Properties

Label 150.6.a.j.1.1
Level $150$
Weight $6$
Character 150.1
Self dual yes
Analytic conductor $24.058$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.0575729719\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -4.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -4.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -500.000 q^{11} -144.000 q^{12} -288.000 q^{13} -16.0000 q^{14} +256.000 q^{16} +1516.00 q^{17} +324.000 q^{18} -1344.00 q^{19} +36.0000 q^{21} -2000.00 q^{22} -4100.00 q^{23} -576.000 q^{24} -1152.00 q^{26} -729.000 q^{27} -64.0000 q^{28} -2646.00 q^{29} -5612.00 q^{31} +1024.00 q^{32} +4500.00 q^{33} +6064.00 q^{34} +1296.00 q^{36} -7288.00 q^{37} -5376.00 q^{38} +2592.00 q^{39} -18986.0 q^{41} +144.000 q^{42} -2404.00 q^{43} -8000.00 q^{44} -16400.0 q^{46} +8900.00 q^{47} -2304.00 q^{48} -16791.0 q^{49} -13644.0 q^{51} -4608.00 q^{52} +39804.0 q^{53} -2916.00 q^{54} -256.000 q^{56} +12096.0 q^{57} -10584.0 q^{58} -28300.0 q^{59} +18290.0 q^{61} -22448.0 q^{62} -324.000 q^{63} +4096.00 q^{64} +18000.0 q^{66} +65956.0 q^{67} +24256.0 q^{68} +36900.0 q^{69} -28800.0 q^{71} +5184.00 q^{72} -30808.0 q^{73} -29152.0 q^{74} -21504.0 q^{76} +2000.00 q^{77} +10368.0 q^{78} +60228.0 q^{79} +6561.00 q^{81} -75944.0 q^{82} -2468.00 q^{83} +576.000 q^{84} -9616.00 q^{86} +23814.0 q^{87} -32000.0 q^{88} +22678.0 q^{89} +1152.00 q^{91} -65600.0 q^{92} +50508.0 q^{93} +35600.0 q^{94} -9216.00 q^{96} -36968.0 q^{97} -67164.0 q^{98} -40500.0 q^{99} +O(q^{100})\)

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
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) −36.0000 −0.408248
\(7\) −4.00000 −0.0308542 −0.0154271 0.999881i \(-0.504911\pi\)
−0.0154271 + 0.999881i \(0.504911\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −500.000 −1.24591 −0.622957 0.782256i \(-0.714069\pi\)
−0.622957 + 0.782256i \(0.714069\pi\)
\(12\) −144.000 −0.288675
\(13\) −288.000 −0.472644 −0.236322 0.971675i \(-0.575942\pi\)
−0.236322 + 0.971675i \(0.575942\pi\)
\(14\) −16.0000 −0.0218172
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1516.00 1.27226 0.636132 0.771581i \(-0.280534\pi\)
0.636132 + 0.771581i \(0.280534\pi\)
\(18\) 324.000 0.235702
\(19\) −1344.00 −0.854113 −0.427056 0.904225i \(-0.640449\pi\)
−0.427056 + 0.904225i \(0.640449\pi\)
\(20\) 0 0
\(21\) 36.0000 0.0178137
\(22\) −2000.00 −0.880995
\(23\) −4100.00 −1.61609 −0.808043 0.589124i \(-0.799473\pi\)
−0.808043 + 0.589124i \(0.799473\pi\)
\(24\) −576.000 −0.204124
\(25\) 0 0
\(26\) −1152.00 −0.334210
\(27\) −729.000 −0.192450
\(28\) −64.0000 −0.0154271
\(29\) −2646.00 −0.584245 −0.292122 0.956381i \(-0.594361\pi\)
−0.292122 + 0.956381i \(0.594361\pi\)
\(30\) 0 0
\(31\) −5612.00 −1.04885 −0.524425 0.851457i \(-0.675720\pi\)
−0.524425 + 0.851457i \(0.675720\pi\)
\(32\) 1024.00 0.176777
\(33\) 4500.00 0.719329
\(34\) 6064.00 0.899626
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −7288.00 −0.875193 −0.437597 0.899171i \(-0.644170\pi\)
−0.437597 + 0.899171i \(0.644170\pi\)
\(38\) −5376.00 −0.603949
\(39\) 2592.00 0.272881
\(40\) 0 0
\(41\) −18986.0 −1.76390 −0.881950 0.471343i \(-0.843769\pi\)
−0.881950 + 0.471343i \(0.843769\pi\)
\(42\) 144.000 0.0125962
\(43\) −2404.00 −0.198273 −0.0991364 0.995074i \(-0.531608\pi\)
−0.0991364 + 0.995074i \(0.531608\pi\)
\(44\) −8000.00 −0.622957
\(45\) 0 0
\(46\) −16400.0 −1.14274
\(47\) 8900.00 0.587686 0.293843 0.955854i \(-0.405066\pi\)
0.293843 + 0.955854i \(0.405066\pi\)
\(48\) −2304.00 −0.144338
\(49\) −16791.0 −0.999048
\(50\) 0 0
\(51\) −13644.0 −0.734541
\(52\) −4608.00 −0.236322
\(53\) 39804.0 1.94642 0.973211 0.229913i \(-0.0738443\pi\)
0.973211 + 0.229913i \(0.0738443\pi\)
\(54\) −2916.00 −0.136083
\(55\) 0 0
\(56\) −256.000 −0.0109086
\(57\) 12096.0 0.493122
\(58\) −10584.0 −0.413123
\(59\) −28300.0 −1.05842 −0.529208 0.848492i \(-0.677511\pi\)
−0.529208 + 0.848492i \(0.677511\pi\)
\(60\) 0 0
\(61\) 18290.0 0.629345 0.314673 0.949200i \(-0.398105\pi\)
0.314673 + 0.949200i \(0.398105\pi\)
\(62\) −22448.0 −0.741649
\(63\) −324.000 −0.0102847
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 18000.0 0.508643
\(67\) 65956.0 1.79501 0.897506 0.441002i \(-0.145377\pi\)
0.897506 + 0.441002i \(0.145377\pi\)
\(68\) 24256.0 0.636132
\(69\) 36900.0 0.933047
\(70\) 0 0
\(71\) −28800.0 −0.678026 −0.339013 0.940782i \(-0.610093\pi\)
−0.339013 + 0.940782i \(0.610093\pi\)
\(72\) 5184.00 0.117851
\(73\) −30808.0 −0.676638 −0.338319 0.941031i \(-0.609858\pi\)
−0.338319 + 0.941031i \(0.609858\pi\)
\(74\) −29152.0 −0.618855
\(75\) 0 0
\(76\) −21504.0 −0.427056
\(77\) 2000.00 0.0384418
\(78\) 10368.0 0.192956
\(79\) 60228.0 1.08575 0.542876 0.839813i \(-0.317335\pi\)
0.542876 + 0.839813i \(0.317335\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −75944.0 −1.24727
\(83\) −2468.00 −0.0393233 −0.0196616 0.999807i \(-0.506259\pi\)
−0.0196616 + 0.999807i \(0.506259\pi\)
\(84\) 576.000 0.00890685
\(85\) 0 0
\(86\) −9616.00 −0.140200
\(87\) 23814.0 0.337314
\(88\) −32000.0 −0.440497
\(89\) 22678.0 0.303480 0.151740 0.988420i \(-0.451512\pi\)
0.151740 + 0.988420i \(0.451512\pi\)
\(90\) 0 0
\(91\) 1152.00 0.0145831
\(92\) −65600.0 −0.808043
\(93\) 50508.0 0.605554
\(94\) 35600.0 0.415557
\(95\) 0 0
\(96\) −9216.00 −0.102062
\(97\) −36968.0 −0.398930 −0.199465 0.979905i \(-0.563920\pi\)
−0.199465 + 0.979905i \(0.563920\pi\)
\(98\) −67164.0 −0.706434
\(99\) −40500.0 −0.415305
\(100\) 0 0
\(101\) 167918. 1.63792 0.818962 0.573848i \(-0.194550\pi\)
0.818962 + 0.573848i \(0.194550\pi\)
\(102\) −54576.0 −0.519399
\(103\) −154364. −1.43368 −0.716841 0.697236i \(-0.754413\pi\)
−0.716841 + 0.697236i \(0.754413\pi\)
\(104\) −18432.0 −0.167105
\(105\) 0 0
\(106\) 159216. 1.37633
\(107\) 136788. 1.15502 0.577509 0.816385i \(-0.304025\pi\)
0.577509 + 0.816385i \(0.304025\pi\)
\(108\) −11664.0 −0.0962250
\(109\) −53810.0 −0.433807 −0.216904 0.976193i \(-0.569596\pi\)
−0.216904 + 0.976193i \(0.569596\pi\)
\(110\) 0 0
\(111\) 65592.0 0.505293
\(112\) −1024.00 −0.00771356
\(113\) 82692.0 0.609211 0.304605 0.952479i \(-0.401475\pi\)
0.304605 + 0.952479i \(0.401475\pi\)
\(114\) 48384.0 0.348690
\(115\) 0 0
\(116\) −42336.0 −0.292122
\(117\) −23328.0 −0.157548
\(118\) −113200. −0.748413
\(119\) −6064.00 −0.0392547
\(120\) 0 0
\(121\) 88949.0 0.552303
\(122\) 73160.0 0.445014
\(123\) 170874. 1.01839
\(124\) −89792.0 −0.524425
\(125\) 0 0
\(126\) −1296.00 −0.00727241
\(127\) −211780. −1.16513 −0.582567 0.812783i \(-0.697952\pi\)
−0.582567 + 0.812783i \(0.697952\pi\)
\(128\) 16384.0 0.0883883
\(129\) 21636.0 0.114473
\(130\) 0 0
\(131\) 169500. 0.862962 0.431481 0.902122i \(-0.357991\pi\)
0.431481 + 0.902122i \(0.357991\pi\)
\(132\) 72000.0 0.359665
\(133\) 5376.00 0.0263530
\(134\) 263824. 1.26927
\(135\) 0 0
\(136\) 97024.0 0.449813
\(137\) 252036. 1.14726 0.573629 0.819115i \(-0.305535\pi\)
0.573629 + 0.819115i \(0.305535\pi\)
\(138\) 147600. 0.659764
\(139\) 192016. 0.842947 0.421474 0.906841i \(-0.361513\pi\)
0.421474 + 0.906841i \(0.361513\pi\)
\(140\) 0 0
\(141\) −80100.0 −0.339301
\(142\) −115200. −0.479437
\(143\) 144000. 0.588874
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) −123232. −0.478455
\(147\) 151119. 0.576801
\(148\) −116608. −0.437597
\(149\) 235694. 0.869727 0.434863 0.900496i \(-0.356797\pi\)
0.434863 + 0.900496i \(0.356797\pi\)
\(150\) 0 0
\(151\) −371492. −1.32589 −0.662944 0.748669i \(-0.730693\pi\)
−0.662944 + 0.748669i \(0.730693\pi\)
\(152\) −86016.0 −0.301975
\(153\) 122796. 0.424088
\(154\) 8000.00 0.0271824
\(155\) 0 0
\(156\) 41472.0 0.136441
\(157\) 264952. 0.857863 0.428932 0.903337i \(-0.358890\pi\)
0.428932 + 0.903337i \(0.358890\pi\)
\(158\) 240912. 0.767743
\(159\) −358236. −1.12377
\(160\) 0 0
\(161\) 16400.0 0.0498631
\(162\) 26244.0 0.0785674
\(163\) −403124. −1.18842 −0.594210 0.804310i \(-0.702535\pi\)
−0.594210 + 0.804310i \(0.702535\pi\)
\(164\) −303776. −0.881950
\(165\) 0 0
\(166\) −9872.00 −0.0278058
\(167\) −261900. −0.726682 −0.363341 0.931656i \(-0.618364\pi\)
−0.363341 + 0.931656i \(0.618364\pi\)
\(168\) 2304.00 0.00629810
\(169\) −288349. −0.776608
\(170\) 0 0
\(171\) −108864. −0.284704
\(172\) −38464.0 −0.0991364
\(173\) 326228. 0.828716 0.414358 0.910114i \(-0.364006\pi\)
0.414358 + 0.910114i \(0.364006\pi\)
\(174\) 95256.0 0.238517
\(175\) 0 0
\(176\) −128000. −0.311479
\(177\) 254700. 0.611077
\(178\) 90712.0 0.214593
\(179\) −109516. −0.255473 −0.127736 0.991808i \(-0.540771\pi\)
−0.127736 + 0.991808i \(0.540771\pi\)
\(180\) 0 0
\(181\) −53146.0 −0.120580 −0.0602898 0.998181i \(-0.519202\pi\)
−0.0602898 + 0.998181i \(0.519202\pi\)
\(182\) 4608.00 0.0103118
\(183\) −164610. −0.363353
\(184\) −262400. −0.571372
\(185\) 0 0
\(186\) 202032. 0.428191
\(187\) −758000. −1.58513
\(188\) 142400. 0.293843
\(189\) 2916.00 0.00593790
\(190\) 0 0
\(191\) 232056. 0.460267 0.230133 0.973159i \(-0.426084\pi\)
0.230133 + 0.973159i \(0.426084\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −1.03067e6 −1.99172 −0.995858 0.0909274i \(-0.971017\pi\)
−0.995858 + 0.0909274i \(0.971017\pi\)
\(194\) −147872. −0.282086
\(195\) 0 0
\(196\) −268656. −0.499524
\(197\) −522796. −0.959769 −0.479884 0.877332i \(-0.659321\pi\)
−0.479884 + 0.877332i \(0.659321\pi\)
\(198\) −162000. −0.293665
\(199\) −215292. −0.385385 −0.192693 0.981259i \(-0.561722\pi\)
−0.192693 + 0.981259i \(0.561722\pi\)
\(200\) 0 0
\(201\) −593604. −1.03635
\(202\) 671672. 1.15819
\(203\) 10584.0 0.0180264
\(204\) −218304. −0.367271
\(205\) 0 0
\(206\) −617456. −1.01377
\(207\) −332100. −0.538695
\(208\) −73728.0 −0.118161
\(209\) 672000. 1.06415
\(210\) 0 0
\(211\) −1.03008e6 −1.59281 −0.796407 0.604762i \(-0.793268\pi\)
−0.796407 + 0.604762i \(0.793268\pi\)
\(212\) 636864. 0.973211
\(213\) 259200. 0.391459
\(214\) 547152. 0.816721
\(215\) 0 0
\(216\) −46656.0 −0.0680414
\(217\) 22448.0 0.0323615
\(218\) −215240. −0.306748
\(219\) 277272. 0.390657
\(220\) 0 0
\(221\) −436608. −0.601327
\(222\) 262368. 0.357296
\(223\) 456020. 0.614075 0.307038 0.951697i \(-0.400662\pi\)
0.307038 + 0.951697i \(0.400662\pi\)
\(224\) −4096.00 −0.00545431
\(225\) 0 0
\(226\) 330768. 0.430777
\(227\) −434252. −0.559342 −0.279671 0.960096i \(-0.590225\pi\)
−0.279671 + 0.960096i \(0.590225\pi\)
\(228\) 193536. 0.246561
\(229\) −722710. −0.910700 −0.455350 0.890313i \(-0.650486\pi\)
−0.455350 + 0.890313i \(0.650486\pi\)
\(230\) 0 0
\(231\) −18000.0 −0.0221944
\(232\) −169344. −0.206562
\(233\) −565348. −0.682223 −0.341111 0.940023i \(-0.610803\pi\)
−0.341111 + 0.940023i \(0.610803\pi\)
\(234\) −93312.0 −0.111403
\(235\) 0 0
\(236\) −452800. −0.529208
\(237\) −542052. −0.626859
\(238\) −24256.0 −0.0277573
\(239\) 324904. 0.367926 0.183963 0.982933i \(-0.441107\pi\)
0.183963 + 0.982933i \(0.441107\pi\)
\(240\) 0 0
\(241\) 915262. 1.01509 0.507543 0.861626i \(-0.330554\pi\)
0.507543 + 0.861626i \(0.330554\pi\)
\(242\) 355796. 0.390537
\(243\) −59049.0 −0.0641500
\(244\) 292640. 0.314673
\(245\) 0 0
\(246\) 683496. 0.720109
\(247\) 387072. 0.403691
\(248\) −359168. −0.370825
\(249\) 22212.0 0.0227033
\(250\) 0 0
\(251\) 1.36708e6 1.36965 0.684823 0.728709i \(-0.259879\pi\)
0.684823 + 0.728709i \(0.259879\pi\)
\(252\) −5184.00 −0.00514237
\(253\) 2.05000e6 2.01350
\(254\) −847120. −0.823874
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 892932. 0.843307 0.421653 0.906757i \(-0.361450\pi\)
0.421653 + 0.906757i \(0.361450\pi\)
\(258\) 86544.0 0.0809446
\(259\) 29152.0 0.0270034
\(260\) 0 0
\(261\) −214326. −0.194748
\(262\) 678000. 0.610206
\(263\) −1.86650e6 −1.66394 −0.831972 0.554818i \(-0.812788\pi\)
−0.831972 + 0.554818i \(0.812788\pi\)
\(264\) 288000. 0.254321
\(265\) 0 0
\(266\) 21504.0 0.0186344
\(267\) −204102. −0.175214
\(268\) 1.05530e6 0.897506
\(269\) −1.37227e6 −1.15627 −0.578133 0.815943i \(-0.696218\pi\)
−0.578133 + 0.815943i \(0.696218\pi\)
\(270\) 0 0
\(271\) 458644. 0.379361 0.189680 0.981846i \(-0.439255\pi\)
0.189680 + 0.981846i \(0.439255\pi\)
\(272\) 388096. 0.318066
\(273\) −10368.0 −0.00841954
\(274\) 1.00814e6 0.811234
\(275\) 0 0
\(276\) 590400. 0.466524
\(277\) −985408. −0.771643 −0.385822 0.922573i \(-0.626082\pi\)
−0.385822 + 0.922573i \(0.626082\pi\)
\(278\) 768064. 0.596054
\(279\) −454572. −0.349617
\(280\) 0 0
\(281\) 165798. 0.125260 0.0626302 0.998037i \(-0.480051\pi\)
0.0626302 + 0.998037i \(0.480051\pi\)
\(282\) −320400. −0.239922
\(283\) −1.66471e6 −1.23558 −0.617792 0.786342i \(-0.711972\pi\)
−0.617792 + 0.786342i \(0.711972\pi\)
\(284\) −460800. −0.339013
\(285\) 0 0
\(286\) 576000. 0.416397
\(287\) 75944.0 0.0544238
\(288\) 82944.0 0.0589256
\(289\) 878399. 0.618653
\(290\) 0 0
\(291\) 332712. 0.230322
\(292\) −492928. −0.338319
\(293\) 2.55104e6 1.73600 0.867998 0.496567i \(-0.165406\pi\)
0.867998 + 0.496567i \(0.165406\pi\)
\(294\) 604476. 0.407860
\(295\) 0 0
\(296\) −466432. −0.309428
\(297\) 364500. 0.239776
\(298\) 942776. 0.614990
\(299\) 1.18080e6 0.763833
\(300\) 0 0
\(301\) 9616.00 0.00611756
\(302\) −1.48597e6 −0.937545
\(303\) −1.51126e6 −0.945656
\(304\) −344064. −0.213528
\(305\) 0 0
\(306\) 491184. 0.299875
\(307\) 736020. 0.445701 0.222851 0.974853i \(-0.428464\pi\)
0.222851 + 0.974853i \(0.428464\pi\)
\(308\) 32000.0 0.0192209
\(309\) 1.38928e6 0.827737
\(310\) 0 0
\(311\) 1.71660e6 1.00639 0.503197 0.864172i \(-0.332157\pi\)
0.503197 + 0.864172i \(0.332157\pi\)
\(312\) 165888. 0.0964780
\(313\) 2.83851e6 1.63768 0.818842 0.574020i \(-0.194617\pi\)
0.818842 + 0.574020i \(0.194617\pi\)
\(314\) 1.05981e6 0.606601
\(315\) 0 0
\(316\) 963648. 0.542876
\(317\) −1.27605e6 −0.713215 −0.356607 0.934254i \(-0.616067\pi\)
−0.356607 + 0.934254i \(0.616067\pi\)
\(318\) −1.43294e6 −0.794624
\(319\) 1.32300e6 0.727919
\(320\) 0 0
\(321\) −1.23109e6 −0.666850
\(322\) 65600.0 0.0352585
\(323\) −2.03750e6 −1.08666
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) −1.61250e6 −0.840339
\(327\) 484290. 0.250459
\(328\) −1.21510e6 −0.623633
\(329\) −35600.0 −0.0181326
\(330\) 0 0
\(331\) 443992. 0.222744 0.111372 0.993779i \(-0.464476\pi\)
0.111372 + 0.993779i \(0.464476\pi\)
\(332\) −39488.0 −0.0196616
\(333\) −590328. −0.291731
\(334\) −1.04760e6 −0.513842
\(335\) 0 0
\(336\) 9216.00 0.00445343
\(337\) 2.71326e6 1.30142 0.650708 0.759328i \(-0.274472\pi\)
0.650708 + 0.759328i \(0.274472\pi\)
\(338\) −1.15340e6 −0.549145
\(339\) −744228. −0.351728
\(340\) 0 0
\(341\) 2.80600e6 1.30678
\(342\) −435456. −0.201316
\(343\) 134392. 0.0616791
\(344\) −153856. −0.0701001
\(345\) 0 0
\(346\) 1.30491e6 0.585991
\(347\) −1.31051e6 −0.584273 −0.292137 0.956377i \(-0.594366\pi\)
−0.292137 + 0.956377i \(0.594366\pi\)
\(348\) 381024. 0.168657
\(349\) −298910. −0.131364 −0.0656821 0.997841i \(-0.520922\pi\)
−0.0656821 + 0.997841i \(0.520922\pi\)
\(350\) 0 0
\(351\) 209952. 0.0909604
\(352\) −512000. −0.220249
\(353\) 737996. 0.315223 0.157611 0.987501i \(-0.449621\pi\)
0.157611 + 0.987501i \(0.449621\pi\)
\(354\) 1.01880e6 0.432097
\(355\) 0 0
\(356\) 362848. 0.151740
\(357\) 54576.0 0.0226637
\(358\) −438064. −0.180647
\(359\) −2.34074e6 −0.958557 −0.479278 0.877663i \(-0.659102\pi\)
−0.479278 + 0.877663i \(0.659102\pi\)
\(360\) 0 0
\(361\) −669763. −0.270491
\(362\) −212584. −0.0852627
\(363\) −800541. −0.318872
\(364\) 18432.0 0.00729154
\(365\) 0 0
\(366\) −658440. −0.256929
\(367\) 127292. 0.0493328 0.0246664 0.999696i \(-0.492148\pi\)
0.0246664 + 0.999696i \(0.492148\pi\)
\(368\) −1.04960e6 −0.404021
\(369\) −1.53787e6 −0.587966
\(370\) 0 0
\(371\) −159216. −0.0600554
\(372\) 808128. 0.302777
\(373\) 4.03870e6 1.50303 0.751517 0.659713i \(-0.229322\pi\)
0.751517 + 0.659713i \(0.229322\pi\)
\(374\) −3.03200e6 −1.12086
\(375\) 0 0
\(376\) 569600. 0.207778
\(377\) 762048. 0.276140
\(378\) 11664.0 0.00419873
\(379\) 1.01214e6 0.361944 0.180972 0.983488i \(-0.442076\pi\)
0.180972 + 0.983488i \(0.442076\pi\)
\(380\) 0 0
\(381\) 1.90602e6 0.672690
\(382\) 928224. 0.325458
\(383\) −2.37610e6 −0.827690 −0.413845 0.910347i \(-0.635814\pi\)
−0.413845 + 0.910347i \(0.635814\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −4.12269e6 −1.40836
\(387\) −194724. −0.0660910
\(388\) −591488. −0.199465
\(389\) 1.42497e6 0.477456 0.238728 0.971087i \(-0.423270\pi\)
0.238728 + 0.971087i \(0.423270\pi\)
\(390\) 0 0
\(391\) −6.21560e6 −2.05609
\(392\) −1.07462e6 −0.353217
\(393\) −1.52550e6 −0.498231
\(394\) −2.09118e6 −0.678659
\(395\) 0 0
\(396\) −648000. −0.207652
\(397\) 1.69345e6 0.539257 0.269628 0.962964i \(-0.413099\pi\)
0.269628 + 0.962964i \(0.413099\pi\)
\(398\) −861168. −0.272509
\(399\) −48384.0 −0.0152149
\(400\) 0 0
\(401\) −2.84501e6 −0.883532 −0.441766 0.897130i \(-0.645648\pi\)
−0.441766 + 0.897130i \(0.645648\pi\)
\(402\) −2.37442e6 −0.732810
\(403\) 1.61626e6 0.495733
\(404\) 2.68669e6 0.818962
\(405\) 0 0
\(406\) 42336.0 0.0127466
\(407\) 3.64400e6 1.09042
\(408\) −873216. −0.259700
\(409\) −1.89069e6 −0.558873 −0.279436 0.960164i \(-0.590148\pi\)
−0.279436 + 0.960164i \(0.590148\pi\)
\(410\) 0 0
\(411\) −2.26832e6 −0.662370
\(412\) −2.46982e6 −0.716841
\(413\) 113200. 0.0326566
\(414\) −1.32840e6 −0.380915
\(415\) 0 0
\(416\) −294912. −0.0835524
\(417\) −1.72814e6 −0.486676
\(418\) 2.68800e6 0.752469
\(419\) 4.60930e6 1.28263 0.641313 0.767280i \(-0.278390\pi\)
0.641313 + 0.767280i \(0.278390\pi\)
\(420\) 0 0
\(421\) −6.04151e6 −1.66127 −0.830635 0.556817i \(-0.812022\pi\)
−0.830635 + 0.556817i \(0.812022\pi\)
\(422\) −4.12032e6 −1.12629
\(423\) 720900. 0.195895
\(424\) 2.54746e6 0.688164
\(425\) 0 0
\(426\) 1.03680e6 0.276803
\(427\) −73160.0 −0.0194180
\(428\) 2.18861e6 0.577509
\(429\) −1.29600e6 −0.339987
\(430\) 0 0
\(431\) −3800.00 −0.000985350 0 −0.000492675 1.00000i \(-0.500157\pi\)
−0.000492675 1.00000i \(0.500157\pi\)
\(432\) −186624. −0.0481125
\(433\) 250736. 0.0642683 0.0321342 0.999484i \(-0.489770\pi\)
0.0321342 + 0.999484i \(0.489770\pi\)
\(434\) 89792.0 0.0228830
\(435\) 0 0
\(436\) −860960. −0.216904
\(437\) 5.51040e6 1.38032
\(438\) 1.10909e6 0.276236
\(439\) −3.58873e6 −0.888750 −0.444375 0.895841i \(-0.646574\pi\)
−0.444375 + 0.895841i \(0.646574\pi\)
\(440\) 0 0
\(441\) −1.36007e6 −0.333016
\(442\) −1.74643e6 −0.425203
\(443\) −1.41479e6 −0.342517 −0.171258 0.985226i \(-0.554783\pi\)
−0.171258 + 0.985226i \(0.554783\pi\)
\(444\) 1.04947e6 0.252647
\(445\) 0 0
\(446\) 1.82408e6 0.434217
\(447\) −2.12125e6 −0.502137
\(448\) −16384.0 −0.00385678
\(449\) −829806. −0.194250 −0.0971249 0.995272i \(-0.530965\pi\)
−0.0971249 + 0.995272i \(0.530965\pi\)
\(450\) 0 0
\(451\) 9.49300e6 2.19767
\(452\) 1.32307e6 0.304605
\(453\) 3.34343e6 0.765502
\(454\) −1.73701e6 −0.395514
\(455\) 0 0
\(456\) 774144. 0.174345
\(457\) 4.68198e6 1.04867 0.524335 0.851512i \(-0.324314\pi\)
0.524335 + 0.851512i \(0.324314\pi\)
\(458\) −2.89084e6 −0.643962
\(459\) −1.10516e6 −0.244847
\(460\) 0 0
\(461\) −141930. −0.0311044 −0.0155522 0.999879i \(-0.504951\pi\)
−0.0155522 + 0.999879i \(0.504951\pi\)
\(462\) −72000.0 −0.0156938
\(463\) 727476. 0.157713 0.0788563 0.996886i \(-0.474873\pi\)
0.0788563 + 0.996886i \(0.474873\pi\)
\(464\) −677376. −0.146061
\(465\) 0 0
\(466\) −2.26139e6 −0.482404
\(467\) −4.47640e6 −0.949809 −0.474905 0.880037i \(-0.657517\pi\)
−0.474905 + 0.880037i \(0.657517\pi\)
\(468\) −373248. −0.0787740
\(469\) −263824. −0.0553837
\(470\) 0 0
\(471\) −2.38457e6 −0.495288
\(472\) −1.81120e6 −0.374207
\(473\) 1.20200e6 0.247031
\(474\) −2.16821e6 −0.443256
\(475\) 0 0
\(476\) −97024.0 −0.0196274
\(477\) 3.22412e6 0.648807
\(478\) 1.29962e6 0.260163
\(479\) −1.32718e6 −0.264297 −0.132149 0.991230i \(-0.542188\pi\)
−0.132149 + 0.991230i \(0.542188\pi\)
\(480\) 0 0
\(481\) 2.09894e6 0.413655
\(482\) 3.66105e6 0.717774
\(483\) −147600. −0.0287885
\(484\) 1.42318e6 0.276152
\(485\) 0 0
\(486\) −236196. −0.0453609
\(487\) −4.11647e6 −0.786507 −0.393253 0.919430i \(-0.628650\pi\)
−0.393253 + 0.919430i \(0.628650\pi\)
\(488\) 1.17056e6 0.222507
\(489\) 3.62812e6 0.686134
\(490\) 0 0
\(491\) −6.12316e6 −1.14623 −0.573115 0.819475i \(-0.694265\pi\)
−0.573115 + 0.819475i \(0.694265\pi\)
\(492\) 2.73398e6 0.509194
\(493\) −4.01134e6 −0.743313
\(494\) 1.54829e6 0.285453
\(495\) 0 0
\(496\) −1.43667e6 −0.262213
\(497\) 115200. 0.0209200
\(498\) 88848.0 0.0160537
\(499\) −7.90490e6 −1.42117 −0.710584 0.703613i \(-0.751569\pi\)
−0.710584 + 0.703613i \(0.751569\pi\)
\(500\) 0 0
\(501\) 2.35710e6 0.419550
\(502\) 5.46830e6 0.968486
\(503\) 3.97628e6 0.700741 0.350370 0.936611i \(-0.386056\pi\)
0.350370 + 0.936611i \(0.386056\pi\)
\(504\) −20736.0 −0.00363621
\(505\) 0 0
\(506\) 8.20000e6 1.42376
\(507\) 2.59514e6 0.448375
\(508\) −3.38848e6 −0.582567
\(509\) 781914. 0.133772 0.0668859 0.997761i \(-0.478694\pi\)
0.0668859 + 0.997761i \(0.478694\pi\)
\(510\) 0 0
\(511\) 123232. 0.0208772
\(512\) 262144. 0.0441942
\(513\) 979776. 0.164374
\(514\) 3.57173e6 0.596308
\(515\) 0 0
\(516\) 346176. 0.0572365
\(517\) −4.45000e6 −0.732207
\(518\) 116608. 0.0190943
\(519\) −2.93605e6 −0.478460
\(520\) 0 0
\(521\) 5.82694e6 0.940472 0.470236 0.882541i \(-0.344169\pi\)
0.470236 + 0.882541i \(0.344169\pi\)
\(522\) −857304. −0.137708
\(523\) −9.78938e6 −1.56495 −0.782476 0.622681i \(-0.786043\pi\)
−0.782476 + 0.622681i \(0.786043\pi\)
\(524\) 2.71200e6 0.431481
\(525\) 0 0
\(526\) −7.46600e6 −1.17659
\(527\) −8.50779e6 −1.33441
\(528\) 1.15200e6 0.179832
\(529\) 1.03737e7 1.61173
\(530\) 0 0
\(531\) −2.29230e6 −0.352805
\(532\) 86016.0 0.0131765
\(533\) 5.46797e6 0.833696
\(534\) −816408. −0.123895
\(535\) 0 0
\(536\) 4.22118e6 0.634633
\(537\) 985644. 0.147497
\(538\) −5.48906e6 −0.817603
\(539\) 8.39550e6 1.24473
\(540\) 0 0
\(541\) 4.76059e6 0.699307 0.349653 0.936879i \(-0.386299\pi\)
0.349653 + 0.936879i \(0.386299\pi\)
\(542\) 1.83458e6 0.268249
\(543\) 478314. 0.0696167
\(544\) 1.55238e6 0.224906
\(545\) 0 0
\(546\) −41472.0 −0.00595351
\(547\) −1.16595e6 −0.166614 −0.0833069 0.996524i \(-0.526548\pi\)
−0.0833069 + 0.996524i \(0.526548\pi\)
\(548\) 4.03258e6 0.573629
\(549\) 1.48149e6 0.209782
\(550\) 0 0
\(551\) 3.55622e6 0.499011
\(552\) 2.36160e6 0.329882
\(553\) −240912. −0.0335001
\(554\) −3.94163e6 −0.545634
\(555\) 0 0
\(556\) 3.07226e6 0.421474
\(557\) −1.61293e6 −0.220282 −0.110141 0.993916i \(-0.535130\pi\)
−0.110141 + 0.993916i \(0.535130\pi\)
\(558\) −1.81829e6 −0.247216
\(559\) 692352. 0.0937125
\(560\) 0 0
\(561\) 6.82200e6 0.915176
\(562\) 663192. 0.0885724
\(563\) 3.40603e6 0.452874 0.226437 0.974026i \(-0.427292\pi\)
0.226437 + 0.974026i \(0.427292\pi\)
\(564\) −1.28160e6 −0.169650
\(565\) 0 0
\(566\) −6.65883e6 −0.873689
\(567\) −26244.0 −0.00342825
\(568\) −1.84320e6 −0.239719
\(569\) −1.44009e7 −1.86470 −0.932350 0.361557i \(-0.882245\pi\)
−0.932350 + 0.361557i \(0.882245\pi\)
\(570\) 0 0
\(571\) 4.74772e6 0.609389 0.304695 0.952450i \(-0.401446\pi\)
0.304695 + 0.952450i \(0.401446\pi\)
\(572\) 2.30400e6 0.294437
\(573\) −2.08850e6 −0.265735
\(574\) 303776. 0.0384834
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) −1.09094e7 −1.36415 −0.682074 0.731283i \(-0.738922\pi\)
−0.682074 + 0.731283i \(0.738922\pi\)
\(578\) 3.51360e6 0.437454
\(579\) 9.27605e6 1.14992
\(580\) 0 0
\(581\) 9872.00 0.00121329
\(582\) 1.33085e6 0.162862
\(583\) −1.99020e7 −2.42508
\(584\) −1.97171e6 −0.239228
\(585\) 0 0
\(586\) 1.02042e7 1.22754
\(587\) 8.53223e6 1.02204 0.511019 0.859569i \(-0.329268\pi\)
0.511019 + 0.859569i \(0.329268\pi\)
\(588\) 2.41790e6 0.288400
\(589\) 7.54253e6 0.895836
\(590\) 0 0
\(591\) 4.70516e6 0.554123
\(592\) −1.86573e6 −0.218798
\(593\) −4.63182e6 −0.540897 −0.270449 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(594\) 1.45800e6 0.169548
\(595\) 0 0
\(596\) 3.77110e6 0.434863
\(597\) 1.93763e6 0.222502
\(598\) 4.72320e6 0.540111
\(599\) 6.27598e6 0.714684 0.357342 0.933974i \(-0.383683\pi\)
0.357342 + 0.933974i \(0.383683\pi\)
\(600\) 0 0
\(601\) 7.71988e6 0.871815 0.435907 0.899992i \(-0.356428\pi\)
0.435907 + 0.899992i \(0.356428\pi\)
\(602\) 38464.0 0.00432577
\(603\) 5.34244e6 0.598337
\(604\) −5.94387e6 −0.662944
\(605\) 0 0
\(606\) −6.04505e6 −0.668680
\(607\) −6.06160e6 −0.667753 −0.333876 0.942617i \(-0.608357\pi\)
−0.333876 + 0.942617i \(0.608357\pi\)
\(608\) −1.37626e6 −0.150987
\(609\) −95256.0 −0.0104076
\(610\) 0 0
\(611\) −2.56320e6 −0.277766
\(612\) 1.96474e6 0.212044
\(613\) −3.66489e6 −0.393921 −0.196961 0.980411i \(-0.563107\pi\)
−0.196961 + 0.980411i \(0.563107\pi\)
\(614\) 2.94408e6 0.315158
\(615\) 0 0
\(616\) 128000. 0.0135912
\(617\) −9.32522e6 −0.986157 −0.493079 0.869985i \(-0.664129\pi\)
−0.493079 + 0.869985i \(0.664129\pi\)
\(618\) 5.55710e6 0.585298
\(619\) −7.40162e6 −0.776426 −0.388213 0.921570i \(-0.626907\pi\)
−0.388213 + 0.921570i \(0.626907\pi\)
\(620\) 0 0
\(621\) 2.98890e6 0.311016
\(622\) 6.86640e6 0.711628
\(623\) −90712.0 −0.00936364
\(624\) 663552. 0.0682203
\(625\) 0 0
\(626\) 1.13540e7 1.15802
\(627\) −6.04800e6 −0.614388
\(628\) 4.23923e6 0.428932
\(629\) −1.10486e7 −1.11348
\(630\) 0 0
\(631\) 160052. 0.0160025 0.00800125 0.999968i \(-0.497453\pi\)
0.00800125 + 0.999968i \(0.497453\pi\)
\(632\) 3.85459e6 0.383871
\(633\) 9.27072e6 0.919611
\(634\) −5.10421e6 −0.504319
\(635\) 0 0
\(636\) −5.73178e6 −0.561884
\(637\) 4.83581e6 0.472194
\(638\) 5.29200e6 0.514717
\(639\) −2.33280e6 −0.226009
\(640\) 0 0
\(641\) −1.69565e7 −1.63002 −0.815008 0.579450i \(-0.803268\pi\)
−0.815008 + 0.579450i \(0.803268\pi\)
\(642\) −4.92437e6 −0.471534
\(643\) 1.10128e7 1.05044 0.525219 0.850967i \(-0.323984\pi\)
0.525219 + 0.850967i \(0.323984\pi\)
\(644\) 262400. 0.0249315
\(645\) 0 0
\(646\) −8.15002e6 −0.768382
\(647\) 3.33848e6 0.313537 0.156768 0.987635i \(-0.449892\pi\)
0.156768 + 0.987635i \(0.449892\pi\)
\(648\) 419904. 0.0392837
\(649\) 1.41500e7 1.31870
\(650\) 0 0
\(651\) −202032. −0.0186839
\(652\) −6.44998e6 −0.594210
\(653\) −4.76181e6 −0.437008 −0.218504 0.975836i \(-0.570118\pi\)
−0.218504 + 0.975836i \(0.570118\pi\)
\(654\) 1.93716e6 0.177101
\(655\) 0 0
\(656\) −4.86042e6 −0.440975
\(657\) −2.49545e6 −0.225546
\(658\) −142400. −0.0128217
\(659\) 798188. 0.0715965 0.0357982 0.999359i \(-0.488603\pi\)
0.0357982 + 0.999359i \(0.488603\pi\)
\(660\) 0 0
\(661\) −1.54048e7 −1.37136 −0.685682 0.727901i \(-0.740496\pi\)
−0.685682 + 0.727901i \(0.740496\pi\)
\(662\) 1.77597e6 0.157503
\(663\) 3.92947e6 0.347177
\(664\) −157952. −0.0139029
\(665\) 0 0
\(666\) −2.36131e6 −0.206285
\(667\) 1.08486e7 0.944189
\(668\) −4.19040e6 −0.363341
\(669\) −4.10418e6 −0.354537
\(670\) 0 0
\(671\) −9.14500e6 −0.784111
\(672\) 36864.0 0.00314905
\(673\) −976704. −0.0831238 −0.0415619 0.999136i \(-0.513233\pi\)
−0.0415619 + 0.999136i \(0.513233\pi\)
\(674\) 1.08530e7 0.920240
\(675\) 0 0
\(676\) −4.61358e6 −0.388304
\(677\) 1.93885e7 1.62582 0.812911 0.582388i \(-0.197881\pi\)
0.812911 + 0.582388i \(0.197881\pi\)
\(678\) −2.97691e6 −0.248709
\(679\) 147872. 0.0123087
\(680\) 0 0
\(681\) 3.90827e6 0.322936
\(682\) 1.12240e7 0.924031
\(683\) −5.25573e6 −0.431103 −0.215552 0.976492i \(-0.569155\pi\)
−0.215552 + 0.976492i \(0.569155\pi\)
\(684\) −1.74182e6 −0.142352
\(685\) 0 0
\(686\) 537568. 0.0436137
\(687\) 6.50439e6 0.525793
\(688\) −615424. −0.0495682
\(689\) −1.14636e7 −0.919965
\(690\) 0 0
\(691\) −5.45034e6 −0.434238 −0.217119 0.976145i \(-0.569666\pi\)
−0.217119 + 0.976145i \(0.569666\pi\)
\(692\) 5.21965e6 0.414358
\(693\) 162000. 0.0128139
\(694\) −5.24203e6 −0.413144
\(695\) 0 0
\(696\) 1.52410e6 0.119258
\(697\) −2.87828e7 −2.24414
\(698\) −1.19564e6 −0.0928885
\(699\) 5.08813e6 0.393881
\(700\) 0 0
\(701\) −4.43961e6 −0.341232 −0.170616 0.985338i \(-0.554576\pi\)
−0.170616 + 0.985338i \(0.554576\pi\)
\(702\) 839808. 0.0643187
\(703\) 9.79507e6 0.747514
\(704\) −2.04800e6 −0.155739
\(705\) 0 0
\(706\) 2.95198e6 0.222896
\(707\) −671672. −0.0505369
\(708\) 4.07520e6 0.305538
\(709\) 4.55918e6 0.340621 0.170310 0.985390i \(-0.445523\pi\)
0.170310 + 0.985390i \(0.445523\pi\)
\(710\) 0 0
\(711\) 4.87847e6 0.361917
\(712\) 1.45139e6 0.107296
\(713\) 2.30092e7 1.69503
\(714\) 218304. 0.0160257
\(715\) 0 0
\(716\) −1.75226e6 −0.127736
\(717\) −2.92414e6 −0.212422
\(718\) −9.36298e6 −0.677802
\(719\) −2.06630e7 −1.49063 −0.745317 0.666710i \(-0.767702\pi\)
−0.745317 + 0.666710i \(0.767702\pi\)
\(720\) 0 0
\(721\) 617456. 0.0442352
\(722\) −2.67905e6 −0.191266
\(723\) −8.23736e6 −0.586060
\(724\) −850336. −0.0602898
\(725\) 0 0
\(726\) −3.20216e6 −0.225477
\(727\) 5.48161e6 0.384656 0.192328 0.981331i \(-0.438396\pi\)
0.192328 + 0.981331i \(0.438396\pi\)
\(728\) 73728.0 0.00515589
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −3.64446e6 −0.252255
\(732\) −2.63376e6 −0.181676
\(733\) −8.55579e6 −0.588166 −0.294083 0.955780i \(-0.595014\pi\)
−0.294083 + 0.955780i \(0.595014\pi\)
\(734\) 509168. 0.0348836
\(735\) 0 0
\(736\) −4.19840e6 −0.285686
\(737\) −3.29780e7 −2.23643
\(738\) −6.15146e6 −0.415755
\(739\) −5.29119e6 −0.356404 −0.178202 0.983994i \(-0.557028\pi\)
−0.178202 + 0.983994i \(0.557028\pi\)
\(740\) 0 0
\(741\) −3.48365e6 −0.233071
\(742\) −636864. −0.0424656
\(743\) −2.36432e6 −0.157121 −0.0785606 0.996909i \(-0.525032\pi\)
−0.0785606 + 0.996909i \(0.525032\pi\)
\(744\) 3.23251e6 0.214096
\(745\) 0 0
\(746\) 1.61548e7 1.06281
\(747\) −199908. −0.0131078
\(748\) −1.21280e7 −0.792566
\(749\) −547152. −0.0356372
\(750\) 0 0
\(751\) −8.79694e6 −0.569157 −0.284578 0.958653i \(-0.591854\pi\)
−0.284578 + 0.958653i \(0.591854\pi\)
\(752\) 2.27840e6 0.146922
\(753\) −1.23037e7 −0.790766
\(754\) 3.04819e6 0.195260
\(755\) 0 0
\(756\) 46656.0 0.00296895
\(757\) 2.95808e7 1.87616 0.938079 0.346421i \(-0.112603\pi\)
0.938079 + 0.346421i \(0.112603\pi\)
\(758\) 4.04854e6 0.255933
\(759\) −1.84500e7 −1.16250
\(760\) 0 0
\(761\) −1.26296e7 −0.790549 −0.395274 0.918563i \(-0.629351\pi\)
−0.395274 + 0.918563i \(0.629351\pi\)
\(762\) 7.62408e6 0.475664
\(763\) 215240. 0.0133848
\(764\) 3.71290e6 0.230133
\(765\) 0 0
\(766\) −9.50440e6 −0.585265
\(767\) 8.15040e6 0.500254
\(768\) −589824. −0.0360844
\(769\) −2.32186e7 −1.41586 −0.707929 0.706283i \(-0.750370\pi\)
−0.707929 + 0.706283i \(0.750370\pi\)
\(770\) 0 0
\(771\) −8.03639e6 −0.486883
\(772\) −1.64908e7 −0.995858
\(773\) −1.73201e7 −1.04256 −0.521280 0.853386i \(-0.674545\pi\)
−0.521280 + 0.853386i \(0.674545\pi\)
\(774\) −778896. −0.0467334
\(775\) 0 0
\(776\) −2.36595e6 −0.141043
\(777\) −262368. −0.0155904
\(778\) 5.69990e6 0.337612
\(779\) 2.55172e7 1.50657
\(780\) 0 0
\(781\) 1.44000e7 0.844763
\(782\) −2.48624e7 −1.45387
\(783\) 1.92893e6 0.112438
\(784\) −4.29850e6 −0.249762
\(785\) 0 0
\(786\) −6.10200e6 −0.352303
\(787\) 556676. 0.0320380 0.0160190 0.999872i \(-0.494901\pi\)
0.0160190 + 0.999872i \(0.494901\pi\)
\(788\) −8.36474e6 −0.479884
\(789\) 1.67985e7 0.960678
\(790\) 0 0
\(791\) −330768. −0.0187967
\(792\) −2.59200e6 −0.146832
\(793\) −5.26752e6 −0.297456
\(794\) 6.77379e6 0.381312
\(795\) 0 0
\(796\) −3.44467e6 −0.192693
\(797\) 3.00562e6 0.167606 0.0838028 0.996482i \(-0.473293\pi\)
0.0838028 + 0.996482i \(0.473293\pi\)
\(798\) −193536. −0.0107586
\(799\) 1.34924e7 0.747691
\(800\) 0 0
\(801\) 1.83692e6 0.101160
\(802\) −1.13800e7 −0.624751
\(803\) 1.54040e7 0.843033
\(804\) −9.49766e6 −0.518175
\(805\) 0 0
\(806\) 6.46502e6 0.350536
\(807\) 1.23504e7 0.667570
\(808\) 1.07468e7 0.579094
\(809\) 2.23153e6 0.119876 0.0599378 0.998202i \(-0.480910\pi\)
0.0599378 + 0.998202i \(0.480910\pi\)
\(810\) 0 0
\(811\) 2.24862e7 1.20051 0.600253 0.799810i \(-0.295067\pi\)
0.600253 + 0.799810i \(0.295067\pi\)
\(812\) 169344. 0.00901322
\(813\) −4.12780e6 −0.219024
\(814\) 1.45760e7 0.771041
\(815\) 0 0
\(816\) −3.49286e6 −0.183635
\(817\) 3.23098e6 0.169347
\(818\) −7.56278e6 −0.395183
\(819\) 93312.0 0.00486102
\(820\) 0 0
\(821\) −1.65921e7 −0.859098 −0.429549 0.903044i \(-0.641327\pi\)
−0.429549 + 0.903044i \(0.641327\pi\)
\(822\) −9.07330e6 −0.468366
\(823\) 1.47544e7 0.759316 0.379658 0.925127i \(-0.376042\pi\)
0.379658 + 0.925127i \(0.376042\pi\)
\(824\) −9.87930e6 −0.506883
\(825\) 0 0
\(826\) 452800. 0.0230917
\(827\) −3.39475e6 −0.172601 −0.0863006 0.996269i \(-0.527505\pi\)
−0.0863006 + 0.996269i \(0.527505\pi\)
\(828\) −5.31360e6 −0.269348
\(829\) −509442. −0.0257459 −0.0128730 0.999917i \(-0.504098\pi\)
−0.0128730 + 0.999917i \(0.504098\pi\)
\(830\) 0 0
\(831\) 8.86867e6 0.445509
\(832\) −1.17965e6 −0.0590805
\(833\) −2.54552e7 −1.27105
\(834\) −6.91258e6 −0.344132
\(835\) 0 0
\(836\) 1.07520e7 0.532076
\(837\) 4.09115e6 0.201851
\(838\) 1.84372e7 0.906953
\(839\) 4.00609e7 1.96479 0.982394 0.186819i \(-0.0598178\pi\)
0.982394 + 0.186819i \(0.0598178\pi\)
\(840\) 0 0
\(841\) −1.35098e7 −0.658658
\(842\) −2.41660e7 −1.17470
\(843\) −1.49218e6 −0.0723191
\(844\) −1.64813e7 −0.796407
\(845\) 0 0
\(846\) 2.88360e6 0.138519
\(847\) −355796. −0.0170409
\(848\) 1.01898e7 0.486606
\(849\) 1.49824e7 0.713364
\(850\) 0 0
\(851\) 2.98808e7 1.41439
\(852\) 4.14720e6 0.195729
\(853\) 9.67506e6 0.455283 0.227641 0.973745i \(-0.426899\pi\)
0.227641 + 0.973745i \(0.426899\pi\)
\(854\) −292640. −0.0137306
\(855\) 0 0
\(856\) 8.75443e6 0.408360
\(857\) 3.27535e7 1.52337 0.761686 0.647946i \(-0.224372\pi\)
0.761686 + 0.647946i \(0.224372\pi\)
\(858\) −5.18400e6 −0.240407
\(859\) −2.17420e7 −1.00535 −0.502675 0.864476i \(-0.667651\pi\)
−0.502675 + 0.864476i \(0.667651\pi\)
\(860\) 0 0
\(861\) −683496. −0.0314216
\(862\) −15200.0 −0.000696748 0
\(863\) 2.08744e7 0.954087 0.477043 0.878880i \(-0.341708\pi\)
0.477043 + 0.878880i \(0.341708\pi\)
\(864\) −746496. −0.0340207
\(865\) 0 0
\(866\) 1.00294e6 0.0454446
\(867\) −7.90559e6 −0.357180
\(868\) 359168. 0.0161807
\(869\) −3.01140e7 −1.35275
\(870\) 0 0
\(871\) −1.89953e7 −0.848401
\(872\) −3.44384e6 −0.153374
\(873\) −2.99441e6 −0.132977
\(874\) 2.20416e7 0.976033
\(875\) 0 0
\(876\) 4.43635e6 0.195329
\(877\) 3.96804e7 1.74212 0.871058 0.491181i \(-0.163434\pi\)
0.871058 + 0.491181i \(0.163434\pi\)
\(878\) −1.43549e7 −0.628441
\(879\) −2.29594e7 −1.00228
\(880\) 0 0
\(881\) 2.60742e7 1.13180 0.565902 0.824472i \(-0.308528\pi\)
0.565902 + 0.824472i \(0.308528\pi\)
\(882\) −5.44028e6 −0.235478
\(883\) 4.10486e7 1.77172 0.885862 0.463949i \(-0.153568\pi\)
0.885862 + 0.463949i \(0.153568\pi\)
\(884\) −6.98573e6 −0.300664
\(885\) 0 0
\(886\) −5.65915e6 −0.242196
\(887\) 1.37553e7 0.587031 0.293515 0.955954i \(-0.405175\pi\)
0.293515 + 0.955954i \(0.405175\pi\)
\(888\) 4.19789e6 0.178648
\(889\) 847120. 0.0359493
\(890\) 0 0
\(891\) −3.28050e6 −0.138435
\(892\) 7.29632e6 0.307038
\(893\) −1.19616e7 −0.501950
\(894\) −8.48498e6 −0.355064
\(895\) 0 0
\(896\) −65536.0 −0.00272716
\(897\) −1.06272e7 −0.440999
\(898\) −3.31922e6 −0.137355
\(899\) 1.48494e7 0.612785
\(900\) 0 0
\(901\) 6.03429e7 2.47636
\(902\) 3.79720e7 1.55399
\(903\) −86544.0 −0.00353197
\(904\) 5.29229e6 0.215388
\(905\) 0 0
\(906\) 1.33737e7 0.541292
\(907\) −5.86936e6 −0.236904 −0.118452 0.992960i \(-0.537793\pi\)
−0.118452 + 0.992960i \(0.537793\pi\)
\(908\) −6.94803e6 −0.279671
\(909\) 1.36014e7 0.545975
\(910\) 0 0
\(911\) 4.63982e7 1.85227 0.926137 0.377188i \(-0.123109\pi\)
0.926137 + 0.377188i \(0.123109\pi\)
\(912\) 3.09658e6 0.123281
\(913\) 1.23400e6 0.0489935
\(914\) 1.87279e7 0.741521
\(915\) 0 0
\(916\) −1.15634e7 −0.455350
\(917\) −678000. −0.0266260
\(918\) −4.42066e6 −0.173133
\(919\) 2.27859e7 0.889975 0.444988 0.895537i \(-0.353208\pi\)
0.444988 + 0.895537i \(0.353208\pi\)
\(920\) 0 0
\(921\) −6.62418e6 −0.257326
\(922\) −567720. −0.0219941
\(923\) 8.29440e6 0.320465
\(924\) −288000. −0.0110972
\(925\) 0 0
\(926\) 2.90990e6 0.111520
\(927\) −1.25035e7 −0.477894
\(928\) −2.70950e6 −0.103281
\(929\) 2.70352e7 1.02775 0.513877 0.857864i \(-0.328209\pi\)
0.513877 + 0.857864i \(0.328209\pi\)
\(930\) 0 0
\(931\) 2.25671e7 0.853300
\(932\) −9.04557e6 −0.341111
\(933\) −1.54494e7 −0.581042
\(934\) −1.79056e7 −0.671616
\(935\) 0 0
\(936\) −1.49299e6 −0.0557016
\(937\) −2.86149e7 −1.06474 −0.532370 0.846512i \(-0.678699\pi\)
−0.532370 + 0.846512i \(0.678699\pi\)
\(938\) −1.05530e6 −0.0391622
\(939\) −2.55466e7 −0.945517
\(940\) 0 0
\(941\) −3.67892e7 −1.35440 −0.677200 0.735799i \(-0.736807\pi\)
−0.677200 + 0.735799i \(0.736807\pi\)
\(942\) −9.53827e6 −0.350221
\(943\) 7.78426e7 2.85061
\(944\) −7.24480e6 −0.264604
\(945\) 0 0
\(946\) 4.80800e6 0.174677
\(947\) 7.96828e6 0.288728 0.144364 0.989525i \(-0.453886\pi\)
0.144364 + 0.989525i \(0.453886\pi\)
\(948\) −8.67283e6 −0.313430
\(949\) 8.87270e6 0.319809
\(950\) 0 0
\(951\) 1.14845e7 0.411775
\(952\) −388096. −0.0138786
\(953\) −4.82202e7 −1.71987 −0.859937 0.510400i \(-0.829497\pi\)
−0.859937 + 0.510400i \(0.829497\pi\)
\(954\) 1.28965e7 0.458776
\(955\) 0 0
\(956\) 5.19846e6 0.183963
\(957\) −1.19070e7 −0.420264
\(958\) −5.30874e6 −0.186886
\(959\) −1.00814e6 −0.0353978
\(960\) 0 0
\(961\) 2.86539e6 0.100087
\(962\) 8.39578e6 0.292498
\(963\) 1.10798e7 0.385006
\(964\) 1.46442e7 0.507543
\(965\) 0 0
\(966\) −590400. −0.0203565
\(967\) 4.83510e7 1.66280 0.831398 0.555678i \(-0.187541\pi\)
0.831398 + 0.555678i \(0.187541\pi\)
\(968\) 5.69274e6 0.195269
\(969\) 1.83375e7 0.627381
\(970\) 0 0
\(971\) −4.05515e7 −1.38025 −0.690127 0.723688i \(-0.742445\pi\)
−0.690127 + 0.723688i \(0.742445\pi\)
\(972\) −944784. −0.0320750
\(973\) −768064. −0.0260085
\(974\) −1.64659e7 −0.556144
\(975\) 0 0
\(976\) 4.68224e6 0.157336
\(977\) −4.34929e7 −1.45775 −0.728874 0.684648i \(-0.759956\pi\)
−0.728874 + 0.684648i \(0.759956\pi\)
\(978\) 1.45125e7 0.485170
\(979\) −1.13390e7 −0.378110
\(980\) 0 0
\(981\) −4.35861e6 −0.144602
\(982\) −2.44926e7 −0.810507
\(983\) −3.34896e6 −0.110542 −0.0552709 0.998471i \(-0.517602\pi\)
−0.0552709 + 0.998471i \(0.517602\pi\)
\(984\) 1.09359e7 0.360054
\(985\) 0 0
\(986\) −1.60453e7 −0.525602
\(987\) 320400. 0.0104689
\(988\) 6.19315e6 0.201846
\(989\) 9.85640e6 0.320426
\(990\) 0 0
\(991\) −5.55726e7 −1.79753 −0.898766 0.438429i \(-0.855535\pi\)
−0.898766 + 0.438429i \(0.855535\pi\)
\(992\) −5.74669e6 −0.185412
\(993\) −3.99593e6 −0.128601
\(994\) 460800. 0.0147927
\(995\) 0 0
\(996\) 355392. 0.0113517
\(997\) −1.27342e7 −0.405726 −0.202863 0.979207i \(-0.565025\pi\)
−0.202863 + 0.979207i \(0.565025\pi\)
\(998\) −3.16196e7 −1.00492
\(999\) 5.31295e6 0.168431
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 150.6.a.j.1.1 1
3.2 odd 2 450.6.a.f.1.1 1
5.2 odd 4 30.6.c.a.19.2 yes 2
5.3 odd 4 30.6.c.a.19.1 2
5.4 even 2 150.6.a.f.1.1 1
15.2 even 4 90.6.c.b.19.1 2
15.8 even 4 90.6.c.b.19.2 2
15.14 odd 2 450.6.a.s.1.1 1
20.3 even 4 240.6.f.a.49.2 2
20.7 even 4 240.6.f.a.49.1 2
60.23 odd 4 720.6.f.g.289.2 2
60.47 odd 4 720.6.f.g.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.6.c.a.19.1 2 5.3 odd 4
30.6.c.a.19.2 yes 2 5.2 odd 4
90.6.c.b.19.1 2 15.2 even 4
90.6.c.b.19.2 2 15.8 even 4
150.6.a.f.1.1 1 5.4 even 2
150.6.a.j.1.1 1 1.1 even 1 trivial
240.6.f.a.49.1 2 20.7 even 4
240.6.f.a.49.2 2 20.3 even 4
450.6.a.f.1.1 1 3.2 odd 2
450.6.a.s.1.1 1 15.14 odd 2
720.6.f.g.289.1 2 60.47 odd 4
720.6.f.g.289.2 2 60.23 odd 4