) −2.22817e7 −1.01841 −0.509204 0.860646i \(-0.670060\pi\)
−0.509204 + 0.860646i \(0.670060\pi\)
\(864\) 0 0
\(865\) 9.36135e6 0.425401
\(866\) 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
\(875\) −1.84375e6 −0.0814108
\(876\) 0 0
\(877\) −3.46748e7 −1.52235 −0.761177 0.648545i \(-0.775378\pi\)
−0.761177 + 0.648545i \(0.775378\pi\)
\(878\) −1.94485e7 −0.851431
\(879\) 0 0
\(880\) −1.22880e6 −0.0534902
\(881\) −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 \(-0.800350\pi\)
−0.809663 + 0.586895i \(0.800350\pi\)
\(884\) −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 \(0.835800\pi\)
\(888\) 0 0
\(889\) −3.53861e7 −1.50168
\(890\) 3.84900e6 0.162882
\(891\) 0 0
\(892\) −4.61238e6 −0.194095
\(893\) 3.59543e7 1.50877
\(894\) 0 0
\(895\) −6.80250e6 −0.283864
\(896\) 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
\(905\) −1.88545e6 −0.0765233
\(906\) 0 0
\(907\) −3.57116e7 −1.44142 −0.720712 0.693235i \(-0.756185\pi\)
−0.720712 + 0.693235i \(0.756185\pi\)
\(908\) −1.80084e7 −0.724869
\(909\) 0 0
\(910\) 1.30508e7 0.522437
\(911\) 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 \(0.381673\pi\)
\(920\) 2.50560e6 0.0975983
\(921\) 0 0
\(922\) 6.03641e6 0.233857
\(923\) −7.76545e7 −3.00028
\(924\) 0 0
\(925\) −3.44875e6 −0.132528
\(926\) −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
\(935\) 3.65760e6 0.136826
\(936\) 0 0
\(937\) −2.19419e7 −0.816441 −0.408221 0.912883i \(-0.633851\pi\)
−0.408221 + 0.912883i \(0.633851\pi\)
\(938\) 1.59168e7 0.590675
\(939\) 0 0
\(940\) −5.24880e6 −0.193749
\(941\) 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 \(-0.694874\pi\)
−0.574681 + 0.818378i \(0.694874\pi\)
\(948\) 0 0
\(949\) 2.43165e7 0.876468
\(950\) 6.85000e6 0.246253
\(951\) 0 0
\(952\) −5.75462e6 −0.205790
\(953\) 5.60285e6 0.199838 0.0999188 0.994996i \(-0.468142\pi\)
0.0999188 + 0.994996i \(0.468142\pi\)
\(954\) 0 0
\(955\) 8.92470e6 0.316654
\(956\) 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
\(965\) −1.09673e7 −0.379126
\(966\) 0 0
\(967\) −2.03532e7 −0.699949 −0.349975 0.936759i \(-0.613810\pi\)
−0.349975 + 0.936759i \(0.613810\pi\)
\(968\) 7.94797e6 0.272626
\(969\) 0 0
\(970\) 191800. 0.00654514
\(971\) 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 \(-0.243543\pi\)
0.721303 + 0.692619i \(0.243543\pi\)
\(978\) 0 0
\(979\) 7.39008e6 0.246429
\(980\) −1.15320e6 −0.0383565
\(981\) 0 0
\(982\) −6.05107e6 −0.200241
\(983\) 4.75003e7 1.56788 0.783940 0.620837i \(-0.213207\pi\)
0.783940 + 0.620837i \(0.213207\pi\)
\(984\) 0 0
\(985\) 3.91995e6 0.128733
\(986\) −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
\(995\) −4.06300e6 −0.130104
\(996\) 0 0
\(997\) 2.96332e7 0.944148 0.472074 0.881559i \(-0.343505\pi\)
0.472074 + 0.881559i \(0.343505\pi\)
\(998\) 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 90.6.a.b.1.1 1
3.2 odd 2 10.6.a.c.1.1 1
4.3 odd 2 720.6.a.v.1.1 1
5.2 odd 4 450.6.c.f.199.1 2
5.3 odd 4 450.6.c.f.199.2 2
5.4 even 2 450.6.a.u.1.1 1
12.11 even 2 80.6.a.c.1.1 1
15.2 even 4 50.6.b.b.49.2 2
15.8 even 4 50.6.b.b.49.1 2
15.14 odd 2 50.6.a.b.1.1 1
21.20 even 2 490.6.a.k.1.1 1
24.5 odd 2 320.6.a.f.1.1 1
24.11 even 2 320.6.a.k.1.1 1
60.23 odd 4 400.6.c.i.49.1 2
60.47 odd 4 400.6.c.i.49.2 2
60.59 even 2 400.6.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 3.2 odd 2
50.6.a.b.1.1 1 15.14 odd 2
50.6.b.b.49.1 2 15.8 even 4
50.6.b.b.49.2 2 15.2 even 4
80.6.a.c.1.1 1 12.11 even 2
90.6.a.b.1.1 1 1.1 even 1 trivial
320.6.a.f.1.1 1 24.5 odd 2
320.6.a.k.1.1 1 24.11 even 2
400.6.a.i.1.1 1 60.59 even 2
400.6.c.i.49.1 2 60.23 odd 4
400.6.c.i.49.2 2 60.47 odd 4
450.6.a.u.1.1 1 5.4 even 2
450.6.c.f.199.1 2 5.2 odd 4
450.6.c.f.199.2 2 5.3 odd 4
490.6.a.k.1.1 1 21.20 even 2
720.6.a.v.1.1 1 4.3 odd 2