Properties

Label 90.6.a.f.1.1
Level 90
Weight 6
Character 90.1
Self dual yes
Analytic conductor 14.435
Analytic rank 0
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: \(0\)
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
Root \(0\)
Character \(\chi\) = 90.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -22.0000 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} -22.0000 q^{7} +64.0000 q^{8} +100.000 q^{10} +768.000 q^{11} -46.0000 q^{13} -88.0000 q^{14} +256.000 q^{16} -378.000 q^{17} +1100.00 q^{19} +400.000 q^{20} +3072.00 q^{22} +1986.00 q^{23} +625.000 q^{25} -184.000 q^{26} -352.000 q^{28} +5610.00 q^{29} -3988.00 q^{31} +1024.00 q^{32} -1512.00 q^{34} -550.000 q^{35} -142.000 q^{37} +4400.00 q^{38} +1600.00 q^{40} -1542.00 q^{41} -5026.00 q^{43} +12288.0 q^{44} +7944.00 q^{46} -24738.0 q^{47} -16323.0 q^{49} +2500.00 q^{50} -736.000 q^{52} +14166.0 q^{53} +19200.0 q^{55} -1408.00 q^{56} +22440.0 q^{58} -28380.0 q^{59} +5522.00 q^{61} -15952.0 q^{62} +4096.00 q^{64} -1150.00 q^{65} -24742.0 q^{67} -6048.00 q^{68} -2200.00 q^{70} -42372.0 q^{71} -52126.0 q^{73} -568.000 q^{74} +17600.0 q^{76} -16896.0 q^{77} -39640.0 q^{79} +6400.00 q^{80} -6168.00 q^{82} +59826.0 q^{83} -9450.00 q^{85} -20104.0 q^{86} +49152.0 q^{88} -57690.0 q^{89} +1012.00 q^{91} +31776.0 q^{92} -98952.0 q^{94} +27500.0 q^{95} -144382. q^{97} -65292.0 q^{98} +O(q^{100})\)

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
\(7\) −22.0000 −0.169698 −0.0848492 0.996394i \(-0.527041\pi\)
−0.0848492 + 0.996394i \(0.527041\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 100.000 0.316228
\(11\) 768.000 1.91372 0.956862 0.290541i \(-0.0938354\pi\)
0.956862 + 0.290541i \(0.0938354\pi\)
\(12\) 0 0
\(13\) −46.0000 −0.0754917 −0.0377459 0.999287i \(-0.512018\pi\)
−0.0377459 + 0.999287i \(0.512018\pi\)
\(14\) −88.0000 −0.119995
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −378.000 −0.317227 −0.158613 0.987341i \(-0.550702\pi\)
−0.158613 + 0.987341i \(0.550702\pi\)
\(18\) 0 0
\(19\) 1100.00 0.699051 0.349525 0.936927i \(-0.386343\pi\)
0.349525 + 0.936927i \(0.386343\pi\)
\(20\) 400.000 0.223607
\(21\) 0 0
\(22\) 3072.00 1.35321
\(23\) 1986.00 0.782816 0.391408 0.920217i \(-0.371988\pi\)
0.391408 + 0.920217i \(0.371988\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −184.000 −0.0533807
\(27\) 0 0
\(28\) −352.000 −0.0848492
\(29\) 5610.00 1.23870 0.619352 0.785113i \(-0.287395\pi\)
0.619352 + 0.785113i \(0.287395\pi\)
\(30\) 0 0
\(31\) −3988.00 −0.745334 −0.372667 0.927965i \(-0.621557\pi\)
−0.372667 + 0.927965i \(0.621557\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) −1512.00 −0.224313
\(35\) −550.000 −0.0758914
\(36\) 0 0
\(37\) −142.000 −0.0170523 −0.00852617 0.999964i \(-0.502714\pi\)
−0.00852617 + 0.999964i \(0.502714\pi\)
\(38\) 4400.00 0.494303
\(39\) 0 0
\(40\) 1600.00 0.158114
\(41\) −1542.00 −0.143260 −0.0716300 0.997431i \(-0.522820\pi\)
−0.0716300 + 0.997431i \(0.522820\pi\)
\(42\) 0 0
\(43\) −5026.00 −0.414526 −0.207263 0.978285i \(-0.566456\pi\)
−0.207263 + 0.978285i \(0.566456\pi\)
\(44\) 12288.0 0.956862
\(45\) 0 0
\(46\) 7944.00 0.553534
\(47\) −24738.0 −1.63350 −0.816752 0.576990i \(-0.804227\pi\)
−0.816752 + 0.576990i \(0.804227\pi\)
\(48\) 0 0
\(49\) −16323.0 −0.971202
\(50\) 2500.00 0.141421
\(51\) 0 0
\(52\) −736.000 −0.0377459
\(53\) 14166.0 0.692720 0.346360 0.938102i \(-0.387418\pi\)
0.346360 + 0.938102i \(0.387418\pi\)
\(54\) 0 0
\(55\) 19200.0 0.855844
\(56\) −1408.00 −0.0599974
\(57\) 0 0
\(58\) 22440.0 0.875897
\(59\) −28380.0 −1.06141 −0.530704 0.847557i \(-0.678072\pi\)
−0.530704 + 0.847557i \(0.678072\pi\)
\(60\) 0 0
\(61\) 5522.00 0.190008 0.0950040 0.995477i \(-0.469714\pi\)
0.0950040 + 0.995477i \(0.469714\pi\)
\(62\) −15952.0 −0.527031
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −1150.00 −0.0337609
\(66\) 0 0
\(67\) −24742.0 −0.673361 −0.336680 0.941619i \(-0.609304\pi\)
−0.336680 + 0.941619i \(0.609304\pi\)
\(68\) −6048.00 −0.158613
\(69\) 0 0
\(70\) −2200.00 −0.0536633
\(71\) −42372.0 −0.997546 −0.498773 0.866733i \(-0.666216\pi\)
−0.498773 + 0.866733i \(0.666216\pi\)
\(72\) 0 0
\(73\) −52126.0 −1.14485 −0.572423 0.819958i \(-0.693997\pi\)
−0.572423 + 0.819958i \(0.693997\pi\)
\(74\) −568.000 −0.0120578
\(75\) 0 0
\(76\) 17600.0 0.349525
\(77\) −16896.0 −0.324756
\(78\) 0 0
\(79\) −39640.0 −0.714605 −0.357302 0.933989i \(-0.616303\pi\)
−0.357302 + 0.933989i \(0.616303\pi\)
\(80\) 6400.00 0.111803
\(81\) 0 0
\(82\) −6168.00 −0.101300
\(83\) 59826.0 0.953223 0.476612 0.879114i \(-0.341865\pi\)
0.476612 + 0.879114i \(0.341865\pi\)
\(84\) 0 0
\(85\) −9450.00 −0.141868
\(86\) −20104.0 −0.293114
\(87\) 0 0
\(88\) 49152.0 0.676604
\(89\) −57690.0 −0.772015 −0.386007 0.922496i \(-0.626146\pi\)
−0.386007 + 0.922496i \(0.626146\pi\)
\(90\) 0 0
\(91\) 1012.00 0.0128108
\(92\) 31776.0 0.391408
\(93\) 0 0
\(94\) −98952.0 −1.15506
\(95\) 27500.0 0.312625
\(96\) 0 0
\(97\) −144382. −1.55806 −0.779029 0.626988i \(-0.784288\pi\)
−0.779029 + 0.626988i \(0.784288\pi\)
\(98\) −65292.0 −0.686744
\(99\) 0 0
\(100\) 10000.0 0.100000
\(101\) 141258. 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(102\) 0 0
\(103\) 139814. 1.29855 0.649273 0.760555i \(-0.275073\pi\)
0.649273 + 0.760555i \(0.275073\pi\)
\(104\) −2944.00 −0.0266904
\(105\) 0 0
\(106\) 56664.0 0.489827
\(107\) −86418.0 −0.729701 −0.364850 0.931066i \(-0.618880\pi\)
−0.364850 + 0.931066i \(0.618880\pi\)
\(108\) 0 0
\(109\) 218450. 1.76111 0.880554 0.473947i \(-0.157171\pi\)
0.880554 + 0.473947i \(0.157171\pi\)
\(110\) 76800.0 0.605173
\(111\) 0 0
\(112\) −5632.00 −0.0424246
\(113\) 28806.0 0.212220 0.106110 0.994354i \(-0.466160\pi\)
0.106110 + 0.994354i \(0.466160\pi\)
\(114\) 0 0
\(115\) 49650.0 0.350086
\(116\) 89760.0 0.619352
\(117\) 0 0
\(118\) −113520. −0.750529
\(119\) 8316.00 0.0538328
\(120\) 0 0
\(121\) 428773. 2.66234
\(122\) 22088.0 0.134356
\(123\) 0 0
\(124\) −63808.0 −0.372667
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −216502. −1.19111 −0.595556 0.803314i \(-0.703068\pi\)
−0.595556 + 0.803314i \(0.703068\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) −4600.00 −0.0238726
\(131\) 244608. 1.24535 0.622676 0.782479i \(-0.286045\pi\)
0.622676 + 0.782479i \(0.286045\pi\)
\(132\) 0 0
\(133\) −24200.0 −0.118628
\(134\) −98968.0 −0.476138
\(135\) 0 0
\(136\) −24192.0 −0.112157
\(137\) 239502. 1.09020 0.545102 0.838370i \(-0.316491\pi\)
0.545102 + 0.838370i \(0.316491\pi\)
\(138\) 0 0
\(139\) 30860.0 0.135475 0.0677375 0.997703i \(-0.478422\pi\)
0.0677375 + 0.997703i \(0.478422\pi\)
\(140\) −8800.00 −0.0379457
\(141\) 0 0
\(142\) −169488. −0.705372
\(143\) −35328.0 −0.144470
\(144\) 0 0
\(145\) 140250. 0.553966
\(146\) −208504. −0.809529
\(147\) 0 0
\(148\) −2272.00 −0.00852617
\(149\) 100950. 0.372512 0.186256 0.982501i \(-0.440365\pi\)
0.186256 + 0.982501i \(0.440365\pi\)
\(150\) 0 0
\(151\) 12452.0 0.0444423 0.0222212 0.999753i \(-0.492926\pi\)
0.0222212 + 0.999753i \(0.492926\pi\)
\(152\) 70400.0 0.247152
\(153\) 0 0
\(154\) −67584.0 −0.229637
\(155\) −99700.0 −0.333323
\(156\) 0 0
\(157\) −6022.00 −0.0194981 −0.00974903 0.999952i \(-0.503103\pi\)
−0.00974903 + 0.999952i \(0.503103\pi\)
\(158\) −158560. −0.505302
\(159\) 0 0
\(160\) 25600.0 0.0790569
\(161\) −43692.0 −0.132843
\(162\) 0 0
\(163\) −500866. −1.47656 −0.738282 0.674492i \(-0.764363\pi\)
−0.738282 + 0.674492i \(0.764363\pi\)
\(164\) −24672.0 −0.0716300
\(165\) 0 0
\(166\) 239304. 0.674031
\(167\) −555258. −1.54065 −0.770324 0.637652i \(-0.779906\pi\)
−0.770324 + 0.637652i \(0.779906\pi\)
\(168\) 0 0
\(169\) −369177. −0.994301
\(170\) −37800.0 −0.100316
\(171\) 0 0
\(172\) −80416.0 −0.207263
\(173\) −417354. −1.06020 −0.530102 0.847934i \(-0.677846\pi\)
−0.530102 + 0.847934i \(0.677846\pi\)
\(174\) 0 0
\(175\) −13750.0 −0.0339397
\(176\) 196608. 0.478431
\(177\) 0 0
\(178\) −230760. −0.545897
\(179\) 52380.0 0.122189 0.0610946 0.998132i \(-0.480541\pi\)
0.0610946 + 0.998132i \(0.480541\pi\)
\(180\) 0 0
\(181\) 546662. 1.24029 0.620144 0.784488i \(-0.287074\pi\)
0.620144 + 0.784488i \(0.287074\pi\)
\(182\) 4048.00 0.00905862
\(183\) 0 0
\(184\) 127104. 0.276767
\(185\) −3550.00 −0.00762604
\(186\) 0 0
\(187\) −290304. −0.607084
\(188\) −395808. −0.816752
\(189\) 0 0
\(190\) 110000. 0.221059
\(191\) 452028. 0.896565 0.448283 0.893892i \(-0.352036\pi\)
0.448283 + 0.893892i \(0.352036\pi\)
\(192\) 0 0
\(193\) 485594. 0.938383 0.469191 0.883097i \(-0.344545\pi\)
0.469191 + 0.883097i \(0.344545\pi\)
\(194\) −577528. −1.10171
\(195\) 0 0
\(196\) −261168. −0.485601
\(197\) −1.01018e6 −1.85452 −0.927262 0.374414i \(-0.877844\pi\)
−0.927262 + 0.374414i \(0.877844\pi\)
\(198\) 0 0
\(199\) −807640. −1.44572 −0.722862 0.690993i \(-0.757174\pi\)
−0.722862 + 0.690993i \(0.757174\pi\)
\(200\) 40000.0 0.0707107
\(201\) 0 0
\(202\) 565032. 0.974304
\(203\) −123420. −0.210206
\(204\) 0 0
\(205\) −38550.0 −0.0640678
\(206\) 559256. 0.918211
\(207\) 0 0
\(208\) −11776.0 −0.0188729
\(209\) 844800. 1.33779
\(210\) 0 0
\(211\) 149552. 0.231252 0.115626 0.993293i \(-0.463113\pi\)
0.115626 + 0.993293i \(0.463113\pi\)
\(212\) 226656. 0.346360
\(213\) 0 0
\(214\) −345672. −0.515976
\(215\) −125650. −0.185381
\(216\) 0 0
\(217\) 87736.0 0.126482
\(218\) 873800. 1.24529
\(219\) 0 0
\(220\) 307200. 0.427922
\(221\) 17388.0 0.0239480
\(222\) 0 0
\(223\) −443506. −0.597224 −0.298612 0.954375i \(-0.596524\pi\)
−0.298612 + 0.954375i \(0.596524\pi\)
\(224\) −22528.0 −0.0299987
\(225\) 0 0
\(226\) 115224. 0.150062
\(227\) −420018. −0.541007 −0.270504 0.962719i \(-0.587190\pi\)
−0.270504 + 0.962719i \(0.587190\pi\)
\(228\) 0 0
\(229\) 1.05875e6 1.33415 0.667075 0.744990i \(-0.267546\pi\)
0.667075 + 0.744990i \(0.267546\pi\)
\(230\) 198600. 0.247548
\(231\) 0 0
\(232\) 359040. 0.437948
\(233\) 1.27345e6 1.53671 0.768353 0.640026i \(-0.221077\pi\)
0.768353 + 0.640026i \(0.221077\pi\)
\(234\) 0 0
\(235\) −618450. −0.730525
\(236\) −454080. −0.530704
\(237\) 0 0
\(238\) 33264.0 0.0380655
\(239\) 370680. 0.419763 0.209882 0.977727i \(-0.432692\pi\)
0.209882 + 0.977727i \(0.432692\pi\)
\(240\) 0 0
\(241\) −561298. −0.622517 −0.311258 0.950325i \(-0.600750\pi\)
−0.311258 + 0.950325i \(0.600750\pi\)
\(242\) 1.71509e6 1.88256
\(243\) 0 0
\(244\) 88352.0 0.0950040
\(245\) −408075. −0.434335
\(246\) 0 0
\(247\) −50600.0 −0.0527726
\(248\) −255232. −0.263515
\(249\) 0 0
\(250\) 62500.0 0.0632456
\(251\) −577152. −0.578237 −0.289119 0.957293i \(-0.593362\pi\)
−0.289119 + 0.957293i \(0.593362\pi\)
\(252\) 0 0
\(253\) 1.52525e6 1.49809
\(254\) −866008. −0.842243
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 651462. 0.615257 0.307628 0.951507i \(-0.400465\pi\)
0.307628 + 0.951507i \(0.400465\pi\)
\(258\) 0 0
\(259\) 3124.00 0.00289375
\(260\) −18400.0 −0.0168805
\(261\) 0 0
\(262\) 978432. 0.880597
\(263\) −917574. −0.817997 −0.408999 0.912535i \(-0.634122\pi\)
−0.408999 + 0.912535i \(0.634122\pi\)
\(264\) 0 0
\(265\) 354150. 0.309794
\(266\) −96800.0 −0.0838825
\(267\) 0 0
\(268\) −395872. −0.336680
\(269\) 735390. 0.619637 0.309818 0.950796i \(-0.399732\pi\)
0.309818 + 0.950796i \(0.399732\pi\)
\(270\) 0 0
\(271\) −1.12131e6 −0.927474 −0.463737 0.885973i \(-0.653492\pi\)
−0.463737 + 0.885973i \(0.653492\pi\)
\(272\) −96768.0 −0.0793066
\(273\) 0 0
\(274\) 958008. 0.770891
\(275\) 480000. 0.382745
\(276\) 0 0
\(277\) −1.66034e6 −1.30016 −0.650082 0.759864i \(-0.725265\pi\)
−0.650082 + 0.759864i \(0.725265\pi\)
\(278\) 123440. 0.0957952
\(279\) 0 0
\(280\) −35200.0 −0.0268317
\(281\) −1.45210e6 −1.09706 −0.548531 0.836130i \(-0.684813\pi\)
−0.548531 + 0.836130i \(0.684813\pi\)
\(282\) 0 0
\(283\) 309014. 0.229357 0.114679 0.993403i \(-0.463416\pi\)
0.114679 + 0.993403i \(0.463416\pi\)
\(284\) −677952. −0.498773
\(285\) 0 0
\(286\) −141312. −0.102156
\(287\) 33924.0 0.0243110
\(288\) 0 0
\(289\) −1.27697e6 −0.899367
\(290\) 561000. 0.391713
\(291\) 0 0
\(292\) −834016. −0.572423
\(293\) 1.59301e6 1.08405 0.542024 0.840363i \(-0.317658\pi\)
0.542024 + 0.840363i \(0.317658\pi\)
\(294\) 0 0
\(295\) −709500. −0.474676
\(296\) −9088.00 −0.00602891
\(297\) 0 0
\(298\) 403800. 0.263406
\(299\) −91356.0 −0.0590961
\(300\) 0 0
\(301\) 110572. 0.0703443
\(302\) 49808.0 0.0314255
\(303\) 0 0
\(304\) 281600. 0.174763
\(305\) 138050. 0.0849741
\(306\) 0 0
\(307\) 1.24726e6 0.755284 0.377642 0.925952i \(-0.376735\pi\)
0.377642 + 0.925952i \(0.376735\pi\)
\(308\) −270336. −0.162378
\(309\) 0 0
\(310\) −398800. −0.235695
\(311\) 665988. 0.390450 0.195225 0.980758i \(-0.437456\pi\)
0.195225 + 0.980758i \(0.437456\pi\)
\(312\) 0 0
\(313\) −591286. −0.341143 −0.170572 0.985345i \(-0.554561\pi\)
−0.170572 + 0.985345i \(0.554561\pi\)
\(314\) −24088.0 −0.0137872
\(315\) 0 0
\(316\) −634240. −0.357302
\(317\) 516342. 0.288595 0.144298 0.989534i \(-0.453908\pi\)
0.144298 + 0.989534i \(0.453908\pi\)
\(318\) 0 0
\(319\) 4.30848e6 2.37054
\(320\) 102400. 0.0559017
\(321\) 0 0
\(322\) −174768. −0.0939339
\(323\) −415800. −0.221757
\(324\) 0 0
\(325\) −28750.0 −0.0150983
\(326\) −2.00346e6 −1.04409
\(327\) 0 0
\(328\) −98688.0 −0.0506500
\(329\) 544236. 0.277203
\(330\) 0 0
\(331\) −3.29577e6 −1.65343 −0.826717 0.562619i \(-0.809794\pi\)
−0.826717 + 0.562619i \(0.809794\pi\)
\(332\) 957216. 0.476612
\(333\) 0 0
\(334\) −2.22103e6 −1.08940
\(335\) −618550. −0.301136
\(336\) 0 0
\(337\) 1.91098e6 0.916602 0.458301 0.888797i \(-0.348458\pi\)
0.458301 + 0.888797i \(0.348458\pi\)
\(338\) −1.47671e6 −0.703077
\(339\) 0 0
\(340\) −151200. −0.0709340
\(341\) −3.06278e6 −1.42636
\(342\) 0 0
\(343\) 728860. 0.334510
\(344\) −321664. −0.146557
\(345\) 0 0
\(346\) −1.66942e6 −0.749677
\(347\) −2.42006e6 −1.07895 −0.539476 0.842001i \(-0.681378\pi\)
−0.539476 + 0.842001i \(0.681378\pi\)
\(348\) 0 0
\(349\) 2.50727e6 1.10189 0.550944 0.834542i \(-0.314268\pi\)
0.550944 + 0.834542i \(0.314268\pi\)
\(350\) −55000.0 −0.0239990
\(351\) 0 0
\(352\) 786432. 0.338302
\(353\) 413166. 0.176477 0.0882384 0.996099i \(-0.471876\pi\)
0.0882384 + 0.996099i \(0.471876\pi\)
\(354\) 0 0
\(355\) −1.05930e6 −0.446116
\(356\) −923040. −0.386007
\(357\) 0 0
\(358\) 209520. 0.0864008
\(359\) −1.73772e6 −0.711613 −0.355806 0.934560i \(-0.615794\pi\)
−0.355806 + 0.934560i \(0.615794\pi\)
\(360\) 0 0
\(361\) −1.26610e6 −0.511328
\(362\) 2.18665e6 0.877016
\(363\) 0 0
\(364\) 16192.0 0.00640541
\(365\) −1.30315e6 −0.511991
\(366\) 0 0
\(367\) 1.16098e6 0.449944 0.224972 0.974365i \(-0.427771\pi\)
0.224972 + 0.974365i \(0.427771\pi\)
\(368\) 508416. 0.195704
\(369\) 0 0
\(370\) −14200.0 −0.00539242
\(371\) −311652. −0.117553
\(372\) 0 0
\(373\) 343754. 0.127931 0.0639655 0.997952i \(-0.479625\pi\)
0.0639655 + 0.997952i \(0.479625\pi\)
\(374\) −1.16122e6 −0.429273
\(375\) 0 0
\(376\) −1.58323e6 −0.577531
\(377\) −258060. −0.0935120
\(378\) 0 0
\(379\) 573140. 0.204957 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(380\) 440000. 0.156312
\(381\) 0 0
\(382\) 1.80811e6 0.633967
\(383\) 2.88055e6 1.00341 0.501704 0.865039i \(-0.332707\pi\)
0.501704 + 0.865039i \(0.332707\pi\)
\(384\) 0 0
\(385\) −422400. −0.145235
\(386\) 1.94238e6 0.663537
\(387\) 0 0
\(388\) −2.31011e6 −0.779029
\(389\) 3.08559e6 1.03387 0.516933 0.856026i \(-0.327074\pi\)
0.516933 + 0.856026i \(0.327074\pi\)
\(390\) 0 0
\(391\) −750708. −0.248330
\(392\) −1.04467e6 −0.343372
\(393\) 0 0
\(394\) −4.04071e6 −1.31135
\(395\) −991000. −0.319581
\(396\) 0 0
\(397\) 885458. 0.281963 0.140981 0.990012i \(-0.454974\pi\)
0.140981 + 0.990012i \(0.454974\pi\)
\(398\) −3.23056e6 −1.02228
\(399\) 0 0
\(400\) 160000. 0.0500000
\(401\) 3.75344e6 1.16565 0.582825 0.812598i \(-0.301947\pi\)
0.582825 + 0.812598i \(0.301947\pi\)
\(402\) 0 0
\(403\) 183448. 0.0562666
\(404\) 2.26013e6 0.688937
\(405\) 0 0
\(406\) −493680. −0.148638
\(407\) −109056. −0.0326335
\(408\) 0 0
\(409\) −1.94653e6 −0.575377 −0.287689 0.957724i \(-0.592887\pi\)
−0.287689 + 0.957724i \(0.592887\pi\)
\(410\) −154200. −0.0453028
\(411\) 0 0
\(412\) 2.23702e6 0.649273
\(413\) 624360. 0.180119
\(414\) 0 0
\(415\) 1.49565e6 0.426295
\(416\) −47104.0 −0.0133452
\(417\) 0 0
\(418\) 3.37920e6 0.945961
\(419\) 2.99166e6 0.832486 0.416243 0.909253i \(-0.363346\pi\)
0.416243 + 0.909253i \(0.363346\pi\)
\(420\) 0 0
\(421\) 3.96660e6 1.09072 0.545360 0.838202i \(-0.316393\pi\)
0.545360 + 0.838202i \(0.316393\pi\)
\(422\) 598208. 0.163520
\(423\) 0 0
\(424\) 906624. 0.244913
\(425\) −236250. −0.0634453
\(426\) 0 0
\(427\) −121484. −0.0322440
\(428\) −1.38269e6 −0.364850
\(429\) 0 0
\(430\) −502600. −0.131085
\(431\) 5.17115e6 1.34089 0.670446 0.741958i \(-0.266103\pi\)
0.670446 + 0.741958i \(0.266103\pi\)
\(432\) 0 0
\(433\) −4.53485e6 −1.16237 −0.581183 0.813773i \(-0.697410\pi\)
−0.581183 + 0.813773i \(0.697410\pi\)
\(434\) 350944. 0.0894362
\(435\) 0 0
\(436\) 3.49520e6 0.880554
\(437\) 2.18460e6 0.547228
\(438\) 0 0
\(439\) −1.08220e6 −0.268007 −0.134004 0.990981i \(-0.542783\pi\)
−0.134004 + 0.990981i \(0.542783\pi\)
\(440\) 1.22880e6 0.302586
\(441\) 0 0
\(442\) 69552.0 0.0169338
\(443\) 1.08079e6 0.261656 0.130828 0.991405i \(-0.458236\pi\)
0.130828 + 0.991405i \(0.458236\pi\)
\(444\) 0 0
\(445\) −1.44225e6 −0.345255
\(446\) −1.77402e6 −0.422301
\(447\) 0 0
\(448\) −90112.0 −0.0212123
\(449\) −2.61783e6 −0.612810 −0.306405 0.951901i \(-0.599126\pi\)
−0.306405 + 0.951901i \(0.599126\pi\)
\(450\) 0 0
\(451\) −1.18426e6 −0.274160
\(452\) 460896. 0.106110
\(453\) 0 0
\(454\) −1.68007e6 −0.382550
\(455\) 25300.0 0.00572917
\(456\) 0 0
\(457\) 1.59046e6 0.356231 0.178115 0.984010i \(-0.443000\pi\)
0.178115 + 0.984010i \(0.443000\pi\)
\(458\) 4.23500e6 0.943387
\(459\) 0 0
\(460\) 794400. 0.175043
\(461\) −4.25470e6 −0.932431 −0.466216 0.884671i \(-0.654383\pi\)
−0.466216 + 0.884671i \(0.654383\pi\)
\(462\) 0 0
\(463\) 3.26605e6 0.708061 0.354031 0.935234i \(-0.384811\pi\)
0.354031 + 0.935234i \(0.384811\pi\)
\(464\) 1.43616e6 0.309676
\(465\) 0 0
\(466\) 5.09378e6 1.08662
\(467\) 601542. 0.127636 0.0638181 0.997962i \(-0.479672\pi\)
0.0638181 + 0.997962i \(0.479672\pi\)
\(468\) 0 0
\(469\) 544324. 0.114268
\(470\) −2.47380e6 −0.516559
\(471\) 0 0
\(472\) −1.81632e6 −0.375264
\(473\) −3.85997e6 −0.793288
\(474\) 0 0
\(475\) 687500. 0.139810
\(476\) 133056. 0.0269164
\(477\) 0 0
\(478\) 1.48272e6 0.296817
\(479\) 4.57932e6 0.911931 0.455966 0.889997i \(-0.349294\pi\)
0.455966 + 0.889997i \(0.349294\pi\)
\(480\) 0 0
\(481\) 6532.00 0.00128731
\(482\) −2.24519e6 −0.440186
\(483\) 0 0
\(484\) 6.86037e6 1.33117
\(485\) −3.60955e6 −0.696785
\(486\) 0 0
\(487\) 7.05226e6 1.34743 0.673714 0.738992i \(-0.264698\pi\)
0.673714 + 0.738992i \(0.264698\pi\)
\(488\) 353408. 0.0671780
\(489\) 0 0
\(490\) −1.63230e6 −0.307121
\(491\) 2.62349e6 0.491106 0.245553 0.969383i \(-0.421030\pi\)
0.245553 + 0.969383i \(0.421030\pi\)
\(492\) 0 0
\(493\) −2.12058e6 −0.392950
\(494\) −202400. −0.0373158
\(495\) 0 0
\(496\) −1.02093e6 −0.186333
\(497\) 932184. 0.169282
\(498\) 0 0
\(499\) −3.61234e6 −0.649437 −0.324719 0.945811i \(-0.605270\pi\)
−0.324719 + 0.945811i \(0.605270\pi\)
\(500\) 250000. 0.0447214
\(501\) 0 0
\(502\) −2.30861e6 −0.408875
\(503\) −9.15629e6 −1.61361 −0.806807 0.590815i \(-0.798806\pi\)
−0.806807 + 0.590815i \(0.798806\pi\)
\(504\) 0 0
\(505\) 3.53145e6 0.616204
\(506\) 6.10099e6 1.05931
\(507\) 0 0
\(508\) −3.46403e6 −0.595556
\(509\) −7.26159e6 −1.24233 −0.621165 0.783679i \(-0.713340\pi\)
−0.621165 + 0.783679i \(0.713340\pi\)
\(510\) 0 0
\(511\) 1.14677e6 0.194279
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) 2.60585e6 0.435052
\(515\) 3.49535e6 0.580728
\(516\) 0 0
\(517\) −1.89988e7 −3.12608
\(518\) 12496.0 0.00204619
\(519\) 0 0
\(520\) −73600.0 −0.0119363
\(521\) −5.81020e6 −0.937771 −0.468886 0.883259i \(-0.655344\pi\)
−0.468886 + 0.883259i \(0.655344\pi\)
\(522\) 0 0
\(523\) −8.17067e6 −1.30618 −0.653090 0.757280i \(-0.726528\pi\)
−0.653090 + 0.757280i \(0.726528\pi\)
\(524\) 3.91373e6 0.622676
\(525\) 0 0
\(526\) −3.67030e6 −0.578411
\(527\) 1.50746e6 0.236440
\(528\) 0 0
\(529\) −2.49215e6 −0.387199
\(530\) 1.41660e6 0.219057
\(531\) 0 0
\(532\) −387200. −0.0593139
\(533\) 70932.0 0.0108149
\(534\) 0 0
\(535\) −2.16045e6 −0.326332
\(536\) −1.58349e6 −0.238069
\(537\) 0 0
\(538\) 2.94156e6 0.438149
\(539\) −1.25361e7 −1.85861
\(540\) 0 0
\(541\) −817378. −0.120069 −0.0600343 0.998196i \(-0.519121\pi\)
−0.0600343 + 0.998196i \(0.519121\pi\)
\(542\) −4.48523e6 −0.655823
\(543\) 0 0
\(544\) −387072. −0.0560783
\(545\) 5.46125e6 0.787591
\(546\) 0 0
\(547\) −3.50750e6 −0.501221 −0.250611 0.968088i \(-0.580631\pi\)
−0.250611 + 0.968088i \(0.580631\pi\)
\(548\) 3.83203e6 0.545102
\(549\) 0 0
\(550\) 1.92000e6 0.270642
\(551\) 6.17100e6 0.865918
\(552\) 0 0
\(553\) 872080. 0.121267
\(554\) −6.64137e6 −0.919355
\(555\) 0 0
\(556\) 493760. 0.0677375
\(557\) −9.61490e6 −1.31313 −0.656563 0.754271i \(-0.727991\pi\)
−0.656563 + 0.754271i \(0.727991\pi\)
\(558\) 0 0
\(559\) 231196. 0.0312933
\(560\) −140800. −0.0189729
\(561\) 0 0
\(562\) −5.80841e6 −0.775740
\(563\) −2.01941e6 −0.268506 −0.134253 0.990947i \(-0.542864\pi\)
−0.134253 + 0.990947i \(0.542864\pi\)
\(564\) 0 0
\(565\) 720150. 0.0949078
\(566\) 1.23606e6 0.162180
\(567\) 0 0
\(568\) −2.71181e6 −0.352686
\(569\) −1.37859e6 −0.178507 −0.0892533 0.996009i \(-0.528448\pi\)
−0.0892533 + 0.996009i \(0.528448\pi\)
\(570\) 0 0
\(571\) 8.54295e6 1.09652 0.548261 0.836307i \(-0.315290\pi\)
0.548261 + 0.836307i \(0.315290\pi\)
\(572\) −565248. −0.0722352
\(573\) 0 0
\(574\) 135696. 0.0171905
\(575\) 1.24125e6 0.156563
\(576\) 0 0
\(577\) −2.31458e6 −0.289423 −0.144711 0.989474i \(-0.546225\pi\)
−0.144711 + 0.989474i \(0.546225\pi\)
\(578\) −5.10789e6 −0.635949
\(579\) 0 0
\(580\) 2.24400e6 0.276983
\(581\) −1.31617e6 −0.161760
\(582\) 0 0
\(583\) 1.08795e7 1.32568
\(584\) −3.33606e6 −0.404764
\(585\) 0 0
\(586\) 6.37202e6 0.766537
\(587\) −928338. −0.111202 −0.0556008 0.998453i \(-0.517707\pi\)
−0.0556008 + 0.998453i \(0.517707\pi\)
\(588\) 0 0
\(589\) −4.38680e6 −0.521026
\(590\) −2.83800e6 −0.335647
\(591\) 0 0
\(592\) −36352.0 −0.00426309
\(593\) 909486. 0.106209 0.0531043 0.998589i \(-0.483088\pi\)
0.0531043 + 0.998589i \(0.483088\pi\)
\(594\) 0 0
\(595\) 207900. 0.0240748
\(596\) 1.61520e6 0.186256
\(597\) 0 0
\(598\) −365424. −0.0417873
\(599\) 8.51136e6 0.969241 0.484621 0.874724i \(-0.338958\pi\)
0.484621 + 0.874724i \(0.338958\pi\)
\(600\) 0 0
\(601\) 6.12498e6 0.691701 0.345851 0.938290i \(-0.387590\pi\)
0.345851 + 0.938290i \(0.387590\pi\)
\(602\) 442288. 0.0497409
\(603\) 0 0
\(604\) 199232. 0.0222212
\(605\) 1.07193e7 1.19064
\(606\) 0 0
\(607\) −4.51646e6 −0.497538 −0.248769 0.968563i \(-0.580026\pi\)
−0.248769 + 0.968563i \(0.580026\pi\)
\(608\) 1.12640e6 0.123576
\(609\) 0 0
\(610\) 552200. 0.0600858
\(611\) 1.13795e6 0.123316
\(612\) 0 0
\(613\) 9.63979e6 1.03614 0.518068 0.855340i \(-0.326651\pi\)
0.518068 + 0.855340i \(0.326651\pi\)
\(614\) 4.98903e6 0.534067
\(615\) 0 0
\(616\) −1.08134e6 −0.114819
\(617\) 9.92650e6 1.04974 0.524872 0.851181i \(-0.324113\pi\)
0.524872 + 0.851181i \(0.324113\pi\)
\(618\) 0 0
\(619\) 7.63322e6 0.800721 0.400360 0.916358i \(-0.368885\pi\)
0.400360 + 0.916358i \(0.368885\pi\)
\(620\) −1.59520e6 −0.166662
\(621\) 0 0
\(622\) 2.66395e6 0.276090
\(623\) 1.26918e6 0.131010
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) −2.36514e6 −0.241225
\(627\) 0 0
\(628\) −96352.0 −0.00974903
\(629\) 53676.0 0.00540946
\(630\) 0 0
\(631\) 1.80314e7 1.80284 0.901418 0.432949i \(-0.142527\pi\)
0.901418 + 0.432949i \(0.142527\pi\)
\(632\) −2.53696e6 −0.252651
\(633\) 0 0
\(634\) 2.06537e6 0.204068
\(635\) −5.41255e6 −0.532681
\(636\) 0 0
\(637\) 750858. 0.0733178
\(638\) 1.72339e7 1.67623
\(639\) 0 0
\(640\) 409600. 0.0395285
\(641\) −9.30190e6 −0.894184 −0.447092 0.894488i \(-0.647540\pi\)
−0.447092 + 0.894488i \(0.647540\pi\)
\(642\) 0 0
\(643\) −1.38332e7 −1.31946 −0.659730 0.751503i \(-0.729329\pi\)
−0.659730 + 0.751503i \(0.729329\pi\)
\(644\) −699072. −0.0664213
\(645\) 0 0
\(646\) −1.66320e6 −0.156806
\(647\) 1.48997e7 1.39932 0.699658 0.714478i \(-0.253336\pi\)
0.699658 + 0.714478i \(0.253336\pi\)
\(648\) 0 0
\(649\) −2.17958e7 −2.03124
\(650\) −115000. −0.0106761
\(651\) 0 0
\(652\) −8.01386e6 −0.738282
\(653\) 1.93306e7 1.77403 0.887016 0.461738i \(-0.152774\pi\)
0.887016 + 0.461738i \(0.152774\pi\)
\(654\) 0 0
\(655\) 6.11520e6 0.556939
\(656\) −394752. −0.0358150
\(657\) 0 0
\(658\) 2.17694e6 0.196012
\(659\) 4.06110e6 0.364276 0.182138 0.983273i \(-0.441698\pi\)
0.182138 + 0.983273i \(0.441698\pi\)
\(660\) 0 0
\(661\) −1.35152e7 −1.20315 −0.601575 0.798816i \(-0.705460\pi\)
−0.601575 + 0.798816i \(0.705460\pi\)
\(662\) −1.31831e7 −1.16915
\(663\) 0 0
\(664\) 3.82886e6 0.337015
\(665\) −605000. −0.0530519
\(666\) 0 0
\(667\) 1.11415e7 0.969678
\(668\) −8.88413e6 −0.770324
\(669\) 0 0
\(670\) −2.47420e6 −0.212935
\(671\) 4.24090e6 0.363623
\(672\) 0 0
\(673\) 1.43520e7 1.22144 0.610722 0.791845i \(-0.290879\pi\)
0.610722 + 0.791845i \(0.290879\pi\)
\(674\) 7.64391e6 0.648136
\(675\) 0 0
\(676\) −5.90683e6 −0.497150
\(677\) −1.89530e6 −0.158930 −0.0794650 0.996838i \(-0.525321\pi\)
−0.0794650 + 0.996838i \(0.525321\pi\)
\(678\) 0 0
\(679\) 3.17640e6 0.264400
\(680\) −604800. −0.0501579
\(681\) 0 0
\(682\) −1.22511e7 −1.00859
\(683\) −2.91641e6 −0.239220 −0.119610 0.992821i \(-0.538164\pi\)
−0.119610 + 0.992821i \(0.538164\pi\)
\(684\) 0 0
\(685\) 5.98755e6 0.487554
\(686\) 2.91544e6 0.236534
\(687\) 0 0
\(688\) −1.28666e6 −0.103631
\(689\) −651636. −0.0522946
\(690\) 0 0
\(691\) 1.44278e7 1.14949 0.574743 0.818334i \(-0.305102\pi\)
0.574743 + 0.818334i \(0.305102\pi\)
\(692\) −6.67766e6 −0.530102
\(693\) 0 0
\(694\) −9.68023e6 −0.762934
\(695\) 771500. 0.0605862
\(696\) 0 0
\(697\) 582876. 0.0454458
\(698\) 1.00291e7 0.779153
\(699\) 0 0
\(700\) −220000. −0.0169698
\(701\) 1.58679e7 1.21962 0.609811 0.792547i \(-0.291246\pi\)
0.609811 + 0.792547i \(0.291246\pi\)
\(702\) 0 0
\(703\) −156200. −0.0119205
\(704\) 3.14573e6 0.239216
\(705\) 0 0
\(706\) 1.65266e6 0.124788
\(707\) −3.10768e6 −0.233823
\(708\) 0 0
\(709\) −301810. −0.0225485 −0.0112743 0.999936i \(-0.503589\pi\)
−0.0112743 + 0.999936i \(0.503589\pi\)
\(710\) −4.23720e6 −0.315452
\(711\) 0 0
\(712\) −3.69216e6 −0.272948
\(713\) −7.92017e6 −0.583459
\(714\) 0 0
\(715\) −883200. −0.0646091
\(716\) 838080. 0.0610946
\(717\) 0 0
\(718\) −6.95088e6 −0.503186
\(719\) −2.12677e7 −1.53426 −0.767130 0.641492i \(-0.778316\pi\)
−0.767130 + 0.641492i \(0.778316\pi\)
\(720\) 0 0
\(721\) −3.07591e6 −0.220361
\(722\) −5.06440e6 −0.361564
\(723\) 0 0
\(724\) 8.74659e6 0.620144
\(725\) 3.50625e6 0.247741
\(726\) 0 0
\(727\) 1.55009e7 1.08773 0.543863 0.839174i \(-0.316961\pi\)
0.543863 + 0.839174i \(0.316961\pi\)
\(728\) 64768.0 0.00452931
\(729\) 0 0
\(730\) −5.21260e6 −0.362032
\(731\) 1.89983e6 0.131499
\(732\) 0 0
\(733\) −1.21850e7 −0.837653 −0.418827 0.908066i \(-0.637559\pi\)
−0.418827 + 0.908066i \(0.637559\pi\)
\(734\) 4.64391e6 0.318159
\(735\) 0 0
\(736\) 2.03366e6 0.138384
\(737\) −1.90019e7 −1.28863
\(738\) 0 0
\(739\) −2.90282e7 −1.95528 −0.977641 0.210282i \(-0.932562\pi\)
−0.977641 + 0.210282i \(0.932562\pi\)
\(740\) −56800.0 −0.00381302
\(741\) 0 0
\(742\) −1.24661e6 −0.0831228
\(743\) −1.61145e7 −1.07089 −0.535445 0.844570i \(-0.679856\pi\)
−0.535445 + 0.844570i \(0.679856\pi\)
\(744\) 0 0
\(745\) 2.52375e6 0.166593
\(746\) 1.37502e6 0.0904609
\(747\) 0 0
\(748\) −4.64486e6 −0.303542
\(749\) 1.90120e6 0.123829
\(750\) 0 0
\(751\) −2.92431e6 −0.189201 −0.0946005 0.995515i \(-0.530157\pi\)
−0.0946005 + 0.995515i \(0.530157\pi\)
\(752\) −6.33293e6 −0.408376
\(753\) 0 0
\(754\) −1.03224e6 −0.0661230
\(755\) 311300. 0.0198752
\(756\) 0 0
\(757\) 2.60325e7 1.65111 0.825557 0.564319i \(-0.190861\pi\)
0.825557 + 0.564319i \(0.190861\pi\)
\(758\) 2.29256e6 0.144926
\(759\) 0 0
\(760\) 1.76000e6 0.110530
\(761\) −1.63263e7 −1.02194 −0.510970 0.859598i \(-0.670714\pi\)
−0.510970 + 0.859598i \(0.670714\pi\)
\(762\) 0 0
\(763\) −4.80590e6 −0.298857
\(764\) 7.23245e6 0.448283
\(765\) 0 0
\(766\) 1.15222e7 0.709517
\(767\) 1.30548e6 0.0801275
\(768\) 0 0
\(769\) 2.58132e7 1.57408 0.787040 0.616902i \(-0.211612\pi\)
0.787040 + 0.616902i \(0.211612\pi\)
\(770\) −1.68960e6 −0.102697
\(771\) 0 0
\(772\) 7.76950e6 0.469191
\(773\) 1.90592e7 1.14725 0.573624 0.819119i \(-0.305537\pi\)
0.573624 + 0.819119i \(0.305537\pi\)
\(774\) 0 0
\(775\) −2.49250e6 −0.149067
\(776\) −9.24045e6 −0.550857
\(777\) 0 0
\(778\) 1.23424e7 0.731054
\(779\) −1.69620e6 −0.100146
\(780\) 0 0
\(781\) −3.25417e7 −1.90903
\(782\) −3.00283e6 −0.175596
\(783\) 0 0
\(784\) −4.17869e6 −0.242801
\(785\) −150550. −0.00871980
\(786\) 0 0
\(787\) −1.73411e7 −0.998021 −0.499011 0.866596i \(-0.666303\pi\)
−0.499011 + 0.866596i \(0.666303\pi\)
\(788\) −1.61628e7 −0.927262
\(789\) 0 0
\(790\) −3.96400e6 −0.225978
\(791\) −633732. −0.0360134
\(792\) 0 0
\(793\) −254012. −0.0143440
\(794\) 3.54183e6 0.199378
\(795\) 0 0
\(796\) −1.29222e7 −0.722862
\(797\) 2.58169e7 1.43965 0.719827 0.694153i \(-0.244221\pi\)
0.719827 + 0.694153i \(0.244221\pi\)
\(798\) 0 0
\(799\) 9.35096e6 0.518190
\(800\) 640000. 0.0353553
\(801\) 0 0
\(802\) 1.50138e7 0.824239
\(803\) −4.00328e7 −2.19092
\(804\) 0 0
\(805\) −1.09230e6 −0.0594090
\(806\) 733792. 0.0397865
\(807\) 0 0
\(808\) 9.04051e6 0.487152
\(809\) −8.88489e6 −0.477288 −0.238644 0.971107i \(-0.576703\pi\)
−0.238644 + 0.971107i \(0.576703\pi\)
\(810\) 0 0
\(811\) −2.46396e7 −1.31547 −0.657735 0.753249i \(-0.728485\pi\)
−0.657735 + 0.753249i \(0.728485\pi\)
\(812\) −1.97472e6 −0.105103
\(813\) 0 0
\(814\) −436224. −0.0230754
\(815\) −1.25216e7 −0.660340
\(816\) 0 0
\(817\) −5.52860e6 −0.289774
\(818\) −7.78612e6 −0.406853
\(819\) 0 0
\(820\) −616800. −0.0320339
\(821\) −1.13768e7 −0.589062 −0.294531 0.955642i \(-0.595163\pi\)
−0.294531 + 0.955642i \(0.595163\pi\)
\(822\) 0 0
\(823\) −1.30783e7 −0.673057 −0.336529 0.941673i \(-0.609253\pi\)
−0.336529 + 0.941673i \(0.609253\pi\)
\(824\) 8.94810e6 0.459106
\(825\) 0 0
\(826\) 2.49744e6 0.127363
\(827\) 3.57188e7 1.81607 0.908037 0.418891i \(-0.137581\pi\)
0.908037 + 0.418891i \(0.137581\pi\)
\(828\) 0 0
\(829\) 1.61880e7 0.818103 0.409052 0.912511i \(-0.365860\pi\)
0.409052 + 0.912511i \(0.365860\pi\)
\(830\) 5.98260e6 0.301436
\(831\) 0 0
\(832\) −188416. −0.00943647
\(833\) 6.17009e6 0.308091
\(834\) 0 0
\(835\) −1.38814e7 −0.688999
\(836\) 1.35168e7 0.668895
\(837\) 0 0
\(838\) 1.19666e7 0.588657
\(839\) 2.55497e7 1.25309 0.626543 0.779387i \(-0.284469\pi\)
0.626543 + 0.779387i \(0.284469\pi\)
\(840\) 0 0
\(841\) 1.09610e7 0.534390
\(842\) 1.58664e7 0.771256
\(843\) 0 0
\(844\) 2.39283e6 0.115626
\(845\) −9.22943e6 −0.444665
\(846\) 0 0
\(847\) −9.43301e6 −0.451795
\(848\) 3.62650e6 0.173180
\(849\) 0 0
\(850\) −945000. −0.0448626
\(851\) −282012. −0.0133488
\(852\) 0 0
\(853\) −2.22953e7 −1.04916 −0.524579 0.851362i \(-0.675777\pi\)
−0.524579 + 0.851362i \(0.675777\pi\)
\(854\) −485936. −0.0228000
\(855\) 0 0
\(856\) −5.53075e6 −0.257988
\(857\) −1.96872e7 −0.915656 −0.457828 0.889041i \(-0.651372\pi\)
−0.457828 + 0.889041i \(0.651372\pi\)
\(858\) 0 0
\(859\) 6.77582e6 0.313313 0.156657 0.987653i \(-0.449928\pi\)
0.156657 + 0.987653i \(0.449928\pi\)
\(860\) −2.01040e6 −0.0926907
\(861\) 0 0
\(862\) 2.06846e7 0.948154
\(863\) 2.63804e7 1.20574 0.602871 0.797839i \(-0.294023\pi\)
0.602871 + 0.797839i \(0.294023\pi\)
\(864\) 0 0
\(865\) −1.04338e7 −0.474138
\(866\) −1.81394e7 −0.821917
\(867\) 0 0
\(868\) 1.40378e6 0.0632410
\(869\) −3.04435e7 −1.36756
\(870\) 0 0
\(871\) 1.13813e6 0.0508332
\(872\) 1.39808e7 0.622645
\(873\) 0 0
\(874\) 8.73840e6 0.386949
\(875\) −343750. −0.0151783
\(876\) 0 0
\(877\) 2.95161e7 1.29587 0.647934 0.761697i \(-0.275633\pi\)
0.647934 + 0.761697i \(0.275633\pi\)
\(878\) −4.32880e6 −0.189510
\(879\) 0 0
\(880\) 4.91520e6 0.213961
\(881\) 1.48565e7 0.644877 0.322438 0.946590i \(-0.395498\pi\)
0.322438 + 0.946590i \(0.395498\pi\)
\(882\) 0 0
\(883\) −1.45340e7 −0.627313 −0.313656 0.949537i \(-0.601554\pi\)
−0.313656 + 0.949537i \(0.601554\pi\)
\(884\) 278208. 0.0119740
\(885\) 0 0
\(886\) 4.32314e6 0.185019
\(887\) 1.72028e7 0.734160 0.367080 0.930189i \(-0.380358\pi\)
0.367080 + 0.930189i \(0.380358\pi\)
\(888\) 0 0
\(889\) 4.76304e6 0.202130
\(890\) −5.76900e6 −0.244132
\(891\) 0 0
\(892\) −7.09610e6 −0.298612
\(893\) −2.72118e7 −1.14190
\(894\) 0 0
\(895\) 1.30950e6 0.0546447
\(896\) −360448. −0.0149994
\(897\) 0 0
\(898\) −1.04713e7 −0.433322
\(899\) −2.23727e7 −0.923249
\(900\) 0 0
\(901\) −5.35475e6 −0.219749
\(902\) −4.73702e6 −0.193860
\(903\) 0 0
\(904\) 1.84358e6 0.0750312
\(905\) 1.36665e7 0.554674
\(906\) 0 0
\(907\) −3.44434e7 −1.39023 −0.695116 0.718897i \(-0.744647\pi\)
−0.695116 + 0.718897i \(0.744647\pi\)
\(908\) −6.72029e6 −0.270504
\(909\) 0 0
\(910\) 101200. 0.00405114
\(911\) 983748. 0.0392724 0.0196362 0.999807i \(-0.493749\pi\)
0.0196362 + 0.999807i \(0.493749\pi\)
\(912\) 0 0
\(913\) 4.59464e7 1.82421
\(914\) 6.36183e6 0.251893
\(915\) 0 0
\(916\) 1.69400e7 0.667075
\(917\) −5.38138e6 −0.211334
\(918\) 0 0
\(919\) 3.08857e7 1.20634 0.603168 0.797614i \(-0.293905\pi\)
0.603168 + 0.797614i \(0.293905\pi\)
\(920\) 3.17760e6 0.123774
\(921\) 0 0
\(922\) −1.70188e7 −0.659328
\(923\) 1.94911e6 0.0753065
\(924\) 0 0
\(925\) −88750.0 −0.00341047
\(926\) 1.30642e7 0.500675
\(927\) 0 0
\(928\) 5.74464e6 0.218974
\(929\) 3.20874e7 1.21982 0.609909 0.792472i \(-0.291206\pi\)
0.609909 + 0.792472i \(0.291206\pi\)
\(930\) 0 0
\(931\) −1.79553e7 −0.678920
\(932\) 2.03751e7 0.768353
\(933\) 0 0
\(934\) 2.40617e6 0.0902524
\(935\) −7.25760e6 −0.271496
\(936\) 0 0
\(937\) 1.52520e7 0.567515 0.283757 0.958896i \(-0.408419\pi\)
0.283757 + 0.958896i \(0.408419\pi\)
\(938\) 2.17730e6 0.0807998
\(939\) 0 0
\(940\) −9.89520e6 −0.365262
\(941\) −3.48166e6 −0.128178 −0.0640889 0.997944i \(-0.520414\pi\)
−0.0640889 + 0.997944i \(0.520414\pi\)
\(942\) 0 0
\(943\) −3.06241e6 −0.112146
\(944\) −7.26528e6 −0.265352
\(945\) 0 0
\(946\) −1.54399e7 −0.560939
\(947\) 2.54010e7 0.920398 0.460199 0.887816i \(-0.347778\pi\)
0.460199 + 0.887816i \(0.347778\pi\)
\(948\) 0 0
\(949\) 2.39780e6 0.0864265
\(950\) 2.75000e6 0.0988607
\(951\) 0 0
\(952\) 532224. 0.0190328
\(953\) 4.97352e7 1.77391 0.886955 0.461856i \(-0.152816\pi\)
0.886955 + 0.461856i \(0.152816\pi\)
\(954\) 0 0
\(955\) 1.13007e7 0.400956
\(956\) 5.93088e6 0.209882
\(957\) 0 0
\(958\) 1.83173e7 0.644833
\(959\) −5.26904e6 −0.185006
\(960\) 0 0
\(961\) −1.27250e7 −0.444477
\(962\) 26128.0 0.000910266 0
\(963\) 0 0
\(964\) −8.98077e6 −0.311258
\(965\) 1.21399e7 0.419658
\(966\) 0 0
\(967\) 3.05173e7 1.04949 0.524747 0.851258i \(-0.324160\pi\)
0.524747 + 0.851258i \(0.324160\pi\)
\(968\) 2.74415e7 0.941280
\(969\) 0 0
\(970\) −1.44382e7 −0.492701
\(971\) −3.19854e7 −1.08869 −0.544344 0.838862i \(-0.683221\pi\)
−0.544344 + 0.838862i \(0.683221\pi\)
\(972\) 0 0
\(973\) −678920. −0.0229899
\(974\) 2.82090e7 0.952776
\(975\) 0 0
\(976\) 1.41363e6 0.0475020
\(977\) −2.90786e6 −0.0974623 −0.0487312 0.998812i \(-0.515518\pi\)
−0.0487312 + 0.998812i \(0.515518\pi\)
\(978\) 0 0
\(979\) −4.43059e7 −1.47742
\(980\) −6.52920e6 −0.217167
\(981\) 0 0
\(982\) 1.04940e7 0.347264
\(983\) −3.49621e7 −1.15402 −0.577010 0.816737i \(-0.695781\pi\)
−0.577010 + 0.816737i \(0.695781\pi\)
\(984\) 0 0
\(985\) −2.52544e7 −0.829368
\(986\) −8.48232e6 −0.277858
\(987\) 0 0
\(988\) −809600. −0.0263863
\(989\) −9.98164e6 −0.324497
\(990\) 0 0
\(991\) 3.00465e6 0.0971874 0.0485937 0.998819i \(-0.484526\pi\)
0.0485937 + 0.998819i \(0.484526\pi\)
\(992\) −4.08371e6 −0.131758
\(993\) 0 0
\(994\) 3.72874e6 0.119700
\(995\) −2.01910e7 −0.646547
\(996\) 0 0
\(997\) 3.20789e7 1.02207 0.511035 0.859560i \(-0.329262\pi\)
0.511035 + 0.859560i \(0.329262\pi\)
\(998\) −1.44494e7 −0.459222
\(999\) 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.f.1.1 1
3.2 odd 2 10.6.a.a.1.1 1
4.3 odd 2 720.6.a.r.1.1 1
5.2 odd 4 450.6.c.o.199.2 2
5.3 odd 4 450.6.c.o.199.1 2
5.4 even 2 450.6.a.h.1.1 1
12.11 even 2 80.6.a.h.1.1 1
15.2 even 4 50.6.b.d.49.1 2
15.8 even 4 50.6.b.d.49.2 2
15.14 odd 2 50.6.a.g.1.1 1
21.20 even 2 490.6.a.j.1.1 1
24.5 odd 2 320.6.a.p.1.1 1
24.11 even 2 320.6.a.a.1.1 1
60.23 odd 4 400.6.c.a.49.2 2
60.47 odd 4 400.6.c.a.49.1 2
60.59 even 2 400.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.a.1.1 1 3.2 odd 2
50.6.a.g.1.1 1 15.14 odd 2
50.6.b.d.49.1 2 15.2 even 4
50.6.b.d.49.2 2 15.8 even 4
80.6.a.h.1.1 1 12.11 even 2
90.6.a.f.1.1 1 1.1 even 1 trivial
320.6.a.a.1.1 1 24.11 even 2
320.6.a.p.1.1 1 24.5 odd 2
400.6.a.a.1.1 1 60.59 even 2
400.6.c.a.49.1 2 60.47 odd 4
400.6.c.a.49.2 2 60.23 odd 4
450.6.a.h.1.1 1 5.4 even 2
450.6.c.o.199.1 2 5.3 odd 4
450.6.c.o.199.2 2 5.2 odd 4
490.6.a.j.1.1 1 21.20 even 2
720.6.a.r.1.1 1 4.3 odd 2