Properties

Label 570.6.a.k.1.1
Level $570$
Weight $6$
Character 570.1
Self dual yes
Analytic conductor $91.419$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.4187772934\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 26355 x^{2} - 7203 x + 128450070\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(79.8406\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} -188.420 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} -188.420 q^{7} -64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} +163.279 q^{11} +144.000 q^{12} -1068.02 q^{13} +753.678 q^{14} -225.000 q^{15} +256.000 q^{16} +726.618 q^{17} -324.000 q^{18} -361.000 q^{19} -400.000 q^{20} -1695.78 q^{21} -653.118 q^{22} -1895.86 q^{23} -576.000 q^{24} +625.000 q^{25} +4272.08 q^{26} +729.000 q^{27} -3014.71 q^{28} -5086.12 q^{29} +900.000 q^{30} +1140.90 q^{31} -1024.00 q^{32} +1469.52 q^{33} -2906.47 q^{34} +4710.49 q^{35} +1296.00 q^{36} -13834.9 q^{37} +1444.00 q^{38} -9612.19 q^{39} +1600.00 q^{40} +7307.01 q^{41} +6783.11 q^{42} -7620.27 q^{43} +2612.47 q^{44} -2025.00 q^{45} +7583.44 q^{46} -2641.80 q^{47} +2304.00 q^{48} +18694.9 q^{49} -2500.00 q^{50} +6539.56 q^{51} -17088.3 q^{52} +23774.5 q^{53} -2916.00 q^{54} -4081.99 q^{55} +12058.9 q^{56} -3249.00 q^{57} +20344.5 q^{58} +25672.6 q^{59} -3600.00 q^{60} +30206.0 q^{61} -4563.60 q^{62} -15262.0 q^{63} +4096.00 q^{64} +26700.5 q^{65} -5878.06 q^{66} -16793.6 q^{67} +11625.9 q^{68} -17062.7 q^{69} -18842.0 q^{70} -18718.5 q^{71} -5184.00 q^{72} +34162.5 q^{73} +55339.6 q^{74} +5625.00 q^{75} -5776.00 q^{76} -30765.1 q^{77} +38448.8 q^{78} +70971.7 q^{79} -6400.00 q^{80} +6561.00 q^{81} -29228.0 q^{82} -79869.8 q^{83} -27132.4 q^{84} -18165.5 q^{85} +30481.1 q^{86} -45775.0 q^{87} -10449.9 q^{88} -19749.0 q^{89} +8100.00 q^{90} +201236. q^{91} -30333.8 q^{92} +10268.1 q^{93} +10567.2 q^{94} +9025.00 q^{95} -9216.00 q^{96} +76942.8 q^{97} -74779.8 q^{98} +13225.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + 400q^{10} - 472q^{11} + 576q^{12} - 482q^{13} - 104q^{14} - 900q^{15} + 1024q^{16} + 1816q^{17} - 1296q^{18} - 1444q^{19} - 1600q^{20} + 234q^{21} + 1888q^{22} - 418q^{23} - 2304q^{24} + 2500q^{25} + 1928q^{26} + 2916q^{27} + 416q^{28} - 10396q^{29} + 3600q^{30} + 528q^{31} - 4096q^{32} - 4248q^{33} - 7264q^{34} - 650q^{35} + 5184q^{36} + 5774q^{37} + 5776q^{38} - 4338q^{39} + 6400q^{40} + 9620q^{41} - 936q^{42} + 21098q^{43} - 7552q^{44} - 8100q^{45} + 1672q^{46} - 18858q^{47} + 9216q^{48} + 2148q^{49} - 10000q^{50} + 16344q^{51} - 7712q^{52} + 822q^{53} - 11664q^{54} + 11800q^{55} - 1664q^{56} - 12996q^{57} + 41584q^{58} + 28672q^{59} - 14400q^{60} + 77748q^{61} - 2112q^{62} + 2106q^{63} + 16384q^{64} + 12050q^{65} + 16992q^{66} + 82400q^{67} + 29056q^{68} - 3762q^{69} + 2600q^{70} + 928q^{71} - 20736q^{72} + 121100q^{73} - 23096q^{74} + 22500q^{75} - 23104q^{76} - 120220q^{77} + 17352q^{78} + 144284q^{79} - 25600q^{80} + 26244q^{81} - 38480q^{82} - 6082q^{83} + 3744q^{84} - 45400q^{85} - 84392q^{86} - 93564q^{87} + 30208q^{88} + 43260q^{89} + 32400q^{90} + 135148q^{91} - 6688q^{92} + 4752q^{93} + 75432q^{94} + 36100q^{95} - 36864q^{96} - 6862q^{97} - 8592q^{98} - 38232q^{99} + O(q^{100}) \)

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
\(5\) −25.0000 −0.447214
\(6\) −36.0000 −0.408248
\(7\) −188.420 −1.45339 −0.726693 0.686962i \(-0.758944\pi\)
−0.726693 + 0.686962i \(0.758944\pi\)
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 100.000 0.316228
\(11\) 163.279 0.406865 0.203432 0.979089i \(-0.434790\pi\)
0.203432 + 0.979089i \(0.434790\pi\)
\(12\) 144.000 0.288675
\(13\) −1068.02 −1.75276 −0.876378 0.481624i \(-0.840047\pi\)
−0.876378 + 0.481624i \(0.840047\pi\)
\(14\) 753.678 1.02770
\(15\) −225.000 −0.258199
\(16\) 256.000 0.250000
\(17\) 726.618 0.609795 0.304898 0.952385i \(-0.401378\pi\)
0.304898 + 0.952385i \(0.401378\pi\)
\(18\) −324.000 −0.235702
\(19\) −361.000 −0.229416
\(20\) −400.000 −0.223607
\(21\) −1695.78 −0.839113
\(22\) −653.118 −0.287697
\(23\) −1895.86 −0.747285 −0.373643 0.927573i \(-0.621891\pi\)
−0.373643 + 0.927573i \(0.621891\pi\)
\(24\) −576.000 −0.204124
\(25\) 625.000 0.200000
\(26\) 4272.08 1.23939
\(27\) 729.000 0.192450
\(28\) −3014.71 −0.726693
\(29\) −5086.12 −1.12303 −0.561515 0.827467i \(-0.689781\pi\)
−0.561515 + 0.827467i \(0.689781\pi\)
\(30\) 900.000 0.182574
\(31\) 1140.90 0.213228 0.106614 0.994301i \(-0.465999\pi\)
0.106614 + 0.994301i \(0.465999\pi\)
\(32\) −1024.00 −0.176777
\(33\) 1469.52 0.234903
\(34\) −2906.47 −0.431190
\(35\) 4710.49 0.649974
\(36\) 1296.00 0.166667
\(37\) −13834.9 −1.66139 −0.830695 0.556728i \(-0.812057\pi\)
−0.830695 + 0.556728i \(0.812057\pi\)
\(38\) 1444.00 0.162221
\(39\) −9612.19 −1.01195
\(40\) 1600.00 0.158114
\(41\) 7307.01 0.678860 0.339430 0.940631i \(-0.389766\pi\)
0.339430 + 0.940631i \(0.389766\pi\)
\(42\) 6783.11 0.593342
\(43\) −7620.27 −0.628491 −0.314246 0.949342i \(-0.601752\pi\)
−0.314246 + 0.949342i \(0.601752\pi\)
\(44\) 2612.47 0.203432
\(45\) −2025.00 −0.149071
\(46\) 7583.44 0.528411
\(47\) −2641.80 −0.174444 −0.0872218 0.996189i \(-0.527799\pi\)
−0.0872218 + 0.996189i \(0.527799\pi\)
\(48\) 2304.00 0.144338
\(49\) 18694.9 1.11233
\(50\) −2500.00 −0.141421
\(51\) 6539.56 0.352065
\(52\) −17088.3 −0.876378
\(53\) 23774.5 1.16258 0.581289 0.813697i \(-0.302549\pi\)
0.581289 + 0.813697i \(0.302549\pi\)
\(54\) −2916.00 −0.136083
\(55\) −4081.99 −0.181955
\(56\) 12058.9 0.513850
\(57\) −3249.00 −0.132453
\(58\) 20344.5 0.794102
\(59\) 25672.6 0.960152 0.480076 0.877227i \(-0.340609\pi\)
0.480076 + 0.877227i \(0.340609\pi\)
\(60\) −3600.00 −0.129099
\(61\) 30206.0 1.03937 0.519683 0.854359i \(-0.326050\pi\)
0.519683 + 0.854359i \(0.326050\pi\)
\(62\) −4563.60 −0.150775
\(63\) −15262.0 −0.484462
\(64\) 4096.00 0.125000
\(65\) 26700.5 0.783856
\(66\) −5878.06 −0.166102
\(67\) −16793.6 −0.457043 −0.228522 0.973539i \(-0.573389\pi\)
−0.228522 + 0.973539i \(0.573389\pi\)
\(68\) 11625.9 0.304898
\(69\) −17062.7 −0.431445
\(70\) −18842.0 −0.459601
\(71\) −18718.5 −0.440683 −0.220341 0.975423i \(-0.570717\pi\)
−0.220341 + 0.975423i \(0.570717\pi\)
\(72\) −5184.00 −0.117851
\(73\) 34162.5 0.750312 0.375156 0.926962i \(-0.377589\pi\)
0.375156 + 0.926962i \(0.377589\pi\)
\(74\) 55339.6 1.17478
\(75\) 5625.00 0.115470
\(76\) −5776.00 −0.114708
\(77\) −30765.1 −0.591331
\(78\) 38448.8 0.715560
\(79\) 70971.7 1.27943 0.639716 0.768611i \(-0.279052\pi\)
0.639716 + 0.768611i \(0.279052\pi\)
\(80\) −6400.00 −0.111803
\(81\) 6561.00 0.111111
\(82\) −29228.0 −0.480026
\(83\) −79869.8 −1.27259 −0.636294 0.771447i \(-0.719533\pi\)
−0.636294 + 0.771447i \(0.719533\pi\)
\(84\) −27132.4 −0.419556
\(85\) −18165.5 −0.272709
\(86\) 30481.1 0.444411
\(87\) −45775.0 −0.648381
\(88\) −10449.9 −0.143848
\(89\) −19749.0 −0.264284 −0.132142 0.991231i \(-0.542186\pi\)
−0.132142 + 0.991231i \(0.542186\pi\)
\(90\) 8100.00 0.105409
\(91\) 201236. 2.54743
\(92\) −30333.8 −0.373643
\(93\) 10268.1 0.123107
\(94\) 10567.2 0.123350
\(95\) 9025.00 0.102598
\(96\) −9216.00 −0.102062
\(97\) 76942.8 0.830307 0.415154 0.909751i \(-0.363728\pi\)
0.415154 + 0.909751i \(0.363728\pi\)
\(98\) −74779.8 −0.786537
\(99\) 13225.6 0.135622
\(100\) 10000.0 0.100000
\(101\) −171035. −1.66833 −0.834166 0.551514i \(-0.814050\pi\)
−0.834166 + 0.551514i \(0.814050\pi\)
\(102\) −26158.3 −0.248948
\(103\) −99416.2 −0.923345 −0.461672 0.887050i \(-0.652750\pi\)
−0.461672 + 0.887050i \(0.652750\pi\)
\(104\) 68353.3 0.619693
\(105\) 42394.4 0.375263
\(106\) −95098.1 −0.822067
\(107\) 98169.8 0.828932 0.414466 0.910065i \(-0.363968\pi\)
0.414466 + 0.910065i \(0.363968\pi\)
\(108\) 11664.0 0.0962250
\(109\) 145046. 1.16933 0.584667 0.811274i \(-0.301225\pi\)
0.584667 + 0.811274i \(0.301225\pi\)
\(110\) 16327.9 0.128662
\(111\) −124514. −0.959204
\(112\) −48235.4 −0.363347
\(113\) −101916. −0.750841 −0.375421 0.926855i \(-0.622502\pi\)
−0.375421 + 0.926855i \(0.622502\pi\)
\(114\) 12996.0 0.0936586
\(115\) 47396.5 0.334196
\(116\) −81377.8 −0.561515
\(117\) −86509.7 −0.584252
\(118\) −102690. −0.678930
\(119\) −136909. −0.886268
\(120\) 14400.0 0.0912871
\(121\) −134391. −0.834461
\(122\) −120824. −0.734943
\(123\) 65763.1 0.391940
\(124\) 18254.4 0.106614
\(125\) −15625.0 −0.0894427
\(126\) 61048.0 0.342566
\(127\) 38026.0 0.209205 0.104602 0.994514i \(-0.466643\pi\)
0.104602 + 0.994514i \(0.466643\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −68582.5 −0.362860
\(130\) −106802. −0.554270
\(131\) 150745. 0.767476 0.383738 0.923442i \(-0.374637\pi\)
0.383738 + 0.923442i \(0.374637\pi\)
\(132\) 23512.2 0.117452
\(133\) 68019.5 0.333430
\(134\) 67174.5 0.323178
\(135\) −18225.0 −0.0860663
\(136\) −46503.6 −0.215595
\(137\) −32093.7 −0.146089 −0.0730446 0.997329i \(-0.523272\pi\)
−0.0730446 + 0.997329i \(0.523272\pi\)
\(138\) 68250.9 0.305078
\(139\) 110185. 0.483712 0.241856 0.970312i \(-0.422244\pi\)
0.241856 + 0.970312i \(0.422244\pi\)
\(140\) 75367.8 0.324987
\(141\) −23776.2 −0.100715
\(142\) 74874.2 0.311610
\(143\) −174386. −0.713134
\(144\) 20736.0 0.0833333
\(145\) 127153. 0.502234
\(146\) −136650. −0.530551
\(147\) 168255. 0.642205
\(148\) −221358. −0.830695
\(149\) 312360. 1.15263 0.576314 0.817228i \(-0.304490\pi\)
0.576314 + 0.817228i \(0.304490\pi\)
\(150\) −22500.0 −0.0816497
\(151\) −6631.03 −0.0236667 −0.0118334 0.999930i \(-0.503767\pi\)
−0.0118334 + 0.999930i \(0.503767\pi\)
\(152\) 23104.0 0.0811107
\(153\) 58856.1 0.203265
\(154\) 123060. 0.418134
\(155\) −28522.5 −0.0953582
\(156\) −153795. −0.505977
\(157\) 402222. 1.30232 0.651159 0.758942i \(-0.274283\pi\)
0.651159 + 0.758942i \(0.274283\pi\)
\(158\) −283887. −0.904695
\(159\) 213971. 0.671215
\(160\) 25600.0 0.0790569
\(161\) 357217. 1.08609
\(162\) −26244.0 −0.0785674
\(163\) 476491. 1.40471 0.702353 0.711828i \(-0.252133\pi\)
0.702353 + 0.711828i \(0.252133\pi\)
\(164\) 116912. 0.339430
\(165\) −36737.9 −0.105052
\(166\) 319479. 0.899855
\(167\) 679880. 1.88643 0.943216 0.332179i \(-0.107784\pi\)
0.943216 + 0.332179i \(0.107784\pi\)
\(168\) 108530. 0.296671
\(169\) 769376. 2.07215
\(170\) 72661.8 0.192834
\(171\) −29241.0 −0.0764719
\(172\) −121924. −0.314246
\(173\) 483983. 1.22946 0.614731 0.788737i \(-0.289265\pi\)
0.614731 + 0.788737i \(0.289265\pi\)
\(174\) 183100. 0.458475
\(175\) −117762. −0.290677
\(176\) 41799.5 0.101716
\(177\) 231053. 0.554344
\(178\) 78996.2 0.186877
\(179\) 94147.5 0.219622 0.109811 0.993952i \(-0.464975\pi\)
0.109811 + 0.993952i \(0.464975\pi\)
\(180\) −32400.0 −0.0745356
\(181\) −506533. −1.14924 −0.574620 0.818420i \(-0.694850\pi\)
−0.574620 + 0.818420i \(0.694850\pi\)
\(182\) −804944. −1.80131
\(183\) 271854. 0.600079
\(184\) 121335. 0.264205
\(185\) 345872. 0.742996
\(186\) −41072.4 −0.0870498
\(187\) 118642. 0.248104
\(188\) −42268.8 −0.0872218
\(189\) −137358. −0.279704
\(190\) −36100.0 −0.0725476
\(191\) −864695. −1.71506 −0.857530 0.514434i \(-0.828002\pi\)
−0.857530 + 0.514434i \(0.828002\pi\)
\(192\) 36864.0 0.0721688
\(193\) 518274. 1.00153 0.500767 0.865582i \(-0.333051\pi\)
0.500767 + 0.865582i \(0.333051\pi\)
\(194\) −307771. −0.587116
\(195\) 240305. 0.452560
\(196\) 299119. 0.556166
\(197\) −491801. −0.902868 −0.451434 0.892305i \(-0.649087\pi\)
−0.451434 + 0.892305i \(0.649087\pi\)
\(198\) −52902.5 −0.0958989
\(199\) 322500. 0.577293 0.288647 0.957436i \(-0.406795\pi\)
0.288647 + 0.957436i \(0.406795\pi\)
\(200\) −40000.0 −0.0707107
\(201\) −151143. −0.263874
\(202\) 684141. 1.17969
\(203\) 958324. 1.63220
\(204\) 104633. 0.176033
\(205\) −182675. −0.303595
\(206\) 397665. 0.652903
\(207\) −153565. −0.249095
\(208\) −273413. −0.438189
\(209\) −58943.9 −0.0933411
\(210\) −169578. −0.265351
\(211\) −122878. −0.190006 −0.0950028 0.995477i \(-0.530286\pi\)
−0.0950028 + 0.995477i \(0.530286\pi\)
\(212\) 380392. 0.581289
\(213\) −168467. −0.254428
\(214\) −392679. −0.586143
\(215\) 190507. 0.281070
\(216\) −46656.0 −0.0680414
\(217\) −214968. −0.309902
\(218\) −580182. −0.826844
\(219\) 307462. 0.433193
\(220\) −65311.8 −0.0909777
\(221\) −776043. −1.06882
\(222\) 498056. 0.678260
\(223\) −464828. −0.625937 −0.312968 0.949764i \(-0.601323\pi\)
−0.312968 + 0.949764i \(0.601323\pi\)
\(224\) 192942. 0.256925
\(225\) 50625.0 0.0666667
\(226\) 407666. 0.530925
\(227\) 685227. 0.882612 0.441306 0.897357i \(-0.354515\pi\)
0.441306 + 0.897357i \(0.354515\pi\)
\(228\) −51984.0 −0.0662266
\(229\) 1.30108e6 1.63952 0.819760 0.572707i \(-0.194107\pi\)
0.819760 + 0.572707i \(0.194107\pi\)
\(230\) −189586. −0.236312
\(231\) −276885. −0.341405
\(232\) 325511. 0.397051
\(233\) 62729.8 0.0756979 0.0378490 0.999283i \(-0.487949\pi\)
0.0378490 + 0.999283i \(0.487949\pi\)
\(234\) 346039. 0.413129
\(235\) 66044.9 0.0780135
\(236\) 410762. 0.480076
\(237\) 638745. 0.738680
\(238\) 547636. 0.626686
\(239\) −409004. −0.463162 −0.231581 0.972816i \(-0.574390\pi\)
−0.231581 + 0.972816i \(0.574390\pi\)
\(240\) −57600.0 −0.0645497
\(241\) 1.77194e6 1.96520 0.982599 0.185740i \(-0.0594683\pi\)
0.982599 + 0.185740i \(0.0594683\pi\)
\(242\) 537563. 0.590053
\(243\) 59049.0 0.0641500
\(244\) 483296. 0.519683
\(245\) −467374. −0.497450
\(246\) −263052. −0.277143
\(247\) 385556. 0.402110
\(248\) −73017.6 −0.0753873
\(249\) −718829. −0.734729
\(250\) 62500.0 0.0632456
\(251\) −24470.8 −0.0245168 −0.0122584 0.999925i \(-0.503902\pi\)
−0.0122584 + 0.999925i \(0.503902\pi\)
\(252\) −244192. −0.242231
\(253\) −309555. −0.304044
\(254\) −152104. −0.147930
\(255\) −163489. −0.157448
\(256\) 65536.0 0.0625000
\(257\) −1.24459e6 −1.17542 −0.587710 0.809071i \(-0.699971\pi\)
−0.587710 + 0.809071i \(0.699971\pi\)
\(258\) 274330. 0.256581
\(259\) 2.60677e6 2.41464
\(260\) 427208. 0.391928
\(261\) −411975. −0.374343
\(262\) −602980. −0.542688
\(263\) −22175.1 −0.0197686 −0.00988430 0.999951i \(-0.503146\pi\)
−0.00988430 + 0.999951i \(0.503146\pi\)
\(264\) −94049.0 −0.0830509
\(265\) −594363. −0.519921
\(266\) −272078. −0.235770
\(267\) −177741. −0.152585
\(268\) −268698. −0.228522
\(269\) −961092. −0.809812 −0.404906 0.914358i \(-0.632696\pi\)
−0.404906 + 0.914358i \(0.632696\pi\)
\(270\) 72900.0 0.0608581
\(271\) −299383. −0.247630 −0.123815 0.992305i \(-0.539513\pi\)
−0.123815 + 0.992305i \(0.539513\pi\)
\(272\) 186014. 0.152449
\(273\) 1.81112e6 1.47076
\(274\) 128375. 0.103301
\(275\) 102050. 0.0813729
\(276\) −273004. −0.215723
\(277\) −541136. −0.423747 −0.211874 0.977297i \(-0.567957\pi\)
−0.211874 + 0.977297i \(0.567957\pi\)
\(278\) −440741. −0.342036
\(279\) 92412.9 0.0710758
\(280\) −301471. −0.229801
\(281\) 205757. 0.155450 0.0777248 0.996975i \(-0.475234\pi\)
0.0777248 + 0.996975i \(0.475234\pi\)
\(282\) 95104.7 0.0712163
\(283\) 2.61917e6 1.94401 0.972004 0.234966i \(-0.0754980\pi\)
0.972004 + 0.234966i \(0.0754980\pi\)
\(284\) −299497. −0.220341
\(285\) 81225.0 0.0592349
\(286\) 697544. 0.504262
\(287\) −1.37678e6 −0.986645
\(288\) −82944.0 −0.0589256
\(289\) −891883. −0.628150
\(290\) −508612. −0.355133
\(291\) 692485. 0.479378
\(292\) 546599. 0.375156
\(293\) 336223. 0.228801 0.114401 0.993435i \(-0.463505\pi\)
0.114401 + 0.993435i \(0.463505\pi\)
\(294\) −673018. −0.454107
\(295\) −641815. −0.429393
\(296\) 885433. 0.587390
\(297\) 119031. 0.0783011
\(298\) −1.24944e6 −0.815032
\(299\) 2.02482e6 1.30981
\(300\) 90000.0 0.0577350
\(301\) 1.43581e6 0.913441
\(302\) 26524.1 0.0167349
\(303\) −1.53932e6 −0.963212
\(304\) −92416.0 −0.0573539
\(305\) −755150. −0.464819
\(306\) −235424. −0.143730
\(307\) 51557.7 0.0312211 0.0156105 0.999878i \(-0.495031\pi\)
0.0156105 + 0.999878i \(0.495031\pi\)
\(308\) −492241. −0.295666
\(309\) −894746. −0.533093
\(310\) 114090. 0.0674285
\(311\) 2.64594e6 1.55124 0.775621 0.631199i \(-0.217437\pi\)
0.775621 + 0.631199i \(0.217437\pi\)
\(312\) 615180. 0.357780
\(313\) −1.66914e6 −0.963014 −0.481507 0.876442i \(-0.659910\pi\)
−0.481507 + 0.876442i \(0.659910\pi\)
\(314\) −1.60889e6 −0.920877
\(315\) 381550. 0.216658
\(316\) 1.13555e6 0.639716
\(317\) 3.00427e6 1.67916 0.839578 0.543239i \(-0.182802\pi\)
0.839578 + 0.543239i \(0.182802\pi\)
\(318\) −855883. −0.474620
\(319\) −830458. −0.456921
\(320\) −102400. −0.0559017
\(321\) 883529. 0.478584
\(322\) −1.42887e6 −0.767985
\(323\) −262309. −0.139897
\(324\) 104976. 0.0555556
\(325\) −667513. −0.350551
\(326\) −1.90596e6 −0.993278
\(327\) 1.30541e6 0.675115
\(328\) −467648. −0.240013
\(329\) 497767. 0.253534
\(330\) 146952. 0.0742830
\(331\) −34286.3 −0.0172009 −0.00860044 0.999963i \(-0.502738\pi\)
−0.00860044 + 0.999963i \(0.502738\pi\)
\(332\) −1.27792e6 −0.636294
\(333\) −1.12063e6 −0.553797
\(334\) −2.71952e6 −1.33391
\(335\) 419840. 0.204396
\(336\) −434119. −0.209778
\(337\) −2.62069e6 −1.25702 −0.628509 0.777802i \(-0.716334\pi\)
−0.628509 + 0.777802i \(0.716334\pi\)
\(338\) −3.07750e6 −1.46523
\(339\) −917248. −0.433498
\(340\) −290647. −0.136354
\(341\) 186285. 0.0867547
\(342\) 116964. 0.0540738
\(343\) −355727. −0.163260
\(344\) 487697. 0.222205
\(345\) 426568. 0.192948
\(346\) −1.93593e6 −0.869360
\(347\) −2.60007e6 −1.15921 −0.579603 0.814899i \(-0.696793\pi\)
−0.579603 + 0.814899i \(0.696793\pi\)
\(348\) −732401. −0.324191
\(349\) −256184. −0.112587 −0.0562934 0.998414i \(-0.517928\pi\)
−0.0562934 + 0.998414i \(0.517928\pi\)
\(350\) 471049. 0.205540
\(351\) −778587. −0.337318
\(352\) −167198. −0.0719242
\(353\) 301006. 0.128569 0.0642847 0.997932i \(-0.479523\pi\)
0.0642847 + 0.997932i \(0.479523\pi\)
\(354\) −924214. −0.391980
\(355\) 467964. 0.197079
\(356\) −315985. −0.132142
\(357\) −1.23218e6 −0.511687
\(358\) −376590. −0.155296
\(359\) 1.93377e6 0.791897 0.395948 0.918273i \(-0.370416\pi\)
0.395948 + 0.918273i \(0.370416\pi\)
\(360\) 129600. 0.0527046
\(361\) 130321. 0.0526316
\(362\) 2.02613e6 0.812636
\(363\) −1.20952e6 −0.481776
\(364\) 3.21978e6 1.27372
\(365\) −854061. −0.335550
\(366\) −1.08742e6 −0.424320
\(367\) −5.03641e6 −1.95189 −0.975947 0.218009i \(-0.930044\pi\)
−0.975947 + 0.218009i \(0.930044\pi\)
\(368\) −485340. −0.186821
\(369\) 591868. 0.226287
\(370\) −1.38349e6 −0.525378
\(371\) −4.47958e6 −1.68967
\(372\) 164290. 0.0615535
\(373\) 1.43009e6 0.532220 0.266110 0.963943i \(-0.414262\pi\)
0.266110 + 0.963943i \(0.414262\pi\)
\(374\) −474567. −0.175436
\(375\) −140625. −0.0516398
\(376\) 169075. 0.0616751
\(377\) 5.43208e6 1.96840
\(378\) 549432. 0.197781
\(379\) 1.68614e6 0.602970 0.301485 0.953471i \(-0.402518\pi\)
0.301485 + 0.953471i \(0.402518\pi\)
\(380\) 144400. 0.0512989
\(381\) 342234. 0.120785
\(382\) 3.45878e6 1.21273
\(383\) −2.47443e6 −0.861942 −0.430971 0.902366i \(-0.641829\pi\)
−0.430971 + 0.902366i \(0.641829\pi\)
\(384\) −147456. −0.0510310
\(385\) 769126. 0.264451
\(386\) −2.07310e6 −0.708192
\(387\) −617242. −0.209497
\(388\) 1.23108e6 0.415154
\(389\) −930474. −0.311767 −0.155883 0.987775i \(-0.549822\pi\)
−0.155883 + 0.987775i \(0.549822\pi\)
\(390\) −961219. −0.320008
\(391\) −1.37757e6 −0.455691
\(392\) −1.19648e6 −0.393268
\(393\) 1.35671e6 0.443103
\(394\) 1.96721e6 0.638424
\(395\) −1.77429e6 −0.572179
\(396\) 211610. 0.0678108
\(397\) 5.42452e6 1.72737 0.863684 0.504034i \(-0.168151\pi\)
0.863684 + 0.504034i \(0.168151\pi\)
\(398\) −1.29000e6 −0.408208
\(399\) 612175. 0.192506
\(400\) 160000. 0.0500000
\(401\) −5.09922e6 −1.58359 −0.791795 0.610787i \(-0.790853\pi\)
−0.791795 + 0.610787i \(0.790853\pi\)
\(402\) 604570. 0.186587
\(403\) −1.21850e6 −0.373736
\(404\) −2.73656e6 −0.834166
\(405\) −164025. −0.0496904
\(406\) −3.83330e6 −1.15414
\(407\) −2.25895e6 −0.675961
\(408\) −418532. −0.124474
\(409\) 2.79695e6 0.826754 0.413377 0.910560i \(-0.364349\pi\)
0.413377 + 0.910560i \(0.364349\pi\)
\(410\) 730701. 0.214674
\(411\) −288843. −0.0843446
\(412\) −1.59066e6 −0.461672
\(413\) −4.83722e6 −1.39547
\(414\) 614258. 0.176137
\(415\) 1.99675e6 0.569118
\(416\) 1.09365e6 0.309846
\(417\) 991667. 0.279271
\(418\) 235776. 0.0660021
\(419\) −800512. −0.222758 −0.111379 0.993778i \(-0.535527\pi\)
−0.111379 + 0.993778i \(0.535527\pi\)
\(420\) 678311. 0.187631
\(421\) −3.85366e6 −1.05966 −0.529832 0.848103i \(-0.677745\pi\)
−0.529832 + 0.848103i \(0.677745\pi\)
\(422\) 491510. 0.134354
\(423\) −213986. −0.0581479
\(424\) −1.52157e6 −0.411033
\(425\) 454136. 0.121959
\(426\) 673868. 0.179908
\(427\) −5.69141e6 −1.51060
\(428\) 1.57072e6 0.414466
\(429\) −1.56947e6 −0.411728
\(430\) −762027. −0.198746
\(431\) −1.56521e6 −0.405863 −0.202932 0.979193i \(-0.565047\pi\)
−0.202932 + 0.979193i \(0.565047\pi\)
\(432\) 186624. 0.0481125
\(433\) −5.27034e6 −1.35089 −0.675443 0.737413i \(-0.736047\pi\)
−0.675443 + 0.737413i \(0.736047\pi\)
\(434\) 859871. 0.219134
\(435\) 1.14438e6 0.289965
\(436\) 2.32073e6 0.584667
\(437\) 684405. 0.171439
\(438\) −1.22985e6 −0.306314
\(439\) −1.48526e6 −0.367825 −0.183913 0.982943i \(-0.558876\pi\)
−0.183913 + 0.982943i \(0.558876\pi\)
\(440\) 261247. 0.0643309
\(441\) 1.51429e6 0.370777
\(442\) 3.10417e6 0.755771
\(443\) −4.96471e6 −1.20195 −0.600973 0.799270i \(-0.705220\pi\)
−0.600973 + 0.799270i \(0.705220\pi\)
\(444\) −1.99223e6 −0.479602
\(445\) 493726. 0.118191
\(446\) 1.85931e6 0.442604
\(447\) 2.81124e6 0.665471
\(448\) −771767. −0.181673
\(449\) 5.51895e6 1.29194 0.645968 0.763365i \(-0.276454\pi\)
0.645968 + 0.763365i \(0.276454\pi\)
\(450\) −202500. −0.0471405
\(451\) 1.19308e6 0.276204
\(452\) −1.63066e6 −0.375421
\(453\) −59679.3 −0.0136640
\(454\) −2.74091e6 −0.624101
\(455\) −5.03090e6 −1.13925
\(456\) 207936. 0.0468293
\(457\) 3.29166e6 0.737268 0.368634 0.929575i \(-0.379826\pi\)
0.368634 + 0.929575i \(0.379826\pi\)
\(458\) −5.20434e6 −1.15932
\(459\) 529705. 0.117355
\(460\) 758344. 0.167098
\(461\) −4.07683e6 −0.893449 −0.446725 0.894672i \(-0.647410\pi\)
−0.446725 + 0.894672i \(0.647410\pi\)
\(462\) 1.10754e6 0.241410
\(463\) 3.14890e6 0.682662 0.341331 0.939943i \(-0.389122\pi\)
0.341331 + 0.939943i \(0.389122\pi\)
\(464\) −1.30205e6 −0.280757
\(465\) −256702. −0.0550551
\(466\) −250919. −0.0535265
\(467\) 3.06196e6 0.649692 0.324846 0.945767i \(-0.394687\pi\)
0.324846 + 0.945767i \(0.394687\pi\)
\(468\) −1.38416e6 −0.292126
\(469\) 3.16425e6 0.664260
\(470\) −264180. −0.0551639
\(471\) 3.62000e6 0.751893
\(472\) −1.64305e6 −0.339465
\(473\) −1.24423e6 −0.255711
\(474\) −2.55498e6 −0.522326
\(475\) −225625. −0.0458831
\(476\) −2.19055e6 −0.443134
\(477\) 1.92574e6 0.387526
\(478\) 1.63602e6 0.327505
\(479\) −1.57723e6 −0.314092 −0.157046 0.987591i \(-0.550197\pi\)
−0.157046 + 0.987591i \(0.550197\pi\)
\(480\) 230400. 0.0456435
\(481\) 1.47760e7 2.91201
\(482\) −7.08776e6 −1.38960
\(483\) 3.21495e6 0.627057
\(484\) −2.15025e6 −0.417231
\(485\) −1.92357e6 −0.371325
\(486\) −236196. −0.0453609
\(487\) −7.37636e6 −1.40935 −0.704676 0.709529i \(-0.748908\pi\)
−0.704676 + 0.709529i \(0.748908\pi\)
\(488\) −1.93319e6 −0.367472
\(489\) 4.28842e6 0.811008
\(490\) 1.86949e6 0.351750
\(491\) −5.15573e6 −0.965131 −0.482565 0.875860i \(-0.660295\pi\)
−0.482565 + 0.875860i \(0.660295\pi\)
\(492\) 1.05221e6 0.195970
\(493\) −3.69566e6 −0.684818
\(494\) −1.54222e6 −0.284335
\(495\) −330641. −0.0606518
\(496\) 292070. 0.0533069
\(497\) 3.52694e6 0.640483
\(498\) 2.87531e6 0.519532
\(499\) 5.84287e6 1.05045 0.525224 0.850964i \(-0.323981\pi\)
0.525224 + 0.850964i \(0.323981\pi\)
\(500\) −250000. −0.0447214
\(501\) 6.11892e6 1.08913
\(502\) 97883.1 0.0173360
\(503\) −8.39398e6 −1.47927 −0.739636 0.673007i \(-0.765002\pi\)
−0.739636 + 0.673007i \(0.765002\pi\)
\(504\) 976767. 0.171283
\(505\) 4.27588e6 0.746100
\(506\) 1.23822e6 0.214992
\(507\) 6.92438e6 1.19636
\(508\) 608417. 0.104602
\(509\) −2.75953e6 −0.472108 −0.236054 0.971740i \(-0.575854\pi\)
−0.236054 + 0.971740i \(0.575854\pi\)
\(510\) 653956. 0.111333
\(511\) −6.43688e6 −1.09049
\(512\) −262144. −0.0441942
\(513\) −263169. −0.0441511
\(514\) 4.97836e6 0.831148
\(515\) 2.48540e6 0.412932
\(516\) −1.09732e6 −0.181430
\(517\) −431351. −0.0709749
\(518\) −1.04271e7 −1.70741
\(519\) 4.35585e6 0.709830
\(520\) −1.70883e6 −0.277135
\(521\) −224096. −0.0361693 −0.0180847 0.999836i \(-0.505757\pi\)
−0.0180847 + 0.999836i \(0.505757\pi\)
\(522\) 1.64790e6 0.264701
\(523\) 6.57676e6 1.05138 0.525688 0.850678i \(-0.323808\pi\)
0.525688 + 0.850678i \(0.323808\pi\)
\(524\) 2.41192e6 0.383738
\(525\) −1.05986e6 −0.167823
\(526\) 88700.3 0.0139785
\(527\) 828998. 0.130025
\(528\) 376196. 0.0587258
\(529\) −2.84206e6 −0.441564
\(530\) 2.37745e6 0.367639
\(531\) 2.07948e6 0.320051
\(532\) 1.08831e6 0.166715
\(533\) −7.80404e6 −1.18988
\(534\) 710966. 0.107894
\(535\) −2.45425e6 −0.370709
\(536\) 1.07479e6 0.161589
\(537\) 847328. 0.126799
\(538\) 3.84437e6 0.572624
\(539\) 3.05250e6 0.452568
\(540\) −291600. −0.0430331
\(541\) 7.65937e6 1.12512 0.562561 0.826756i \(-0.309816\pi\)
0.562561 + 0.826756i \(0.309816\pi\)
\(542\) 1.19753e6 0.175101
\(543\) −4.55879e6 −0.663515
\(544\) −744057. −0.107798
\(545\) −3.62614e6 −0.522942
\(546\) −7.24450e6 −1.03998
\(547\) −2.99733e6 −0.428318 −0.214159 0.976799i \(-0.568701\pi\)
−0.214159 + 0.976799i \(0.568701\pi\)
\(548\) −513499. −0.0730446
\(549\) 2.44669e6 0.346456
\(550\) −408199. −0.0575393
\(551\) 1.83609e6 0.257641
\(552\) 1.09202e6 0.152539
\(553\) −1.33725e7 −1.85951
\(554\) 2.16454e6 0.299635
\(555\) 3.11285e6 0.428969
\(556\) 1.76296e6 0.241856
\(557\) 7.13044e6 0.973820 0.486910 0.873452i \(-0.338124\pi\)
0.486910 + 0.873452i \(0.338124\pi\)
\(558\) −369651. −0.0502582
\(559\) 8.13861e6 1.10159
\(560\) 1.20589e6 0.162494
\(561\) 1.06778e6 0.143243
\(562\) −823030. −0.109920
\(563\) −1.23452e7 −1.64145 −0.820726 0.571322i \(-0.806431\pi\)
−0.820726 + 0.571322i \(0.806431\pi\)
\(564\) −380419. −0.0503575
\(565\) 2.54791e6 0.335786
\(566\) −1.04767e7 −1.37462
\(567\) −1.23622e6 −0.161487
\(568\) 1.19799e6 0.155805
\(569\) 2.37665e6 0.307740 0.153870 0.988091i \(-0.450826\pi\)
0.153870 + 0.988091i \(0.450826\pi\)
\(570\) −324900. −0.0418854
\(571\) −215347. −0.0276407 −0.0138203 0.999904i \(-0.504399\pi\)
−0.0138203 + 0.999904i \(0.504399\pi\)
\(572\) −2.79017e6 −0.356567
\(573\) −7.78225e6 −0.990190
\(574\) 5.50713e6 0.697663
\(575\) −1.18491e6 −0.149457
\(576\) 331776. 0.0416667
\(577\) −3.05400e6 −0.381883 −0.190941 0.981601i \(-0.561154\pi\)
−0.190941 + 0.981601i \(0.561154\pi\)
\(578\) 3.56753e6 0.444169
\(579\) 4.66446e6 0.578236
\(580\) 2.03445e6 0.251117
\(581\) 1.50490e7 1.84956
\(582\) −2.76994e6 −0.338971
\(583\) 3.88189e6 0.473012
\(584\) −2.18640e6 −0.265275
\(585\) 2.16274e6 0.261285
\(586\) −1.34489e6 −0.161787
\(587\) −1.45039e7 −1.73736 −0.868678 0.495377i \(-0.835030\pi\)
−0.868678 + 0.495377i \(0.835030\pi\)
\(588\) 2.69207e6 0.321102
\(589\) −411865. −0.0489177
\(590\) 2.56726e6 0.303627
\(591\) −4.42621e6 −0.521271
\(592\) −3.54173e6 −0.415348
\(593\) 7.31032e6 0.853689 0.426844 0.904325i \(-0.359625\pi\)
0.426844 + 0.904325i \(0.359625\pi\)
\(594\) −476123. −0.0553673
\(595\) 3.42273e6 0.396351
\(596\) 4.99776e6 0.576314
\(597\) 2.90250e6 0.333300
\(598\) −8.09927e6 −0.926175
\(599\) −5.99827e6 −0.683060 −0.341530 0.939871i \(-0.610945\pi\)
−0.341530 + 0.939871i \(0.610945\pi\)
\(600\) −360000. −0.0408248
\(601\) −2.65975e6 −0.300368 −0.150184 0.988658i \(-0.547987\pi\)
−0.150184 + 0.988658i \(0.547987\pi\)
\(602\) −5.74324e6 −0.645900
\(603\) −1.36028e6 −0.152348
\(604\) −106096. −0.0118334
\(605\) 3.35977e6 0.373182
\(606\) 6.15727e6 0.681093
\(607\) 1.42361e7 1.56826 0.784130 0.620596i \(-0.213109\pi\)
0.784130 + 0.620596i \(0.213109\pi\)
\(608\) 369664. 0.0405554
\(609\) 8.62492e6 0.942349
\(610\) 3.02060e6 0.328677
\(611\) 2.82150e6 0.305757
\(612\) 941697. 0.101633
\(613\) −1.30115e7 −1.39855 −0.699273 0.714854i \(-0.746493\pi\)
−0.699273 + 0.714854i \(0.746493\pi\)
\(614\) −206231. −0.0220766
\(615\) −1.64408e6 −0.175281
\(616\) 1.96896e6 0.209067
\(617\) 8.65260e6 0.915026 0.457513 0.889203i \(-0.348740\pi\)
0.457513 + 0.889203i \(0.348740\pi\)
\(618\) 3.57898e6 0.376954
\(619\) 1.81666e7 1.90567 0.952834 0.303491i \(-0.0981522\pi\)
0.952834 + 0.303491i \(0.0981522\pi\)
\(620\) −456360. −0.0476791
\(621\) −1.38208e6 −0.143815
\(622\) −1.05838e7 −1.09689
\(623\) 3.72111e6 0.384107
\(624\) −2.46072e6 −0.252989
\(625\) 390625. 0.0400000
\(626\) 6.67657e6 0.680954
\(627\) −530495. −0.0538905
\(628\) 6.43555e6 0.651159
\(629\) −1.00527e7 −1.01311
\(630\) −1.52620e6 −0.153200
\(631\) −4.68737e6 −0.468658 −0.234329 0.972157i \(-0.575289\pi\)
−0.234329 + 0.972157i \(0.575289\pi\)
\(632\) −4.54219e6 −0.452348
\(633\) −1.10590e6 −0.109700
\(634\) −1.20171e7 −1.18734
\(635\) −950651. −0.0935593
\(636\) 3.42353e6 0.335607
\(637\) −1.99666e7 −1.94964
\(638\) 3.32183e6 0.323092
\(639\) −1.51620e6 −0.146894
\(640\) 409600. 0.0395285
\(641\) 1.49467e7 1.43681 0.718407 0.695623i \(-0.244872\pi\)
0.718407 + 0.695623i \(0.244872\pi\)
\(642\) −3.53411e6 −0.338410
\(643\) 8.05456e6 0.768271 0.384136 0.923277i \(-0.374499\pi\)
0.384136 + 0.923277i \(0.374499\pi\)
\(644\) 5.71547e6 0.543047
\(645\) 1.71456e6 0.162276
\(646\) 1.04924e6 0.0989218
\(647\) −1.00493e7 −0.943792 −0.471896 0.881654i \(-0.656430\pi\)
−0.471896 + 0.881654i \(0.656430\pi\)
\(648\) −419904. −0.0392837
\(649\) 4.19181e6 0.390652
\(650\) 2.67005e6 0.247877
\(651\) −1.93471e6 −0.178922
\(652\) 7.62386e6 0.702353
\(653\) 8.67881e6 0.796484 0.398242 0.917280i \(-0.369620\pi\)
0.398242 + 0.917280i \(0.369620\pi\)
\(654\) −5.22164e6 −0.477378
\(655\) −3.76863e6 −0.343226
\(656\) 1.87059e6 0.169715
\(657\) 2.76716e6 0.250104
\(658\) −1.99107e6 −0.179275
\(659\) 8.33025e6 0.747213 0.373606 0.927587i \(-0.378121\pi\)
0.373606 + 0.927587i \(0.378121\pi\)
\(660\) −587806. −0.0525260
\(661\) −1.14148e7 −1.01617 −0.508084 0.861307i \(-0.669646\pi\)
−0.508084 + 0.861307i \(0.669646\pi\)
\(662\) 137145. 0.0121629
\(663\) −6.98439e6 −0.617085
\(664\) 5.11167e6 0.449928
\(665\) −1.70049e6 −0.149114
\(666\) 4.48251e6 0.391593
\(667\) 9.64256e6 0.839224
\(668\) 1.08781e7 0.943216
\(669\) −4.18346e6 −0.361385
\(670\) −1.67936e6 −0.144530
\(671\) 4.93202e6 0.422882
\(672\) 1.73648e6 0.148336
\(673\) −1.97834e7 −1.68370 −0.841849 0.539714i \(-0.818532\pi\)
−0.841849 + 0.539714i \(0.818532\pi\)
\(674\) 1.04828e7 0.888846
\(675\) 455625. 0.0384900
\(676\) 1.23100e7 1.03608
\(677\) 1.00766e7 0.844969 0.422484 0.906370i \(-0.361158\pi\)
0.422484 + 0.906370i \(0.361158\pi\)
\(678\) 3.66899e6 0.306530
\(679\) −1.44975e7 −1.20676
\(680\) 1.16259e6 0.0964171
\(681\) 6.16705e6 0.509577
\(682\) −745142. −0.0613448
\(683\) 356129. 0.0292116 0.0146058 0.999893i \(-0.495351\pi\)
0.0146058 + 0.999893i \(0.495351\pi\)
\(684\) −467856. −0.0382360
\(685\) 802342. 0.0653331
\(686\) 1.42291e6 0.115443
\(687\) 1.17098e7 0.946578
\(688\) −1.95079e6 −0.157123
\(689\) −2.53917e7 −2.03772
\(690\) −1.70627e6 −0.136435
\(691\) −5.13402e6 −0.409037 −0.204518 0.978863i \(-0.565563\pi\)
−0.204518 + 0.978863i \(0.565563\pi\)
\(692\) 7.74373e6 0.614731
\(693\) −2.49197e6 −0.197110
\(694\) 1.04003e7 0.819683
\(695\) −2.75463e6 −0.216322
\(696\) 2.92960e6 0.229237
\(697\) 5.30940e6 0.413965
\(698\) 1.02473e6 0.0796110
\(699\) 564568. 0.0437042
\(700\) −1.88420e6 −0.145339
\(701\) 1.47905e7 1.13681 0.568403 0.822750i \(-0.307561\pi\)
0.568403 + 0.822750i \(0.307561\pi\)
\(702\) 3.11435e6 0.238520
\(703\) 4.99440e6 0.381149
\(704\) 668793. 0.0508581
\(705\) 594405. 0.0450411
\(706\) −1.20402e6 −0.0909123
\(707\) 3.22264e7 2.42473
\(708\) 3.69686e6 0.277172
\(709\) −1.53286e7 −1.14522 −0.572609 0.819829i \(-0.694069\pi\)
−0.572609 + 0.819829i \(0.694069\pi\)
\(710\) −1.87185e6 −0.139356
\(711\) 5.74870e6 0.426477
\(712\) 1.26394e6 0.0934386
\(713\) −2.16299e6 −0.159342
\(714\) 4.92873e6 0.361817
\(715\) 4.35965e6 0.318923
\(716\) 1.50636e6 0.109811
\(717\) −3.68104e6 −0.267407
\(718\) −7.73508e6 −0.559956
\(719\) 1.04723e6 0.0755477 0.0377739 0.999286i \(-0.487973\pi\)
0.0377739 + 0.999286i \(0.487973\pi\)
\(720\) −518400. −0.0372678
\(721\) 1.87320e7 1.34198
\(722\) −521284. −0.0372161
\(723\) 1.59475e7 1.13461
\(724\) −8.10452e6 −0.574620
\(725\) −3.17882e6 −0.224606
\(726\) 4.83807e6 0.340667
\(727\) 1.60299e7 1.12485 0.562426 0.826848i \(-0.309868\pi\)
0.562426 + 0.826848i \(0.309868\pi\)
\(728\) −1.28791e7 −0.900653
\(729\) 531441. 0.0370370
\(730\) 3.41625e6 0.237269
\(731\) −5.53703e6 −0.383251
\(732\) 4.34967e6 0.300039
\(733\) −4.83302e6 −0.332245 −0.166123 0.986105i \(-0.553125\pi\)
−0.166123 + 0.986105i \(0.553125\pi\)
\(734\) 2.01456e7 1.38020
\(735\) −4.20636e6 −0.287203
\(736\) 1.94136e6 0.132103
\(737\) −2.74205e6 −0.185955
\(738\) −2.36747e6 −0.160009
\(739\) 2.41465e7 1.62646 0.813229 0.581944i \(-0.197708\pi\)
0.813229 + 0.581944i \(0.197708\pi\)
\(740\) 5.53396e6 0.371498
\(741\) 3.47000e6 0.232158
\(742\) 1.79183e7 1.19478
\(743\) 2.08822e7 1.38772 0.693862 0.720108i \(-0.255908\pi\)
0.693862 + 0.720108i \(0.255908\pi\)
\(744\) −657158. −0.0435249
\(745\) −7.80900e6 −0.515471
\(746\) −5.72036e6 −0.376337
\(747\) −6.46946e6 −0.424196
\(748\) 1.89827e6 0.124052
\(749\) −1.84971e7 −1.20476
\(750\) 562500. 0.0365148
\(751\) 2.05464e7 1.32934 0.664670 0.747137i \(-0.268572\pi\)
0.664670 + 0.747137i \(0.268572\pi\)
\(752\) −676300. −0.0436109
\(753\) −220237. −0.0141548
\(754\) −2.17283e7 −1.39187
\(755\) 165776. 0.0105841
\(756\) −2.19773e6 −0.139852
\(757\) −5.95592e6 −0.377754 −0.188877 0.982001i \(-0.560485\pi\)
−0.188877 + 0.982001i \(0.560485\pi\)
\(758\) −6.74456e6 −0.426364
\(759\) −2.78599e6 −0.175540
\(760\) −577600. −0.0362738
\(761\) −2.30460e7 −1.44256 −0.721280 0.692643i \(-0.756446\pi\)
−0.721280 + 0.692643i \(0.756446\pi\)
\(762\) −1.36894e6 −0.0854075
\(763\) −2.73294e7 −1.69949
\(764\) −1.38351e7 −0.857530
\(765\) −1.47140e6 −0.0909029
\(766\) 9.89771e6 0.609485
\(767\) −2.74189e7 −1.68291
\(768\) 589824. 0.0360844
\(769\) −2.79252e7 −1.70286 −0.851432 0.524464i \(-0.824266\pi\)
−0.851432 + 0.524464i \(0.824266\pi\)
\(770\) −3.07651e6 −0.186995
\(771\) −1.12013e7 −0.678630
\(772\) 8.29238e6 0.500767
\(773\) −1.14071e7 −0.686633 −0.343317 0.939220i \(-0.611550\pi\)
−0.343317 + 0.939220i \(0.611550\pi\)
\(774\) 2.46897e6 0.148137
\(775\) 713062. 0.0426455
\(776\) −4.92434e6 −0.293558
\(777\) 2.34609e7 1.39409
\(778\) 3.72190e6 0.220453
\(779\) −2.63783e6 −0.155741
\(780\) 3.84488e6 0.226280
\(781\) −3.05635e6 −0.179298
\(782\) 5.51026e6 0.322222
\(783\) −3.70778e6 −0.216127
\(784\) 4.78591e6 0.278083
\(785\) −1.00556e7 −0.582414
\(786\) −5.42682e6 −0.313321
\(787\) −2.15807e6 −0.124202 −0.0621011 0.998070i \(-0.519780\pi\)
−0.0621011 + 0.998070i \(0.519780\pi\)
\(788\) −7.86882e6 −0.451434
\(789\) −199576. −0.0114134
\(790\) 7.09717e6 0.404592
\(791\) 1.92031e7 1.09126
\(792\) −846441. −0.0479494
\(793\) −3.22607e7 −1.82176
\(794\) −2.16981e7 −1.22143
\(795\) −5.34927e6 −0.300176
\(796\) 5.15999e6 0.288647
\(797\) −3.37116e7 −1.87989 −0.939947 0.341320i \(-0.889126\pi\)
−0.939947 + 0.341320i \(0.889126\pi\)
\(798\) −2.44870e6 −0.136122
\(799\) −1.91958e6 −0.106375
\(800\) −640000. −0.0353553
\(801\) −1.59967e6 −0.0880947
\(802\) 2.03969e7 1.11977
\(803\) 5.57803e6 0.305275
\(804\) −2.41828e6 −0.131937
\(805\) −8.93043e6 −0.485716
\(806\) 4.87402e6 0.264271
\(807\) −8.64983e6 −0.467545
\(808\) 1.09463e7 0.589844
\(809\) 2.64286e7 1.41972 0.709861 0.704341i \(-0.248758\pi\)
0.709861 + 0.704341i \(0.248758\pi\)
\(810\) 656100. 0.0351364
\(811\) 2.42180e7 1.29296 0.646481 0.762930i \(-0.276240\pi\)
0.646481 + 0.762930i \(0.276240\pi\)
\(812\) 1.53332e7 0.816098
\(813\) −2.69445e6 −0.142970
\(814\) 9.03582e6 0.477976
\(815\) −1.19123e7 −0.628204
\(816\) 1.67413e6 0.0880163
\(817\) 2.75092e6 0.144186
\(818\) −1.11878e7 −0.584603
\(819\) 1.63001e7 0.849144
\(820\) −2.92280e6 −0.151798
\(821\) −8.68308e6 −0.449589 −0.224795 0.974406i \(-0.572171\pi\)
−0.224795 + 0.974406i \(0.572171\pi\)
\(822\) 1.15537e6 0.0596407
\(823\) −4.02059e6 −0.206914 −0.103457 0.994634i \(-0.532990\pi\)
−0.103457 + 0.994634i \(0.532990\pi\)
\(824\) 6.36264e6 0.326452
\(825\) 918447. 0.0469807
\(826\) 1.93489e7 0.986747
\(827\) −2.90473e7 −1.47687 −0.738435 0.674325i \(-0.764435\pi\)
−0.738435 + 0.674325i \(0.764435\pi\)
\(828\) −2.45703e6 −0.124548
\(829\) −3.07354e6 −0.155329 −0.0776644 0.996980i \(-0.524746\pi\)
−0.0776644 + 0.996980i \(0.524746\pi\)
\(830\) −7.98698e6 −0.402427
\(831\) −4.87022e6 −0.244651
\(832\) −4.37461e6 −0.219094
\(833\) 1.35841e7 0.678294
\(834\) −3.96667e6 −0.197474
\(835\) −1.69970e7 −0.843638
\(836\) −943102. −0.0466706
\(837\) 831716. 0.0410357
\(838\) 3.20205e6 0.157513
\(839\) 7.10199e6 0.348317 0.174159 0.984718i \(-0.444279\pi\)
0.174159 + 0.984718i \(0.444279\pi\)
\(840\) −2.71324e6 −0.132675
\(841\) 5.35742e6 0.261196
\(842\) 1.54146e7 0.749295
\(843\) 1.85182e6 0.0897489
\(844\) −1.96604e6 −0.0950028
\(845\) −1.92344e7 −0.926695
\(846\) 855942. 0.0411167
\(847\) 2.53219e7 1.21279
\(848\) 6.08628e6 0.290644
\(849\) 2.35725e7 1.12237
\(850\) −1.81655e6 −0.0862380
\(851\) 2.62290e7 1.24153
\(852\) −2.69547e6 −0.127214
\(853\) −4.00763e7 −1.88588 −0.942942 0.332958i \(-0.891953\pi\)
−0.942942 + 0.332958i \(0.891953\pi\)
\(854\) 2.27656e7 1.06816
\(855\) 731025. 0.0341993
\(856\) −6.28287e6 −0.293072
\(857\) −1.83952e6 −0.0855566 −0.0427783 0.999085i \(-0.513621\pi\)
−0.0427783 + 0.999085i \(0.513621\pi\)
\(858\) 6.27789e6 0.291136
\(859\) −3.63663e7 −1.68157 −0.840787 0.541367i \(-0.817907\pi\)
−0.840787 + 0.541367i \(0.817907\pi\)
\(860\) 3.04811e6 0.140535
\(861\) −1.23911e7 −0.569640
\(862\) 6.26084e6 0.286989
\(863\) 2.65190e7 1.21208 0.606039 0.795435i \(-0.292758\pi\)
0.606039 + 0.795435i \(0.292758\pi\)
\(864\) −746496. −0.0340207
\(865\) −1.20996e7 −0.549832
\(866\) 2.10813e7 0.955220
\(867\) −8.02695e6 −0.362663
\(868\) −3.43949e6 −0.154951
\(869\) 1.15882e7 0.520556
\(870\) −4.57750e6 −0.205036
\(871\) 1.79359e7 0.801085
\(872\) −9.28292e6 −0.413422
\(873\) 6.23237e6 0.276769
\(874\) −2.73762e6 −0.121226
\(875\) 2.94406e6 0.129995
\(876\) 4.91939e6 0.216596
\(877\) −1.40756e7 −0.617971 −0.308985 0.951067i \(-0.599989\pi\)
−0.308985 + 0.951067i \(0.599989\pi\)
\(878\) 5.94105e6 0.260092
\(879\) 3.02601e6 0.132098
\(880\) −1.04499e6 −0.0454888
\(881\) −1.32094e7 −0.573383 −0.286692 0.958023i \(-0.592555\pi\)
−0.286692 + 0.958023i \(0.592555\pi\)
\(882\) −6.05716e6 −0.262179
\(883\) 7.79174e6 0.336304 0.168152 0.985761i \(-0.446220\pi\)
0.168152 + 0.985761i \(0.446220\pi\)
\(884\) −1.24167e7 −0.534411
\(885\) −5.77634e6 −0.247910
\(886\) 1.98588e7 0.849904
\(887\) 1.89333e7 0.808011 0.404005 0.914757i \(-0.367618\pi\)
0.404005 + 0.914757i \(0.367618\pi\)
\(888\) 7.96890e6 0.339130
\(889\) −7.16485e6 −0.304055
\(890\) −1.97490e6 −0.0835740
\(891\) 1.07128e6 0.0452072
\(892\) −7.43725e6 −0.312968
\(893\) 953689. 0.0400201
\(894\) −1.12450e7 −0.470559
\(895\) −2.35369e6 −0.0982181
\(896\) 3.08707e6 0.128462
\(897\) 1.82234e7 0.756219
\(898\) −2.20758e7 −0.913536
\(899\) −5.80275e6 −0.239461
\(900\) 810000. 0.0333333
\(901\) 1.72750e7 0.708934
\(902\) −4.77234e6 −0.195306
\(903\) 1.29223e7 0.527375
\(904\) 6.52265e6 0.265462
\(905\) 1.26633e7 0.513956
\(906\) 238717. 0.00966191
\(907\) 1.56137e7 0.630214 0.315107 0.949056i \(-0.397960\pi\)
0.315107 + 0.949056i \(0.397960\pi\)
\(908\) 1.09636e7 0.441306
\(909\) −1.38539e7 −0.556110
\(910\) 2.01236e7 0.805568
\(911\) −2.46900e7 −0.985656 −0.492828 0.870127i \(-0.664037\pi\)
−0.492828 + 0.870127i \(0.664037\pi\)
\(912\) −831744. −0.0331133
\(913\) −1.30411e7 −0.517771
\(914\) −1.31667e7 −0.521327
\(915\) −6.79635e6 −0.268363
\(916\) 2.08174e7 0.819760
\(917\) −2.84033e7 −1.11544
\(918\) −2.11882e6 −0.0829826
\(919\) 1.62633e7 0.635213 0.317607 0.948223i \(-0.397121\pi\)
0.317607 + 0.948223i \(0.397121\pi\)
\(920\) −3.03338e6 −0.118156
\(921\) 464019. 0.0180255
\(922\) 1.63073e7 0.631764
\(923\) 1.99918e7 0.772410
\(924\) −4.43017e6 −0.170703
\(925\) −8.64681e6 −0.332278
\(926\) −1.25956e7 −0.482715
\(927\) −8.05271e6 −0.307782
\(928\) 5.20818e6 0.198525
\(929\) 4.37853e7 1.66452 0.832260 0.554385i \(-0.187047\pi\)
0.832260 + 0.554385i \(0.187047\pi\)
\(930\) 1.02681e6 0.0389298
\(931\) −6.74888e6 −0.255186
\(932\) 1.00368e6 0.0378490
\(933\) 2.38135e7 0.895610
\(934\) −1.22479e7 −0.459402
\(935\) −2.96605e6 −0.110955
\(936\) 5.53662e6 0.206564
\(937\) 2.22150e7 0.826604 0.413302 0.910594i \(-0.364375\pi\)
0.413302 + 0.910594i \(0.364375\pi\)
\(938\) −1.26570e7 −0.469703
\(939\) −1.50223e7 −0.555996
\(940\) 1.05672e6 0.0390068
\(941\) 2.54661e7 0.937538 0.468769 0.883321i \(-0.344698\pi\)
0.468769 + 0.883321i \(0.344698\pi\)
\(942\) −1.44800e7 −0.531669
\(943\) −1.38531e7 −0.507302
\(944\) 6.57219e6 0.240038
\(945\) 3.43395e6 0.125088
\(946\) 4.97694e6 0.180815
\(947\) −6.44913e6 −0.233683 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(948\) 1.02199e7 0.369340
\(949\) −3.64862e7 −1.31511
\(950\) 902500. 0.0324443
\(951\) 2.70384e7 0.969461
\(952\) 8.76218e6 0.313343
\(953\) −1.12726e7 −0.402060 −0.201030 0.979585i \(-0.564429\pi\)
−0.201030 + 0.979585i \(0.564429\pi\)
\(954\) −7.70294e6 −0.274022
\(955\) 2.16174e7 0.766998
\(956\) −6.54406e6 −0.231581
\(957\) −7.47412e6 −0.263803
\(958\) 6.30894e6 0.222097
\(959\) 6.04708e6 0.212324
\(960\) −921600. −0.0322749
\(961\) −2.73275e7 −0.954534
\(962\) −5.91038e7 −2.05910
\(963\) 7.95176e6 0.276311
\(964\) 2.83510e7 0.982599
\(965\) −1.29568e7 −0.447900
\(966\) −1.28598e7 −0.443396
\(967\) 2.88542e7 0.992300 0.496150 0.868237i \(-0.334747\pi\)
0.496150 + 0.868237i \(0.334747\pi\)
\(968\) 8.60101e6 0.295027
\(969\) −2.36078e6 −0.0807693
\(970\) 7.69428e6 0.262566
\(971\) 3.25551e7 1.10808 0.554041 0.832490i \(-0.313085\pi\)
0.554041 + 0.832490i \(0.313085\pi\)
\(972\) 944784. 0.0320750
\(973\) −2.07611e7 −0.703020
\(974\) 2.95054e7 0.996563
\(975\) −6.00762e6 −0.202391
\(976\) 7.73274e6 0.259842
\(977\) −2.58544e7 −0.866559 −0.433279 0.901260i \(-0.642644\pi\)
−0.433279 + 0.901260i \(0.642644\pi\)
\(978\) −1.71537e7 −0.573469
\(979\) −3.22461e6 −0.107528
\(980\) −7.47798e6 −0.248725
\(981\) 1.17487e7 0.389778
\(982\) 2.06229e7 0.682450
\(983\) 1.48810e7 0.491189 0.245595 0.969373i \(-0.421017\pi\)
0.245595 + 0.969373i \(0.421017\pi\)
\(984\) −4.20884e6 −0.138572
\(985\) 1.22950e7 0.403775
\(986\) 1.47827e7 0.484239
\(987\) 4.47990e6 0.146378
\(988\) 6.16889e6 0.201055
\(989\) 1.44470e7 0.469663
\(990\) 1.32256e6 0.0428873
\(991\) −823632. −0.0266409 −0.0133205 0.999911i \(-0.504240\pi\)
−0.0133205 + 0.999911i \(0.504240\pi\)
\(992\) −1.16828e6 −0.0376937
\(993\) −308577. −0.00993093
\(994\) −1.41078e7 −0.452890
\(995\) −8.06249e6 −0.258173
\(996\) −1.15013e7 −0.367364
\(997\) 1.85076e7 0.589675 0.294838 0.955547i \(-0.404734\pi\)
0.294838 + 0.955547i \(0.404734\pi\)
\(998\) −2.33715e7 −0.742780
\(999\) −1.00856e7 −0.319735
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 570.6.a.k.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.6.a.k.1.1 4 1.1 even 1 trivial