Properties

Label 33.6.a.a.1.1
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +46.0000 q^{5} +18.0000 q^{6} +148.000 q^{7} +120.000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +46.0000 q^{5} +18.0000 q^{6} +148.000 q^{7} +120.000 q^{8} +81.0000 q^{9} -92.0000 q^{10} +121.000 q^{11} +252.000 q^{12} +574.000 q^{13} -296.000 q^{14} -414.000 q^{15} +656.000 q^{16} -722.000 q^{17} -162.000 q^{18} +2160.00 q^{19} -1288.00 q^{20} -1332.00 q^{21} -242.000 q^{22} -2536.00 q^{23} -1080.00 q^{24} -1009.00 q^{25} -1148.00 q^{26} -729.000 q^{27} -4144.00 q^{28} +4650.00 q^{29} +828.000 q^{30} +5032.00 q^{31} -5152.00 q^{32} -1089.00 q^{33} +1444.00 q^{34} +6808.00 q^{35} -2268.00 q^{36} +8118.00 q^{37} -4320.00 q^{38} -5166.00 q^{39} +5520.00 q^{40} -5138.00 q^{41} +2664.00 q^{42} +8304.00 q^{43} -3388.00 q^{44} +3726.00 q^{45} +5072.00 q^{46} +24728.0 q^{47} -5904.00 q^{48} +5097.00 q^{49} +2018.00 q^{50} +6498.00 q^{51} -16072.0 q^{52} -28746.0 q^{53} +1458.00 q^{54} +5566.00 q^{55} +17760.0 q^{56} -19440.0 q^{57} -9300.00 q^{58} -5860.00 q^{59} +11592.0 q^{60} -53658.0 q^{61} -10064.0 q^{62} +11988.0 q^{63} -10688.0 q^{64} +26404.0 q^{65} +2178.00 q^{66} +30908.0 q^{67} +20216.0 q^{68} +22824.0 q^{69} -13616.0 q^{70} -69648.0 q^{71} +9720.00 q^{72} -18446.0 q^{73} -16236.0 q^{74} +9081.00 q^{75} -60480.0 q^{76} +17908.0 q^{77} +10332.0 q^{78} -25300.0 q^{79} +30176.0 q^{80} +6561.00 q^{81} +10276.0 q^{82} -17556.0 q^{83} +37296.0 q^{84} -33212.0 q^{85} -16608.0 q^{86} -41850.0 q^{87} +14520.0 q^{88} +132570. q^{89} -7452.00 q^{90} +84952.0 q^{91} +71008.0 q^{92} -45288.0 q^{93} -49456.0 q^{94} +99360.0 q^{95} +46368.0 q^{96} +70658.0 q^{97} -10194.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −9.00000 −0.577350
\(4\) −28.0000 −0.875000
\(5\) 46.0000 0.822873 0.411437 0.911438i \(-0.365027\pi\)
0.411437 + 0.911438i \(0.365027\pi\)
\(6\) 18.0000 0.204124
\(7\) 148.000 1.14161 0.570803 0.821087i \(-0.306632\pi\)
0.570803 + 0.821087i \(0.306632\pi\)
\(8\) 120.000 0.662913
\(9\) 81.0000 0.333333
\(10\) −92.0000 −0.290930
\(11\) 121.000 0.301511
\(12\) 252.000 0.505181
\(13\) 574.000 0.942006 0.471003 0.882132i \(-0.343892\pi\)
0.471003 + 0.882132i \(0.343892\pi\)
\(14\) −296.000 −0.403619
\(15\) −414.000 −0.475086
\(16\) 656.000 0.640625
\(17\) −722.000 −0.605919 −0.302960 0.953003i \(-0.597975\pi\)
−0.302960 + 0.953003i \(0.597975\pi\)
\(18\) −162.000 −0.117851
\(19\) 2160.00 1.37268 0.686341 0.727280i \(-0.259216\pi\)
0.686341 + 0.727280i \(0.259216\pi\)
\(20\) −1288.00 −0.720014
\(21\) −1332.00 −0.659107
\(22\) −242.000 −0.106600
\(23\) −2536.00 −0.999608 −0.499804 0.866139i \(-0.666595\pi\)
−0.499804 + 0.866139i \(0.666595\pi\)
\(24\) −1080.00 −0.382733
\(25\) −1009.00 −0.322880
\(26\) −1148.00 −0.333049
\(27\) −729.000 −0.192450
\(28\) −4144.00 −0.998906
\(29\) 4650.00 1.02673 0.513367 0.858169i \(-0.328398\pi\)
0.513367 + 0.858169i \(0.328398\pi\)
\(30\) 828.000 0.167968
\(31\) 5032.00 0.940451 0.470226 0.882546i \(-0.344172\pi\)
0.470226 + 0.882546i \(0.344172\pi\)
\(32\) −5152.00 −0.889408
\(33\) −1089.00 −0.174078
\(34\) 1444.00 0.214225
\(35\) 6808.00 0.939398
\(36\) −2268.00 −0.291667
\(37\) 8118.00 0.974866 0.487433 0.873161i \(-0.337933\pi\)
0.487433 + 0.873161i \(0.337933\pi\)
\(38\) −4320.00 −0.485316
\(39\) −5166.00 −0.543867
\(40\) 5520.00 0.545493
\(41\) −5138.00 −0.477347 −0.238674 0.971100i \(-0.576713\pi\)
−0.238674 + 0.971100i \(0.576713\pi\)
\(42\) 2664.00 0.233030
\(43\) 8304.00 0.684883 0.342441 0.939539i \(-0.388746\pi\)
0.342441 + 0.939539i \(0.388746\pi\)
\(44\) −3388.00 −0.263822
\(45\) 3726.00 0.274291
\(46\) 5072.00 0.353415
\(47\) 24728.0 1.63284 0.816421 0.577457i \(-0.195955\pi\)
0.816421 + 0.577457i \(0.195955\pi\)
\(48\) −5904.00 −0.369865
\(49\) 5097.00 0.303266
\(50\) 2018.00 0.114155
\(51\) 6498.00 0.349828
\(52\) −16072.0 −0.824255
\(53\) −28746.0 −1.40568 −0.702842 0.711346i \(-0.748086\pi\)
−0.702842 + 0.711346i \(0.748086\pi\)
\(54\) 1458.00 0.0680414
\(55\) 5566.00 0.248106
\(56\) 17760.0 0.756786
\(57\) −19440.0 −0.792518
\(58\) −9300.00 −0.363005
\(59\) −5860.00 −0.219163 −0.109582 0.993978i \(-0.534951\pi\)
−0.109582 + 0.993978i \(0.534951\pi\)
\(60\) 11592.0 0.415700
\(61\) −53658.0 −1.84633 −0.923166 0.384401i \(-0.874408\pi\)
−0.923166 + 0.384401i \(0.874408\pi\)
\(62\) −10064.0 −0.332500
\(63\) 11988.0 0.380536
\(64\) −10688.0 −0.326172
\(65\) 26404.0 0.775151
\(66\) 2178.00 0.0615457
\(67\) 30908.0 0.841170 0.420585 0.907253i \(-0.361825\pi\)
0.420585 + 0.907253i \(0.361825\pi\)
\(68\) 20216.0 0.530180
\(69\) 22824.0 0.577124
\(70\) −13616.0 −0.332127
\(71\) −69648.0 −1.63969 −0.819847 0.572583i \(-0.805942\pi\)
−0.819847 + 0.572583i \(0.805942\pi\)
\(72\) 9720.00 0.220971
\(73\) −18446.0 −0.405131 −0.202565 0.979269i \(-0.564928\pi\)
−0.202565 + 0.979269i \(0.564928\pi\)
\(74\) −16236.0 −0.344667
\(75\) 9081.00 0.186415
\(76\) −60480.0 −1.20110
\(77\) 17908.0 0.344207
\(78\) 10332.0 0.192286
\(79\) −25300.0 −0.456092 −0.228046 0.973650i \(-0.573234\pi\)
−0.228046 + 0.973650i \(0.573234\pi\)
\(80\) 30176.0 0.527153
\(81\) 6561.00 0.111111
\(82\) 10276.0 0.168768
\(83\) −17556.0 −0.279724 −0.139862 0.990171i \(-0.544666\pi\)
−0.139862 + 0.990171i \(0.544666\pi\)
\(84\) 37296.0 0.576719
\(85\) −33212.0 −0.498595
\(86\) −16608.0 −0.242143
\(87\) −41850.0 −0.592785
\(88\) 14520.0 0.199876
\(89\) 132570. 1.77407 0.887034 0.461704i \(-0.152762\pi\)
0.887034 + 0.461704i \(0.152762\pi\)
\(90\) −7452.00 −0.0969765
\(91\) 84952.0 1.07540
\(92\) 71008.0 0.874657
\(93\) −45288.0 −0.542970
\(94\) −49456.0 −0.577297
\(95\) 99360.0 1.12954
\(96\) 46368.0 0.513500
\(97\) 70658.0 0.762486 0.381243 0.924475i \(-0.375496\pi\)
0.381243 + 0.924475i \(0.375496\pi\)
\(98\) −10194.0 −0.107221
\(99\) 9801.00 0.100504
\(100\) 28252.0 0.282520
\(101\) −101998. −0.994920 −0.497460 0.867487i \(-0.665734\pi\)
−0.497460 + 0.867487i \(0.665734\pi\)
\(102\) −12996.0 −0.123683
\(103\) 130904. 1.21579 0.607897 0.794016i \(-0.292013\pi\)
0.607897 + 0.794016i \(0.292013\pi\)
\(104\) 68880.0 0.624467
\(105\) −61272.0 −0.542361
\(106\) 57492.0 0.496984
\(107\) −141612. −1.19575 −0.597875 0.801589i \(-0.703988\pi\)
−0.597875 + 0.801589i \(0.703988\pi\)
\(108\) 20412.0 0.168394
\(109\) −239810. −1.93331 −0.966654 0.256086i \(-0.917567\pi\)
−0.966654 + 0.256086i \(0.917567\pi\)
\(110\) −11132.0 −0.0877186
\(111\) −73062.0 −0.562839
\(112\) 97088.0 0.731342
\(113\) −42726.0 −0.314772 −0.157386 0.987537i \(-0.550307\pi\)
−0.157386 + 0.987537i \(0.550307\pi\)
\(114\) 38880.0 0.280197
\(115\) −116656. −0.822550
\(116\) −130200. −0.898392
\(117\) 46494.0 0.314002
\(118\) 11720.0 0.0774859
\(119\) −106856. −0.691722
\(120\) −49680.0 −0.314940
\(121\) 14641.0 0.0909091
\(122\) 107316. 0.652777
\(123\) 46242.0 0.275597
\(124\) −140896. −0.822895
\(125\) −190164. −1.08856
\(126\) −23976.0 −0.134540
\(127\) 51788.0 0.284918 0.142459 0.989801i \(-0.454499\pi\)
0.142459 + 0.989801i \(0.454499\pi\)
\(128\) 186240. 1.00473
\(129\) −74736.0 −0.395417
\(130\) −52808.0 −0.274057
\(131\) 53652.0 0.273154 0.136577 0.990629i \(-0.456390\pi\)
0.136577 + 0.990629i \(0.456390\pi\)
\(132\) 30492.0 0.152318
\(133\) 319680. 1.56706
\(134\) −61816.0 −0.297399
\(135\) −33534.0 −0.158362
\(136\) −86640.0 −0.401672
\(137\) −228862. −1.04177 −0.520886 0.853627i \(-0.674398\pi\)
−0.520886 + 0.853627i \(0.674398\pi\)
\(138\) −45648.0 −0.204044
\(139\) 374920. 1.64589 0.822947 0.568119i \(-0.192329\pi\)
0.822947 + 0.568119i \(0.192329\pi\)
\(140\) −190624. −0.821973
\(141\) −222552. −0.942722
\(142\) 139296. 0.579719
\(143\) 69454.0 0.284025
\(144\) 53136.0 0.213542
\(145\) 213900. 0.844872
\(146\) 36892.0 0.143235
\(147\) −45873.0 −0.175091
\(148\) −227304. −0.853007
\(149\) −65830.0 −0.242917 −0.121459 0.992597i \(-0.538757\pi\)
−0.121459 + 0.992597i \(0.538757\pi\)
\(150\) −18162.0 −0.0659076
\(151\) 154052. 0.549826 0.274913 0.961469i \(-0.411351\pi\)
0.274913 + 0.961469i \(0.411351\pi\)
\(152\) 259200. 0.909968
\(153\) −58482.0 −0.201973
\(154\) −35816.0 −0.121696
\(155\) 231472. 0.773872
\(156\) 144648. 0.475884
\(157\) 287678. 0.931446 0.465723 0.884931i \(-0.345794\pi\)
0.465723 + 0.884931i \(0.345794\pi\)
\(158\) 50600.0 0.161253
\(159\) 258714. 0.811572
\(160\) −236992. −0.731870
\(161\) −375328. −1.14116
\(162\) −13122.0 −0.0392837
\(163\) 105124. 0.309908 0.154954 0.987922i \(-0.450477\pi\)
0.154954 + 0.987922i \(0.450477\pi\)
\(164\) 143864. 0.417679
\(165\) −50094.0 −0.143244
\(166\) 35112.0 0.0988975
\(167\) 150528. 0.417663 0.208832 0.977952i \(-0.433034\pi\)
0.208832 + 0.977952i \(0.433034\pi\)
\(168\) −159840. −0.436930
\(169\) −41817.0 −0.112625
\(170\) 66424.0 0.176280
\(171\) 174960. 0.457560
\(172\) −232512. −0.599272
\(173\) −2166.00 −0.00550229 −0.00275114 0.999996i \(-0.500876\pi\)
−0.00275114 + 0.999996i \(0.500876\pi\)
\(174\) 83700.0 0.209581
\(175\) −149332. −0.368602
\(176\) 79376.0 0.193156
\(177\) 52740.0 0.126534
\(178\) −265140. −0.627228
\(179\) 672780. 1.56942 0.784712 0.619860i \(-0.212811\pi\)
0.784712 + 0.619860i \(0.212811\pi\)
\(180\) −104328. −0.240005
\(181\) −526778. −1.19517 −0.597587 0.801804i \(-0.703874\pi\)
−0.597587 + 0.801804i \(0.703874\pi\)
\(182\) −169904. −0.380211
\(183\) 482922. 1.06598
\(184\) −304320. −0.662653
\(185\) 373428. 0.802191
\(186\) 90576.0 0.191969
\(187\) −87362.0 −0.182692
\(188\) −692384. −1.42874
\(189\) −107892. −0.219702
\(190\) −198720. −0.399354
\(191\) −305608. −0.606152 −0.303076 0.952966i \(-0.598014\pi\)
−0.303076 + 0.952966i \(0.598014\pi\)
\(192\) 96192.0 0.188315
\(193\) 116434. 0.225002 0.112501 0.993652i \(-0.464114\pi\)
0.112501 + 0.993652i \(0.464114\pi\)
\(194\) −141316. −0.269580
\(195\) −237636. −0.447534
\(196\) −142716. −0.265358
\(197\) −247742. −0.454814 −0.227407 0.973800i \(-0.573025\pi\)
−0.227407 + 0.973800i \(0.573025\pi\)
\(198\) −19602.0 −0.0355335
\(199\) −513360. −0.918945 −0.459472 0.888192i \(-0.651961\pi\)
−0.459472 + 0.888192i \(0.651961\pi\)
\(200\) −121080. −0.214041
\(201\) −278172. −0.485650
\(202\) 203996. 0.351757
\(203\) 688200. 1.17213
\(204\) −181944. −0.306099
\(205\) −236348. −0.392796
\(206\) −261808. −0.429848
\(207\) −205416. −0.333203
\(208\) 376544. 0.603472
\(209\) 261360. 0.413879
\(210\) 122544. 0.191754
\(211\) −620688. −0.959770 −0.479885 0.877331i \(-0.659322\pi\)
−0.479885 + 0.877331i \(0.659322\pi\)
\(212\) 804888. 1.22997
\(213\) 626832. 0.946678
\(214\) 283224. 0.422762
\(215\) 381984. 0.563571
\(216\) −87480.0 −0.127578
\(217\) 744736. 1.07363
\(218\) 479620. 0.683528
\(219\) 166014. 0.233902
\(220\) −155848. −0.217092
\(221\) −414428. −0.570780
\(222\) 146124. 0.198994
\(223\) −1.31802e6 −1.77484 −0.887419 0.460964i \(-0.847504\pi\)
−0.887419 + 0.460964i \(0.847504\pi\)
\(224\) −762496. −1.01535
\(225\) −81729.0 −0.107627
\(226\) 85452.0 0.111289
\(227\) −887412. −1.14304 −0.571519 0.820589i \(-0.693646\pi\)
−0.571519 + 0.820589i \(0.693646\pi\)
\(228\) 544320. 0.693453
\(229\) −237450. −0.299215 −0.149608 0.988745i \(-0.547801\pi\)
−0.149608 + 0.988745i \(0.547801\pi\)
\(230\) 233312. 0.290815
\(231\) −161172. −0.198728
\(232\) 558000. 0.680635
\(233\) −914706. −1.10380 −0.551902 0.833909i \(-0.686098\pi\)
−0.551902 + 0.833909i \(0.686098\pi\)
\(234\) −92988.0 −0.111016
\(235\) 1.13749e6 1.34362
\(236\) 164080. 0.191768
\(237\) 227700. 0.263325
\(238\) 213712. 0.244561
\(239\) 1.40892e6 1.59548 0.797740 0.603001i \(-0.206029\pi\)
0.797740 + 0.603001i \(0.206029\pi\)
\(240\) −271584. −0.304352
\(241\) −826358. −0.916486 −0.458243 0.888827i \(-0.651521\pi\)
−0.458243 + 0.888827i \(0.651521\pi\)
\(242\) −29282.0 −0.0321412
\(243\) −59049.0 −0.0641500
\(244\) 1.50242e6 1.61554
\(245\) 234462. 0.249550
\(246\) −92484.0 −0.0974381
\(247\) 1.23984e6 1.29307
\(248\) 603840. 0.623437
\(249\) 158004. 0.161499
\(250\) 380328. 0.384865
\(251\) −1.60387e6 −1.60688 −0.803442 0.595384i \(-0.797000\pi\)
−0.803442 + 0.595384i \(0.797000\pi\)
\(252\) −335664. −0.332969
\(253\) −306856. −0.301393
\(254\) −103576. −0.100734
\(255\) 298908. 0.287864
\(256\) −30464.0 −0.0290527
\(257\) 397618. 0.375520 0.187760 0.982215i \(-0.439877\pi\)
0.187760 + 0.982215i \(0.439877\pi\)
\(258\) 149472. 0.139801
\(259\) 1.20146e6 1.11291
\(260\) −739312. −0.678257
\(261\) 376650. 0.342245
\(262\) −107304. −0.0965745
\(263\) 2.13166e6 1.90033 0.950166 0.311745i \(-0.100913\pi\)
0.950166 + 0.311745i \(0.100913\pi\)
\(264\) −130680. −0.115398
\(265\) −1.32232e6 −1.15670
\(266\) −639360. −0.554040
\(267\) −1.19313e6 −1.02426
\(268\) −865424. −0.736024
\(269\) −725810. −0.611564 −0.305782 0.952101i \(-0.598918\pi\)
−0.305782 + 0.952101i \(0.598918\pi\)
\(270\) 67068.0 0.0559894
\(271\) −1.46787e6 −1.21413 −0.607063 0.794654i \(-0.707652\pi\)
−0.607063 + 0.794654i \(0.707652\pi\)
\(272\) −473632. −0.388167
\(273\) −764568. −0.620883
\(274\) 457724. 0.368322
\(275\) −122089. −0.0973520
\(276\) −639072. −0.504983
\(277\) 1.52100e6 1.19105 0.595524 0.803338i \(-0.296944\pi\)
0.595524 + 0.803338i \(0.296944\pi\)
\(278\) −749840. −0.581911
\(279\) 407592. 0.313484
\(280\) 816960. 0.622738
\(281\) 464382. 0.350840 0.175420 0.984494i \(-0.443872\pi\)
0.175420 + 0.984494i \(0.443872\pi\)
\(282\) 445104. 0.333303
\(283\) −415136. −0.308123 −0.154062 0.988061i \(-0.549235\pi\)
−0.154062 + 0.988061i \(0.549235\pi\)
\(284\) 1.95014e6 1.43473
\(285\) −894240. −0.652142
\(286\) −138908. −0.100418
\(287\) −760424. −0.544943
\(288\) −417312. −0.296469
\(289\) −898573. −0.632862
\(290\) −427800. −0.298707
\(291\) −635922. −0.440222
\(292\) 516488. 0.354489
\(293\) −2.59321e6 −1.76469 −0.882344 0.470605i \(-0.844036\pi\)
−0.882344 + 0.470605i \(0.844036\pi\)
\(294\) 91746.0 0.0619040
\(295\) −269560. −0.180343
\(296\) 974160. 0.646251
\(297\) −88209.0 −0.0580259
\(298\) 131660. 0.0858842
\(299\) −1.45566e6 −0.941636
\(300\) −254268. −0.163113
\(301\) 1.22899e6 0.781867
\(302\) −308104. −0.194393
\(303\) 917982. 0.574417
\(304\) 1.41696e6 0.879374
\(305\) −2.46827e6 −1.51930
\(306\) 116964. 0.0714083
\(307\) −930832. −0.563671 −0.281835 0.959463i \(-0.590943\pi\)
−0.281835 + 0.959463i \(0.590943\pi\)
\(308\) −501424. −0.301182
\(309\) −1.17814e6 −0.701939
\(310\) −462944. −0.273605
\(311\) 2.48527e6 1.45704 0.728522 0.685022i \(-0.240207\pi\)
0.728522 + 0.685022i \(0.240207\pi\)
\(312\) −619920. −0.360536
\(313\) 1.31719e6 0.759957 0.379978 0.924995i \(-0.375931\pi\)
0.379978 + 0.924995i \(0.375931\pi\)
\(314\) −575356. −0.329316
\(315\) 551448. 0.313133
\(316\) 708400. 0.399081
\(317\) 2.25540e6 1.26059 0.630297 0.776354i \(-0.282933\pi\)
0.630297 + 0.776354i \(0.282933\pi\)
\(318\) −517428. −0.286934
\(319\) 562650. 0.309572
\(320\) −491648. −0.268398
\(321\) 1.27451e6 0.690367
\(322\) 750656. 0.403461
\(323\) −1.55952e6 −0.831734
\(324\) −183708. −0.0972222
\(325\) −579166. −0.304155
\(326\) −210248. −0.109569
\(327\) 2.15829e6 1.11620
\(328\) −616560. −0.316440
\(329\) 3.65974e6 1.86406
\(330\) 100188. 0.0506443
\(331\) −3.17071e6 −1.59069 −0.795346 0.606155i \(-0.792711\pi\)
−0.795346 + 0.606155i \(0.792711\pi\)
\(332\) 491568. 0.244759
\(333\) 657558. 0.324955
\(334\) −301056. −0.147666
\(335\) 1.42177e6 0.692176
\(336\) −873792. −0.422240
\(337\) 1.27630e6 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(338\) 83634.0 0.0398191
\(339\) 384534. 0.181734
\(340\) 929936. 0.436270
\(341\) 608872. 0.283557
\(342\) −349920. −0.161772
\(343\) −1.73308e6 −0.795396
\(344\) 996480. 0.454017
\(345\) 1.04990e6 0.474900
\(346\) 4332.00 0.00194535
\(347\) 3.69303e6 1.64649 0.823245 0.567687i \(-0.192162\pi\)
0.823245 + 0.567687i \(0.192162\pi\)
\(348\) 1.17180e6 0.518687
\(349\) 1.70919e6 0.751150 0.375575 0.926792i \(-0.377445\pi\)
0.375575 + 0.926792i \(0.377445\pi\)
\(350\) 298664. 0.130321
\(351\) −418446. −0.181289
\(352\) −623392. −0.268167
\(353\) 4.36859e6 1.86597 0.932986 0.359914i \(-0.117194\pi\)
0.932986 + 0.359914i \(0.117194\pi\)
\(354\) −105480. −0.0447365
\(355\) −3.20381e6 −1.34926
\(356\) −3.71196e6 −1.55231
\(357\) 961704. 0.399366
\(358\) −1.34556e6 −0.554875
\(359\) −3.51284e6 −1.43854 −0.719271 0.694730i \(-0.755524\pi\)
−0.719271 + 0.694730i \(0.755524\pi\)
\(360\) 447120. 0.181831
\(361\) 2.18950e6 0.884254
\(362\) 1.05356e6 0.422558
\(363\) −131769. −0.0524864
\(364\) −2.37866e6 −0.940975
\(365\) −848516. −0.333371
\(366\) −965844. −0.376881
\(367\) −2.15259e6 −0.834251 −0.417125 0.908849i \(-0.636962\pi\)
−0.417125 + 0.908849i \(0.636962\pi\)
\(368\) −1.66362e6 −0.640374
\(369\) −416178. −0.159116
\(370\) −746856. −0.283617
\(371\) −4.25441e6 −1.60474
\(372\) 1.26806e6 0.475099
\(373\) −2.24247e6 −0.834553 −0.417276 0.908780i \(-0.637015\pi\)
−0.417276 + 0.908780i \(0.637015\pi\)
\(374\) 174724. 0.0645912
\(375\) 1.71148e6 0.628482
\(376\) 2.96736e6 1.08243
\(377\) 2.66910e6 0.967189
\(378\) 215784. 0.0776765
\(379\) −2.40986e6 −0.861775 −0.430887 0.902406i \(-0.641799\pi\)
−0.430887 + 0.902406i \(0.641799\pi\)
\(380\) −2.78208e6 −0.988350
\(381\) −466092. −0.164497
\(382\) 611216. 0.214307
\(383\) −1.01066e6 −0.352052 −0.176026 0.984386i \(-0.556324\pi\)
−0.176026 + 0.984386i \(0.556324\pi\)
\(384\) −1.67616e6 −0.580079
\(385\) 823768. 0.283239
\(386\) −232868. −0.0795503
\(387\) 672624. 0.228294
\(388\) −1.97842e6 −0.667175
\(389\) 1.27779e6 0.428140 0.214070 0.976818i \(-0.431328\pi\)
0.214070 + 0.976818i \(0.431328\pi\)
\(390\) 475272. 0.158227
\(391\) 1.83099e6 0.605682
\(392\) 611640. 0.201039
\(393\) −482868. −0.157706
\(394\) 495484. 0.160801
\(395\) −1.16380e6 −0.375306
\(396\) −274428. −0.0879408
\(397\) 5.45400e6 1.73676 0.868378 0.495903i \(-0.165163\pi\)
0.868378 + 0.495903i \(0.165163\pi\)
\(398\) 1.02672e6 0.324896
\(399\) −2.87712e6 −0.904744
\(400\) −661904. −0.206845
\(401\) −1.48980e6 −0.462665 −0.231332 0.972875i \(-0.574308\pi\)
−0.231332 + 0.972875i \(0.574308\pi\)
\(402\) 556344. 0.171703
\(403\) 2.88837e6 0.885911
\(404\) 2.85594e6 0.870555
\(405\) 301806. 0.0914303
\(406\) −1.37640e6 −0.414409
\(407\) 982278. 0.293933
\(408\) 779760. 0.231905
\(409\) −4.39899e6 −1.30030 −0.650152 0.759804i \(-0.725295\pi\)
−0.650152 + 0.759804i \(0.725295\pi\)
\(410\) 472696. 0.138874
\(411\) 2.05976e6 0.601467
\(412\) −3.66531e6 −1.06382
\(413\) −867280. −0.250198
\(414\) 410832. 0.117805
\(415\) −807576. −0.230178
\(416\) −2.95725e6 −0.837827
\(417\) −3.37428e6 −0.950257
\(418\) −522720. −0.146328
\(419\) −280420. −0.0780322 −0.0390161 0.999239i \(-0.512422\pi\)
−0.0390161 + 0.999239i \(0.512422\pi\)
\(420\) 1.71562e6 0.474566
\(421\) 817462. 0.224782 0.112391 0.993664i \(-0.464149\pi\)
0.112391 + 0.993664i \(0.464149\pi\)
\(422\) 1.24138e6 0.339330
\(423\) 2.00297e6 0.544281
\(424\) −3.44952e6 −0.931846
\(425\) 728498. 0.195639
\(426\) −1.25366e6 −0.334701
\(427\) −7.94138e6 −2.10779
\(428\) 3.96514e6 1.04628
\(429\) −625086. −0.163982
\(430\) −763968. −0.199253
\(431\) 1.88599e6 0.489043 0.244521 0.969644i \(-0.421369\pi\)
0.244521 + 0.969644i \(0.421369\pi\)
\(432\) −478224. −0.123288
\(433\) 5.84067e6 1.49707 0.748537 0.663093i \(-0.230757\pi\)
0.748537 + 0.663093i \(0.230757\pi\)
\(434\) −1.48947e6 −0.379584
\(435\) −1.92510e6 −0.487787
\(436\) 6.71468e6 1.69164
\(437\) −5.47776e6 −1.37214
\(438\) −332028. −0.0826969
\(439\) −509540. −0.126188 −0.0630938 0.998008i \(-0.520097\pi\)
−0.0630938 + 0.998008i \(0.520097\pi\)
\(440\) 667920. 0.164472
\(441\) 412857. 0.101089
\(442\) 828856. 0.201801
\(443\) 4.10268e6 0.993250 0.496625 0.867965i \(-0.334572\pi\)
0.496625 + 0.867965i \(0.334572\pi\)
\(444\) 2.04574e6 0.492484
\(445\) 6.09822e6 1.45983
\(446\) 2.63603e6 0.627500
\(447\) 592470. 0.140248
\(448\) −1.58182e6 −0.372360
\(449\) 513410. 0.120185 0.0600923 0.998193i \(-0.480861\pi\)
0.0600923 + 0.998193i \(0.480861\pi\)
\(450\) 163458. 0.0380518
\(451\) −621698. −0.143926
\(452\) 1.19633e6 0.275426
\(453\) −1.38647e6 −0.317442
\(454\) 1.77482e6 0.404125
\(455\) 3.90779e6 0.884918
\(456\) −2.33280e6 −0.525370
\(457\) 1.22738e6 0.274908 0.137454 0.990508i \(-0.456108\pi\)
0.137454 + 0.990508i \(0.456108\pi\)
\(458\) 474900. 0.105789
\(459\) 526338. 0.116609
\(460\) 3.26637e6 0.719732
\(461\) −6.41000e6 −1.40477 −0.702386 0.711797i \(-0.747882\pi\)
−0.702386 + 0.711797i \(0.747882\pi\)
\(462\) 322344. 0.0702611
\(463\) 6.63030e6 1.43741 0.718705 0.695315i \(-0.244735\pi\)
0.718705 + 0.695315i \(0.244735\pi\)
\(464\) 3.05040e6 0.657751
\(465\) −2.08325e6 −0.446795
\(466\) 1.82941e6 0.390253
\(467\) −4.14769e6 −0.880064 −0.440032 0.897982i \(-0.645033\pi\)
−0.440032 + 0.897982i \(0.645033\pi\)
\(468\) −1.30183e6 −0.274752
\(469\) 4.57438e6 0.960286
\(470\) −2.27498e6 −0.475042
\(471\) −2.58910e6 −0.537770
\(472\) −703200. −0.145286
\(473\) 1.00478e6 0.206500
\(474\) −455400. −0.0930995
\(475\) −2.17944e6 −0.443211
\(476\) 2.99197e6 0.605257
\(477\) −2.32843e6 −0.468561
\(478\) −2.81784e6 −0.564088
\(479\) −5.05132e6 −1.00593 −0.502963 0.864308i \(-0.667757\pi\)
−0.502963 + 0.864308i \(0.667757\pi\)
\(480\) 2.13293e6 0.422545
\(481\) 4.65973e6 0.918329
\(482\) 1.65272e6 0.324027
\(483\) 3.37795e6 0.658849
\(484\) −409948. −0.0795455
\(485\) 3.25027e6 0.627429
\(486\) 118098. 0.0226805
\(487\) 2.66221e6 0.508651 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(488\) −6.43896e6 −1.22396
\(489\) −946116. −0.178925
\(490\) −468924. −0.0882292
\(491\) −5.54659e6 −1.03830 −0.519149 0.854684i \(-0.673751\pi\)
−0.519149 + 0.854684i \(0.673751\pi\)
\(492\) −1.29478e6 −0.241147
\(493\) −3.35730e6 −0.622118
\(494\) −2.47968e6 −0.457171
\(495\) 450846. 0.0827018
\(496\) 3.30099e6 0.602477
\(497\) −1.03079e7 −1.87189
\(498\) −316008. −0.0570985
\(499\) −6820.00 −0.00122612 −0.000613060 1.00000i \(-0.500195\pi\)
−0.000613060 1.00000i \(0.500195\pi\)
\(500\) 5.32459e6 0.952492
\(501\) −1.35475e6 −0.241138
\(502\) 3.20774e6 0.568119
\(503\) −451136. −0.0795037 −0.0397519 0.999210i \(-0.512657\pi\)
−0.0397519 + 0.999210i \(0.512657\pi\)
\(504\) 1.43856e6 0.252262
\(505\) −4.69191e6 −0.818693
\(506\) 613712. 0.106559
\(507\) 376353. 0.0650243
\(508\) −1.45006e6 −0.249303
\(509\) 393390. 0.0673021 0.0336511 0.999434i \(-0.489287\pi\)
0.0336511 + 0.999434i \(0.489287\pi\)
\(510\) −597816. −0.101775
\(511\) −2.73001e6 −0.462500
\(512\) −5.89875e6 −0.994455
\(513\) −1.57464e6 −0.264173
\(514\) −795236. −0.132766
\(515\) 6.02158e6 1.00044
\(516\) 2.09261e6 0.345990
\(517\) 2.99209e6 0.492321
\(518\) −2.40293e6 −0.393474
\(519\) 19494.0 0.00317675
\(520\) 3.16848e6 0.513857
\(521\) 3.28432e6 0.530092 0.265046 0.964236i \(-0.414613\pi\)
0.265046 + 0.964236i \(0.414613\pi\)
\(522\) −753300. −0.121002
\(523\) −1.68266e6 −0.268993 −0.134497 0.990914i \(-0.542942\pi\)
−0.134497 + 0.990914i \(0.542942\pi\)
\(524\) −1.50226e6 −0.239010
\(525\) 1.34399e6 0.212812
\(526\) −4.26333e6 −0.671869
\(527\) −3.63310e6 −0.569838
\(528\) −714384. −0.111518
\(529\) −5047.00 −0.000784141 0
\(530\) 2.64463e6 0.408955
\(531\) −474660. −0.0730544
\(532\) −8.95104e6 −1.37118
\(533\) −2.94921e6 −0.449664
\(534\) 2.38626e6 0.362130
\(535\) −6.51415e6 −0.983951
\(536\) 3.70896e6 0.557622
\(537\) −6.05502e6 −0.906108
\(538\) 1.45162e6 0.216221
\(539\) 616737. 0.0914383
\(540\) 938952. 0.138567
\(541\) 9.48158e6 1.39280 0.696398 0.717656i \(-0.254785\pi\)
0.696398 + 0.717656i \(0.254785\pi\)
\(542\) 2.93574e6 0.429258
\(543\) 4.74100e6 0.690034
\(544\) 3.71974e6 0.538909
\(545\) −1.10313e7 −1.59087
\(546\) 1.52914e6 0.219515
\(547\) −6.09239e6 −0.870602 −0.435301 0.900285i \(-0.643358\pi\)
−0.435301 + 0.900285i \(0.643358\pi\)
\(548\) 6.40814e6 0.911550
\(549\) −4.34630e6 −0.615444
\(550\) 244178. 0.0344191
\(551\) 1.00440e7 1.40938
\(552\) 2.73888e6 0.382583
\(553\) −3.74440e6 −0.520678
\(554\) −3.04200e6 −0.421099
\(555\) −3.36085e6 −0.463145
\(556\) −1.04978e7 −1.44016
\(557\) 8.49594e6 1.16031 0.580154 0.814507i \(-0.302992\pi\)
0.580154 + 0.814507i \(0.302992\pi\)
\(558\) −815184. −0.110833
\(559\) 4.76650e6 0.645163
\(560\) 4.46605e6 0.601802
\(561\) 786258. 0.105477
\(562\) −928764. −0.124041
\(563\) −7.02216e6 −0.933683 −0.466842 0.884341i \(-0.654608\pi\)
−0.466842 + 0.884341i \(0.654608\pi\)
\(564\) 6.23146e6 0.824882
\(565\) −1.96540e6 −0.259017
\(566\) 830272. 0.108938
\(567\) 971028. 0.126845
\(568\) −8.35776e6 −1.08697
\(569\) 9.41847e6 1.21955 0.609775 0.792574i \(-0.291260\pi\)
0.609775 + 0.792574i \(0.291260\pi\)
\(570\) 1.78848e6 0.230567
\(571\) 7.29699e6 0.936599 0.468299 0.883570i \(-0.344867\pi\)
0.468299 + 0.883570i \(0.344867\pi\)
\(572\) −1.94471e6 −0.248522
\(573\) 2.75047e6 0.349962
\(574\) 1.52085e6 0.192666
\(575\) 2.55882e6 0.322753
\(576\) −865728. −0.108724
\(577\) −3.29590e6 −0.412131 −0.206065 0.978538i \(-0.566066\pi\)
−0.206065 + 0.978538i \(0.566066\pi\)
\(578\) 1.79715e6 0.223750
\(579\) −1.04791e6 −0.129905
\(580\) −5.98920e6 −0.739263
\(581\) −2.59829e6 −0.319335
\(582\) 1.27184e6 0.155642
\(583\) −3.47827e6 −0.423830
\(584\) −2.21352e6 −0.268566
\(585\) 2.13872e6 0.258384
\(586\) 5.18641e6 0.623911
\(587\) 4.39827e6 0.526849 0.263425 0.964680i \(-0.415148\pi\)
0.263425 + 0.964680i \(0.415148\pi\)
\(588\) 1.28444e6 0.153205
\(589\) 1.08691e7 1.29094
\(590\) 539120. 0.0637610
\(591\) 2.22968e6 0.262587
\(592\) 5.32541e6 0.624523
\(593\) 9.21781e6 1.07644 0.538222 0.842803i \(-0.319096\pi\)
0.538222 + 0.842803i \(0.319096\pi\)
\(594\) 176418. 0.0205152
\(595\) −4.91538e6 −0.569199
\(596\) 1.84324e6 0.212553
\(597\) 4.62024e6 0.530553
\(598\) 2.91133e6 0.332919
\(599\) 3.77140e6 0.429473 0.214736 0.976672i \(-0.431111\pi\)
0.214736 + 0.976672i \(0.431111\pi\)
\(600\) 1.08972e6 0.123577
\(601\) 4.19724e6 0.473999 0.237000 0.971510i \(-0.423836\pi\)
0.237000 + 0.971510i \(0.423836\pi\)
\(602\) −2.45798e6 −0.276432
\(603\) 2.50355e6 0.280390
\(604\) −4.31346e6 −0.481097
\(605\) 673486. 0.0748066
\(606\) −1.83596e6 −0.203087
\(607\) −1.00133e6 −0.110308 −0.0551539 0.998478i \(-0.517565\pi\)
−0.0551539 + 0.998478i \(0.517565\pi\)
\(608\) −1.11283e7 −1.22087
\(609\) −6.19380e6 −0.676728
\(610\) 4.93654e6 0.537153
\(611\) 1.41939e7 1.53815
\(612\) 1.63750e6 0.176727
\(613\) −7.38239e6 −0.793498 −0.396749 0.917927i \(-0.629862\pi\)
−0.396749 + 0.917927i \(0.629862\pi\)
\(614\) 1.86166e6 0.199288
\(615\) 2.12713e6 0.226781
\(616\) 2.14896e6 0.228179
\(617\) −1.54025e7 −1.62884 −0.814418 0.580279i \(-0.802944\pi\)
−0.814418 + 0.580279i \(0.802944\pi\)
\(618\) 2.35627e6 0.248173
\(619\) −7.12402e6 −0.747306 −0.373653 0.927569i \(-0.621895\pi\)
−0.373653 + 0.927569i \(0.621895\pi\)
\(620\) −6.48122e6 −0.677138
\(621\) 1.84874e6 0.192375
\(622\) −4.97054e6 −0.515143
\(623\) 1.96204e7 2.02529
\(624\) −3.38890e6 −0.348415
\(625\) −5.59442e6 −0.572869
\(626\) −2.63439e6 −0.268685
\(627\) −2.35224e6 −0.238953
\(628\) −8.05498e6 −0.815015
\(629\) −5.86120e6 −0.590690
\(630\) −1.10290e6 −0.110709
\(631\) 1.16696e7 1.16677 0.583383 0.812197i \(-0.301729\pi\)
0.583383 + 0.812197i \(0.301729\pi\)
\(632\) −3.03600e6 −0.302349
\(633\) 5.58619e6 0.554124
\(634\) −4.51080e6 −0.445687
\(635\) 2.38225e6 0.234451
\(636\) −7.24399e6 −0.710126
\(637\) 2.92568e6 0.285679
\(638\) −1.12530e6 −0.109450
\(639\) −5.64149e6 −0.546565
\(640\) 8.56704e6 0.826763
\(641\) −1.10271e7 −1.06003 −0.530014 0.847989i \(-0.677813\pi\)
−0.530014 + 0.847989i \(0.677813\pi\)
\(642\) −2.54902e6 −0.244082
\(643\) −9.56024e6 −0.911887 −0.455944 0.890009i \(-0.650698\pi\)
−0.455944 + 0.890009i \(0.650698\pi\)
\(644\) 1.05092e7 0.998514
\(645\) −3.43786e6 −0.325378
\(646\) 3.11904e6 0.294063
\(647\) −1.09942e7 −1.03253 −0.516263 0.856430i \(-0.672677\pi\)
−0.516263 + 0.856430i \(0.672677\pi\)
\(648\) 787320. 0.0736570
\(649\) −709060. −0.0660802
\(650\) 1.15833e6 0.107535
\(651\) −6.70262e6 −0.619858
\(652\) −2.94347e6 −0.271170
\(653\) −295346. −0.0271049 −0.0135525 0.999908i \(-0.504314\pi\)
−0.0135525 + 0.999908i \(0.504314\pi\)
\(654\) −4.31658e6 −0.394635
\(655\) 2.46799e6 0.224771
\(656\) −3.37053e6 −0.305801
\(657\) −1.49413e6 −0.135044
\(658\) −7.31949e6 −0.659046
\(659\) −1.65613e7 −1.48553 −0.742766 0.669551i \(-0.766486\pi\)
−0.742766 + 0.669551i \(0.766486\pi\)
\(660\) 1.40263e6 0.125338
\(661\) 1.97042e6 0.175411 0.0877053 0.996146i \(-0.472047\pi\)
0.0877053 + 0.996146i \(0.472047\pi\)
\(662\) 6.34142e6 0.562395
\(663\) 3.72985e6 0.329540
\(664\) −2.10672e6 −0.185433
\(665\) 1.47053e7 1.28949
\(666\) −1.31512e6 −0.114889
\(667\) −1.17924e7 −1.02633
\(668\) −4.21478e6 −0.365455
\(669\) 1.18621e7 1.02470
\(670\) −2.84354e6 −0.244721
\(671\) −6.49262e6 −0.556690
\(672\) 6.86246e6 0.586215
\(673\) −1.63733e6 −0.139347 −0.0696735 0.997570i \(-0.522196\pi\)
−0.0696735 + 0.997570i \(0.522196\pi\)
\(674\) −2.55260e6 −0.216437
\(675\) 735561. 0.0621383
\(676\) 1.17088e6 0.0985472
\(677\) −6.35878e6 −0.533215 −0.266607 0.963805i \(-0.585903\pi\)
−0.266607 + 0.963805i \(0.585903\pi\)
\(678\) −769068. −0.0642526
\(679\) 1.04574e7 0.870460
\(680\) −3.98544e6 −0.330525
\(681\) 7.98671e6 0.659933
\(682\) −1.21774e6 −0.100252
\(683\) 1.11033e7 0.910751 0.455376 0.890299i \(-0.349505\pi\)
0.455376 + 0.890299i \(0.349505\pi\)
\(684\) −4.89888e6 −0.400365
\(685\) −1.05277e7 −0.857245
\(686\) 3.46616e6 0.281215
\(687\) 2.13705e6 0.172752
\(688\) 5.44742e6 0.438753
\(689\) −1.65002e7 −1.32416
\(690\) −2.09981e6 −0.167902
\(691\) 1.70189e7 1.35592 0.677962 0.735097i \(-0.262864\pi\)
0.677962 + 0.735097i \(0.262864\pi\)
\(692\) 60648.0 0.00481450
\(693\) 1.45055e6 0.114736
\(694\) −7.38606e6 −0.582122
\(695\) 1.72463e7 1.35436
\(696\) −5.02200e6 −0.392965
\(697\) 3.70964e6 0.289234
\(698\) −3.41838e6 −0.265572
\(699\) 8.23235e6 0.637281
\(700\) 4.18130e6 0.322527
\(701\) 1.58021e7 1.21456 0.607280 0.794488i \(-0.292260\pi\)
0.607280 + 0.794488i \(0.292260\pi\)
\(702\) 836892. 0.0640954
\(703\) 1.75349e7 1.33818
\(704\) −1.29325e6 −0.0983445
\(705\) −1.02374e7 −0.775741
\(706\) −8.73719e6 −0.659720
\(707\) −1.50957e7 −1.13581
\(708\) −1.47672e6 −0.110717
\(709\) 1.24834e7 0.932643 0.466322 0.884615i \(-0.345579\pi\)
0.466322 + 0.884615i \(0.345579\pi\)
\(710\) 6.40762e6 0.477035
\(711\) −2.04930e6 −0.152031
\(712\) 1.59084e7 1.17605
\(713\) −1.27612e7 −0.940083
\(714\) −1.92341e6 −0.141197
\(715\) 3.19488e6 0.233717
\(716\) −1.88378e7 −1.37325
\(717\) −1.26803e7 −0.921151
\(718\) 7.02568e6 0.508601
\(719\) 2.00724e6 0.144803 0.0724014 0.997376i \(-0.476934\pi\)
0.0724014 + 0.997376i \(0.476934\pi\)
\(720\) 2.44426e6 0.175718
\(721\) 1.93738e7 1.38796
\(722\) −4.37900e6 −0.312631
\(723\) 7.43722e6 0.529133
\(724\) 1.47498e7 1.04578
\(725\) −4.69185e6 −0.331512
\(726\) 263538. 0.0185567
\(727\) 6.97301e6 0.489310 0.244655 0.969610i \(-0.421325\pi\)
0.244655 + 0.969610i \(0.421325\pi\)
\(728\) 1.01942e7 0.712896
\(729\) 531441. 0.0370370
\(730\) 1.69703e6 0.117864
\(731\) −5.99549e6 −0.414984
\(732\) −1.35218e7 −0.932733
\(733\) −2.34965e7 −1.61527 −0.807633 0.589685i \(-0.799252\pi\)
−0.807633 + 0.589685i \(0.799252\pi\)
\(734\) 4.30518e6 0.294952
\(735\) −2.11016e6 −0.144078
\(736\) 1.30655e7 0.889059
\(737\) 3.73987e6 0.253622
\(738\) 832356. 0.0562559
\(739\) −1.39901e7 −0.942346 −0.471173 0.882041i \(-0.656169\pi\)
−0.471173 + 0.882041i \(0.656169\pi\)
\(740\) −1.04560e7 −0.701917
\(741\) −1.11586e7 −0.746556
\(742\) 8.50882e6 0.567361
\(743\) 2.42745e7 1.61316 0.806582 0.591123i \(-0.201315\pi\)
0.806582 + 0.591123i \(0.201315\pi\)
\(744\) −5.43456e6 −0.359942
\(745\) −3.02818e6 −0.199890
\(746\) 4.48493e6 0.295059
\(747\) −1.42204e6 −0.0932415
\(748\) 2.44614e6 0.159855
\(749\) −2.09586e7 −1.36508
\(750\) −3.42295e6 −0.222202
\(751\) 1.53660e7 0.994170 0.497085 0.867702i \(-0.334404\pi\)
0.497085 + 0.867702i \(0.334404\pi\)
\(752\) 1.62216e7 1.04604
\(753\) 1.44348e7 0.927734
\(754\) −5.33820e6 −0.341953
\(755\) 7.08639e6 0.452437
\(756\) 3.02098e6 0.192240
\(757\) 2.07605e7 1.31674 0.658368 0.752697i \(-0.271247\pi\)
0.658368 + 0.752697i \(0.271247\pi\)
\(758\) 4.81972e6 0.304683
\(759\) 2.76170e6 0.174009
\(760\) 1.19232e7 0.748788
\(761\) 5.83810e6 0.365435 0.182717 0.983165i \(-0.441511\pi\)
0.182717 + 0.983165i \(0.441511\pi\)
\(762\) 932184. 0.0581586
\(763\) −3.54919e7 −2.20708
\(764\) 8.55702e6 0.530383
\(765\) −2.69017e6 −0.166198
\(766\) 2.02131e6 0.124469
\(767\) −3.36364e6 −0.206453
\(768\) 274176. 0.0167736
\(769\) −1.39197e7 −0.848818 −0.424409 0.905471i \(-0.639518\pi\)
−0.424409 + 0.905471i \(0.639518\pi\)
\(770\) −1.64754e6 −0.100140
\(771\) −3.57856e6 −0.216807
\(772\) −3.26015e6 −0.196877
\(773\) −4.17883e6 −0.251539 −0.125770 0.992059i \(-0.540140\pi\)
−0.125770 + 0.992059i \(0.540140\pi\)
\(774\) −1.34525e6 −0.0807142
\(775\) −5.07729e6 −0.303653
\(776\) 8.47896e6 0.505462
\(777\) −1.08132e7 −0.642541
\(778\) −2.55558e6 −0.151370
\(779\) −1.10981e7 −0.655246
\(780\) 6.65381e6 0.391592
\(781\) −8.42741e6 −0.494386
\(782\) −3.66198e6 −0.214141
\(783\) −3.38985e6 −0.197595
\(784\) 3.34363e6 0.194280
\(785\) 1.32332e7 0.766461
\(786\) 965736. 0.0557573
\(787\) 9.66705e6 0.556361 0.278181 0.960529i \(-0.410269\pi\)
0.278181 + 0.960529i \(0.410269\pi\)
\(788\) 6.93678e6 0.397962
\(789\) −1.91850e7 −1.09716
\(790\) 2.32760e6 0.132691
\(791\) −6.32345e6 −0.359346
\(792\) 1.17612e6 0.0666252
\(793\) −3.07997e7 −1.73926
\(794\) −1.09080e7 −0.614036
\(795\) 1.19008e7 0.667821
\(796\) 1.43741e7 0.804077
\(797\) −5.79884e6 −0.323367 −0.161683 0.986843i \(-0.551692\pi\)
−0.161683 + 0.986843i \(0.551692\pi\)
\(798\) 5.75424e6 0.319875
\(799\) −1.78536e7 −0.989371
\(800\) 5.19837e6 0.287172
\(801\) 1.07382e7 0.591356
\(802\) 2.97960e6 0.163577
\(803\) −2.23197e6 −0.122151
\(804\) 7.78882e6 0.424944
\(805\) −1.72651e7 −0.939029
\(806\) −5.77674e6 −0.313217
\(807\) 6.53229e6 0.353087
\(808\) −1.22398e7 −0.659545
\(809\) −1.92543e7 −1.03433 −0.517163 0.855887i \(-0.673012\pi\)
−0.517163 + 0.855887i \(0.673012\pi\)
\(810\) −603612. −0.0323255
\(811\) −1.31938e7 −0.704396 −0.352198 0.935926i \(-0.614566\pi\)
−0.352198 + 0.935926i \(0.614566\pi\)
\(812\) −1.92696e7 −1.02561
\(813\) 1.32108e7 0.700976
\(814\) −1.96456e6 −0.103921
\(815\) 4.83570e6 0.255015
\(816\) 4.26269e6 0.224108
\(817\) 1.79366e7 0.940126
\(818\) 8.79798e6 0.459727
\(819\) 6.88111e6 0.358467
\(820\) 6.61774e6 0.343697
\(821\) 1.33779e7 0.692677 0.346338 0.938110i \(-0.387425\pi\)
0.346338 + 0.938110i \(0.387425\pi\)
\(822\) −4.11952e6 −0.212651
\(823\) −1.88613e7 −0.970673 −0.485336 0.874327i \(-0.661303\pi\)
−0.485336 + 0.874327i \(0.661303\pi\)
\(824\) 1.57085e7 0.805965
\(825\) 1.09880e6 0.0562062
\(826\) 1.73456e6 0.0884584
\(827\) 1.62680e7 0.827123 0.413561 0.910476i \(-0.364285\pi\)
0.413561 + 0.910476i \(0.364285\pi\)
\(828\) 5.75165e6 0.291552
\(829\) −2.18098e7 −1.10221 −0.551107 0.834435i \(-0.685794\pi\)
−0.551107 + 0.834435i \(0.685794\pi\)
\(830\) 1.61515e6 0.0813801
\(831\) −1.36890e7 −0.687652
\(832\) −6.13491e6 −0.307256
\(833\) −3.68003e6 −0.183755
\(834\) 6.74856e6 0.335967
\(835\) 6.92429e6 0.343684
\(836\) −7.31808e6 −0.362144
\(837\) −3.66833e6 −0.180990
\(838\) 560840. 0.0275886
\(839\) 1.17771e7 0.577607 0.288804 0.957388i \(-0.406743\pi\)
0.288804 + 0.957388i \(0.406743\pi\)
\(840\) −7.35264e6 −0.359538
\(841\) 1.11135e6 0.0541828
\(842\) −1.63492e6 −0.0794726
\(843\) −4.17944e6 −0.202558
\(844\) 1.73793e7 0.839799
\(845\) −1.92358e6 −0.0926764
\(846\) −4.00594e6 −0.192432
\(847\) 2.16687e6 0.103782
\(848\) −1.88574e7 −0.900516
\(849\) 3.73622e6 0.177895
\(850\) −1.45700e6 −0.0691689
\(851\) −2.05872e7 −0.974483
\(852\) −1.75513e7 −0.828343
\(853\) −1.43993e7 −0.677591 −0.338796 0.940860i \(-0.610020\pi\)
−0.338796 + 0.940860i \(0.610020\pi\)
\(854\) 1.58828e7 0.745215
\(855\) 8.04816e6 0.376514
\(856\) −1.69934e7 −0.792678
\(857\) 6.27604e6 0.291900 0.145950 0.989292i \(-0.453376\pi\)
0.145950 + 0.989292i \(0.453376\pi\)
\(858\) 1.25017e6 0.0579764
\(859\) −4.71738e6 −0.218131 −0.109066 0.994035i \(-0.534786\pi\)
−0.109066 + 0.994035i \(0.534786\pi\)
\(860\) −1.06956e7 −0.493125
\(861\) 6.84382e6 0.314623
\(862\) −3.77198e6 −0.172903
\(863\) 7.53926e6 0.344589 0.172295 0.985045i \(-0.444882\pi\)
0.172295 + 0.985045i \(0.444882\pi\)
\(864\) 3.75581e6 0.171167
\(865\) −99636.0 −0.00452768
\(866\) −1.16813e7 −0.529296
\(867\) 8.08716e6 0.365383
\(868\) −2.08526e7 −0.939423
\(869\) −3.06130e6 −0.137517
\(870\) 3.85020e6 0.172459
\(871\) 1.77412e7 0.792387
\(872\) −2.87772e7 −1.28161
\(873\) 5.72330e6 0.254162
\(874\) 1.09555e7 0.485126
\(875\) −2.81443e7 −1.24271
\(876\) −4.64839e6 −0.204664
\(877\) −1.04331e7 −0.458051 −0.229025 0.973420i \(-0.573554\pi\)
−0.229025 + 0.973420i \(0.573554\pi\)
\(878\) 1.01908e6 0.0446141
\(879\) 2.33389e7 1.01884
\(880\) 3.65130e6 0.158943
\(881\) 3.91076e7 1.69755 0.848774 0.528756i \(-0.177342\pi\)
0.848774 + 0.528756i \(0.177342\pi\)
\(882\) −825714. −0.0357403
\(883\) 1.29282e7 0.558003 0.279001 0.960291i \(-0.409997\pi\)
0.279001 + 0.960291i \(0.409997\pi\)
\(884\) 1.16040e7 0.499432
\(885\) 2.42604e6 0.104121
\(886\) −8.20537e6 −0.351167
\(887\) 3.36466e7 1.43592 0.717962 0.696082i \(-0.245075\pi\)
0.717962 + 0.696082i \(0.245075\pi\)
\(888\) −8.76744e6 −0.373113
\(889\) 7.66462e6 0.325264
\(890\) −1.21964e7 −0.516129
\(891\) 793881. 0.0335013
\(892\) 3.69044e7 1.55298
\(893\) 5.34125e7 2.24137
\(894\) −1.18494e6 −0.0495853
\(895\) 3.09479e7 1.29144
\(896\) 2.75635e7 1.14700
\(897\) 1.31010e7 0.543654
\(898\) −1.02682e6 −0.0424916
\(899\) 2.33988e7 0.965594
\(900\) 2.28841e6 0.0941733
\(901\) 2.07546e7 0.851731
\(902\) 1.24340e6 0.0508854
\(903\) −1.10609e7 −0.451411
\(904\) −5.12712e6 −0.208666
\(905\) −2.42318e7 −0.983477
\(906\) 2.77294e6 0.112233
\(907\) −4.19629e7 −1.69374 −0.846872 0.531797i \(-0.821517\pi\)
−0.846872 + 0.531797i \(0.821517\pi\)
\(908\) 2.48475e7 1.00016
\(909\) −8.26184e6 −0.331640
\(910\) −7.81558e6 −0.312866
\(911\) −1.92521e6 −0.0768567 −0.0384283 0.999261i \(-0.512235\pi\)
−0.0384283 + 0.999261i \(0.512235\pi\)
\(912\) −1.27526e7 −0.507707
\(913\) −2.12428e6 −0.0843401
\(914\) −2.45476e6 −0.0971948
\(915\) 2.22144e7 0.877167
\(916\) 6.64860e6 0.261813
\(917\) 7.94050e6 0.311835
\(918\) −1.05268e6 −0.0412276
\(919\) −1.72481e7 −0.673678 −0.336839 0.941562i \(-0.609358\pi\)
−0.336839 + 0.941562i \(0.609358\pi\)
\(920\) −1.39987e7 −0.545279
\(921\) 8.37749e6 0.325435
\(922\) 1.28200e7 0.496662
\(923\) −3.99780e7 −1.54460
\(924\) 4.51282e6 0.173887
\(925\) −8.19106e6 −0.314765
\(926\) −1.32606e7 −0.508202
\(927\) 1.06032e7 0.405265
\(928\) −2.39568e7 −0.913185
\(929\) 2.51145e6 0.0954740 0.0477370 0.998860i \(-0.484799\pi\)
0.0477370 + 0.998860i \(0.484799\pi\)
\(930\) 4.16650e6 0.157966
\(931\) 1.10095e7 0.416288
\(932\) 2.56118e7 0.965828
\(933\) −2.23674e7 −0.841225
\(934\) 8.29538e6 0.311150
\(935\) −4.01865e6 −0.150332
\(936\) 5.57928e6 0.208156
\(937\) 1.79853e7 0.669221 0.334611 0.942357i \(-0.391395\pi\)
0.334611 + 0.942357i \(0.391395\pi\)
\(938\) −9.14877e6 −0.339512
\(939\) −1.18547e7 −0.438761
\(940\) −3.18497e7 −1.17567
\(941\) −3.22586e7 −1.18760 −0.593802 0.804611i \(-0.702374\pi\)
−0.593802 + 0.804611i \(0.702374\pi\)
\(942\) 5.17820e6 0.190131
\(943\) 1.30300e7 0.477160
\(944\) −3.84416e6 −0.140401
\(945\) −4.96303e6 −0.180787
\(946\) −2.00957e6 −0.0730087
\(947\) 4.41659e7 1.60034 0.800169 0.599774i \(-0.204743\pi\)
0.800169 + 0.599774i \(0.204743\pi\)
\(948\) −6.37560e6 −0.230409
\(949\) −1.05880e7 −0.381635
\(950\) 4.35888e6 0.156699
\(951\) −2.02986e7 −0.727804
\(952\) −1.28227e7 −0.458551
\(953\) 1.87488e7 0.668714 0.334357 0.942446i \(-0.391481\pi\)
0.334357 + 0.942446i \(0.391481\pi\)
\(954\) 4.65685e6 0.165661
\(955\) −1.40580e7 −0.498786
\(956\) −3.94498e7 −1.39605
\(957\) −5.06385e6 −0.178731
\(958\) 1.01026e7 0.355649
\(959\) −3.38716e7 −1.18929
\(960\) 4.42483e6 0.154960
\(961\) −3.30813e6 −0.115551
\(962\) −9.31946e6 −0.324678
\(963\) −1.14706e7 −0.398584
\(964\) 2.31380e7 0.801925
\(965\) 5.35596e6 0.185148
\(966\) −6.75590e6 −0.232938
\(967\) 1.08673e7 0.373730 0.186865 0.982386i \(-0.440167\pi\)
0.186865 + 0.982386i \(0.440167\pi\)
\(968\) 1.75692e6 0.0602648
\(969\) 1.40357e7 0.480202
\(970\) −6.50054e6 −0.221830
\(971\) −4.79123e7 −1.63079 −0.815397 0.578902i \(-0.803481\pi\)
−0.815397 + 0.578902i \(0.803481\pi\)
\(972\) 1.65337e6 0.0561313
\(973\) 5.54882e7 1.87896
\(974\) −5.32442e6 −0.179835
\(975\) 5.21249e6 0.175604
\(976\) −3.51996e7 −1.18281
\(977\) 4.01385e7 1.34532 0.672658 0.739954i \(-0.265153\pi\)
0.672658 + 0.739954i \(0.265153\pi\)
\(978\) 1.89223e6 0.0632597
\(979\) 1.60410e7 0.534902
\(980\) −6.56494e6 −0.218356
\(981\) −1.94246e7 −0.644436
\(982\) 1.10932e7 0.367094
\(983\) 3.22682e6 0.106510 0.0532551 0.998581i \(-0.483040\pi\)
0.0532551 + 0.998581i \(0.483040\pi\)
\(984\) 5.54904e6 0.182696
\(985\) −1.13961e7 −0.374254
\(986\) 6.71460e6 0.219952
\(987\) −3.29377e7 −1.07622
\(988\) −3.47155e7 −1.13144
\(989\) −2.10589e7 −0.684614
\(990\) −901692. −0.0292395
\(991\) −5.95345e6 −0.192568 −0.0962841 0.995354i \(-0.530696\pi\)
−0.0962841 + 0.995354i \(0.530696\pi\)
\(992\) −2.59249e7 −0.836445
\(993\) 2.85364e7 0.918387
\(994\) 2.06158e7 0.661812
\(995\) −2.36146e7 −0.756175
\(996\) −4.42411e6 −0.141312
\(997\) 3.20783e7 1.02205 0.511027 0.859565i \(-0.329265\pi\)
0.511027 + 0.859565i \(0.329265\pi\)
\(998\) 13640.0 0.000433499 0
\(999\) −5.91802e6 −0.187613
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 33.6.a.a.1.1 1
3.2 odd 2 99.6.a.b.1.1 1
4.3 odd 2 528.6.a.i.1.1 1
5.4 even 2 825.6.a.b.1.1 1
11.10 odd 2 363.6.a.c.1.1 1
33.32 even 2 1089.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.a.1.1 1 1.1 even 1 trivial
99.6.a.b.1.1 1 3.2 odd 2
363.6.a.c.1.1 1 11.10 odd 2
528.6.a.i.1.1 1 4.3 odd 2
825.6.a.b.1.1 1 5.4 even 2
1089.6.a.d.1.1 1 33.32 even 2