Properties

Label 6.6.a.a.1.1
Level 6
Weight 6
Character 6.1
Self dual yes
Analytic conductor 0.962
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.962302918878\)
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 \(\chi\) \(=\) 6.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -66.0000 q^{5} -36.0000 q^{6} +176.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -66.0000 q^{5} -36.0000 q^{6} +176.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -264.000 q^{10} -60.0000 q^{11} -144.000 q^{12} -658.000 q^{13} +704.000 q^{14} +594.000 q^{15} +256.000 q^{16} -414.000 q^{17} +324.000 q^{18} +956.000 q^{19} -1056.00 q^{20} -1584.00 q^{21} -240.000 q^{22} +600.000 q^{23} -576.000 q^{24} +1231.00 q^{25} -2632.00 q^{26} -729.000 q^{27} +2816.00 q^{28} +5574.00 q^{29} +2376.00 q^{30} -3592.00 q^{31} +1024.00 q^{32} +540.000 q^{33} -1656.00 q^{34} -11616.0 q^{35} +1296.00 q^{36} -8458.00 q^{37} +3824.00 q^{38} +5922.00 q^{39} -4224.00 q^{40} +19194.0 q^{41} -6336.00 q^{42} +13316.0 q^{43} -960.000 q^{44} -5346.00 q^{45} +2400.00 q^{46} -19680.0 q^{47} -2304.00 q^{48} +14169.0 q^{49} +4924.00 q^{50} +3726.00 q^{51} -10528.0 q^{52} -31266.0 q^{53} -2916.00 q^{54} +3960.00 q^{55} +11264.0 q^{56} -8604.00 q^{57} +22296.0 q^{58} +26340.0 q^{59} +9504.00 q^{60} -31090.0 q^{61} -14368.0 q^{62} +14256.0 q^{63} +4096.00 q^{64} +43428.0 q^{65} +2160.00 q^{66} -16804.0 q^{67} -6624.00 q^{68} -5400.00 q^{69} -46464.0 q^{70} +6120.00 q^{71} +5184.00 q^{72} -25558.0 q^{73} -33832.0 q^{74} -11079.0 q^{75} +15296.0 q^{76} -10560.0 q^{77} +23688.0 q^{78} +74408.0 q^{79} -16896.0 q^{80} +6561.00 q^{81} +76776.0 q^{82} -6468.00 q^{83} -25344.0 q^{84} +27324.0 q^{85} +53264.0 q^{86} -50166.0 q^{87} -3840.00 q^{88} -32742.0 q^{89} -21384.0 q^{90} -115808. q^{91} +9600.00 q^{92} +32328.0 q^{93} -78720.0 q^{94} -63096.0 q^{95} -9216.00 q^{96} +166082. q^{97} +56676.0 q^{98} -4860.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) −66.0000 −1.18064 −0.590322 0.807168i \(-0.700999\pi\)
−0.590322 + 0.807168i \(0.700999\pi\)
\(6\) −36.0000 −0.408248
\(7\) 176.000 1.35759 0.678793 0.734329i \(-0.262503\pi\)
0.678793 + 0.734329i \(0.262503\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) −264.000 −0.834841
\(11\) −60.0000 −0.149510 −0.0747549 0.997202i \(-0.523817\pi\)
−0.0747549 + 0.997202i \(0.523817\pi\)
\(12\) −144.000 −0.288675
\(13\) −658.000 −1.07986 −0.539930 0.841710i \(-0.681549\pi\)
−0.539930 + 0.841710i \(0.681549\pi\)
\(14\) 704.000 0.959959
\(15\) 594.000 0.681645
\(16\) 256.000 0.250000
\(17\) −414.000 −0.347439 −0.173719 0.984795i \(-0.555579\pi\)
−0.173719 + 0.984795i \(0.555579\pi\)
\(18\) 324.000 0.235702
\(19\) 956.000 0.607539 0.303769 0.952746i \(-0.401755\pi\)
0.303769 + 0.952746i \(0.401755\pi\)
\(20\) −1056.00 −0.590322
\(21\) −1584.00 −0.783803
\(22\) −240.000 −0.105719
\(23\) 600.000 0.236500 0.118250 0.992984i \(-0.462272\pi\)
0.118250 + 0.992984i \(0.462272\pi\)
\(24\) −576.000 −0.204124
\(25\) 1231.00 0.393920
\(26\) −2632.00 −0.763576
\(27\) −729.000 −0.192450
\(28\) 2816.00 0.678793
\(29\) 5574.00 1.23076 0.615378 0.788232i \(-0.289003\pi\)
0.615378 + 0.788232i \(0.289003\pi\)
\(30\) 2376.00 0.481996
\(31\) −3592.00 −0.671324 −0.335662 0.941983i \(-0.608960\pi\)
−0.335662 + 0.941983i \(0.608960\pi\)
\(32\) 1024.00 0.176777
\(33\) 540.000 0.0863195
\(34\) −1656.00 −0.245676
\(35\) −11616.0 −1.60283
\(36\) 1296.00 0.166667
\(37\) −8458.00 −1.01570 −0.507848 0.861447i \(-0.669559\pi\)
−0.507848 + 0.861447i \(0.669559\pi\)
\(38\) 3824.00 0.429595
\(39\) 5922.00 0.623458
\(40\) −4224.00 −0.417421
\(41\) 19194.0 1.78322 0.891612 0.452800i \(-0.149575\pi\)
0.891612 + 0.452800i \(0.149575\pi\)
\(42\) −6336.00 −0.554232
\(43\) 13316.0 1.09825 0.549127 0.835739i \(-0.314960\pi\)
0.549127 + 0.835739i \(0.314960\pi\)
\(44\) −960.000 −0.0747549
\(45\) −5346.00 −0.393548
\(46\) 2400.00 0.167231
\(47\) −19680.0 −1.29951 −0.649756 0.760143i \(-0.725129\pi\)
−0.649756 + 0.760143i \(0.725129\pi\)
\(48\) −2304.00 −0.144338
\(49\) 14169.0 0.843042
\(50\) 4924.00 0.278544
\(51\) 3726.00 0.200594
\(52\) −10528.0 −0.539930
\(53\) −31266.0 −1.52891 −0.764456 0.644676i \(-0.776992\pi\)
−0.764456 + 0.644676i \(0.776992\pi\)
\(54\) −2916.00 −0.136083
\(55\) 3960.00 0.176518
\(56\) 11264.0 0.479979
\(57\) −8604.00 −0.350763
\(58\) 22296.0 0.870276
\(59\) 26340.0 0.985112 0.492556 0.870281i \(-0.336063\pi\)
0.492556 + 0.870281i \(0.336063\pi\)
\(60\) 9504.00 0.340823
\(61\) −31090.0 −1.06978 −0.534892 0.844920i \(-0.679648\pi\)
−0.534892 + 0.844920i \(0.679648\pi\)
\(62\) −14368.0 −0.474698
\(63\) 14256.0 0.452529
\(64\) 4096.00 0.125000
\(65\) 43428.0 1.27493
\(66\) 2160.00 0.0610371
\(67\) −16804.0 −0.457326 −0.228663 0.973506i \(-0.573435\pi\)
−0.228663 + 0.973506i \(0.573435\pi\)
\(68\) −6624.00 −0.173719
\(69\) −5400.00 −0.136544
\(70\) −46464.0 −1.13337
\(71\) 6120.00 0.144081 0.0720403 0.997402i \(-0.477049\pi\)
0.0720403 + 0.997402i \(0.477049\pi\)
\(72\) 5184.00 0.117851
\(73\) −25558.0 −0.561332 −0.280666 0.959806i \(-0.590555\pi\)
−0.280666 + 0.959806i \(0.590555\pi\)
\(74\) −33832.0 −0.718205
\(75\) −11079.0 −0.227430
\(76\) 15296.0 0.303769
\(77\) −10560.0 −0.202972
\(78\) 23688.0 0.440851
\(79\) 74408.0 1.34138 0.670690 0.741738i \(-0.265998\pi\)
0.670690 + 0.741738i \(0.265998\pi\)
\(80\) −16896.0 −0.295161
\(81\) 6561.00 0.111111
\(82\) 76776.0 1.26093
\(83\) −6468.00 −0.103056 −0.0515282 0.998672i \(-0.516409\pi\)
−0.0515282 + 0.998672i \(0.516409\pi\)
\(84\) −25344.0 −0.391902
\(85\) 27324.0 0.410201
\(86\) 53264.0 0.776583
\(87\) −50166.0 −0.710577
\(88\) −3840.00 −0.0528597
\(89\) −32742.0 −0.438157 −0.219079 0.975707i \(-0.570305\pi\)
−0.219079 + 0.975707i \(0.570305\pi\)
\(90\) −21384.0 −0.278280
\(91\) −115808. −1.46600
\(92\) 9600.00 0.118250
\(93\) 32328.0 0.387589
\(94\) −78720.0 −0.918894
\(95\) −63096.0 −0.717287
\(96\) −9216.00 −0.102062
\(97\) 166082. 1.79223 0.896114 0.443824i \(-0.146378\pi\)
0.896114 + 0.443824i \(0.146378\pi\)
\(98\) 56676.0 0.596120
\(99\) −4860.00 −0.0498366
\(100\) 19696.0 0.196960
\(101\) −22002.0 −0.214614 −0.107307 0.994226i \(-0.534223\pi\)
−0.107307 + 0.994226i \(0.534223\pi\)
\(102\) 14904.0 0.141841
\(103\) −79264.0 −0.736178 −0.368089 0.929791i \(-0.619988\pi\)
−0.368089 + 0.929791i \(0.619988\pi\)
\(104\) −42112.0 −0.381788
\(105\) 104544. 0.925392
\(106\) −125064. −1.08110
\(107\) 227988. 1.92510 0.962548 0.271110i \(-0.0873908\pi\)
0.962548 + 0.271110i \(0.0873908\pi\)
\(108\) −11664.0 −0.0962250
\(109\) −8530.00 −0.0687674 −0.0343837 0.999409i \(-0.510947\pi\)
−0.0343837 + 0.999409i \(0.510947\pi\)
\(110\) 15840.0 0.124817
\(111\) 76122.0 0.586412
\(112\) 45056.0 0.339397
\(113\) −195438. −1.43984 −0.719918 0.694059i \(-0.755821\pi\)
−0.719918 + 0.694059i \(0.755821\pi\)
\(114\) −34416.0 −0.248027
\(115\) −39600.0 −0.279223
\(116\) 89184.0 0.615378
\(117\) −53298.0 −0.359953
\(118\) 105360. 0.696580
\(119\) −72864.0 −0.471678
\(120\) 38016.0 0.240998
\(121\) −157451. −0.977647
\(122\) −124360. −0.756452
\(123\) −172746. −1.02954
\(124\) −57472.0 −0.335662
\(125\) 125004. 0.715565
\(126\) 57024.0 0.319986
\(127\) 173000. 0.951780 0.475890 0.879505i \(-0.342126\pi\)
0.475890 + 0.879505i \(0.342126\pi\)
\(128\) 16384.0 0.0883883
\(129\) −119844. −0.634077
\(130\) 173712. 0.901512
\(131\) 151260. 0.770098 0.385049 0.922896i \(-0.374185\pi\)
0.385049 + 0.922896i \(0.374185\pi\)
\(132\) 8640.00 0.0431597
\(133\) 168256. 0.824786
\(134\) −67216.0 −0.323378
\(135\) 48114.0 0.227215
\(136\) −26496.0 −0.122838
\(137\) −128454. −0.584718 −0.292359 0.956309i \(-0.594440\pi\)
−0.292359 + 0.956309i \(0.594440\pi\)
\(138\) −21600.0 −0.0965508
\(139\) 154196. 0.676918 0.338459 0.940981i \(-0.390094\pi\)
0.338459 + 0.940981i \(0.390094\pi\)
\(140\) −185856. −0.801413
\(141\) 177120. 0.750274
\(142\) 24480.0 0.101880
\(143\) 39480.0 0.161450
\(144\) 20736.0 0.0833333
\(145\) −367884. −1.45308
\(146\) −102232. −0.396922
\(147\) −127521. −0.486730
\(148\) −135328. −0.507848
\(149\) 29454.0 0.108687 0.0543436 0.998522i \(-0.482693\pi\)
0.0543436 + 0.998522i \(0.482693\pi\)
\(150\) −44316.0 −0.160817
\(151\) −203872. −0.727638 −0.363819 0.931470i \(-0.618527\pi\)
−0.363819 + 0.931470i \(0.618527\pi\)
\(152\) 61184.0 0.214797
\(153\) −33534.0 −0.115813
\(154\) −42240.0 −0.143523
\(155\) 237072. 0.792594
\(156\) 94752.0 0.311729
\(157\) 136142. 0.440801 0.220401 0.975409i \(-0.429263\pi\)
0.220401 + 0.975409i \(0.429263\pi\)
\(158\) 297632. 0.948499
\(159\) 281394. 0.882718
\(160\) −67584.0 −0.208710
\(161\) 105600. 0.321070
\(162\) 26244.0 0.0785674
\(163\) −171124. −0.504478 −0.252239 0.967665i \(-0.581167\pi\)
−0.252239 + 0.967665i \(0.581167\pi\)
\(164\) 307104. 0.891612
\(165\) −35640.0 −0.101913
\(166\) −25872.0 −0.0728718
\(167\) −676200. −1.87622 −0.938110 0.346336i \(-0.887426\pi\)
−0.938110 + 0.346336i \(0.887426\pi\)
\(168\) −101376. −0.277116
\(169\) 61671.0 0.166098
\(170\) 109296. 0.290056
\(171\) 77436.0 0.202513
\(172\) 213056. 0.549127
\(173\) 133158. 0.338261 0.169131 0.985594i \(-0.445904\pi\)
0.169131 + 0.985594i \(0.445904\pi\)
\(174\) −200664. −0.502454
\(175\) 216656. 0.534781
\(176\) −15360.0 −0.0373774
\(177\) −237060. −0.568755
\(178\) −130968. −0.309824
\(179\) −693396. −1.61752 −0.808758 0.588141i \(-0.799860\pi\)
−0.808758 + 0.588141i \(0.799860\pi\)
\(180\) −85536.0 −0.196774
\(181\) 377174. 0.855747 0.427873 0.903839i \(-0.359263\pi\)
0.427873 + 0.903839i \(0.359263\pi\)
\(182\) −463232. −1.03662
\(183\) 279810. 0.617640
\(184\) 38400.0 0.0836155
\(185\) 558228. 1.19917
\(186\) 129312. 0.274067
\(187\) 24840.0 0.0519455
\(188\) −314880. −0.649756
\(189\) −128304. −0.261268
\(190\) −252384. −0.507198
\(191\) −265344. −0.526291 −0.263145 0.964756i \(-0.584760\pi\)
−0.263145 + 0.964756i \(0.584760\pi\)
\(192\) −36864.0 −0.0721688
\(193\) 295298. 0.570647 0.285323 0.958431i \(-0.407899\pi\)
0.285323 + 0.958431i \(0.407899\pi\)
\(194\) 664328. 1.26730
\(195\) −390852. −0.736081
\(196\) 226704. 0.421521
\(197\) 201294. 0.369543 0.184772 0.982781i \(-0.440845\pi\)
0.184772 + 0.982781i \(0.440845\pi\)
\(198\) −19440.0 −0.0352398
\(199\) 652448. 1.16792 0.583960 0.811782i \(-0.301502\pi\)
0.583960 + 0.811782i \(0.301502\pi\)
\(200\) 78784.0 0.139272
\(201\) 151236. 0.264037
\(202\) −88008.0 −0.151755
\(203\) 981024. 1.67086
\(204\) 59616.0 0.100297
\(205\) −1.26680e6 −2.10535
\(206\) −317056. −0.520557
\(207\) 48600.0 0.0788334
\(208\) −168448. −0.269965
\(209\) −57360.0 −0.0908330
\(210\) 418176. 0.654351
\(211\) −1.14706e6 −1.77370 −0.886850 0.462058i \(-0.847111\pi\)
−0.886850 + 0.462058i \(0.847111\pi\)
\(212\) −500256. −0.764456
\(213\) −55080.0 −0.0831850
\(214\) 911952. 1.36125
\(215\) −878856. −1.29665
\(216\) −46656.0 −0.0680414
\(217\) −632192. −0.911380
\(218\) −34120.0 −0.0486259
\(219\) 230022. 0.324085
\(220\) 63360.0 0.0882589
\(221\) 272412. 0.375185
\(222\) 304488. 0.414656
\(223\) 701960. 0.945258 0.472629 0.881262i \(-0.343305\pi\)
0.472629 + 0.881262i \(0.343305\pi\)
\(224\) 180224. 0.239990
\(225\) 99711.0 0.131307
\(226\) −781752. −1.01812
\(227\) 1.23611e6 1.59218 0.796089 0.605179i \(-0.206899\pi\)
0.796089 + 0.605179i \(0.206899\pi\)
\(228\) −137664. −0.175381
\(229\) 105830. 0.133358 0.0666792 0.997774i \(-0.478760\pi\)
0.0666792 + 0.997774i \(0.478760\pi\)
\(230\) −158400. −0.197440
\(231\) 95040.0 0.117186
\(232\) 356736. 0.435138
\(233\) −438678. −0.529366 −0.264683 0.964335i \(-0.585267\pi\)
−0.264683 + 0.964335i \(0.585267\pi\)
\(234\) −213192. −0.254525
\(235\) 1.29888e6 1.53426
\(236\) 421440. 0.492556
\(237\) −669672. −0.774446
\(238\) −291456. −0.333527
\(239\) 28464.0 0.0322330 0.0161165 0.999870i \(-0.494870\pi\)
0.0161165 + 0.999870i \(0.494870\pi\)
\(240\) 152064. 0.170411
\(241\) 892562. 0.989910 0.494955 0.868919i \(-0.335185\pi\)
0.494955 + 0.868919i \(0.335185\pi\)
\(242\) −629804. −0.691301
\(243\) −59049.0 −0.0641500
\(244\) −497440. −0.534892
\(245\) −935154. −0.995332
\(246\) −690984. −0.727998
\(247\) −629048. −0.656057
\(248\) −229888. −0.237349
\(249\) 58212.0 0.0594996
\(250\) 500016. 0.505981
\(251\) −110124. −0.110331 −0.0551655 0.998477i \(-0.517569\pi\)
−0.0551655 + 0.998477i \(0.517569\pi\)
\(252\) 228096. 0.226264
\(253\) −36000.0 −0.0353591
\(254\) 692000. 0.673010
\(255\) −245916. −0.236830
\(256\) 65536.0 0.0625000
\(257\) 140802. 0.132977 0.0664884 0.997787i \(-0.478820\pi\)
0.0664884 + 0.997787i \(0.478820\pi\)
\(258\) −479376. −0.448360
\(259\) −1.48861e6 −1.37889
\(260\) 694848. 0.637465
\(261\) 451494. 0.410252
\(262\) 605040. 0.544541
\(263\) −938760. −0.836884 −0.418442 0.908244i \(-0.637424\pi\)
−0.418442 + 0.908244i \(0.637424\pi\)
\(264\) 34560.0 0.0305186
\(265\) 2.06356e6 1.80510
\(266\) 673024. 0.583212
\(267\) 294678. 0.252970
\(268\) −268864. −0.228663
\(269\) −1.11451e6 −0.939078 −0.469539 0.882912i \(-0.655580\pi\)
−0.469539 + 0.882912i \(0.655580\pi\)
\(270\) 192456. 0.160665
\(271\) 567704. 0.469568 0.234784 0.972048i \(-0.424562\pi\)
0.234784 + 0.972048i \(0.424562\pi\)
\(272\) −105984. −0.0868596
\(273\) 1.04227e6 0.846398
\(274\) −513816. −0.413458
\(275\) −73860.0 −0.0588949
\(276\) −86400.0 −0.0682718
\(277\) −1.21326e6 −0.950066 −0.475033 0.879968i \(-0.657564\pi\)
−0.475033 + 0.879968i \(0.657564\pi\)
\(278\) 616784. 0.478653
\(279\) −290952. −0.223775
\(280\) −743424. −0.566685
\(281\) 687738. 0.519586 0.259793 0.965664i \(-0.416346\pi\)
0.259793 + 0.965664i \(0.416346\pi\)
\(282\) 708480. 0.530524
\(283\) −830908. −0.616718 −0.308359 0.951270i \(-0.599780\pi\)
−0.308359 + 0.951270i \(0.599780\pi\)
\(284\) 97920.0 0.0720403
\(285\) 567864. 0.414126
\(286\) 157920. 0.114162
\(287\) 3.37814e6 2.42088
\(288\) 82944.0 0.0589256
\(289\) −1.24846e6 −0.879286
\(290\) −1.47154e6 −1.02749
\(291\) −1.49474e6 −1.03474
\(292\) −408928. −0.280666
\(293\) −1.31263e6 −0.893248 −0.446624 0.894722i \(-0.647374\pi\)
−0.446624 + 0.894722i \(0.647374\pi\)
\(294\) −510084. −0.344170
\(295\) −1.73844e6 −1.16307
\(296\) −541312. −0.359102
\(297\) 43740.0 0.0287732
\(298\) 117816. 0.0768535
\(299\) −394800. −0.255387
\(300\) −177264. −0.113715
\(301\) 2.34362e6 1.49097
\(302\) −815488. −0.514518
\(303\) 198018. 0.123908
\(304\) 244736. 0.151885
\(305\) 2.05194e6 1.26303
\(306\) −134136. −0.0818921
\(307\) 1.69022e6 1.02352 0.511761 0.859128i \(-0.328993\pi\)
0.511761 + 0.859128i \(0.328993\pi\)
\(308\) −168960. −0.101486
\(309\) 713376. 0.425033
\(310\) 948288. 0.560449
\(311\) −1.50204e6 −0.880604 −0.440302 0.897850i \(-0.645129\pi\)
−0.440302 + 0.897850i \(0.645129\pi\)
\(312\) 379008. 0.220426
\(313\) 810842. 0.467816 0.233908 0.972259i \(-0.424848\pi\)
0.233908 + 0.972259i \(0.424848\pi\)
\(314\) 544568. 0.311694
\(315\) −940896. −0.534275
\(316\) 1.19053e6 0.670690
\(317\) 903558. 0.505019 0.252510 0.967594i \(-0.418744\pi\)
0.252510 + 0.967594i \(0.418744\pi\)
\(318\) 1.12558e6 0.624176
\(319\) −334440. −0.184010
\(320\) −270336. −0.147580
\(321\) −2.05189e6 −1.11146
\(322\) 422400. 0.227031
\(323\) −395784. −0.211082
\(324\) 104976. 0.0555556
\(325\) −809998. −0.425379
\(326\) −684496. −0.356720
\(327\) 76770.0 0.0397029
\(328\) 1.22842e6 0.630465
\(329\) −3.46368e6 −1.76420
\(330\) −142560. −0.0720631
\(331\) 1.12197e6 0.562875 0.281438 0.959580i \(-0.409189\pi\)
0.281438 + 0.959580i \(0.409189\pi\)
\(332\) −103488. −0.0515282
\(333\) −685098. −0.338565
\(334\) −2.70480e6 −1.32669
\(335\) 1.10906e6 0.539939
\(336\) −405504. −0.195951
\(337\) −2.75217e6 −1.32008 −0.660041 0.751229i \(-0.729461\pi\)
−0.660041 + 0.751229i \(0.729461\pi\)
\(338\) 246684. 0.117449
\(339\) 1.75894e6 0.831289
\(340\) 437184. 0.205101
\(341\) 215520. 0.100369
\(342\) 309744. 0.143198
\(343\) −464288. −0.213085
\(344\) 852224. 0.388291
\(345\) 356400. 0.161209
\(346\) 532632. 0.239187
\(347\) 1.91749e6 0.854889 0.427445 0.904042i \(-0.359414\pi\)
0.427445 + 0.904042i \(0.359414\pi\)
\(348\) −802656. −0.355289
\(349\) 1.83659e6 0.807140 0.403570 0.914949i \(-0.367769\pi\)
0.403570 + 0.914949i \(0.367769\pi\)
\(350\) 866624. 0.378147
\(351\) 479682. 0.207819
\(352\) −61440.0 −0.0264298
\(353\) −622014. −0.265683 −0.132841 0.991137i \(-0.542410\pi\)
−0.132841 + 0.991137i \(0.542410\pi\)
\(354\) −948240. −0.402170
\(355\) −403920. −0.170108
\(356\) −523872. −0.219079
\(357\) 655776. 0.272323
\(358\) −2.77358e6 −1.14376
\(359\) 3.74062e6 1.53182 0.765909 0.642949i \(-0.222289\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(360\) −342144. −0.139140
\(361\) −1.56216e6 −0.630897
\(362\) 1.50870e6 0.605104
\(363\) 1.41706e6 0.564445
\(364\) −1.85293e6 −0.733002
\(365\) 1.68683e6 0.662733
\(366\) 1.11924e6 0.436738
\(367\) 16232.0 0.00629081 0.00314541 0.999995i \(-0.498999\pi\)
0.00314541 + 0.999995i \(0.498999\pi\)
\(368\) 153600. 0.0591251
\(369\) 1.55471e6 0.594408
\(370\) 2.23291e6 0.847944
\(371\) −5.50282e6 −2.07563
\(372\) 517248. 0.193795
\(373\) 293606. 0.109268 0.0546340 0.998506i \(-0.482601\pi\)
0.0546340 + 0.998506i \(0.482601\pi\)
\(374\) 99360.0 0.0367310
\(375\) −1.12504e6 −0.413131
\(376\) −1.25952e6 −0.459447
\(377\) −3.66769e6 −1.32904
\(378\) −513216. −0.184744
\(379\) 3.18012e6 1.13722 0.568611 0.822607i \(-0.307481\pi\)
0.568611 + 0.822607i \(0.307481\pi\)
\(380\) −1.00954e6 −0.358643
\(381\) −1.55700e6 −0.549511
\(382\) −1.06138e6 −0.372144
\(383\) −2.97984e6 −1.03800 −0.518998 0.854775i \(-0.673695\pi\)
−0.518998 + 0.854775i \(0.673695\pi\)
\(384\) −147456. −0.0510310
\(385\) 696960. 0.239638
\(386\) 1.18119e6 0.403508
\(387\) 1.07860e6 0.366085
\(388\) 2.65731e6 0.896114
\(389\) 3.45977e6 1.15924 0.579620 0.814887i \(-0.303201\pi\)
0.579620 + 0.814887i \(0.303201\pi\)
\(390\) −1.56341e6 −0.520488
\(391\) −248400. −0.0821693
\(392\) 906816. 0.298060
\(393\) −1.36134e6 −0.444616
\(394\) 805176. 0.261307
\(395\) −4.91093e6 −1.58369
\(396\) −77760.0 −0.0249183
\(397\) −3.90416e6 −1.24323 −0.621615 0.783323i \(-0.713523\pi\)
−0.621615 + 0.783323i \(0.713523\pi\)
\(398\) 2.60979e6 0.825844
\(399\) −1.51430e6 −0.476191
\(400\) 315136. 0.0984800
\(401\) 5.44115e6 1.68978 0.844890 0.534940i \(-0.179666\pi\)
0.844890 + 0.534940i \(0.179666\pi\)
\(402\) 604944. 0.186702
\(403\) 2.36354e6 0.724936
\(404\) −352032. −0.107307
\(405\) −433026. −0.131183
\(406\) 3.92410e6 1.18148
\(407\) 507480. 0.151856
\(408\) 238464. 0.0709206
\(409\) 1.96995e6 0.582299 0.291150 0.956678i \(-0.405962\pi\)
0.291150 + 0.956678i \(0.405962\pi\)
\(410\) −5.06722e6 −1.48871
\(411\) 1.15609e6 0.337587
\(412\) −1.26822e6 −0.368089
\(413\) 4.63584e6 1.33738
\(414\) 194400. 0.0557437
\(415\) 426888. 0.121673
\(416\) −673792. −0.190894
\(417\) −1.38776e6 −0.390819
\(418\) −229440. −0.0642286
\(419\) 139020. 0.0386850 0.0193425 0.999813i \(-0.493843\pi\)
0.0193425 + 0.999813i \(0.493843\pi\)
\(420\) 1.67270e6 0.462696
\(421\) 4.32743e6 1.18994 0.594970 0.803748i \(-0.297164\pi\)
0.594970 + 0.803748i \(0.297164\pi\)
\(422\) −4.58824e6 −1.25419
\(423\) −1.59408e6 −0.433171
\(424\) −2.00102e6 −0.540552
\(425\) −509634. −0.136863
\(426\) −220320. −0.0588207
\(427\) −5.47184e6 −1.45232
\(428\) 3.64781e6 0.962548
\(429\) −355320. −0.0932130
\(430\) −3.51542e6 −0.916867
\(431\) −2.79936e6 −0.725881 −0.362941 0.931812i \(-0.618227\pi\)
−0.362941 + 0.931812i \(0.618227\pi\)
\(432\) −186624. −0.0481125
\(433\) −5.90241e6 −1.51290 −0.756449 0.654052i \(-0.773068\pi\)
−0.756449 + 0.654052i \(0.773068\pi\)
\(434\) −2.52877e6 −0.644443
\(435\) 3.31096e6 0.838939
\(436\) −136480. −0.0343837
\(437\) 573600. 0.143683
\(438\) 920088. 0.229163
\(439\) −446512. −0.110579 −0.0552894 0.998470i \(-0.517608\pi\)
−0.0552894 + 0.998470i \(0.517608\pi\)
\(440\) 253440. 0.0624085
\(441\) 1.14769e6 0.281014
\(442\) 1.08965e6 0.265296
\(443\) 3.49525e6 0.846193 0.423096 0.906085i \(-0.360943\pi\)
0.423096 + 0.906085i \(0.360943\pi\)
\(444\) 1.21795e6 0.293206
\(445\) 2.16097e6 0.517308
\(446\) 2.80784e6 0.668398
\(447\) −265086. −0.0627506
\(448\) 720896. 0.169698
\(449\) −1.20613e6 −0.282343 −0.141171 0.989985i \(-0.545087\pi\)
−0.141171 + 0.989985i \(0.545087\pi\)
\(450\) 398844. 0.0928478
\(451\) −1.15164e6 −0.266609
\(452\) −3.12701e6 −0.719918
\(453\) 1.83485e6 0.420102
\(454\) 4.94443e6 1.12584
\(455\) 7.64333e6 1.73083
\(456\) −550656. −0.124013
\(457\) 233546. 0.0523097 0.0261548 0.999658i \(-0.491674\pi\)
0.0261548 + 0.999658i \(0.491674\pi\)
\(458\) 423320. 0.0942986
\(459\) 301806. 0.0668646
\(460\) −633600. −0.139611
\(461\) −1.74489e6 −0.382398 −0.191199 0.981551i \(-0.561238\pi\)
−0.191199 + 0.981551i \(0.561238\pi\)
\(462\) 380160. 0.0828632
\(463\) −2.91786e6 −0.632576 −0.316288 0.948663i \(-0.602437\pi\)
−0.316288 + 0.948663i \(0.602437\pi\)
\(464\) 1.42694e6 0.307689
\(465\) −2.13365e6 −0.457605
\(466\) −1.75471e6 −0.374318
\(467\) −5.31076e6 −1.12684 −0.563422 0.826169i \(-0.690516\pi\)
−0.563422 + 0.826169i \(0.690516\pi\)
\(468\) −852768. −0.179977
\(469\) −2.95750e6 −0.620859
\(470\) 5.19552e6 1.08489
\(471\) −1.22528e6 −0.254497
\(472\) 1.68576e6 0.348290
\(473\) −798960. −0.164200
\(474\) −2.67869e6 −0.547616
\(475\) 1.17684e6 0.239322
\(476\) −1.16582e6 −0.235839
\(477\) −2.53255e6 −0.509638
\(478\) 113856. 0.0227922
\(479\) 2.34466e6 0.466918 0.233459 0.972367i \(-0.424996\pi\)
0.233459 + 0.972367i \(0.424996\pi\)
\(480\) 608256. 0.120499
\(481\) 5.56536e6 1.09681
\(482\) 3.57025e6 0.699972
\(483\) −950400. −0.185370
\(484\) −2.51922e6 −0.488823
\(485\) −1.09614e7 −2.11598
\(486\) −236196. −0.0453609
\(487\) 9.81531e6 1.87535 0.937674 0.347517i \(-0.112975\pi\)
0.937674 + 0.347517i \(0.112975\pi\)
\(488\) −1.98976e6 −0.378226
\(489\) 1.54012e6 0.291260
\(490\) −3.74062e6 −0.703806
\(491\) −5.94520e6 −1.11292 −0.556458 0.830876i \(-0.687840\pi\)
−0.556458 + 0.830876i \(0.687840\pi\)
\(492\) −2.76394e6 −0.514772
\(493\) −2.30764e6 −0.427612
\(494\) −2.51619e6 −0.463902
\(495\) 320760. 0.0588393
\(496\) −919552. −0.167831
\(497\) 1.07712e6 0.195602
\(498\) 232848. 0.0420726
\(499\) 6.47832e6 1.16469 0.582346 0.812941i \(-0.302135\pi\)
0.582346 + 0.812941i \(0.302135\pi\)
\(500\) 2.00006e6 0.357782
\(501\) 6.08580e6 1.08324
\(502\) −440496. −0.0780158
\(503\) 4.71794e6 0.831444 0.415722 0.909492i \(-0.363529\pi\)
0.415722 + 0.909492i \(0.363529\pi\)
\(504\) 912384. 0.159993
\(505\) 1.45213e6 0.253383
\(506\) −144000. −0.0250027
\(507\) −555039. −0.0958967
\(508\) 2.76800e6 0.475890
\(509\) −1.90771e6 −0.326375 −0.163188 0.986595i \(-0.552178\pi\)
−0.163188 + 0.986595i \(0.552178\pi\)
\(510\) −983664. −0.167464
\(511\) −4.49821e6 −0.762057
\(512\) 262144. 0.0441942
\(513\) −696924. −0.116921
\(514\) 563208. 0.0940288
\(515\) 5.23142e6 0.869164
\(516\) −1.91750e6 −0.317039
\(517\) 1.18080e6 0.194290
\(518\) −5.95443e6 −0.975025
\(519\) −1.19842e6 −0.195295
\(520\) 2.77939e6 0.450756
\(521\) 8.01974e6 1.29439 0.647196 0.762324i \(-0.275941\pi\)
0.647196 + 0.762324i \(0.275941\pi\)
\(522\) 1.80598e6 0.290092
\(523\) 1.91162e6 0.305596 0.152798 0.988257i \(-0.451172\pi\)
0.152798 + 0.988257i \(0.451172\pi\)
\(524\) 2.42016e6 0.385049
\(525\) −1.94990e6 −0.308756
\(526\) −3.75504e6 −0.591766
\(527\) 1.48709e6 0.233244
\(528\) 138240. 0.0215799
\(529\) −6.07634e6 −0.944068
\(530\) 8.25422e6 1.27640
\(531\) 2.13354e6 0.328371
\(532\) 2.69210e6 0.412393
\(533\) −1.26297e7 −1.92563
\(534\) 1.17871e6 0.178877
\(535\) −1.50472e7 −2.27285
\(536\) −1.07546e6 −0.161689
\(537\) 6.24056e6 0.933874
\(538\) −4.45802e6 −0.664028
\(539\) −850140. −0.126043
\(540\) 769824. 0.113608
\(541\) −1.19900e7 −1.76128 −0.880639 0.473788i \(-0.842886\pi\)
−0.880639 + 0.473788i \(0.842886\pi\)
\(542\) 2.27082e6 0.332035
\(543\) −3.39457e6 −0.494066
\(544\) −423936. −0.0614190
\(545\) 562980. 0.0811898
\(546\) 4.16909e6 0.598494
\(547\) 4.45809e6 0.637061 0.318530 0.947913i \(-0.396811\pi\)
0.318530 + 0.947913i \(0.396811\pi\)
\(548\) −2.05526e6 −0.292359
\(549\) −2.51829e6 −0.356595
\(550\) −295440. −0.0416450
\(551\) 5.32874e6 0.747732
\(552\) −345600. −0.0482754
\(553\) 1.30958e7 1.82104
\(554\) −4.85303e6 −0.671798
\(555\) −5.02405e6 −0.692344
\(556\) 2.46714e6 0.338459
\(557\) 9.02612e6 1.23272 0.616358 0.787466i \(-0.288607\pi\)
0.616358 + 0.787466i \(0.288607\pi\)
\(558\) −1.16381e6 −0.158233
\(559\) −8.76193e6 −1.18596
\(560\) −2.97370e6 −0.400707
\(561\) −223560. −0.0299907
\(562\) 2.75095e6 0.367403
\(563\) 6.84899e6 0.910658 0.455329 0.890323i \(-0.349522\pi\)
0.455329 + 0.890323i \(0.349522\pi\)
\(564\) 2.83392e6 0.375137
\(565\) 1.28989e7 1.69993
\(566\) −3.32363e6 −0.436086
\(567\) 1.15474e6 0.150843
\(568\) 391680. 0.0509402
\(569\) −5.46322e6 −0.707405 −0.353703 0.935358i \(-0.615077\pi\)
−0.353703 + 0.935358i \(0.615077\pi\)
\(570\) 2.27146e6 0.292831
\(571\) −1.02324e7 −1.31337 −0.656684 0.754166i \(-0.728041\pi\)
−0.656684 + 0.754166i \(0.728041\pi\)
\(572\) 631680. 0.0807248
\(573\) 2.38810e6 0.303854
\(574\) 1.35126e7 1.71182
\(575\) 738600. 0.0931622
\(576\) 331776. 0.0416667
\(577\) 1.59437e7 1.99365 0.996825 0.0796186i \(-0.0253702\pi\)
0.996825 + 0.0796186i \(0.0253702\pi\)
\(578\) −4.99384e6 −0.621749
\(579\) −2.65768e6 −0.329463
\(580\) −5.88614e6 −0.726542
\(581\) −1.13837e6 −0.139908
\(582\) −5.97895e6 −0.731674
\(583\) 1.87596e6 0.228587
\(584\) −1.63571e6 −0.198461
\(585\) 3.51767e6 0.424977
\(586\) −5.25050e6 −0.631622
\(587\) −9.47713e6 −1.13522 −0.567612 0.823296i \(-0.692133\pi\)
−0.567612 + 0.823296i \(0.692133\pi\)
\(588\) −2.04034e6 −0.243365
\(589\) −3.43395e6 −0.407855
\(590\) −6.95376e6 −0.822412
\(591\) −1.81165e6 −0.213356
\(592\) −2.16525e6 −0.253924
\(593\) 2.45349e6 0.286515 0.143258 0.989685i \(-0.454242\pi\)
0.143258 + 0.989685i \(0.454242\pi\)
\(594\) 174960. 0.0203457
\(595\) 4.80902e6 0.556884
\(596\) 471264. 0.0543436
\(597\) −5.87203e6 −0.674299
\(598\) −1.57920e6 −0.180586
\(599\) −9.29978e6 −1.05902 −0.529512 0.848302i \(-0.677625\pi\)
−0.529512 + 0.848302i \(0.677625\pi\)
\(600\) −709056. −0.0804086
\(601\) −1.14617e7 −1.29438 −0.647192 0.762327i \(-0.724057\pi\)
−0.647192 + 0.762327i \(0.724057\pi\)
\(602\) 9.37446e6 1.05428
\(603\) −1.36112e6 −0.152442
\(604\) −3.26195e6 −0.363819
\(605\) 1.03918e7 1.15425
\(606\) 792072. 0.0876159
\(607\) 1.12784e7 1.24244 0.621219 0.783637i \(-0.286638\pi\)
0.621219 + 0.783637i \(0.286638\pi\)
\(608\) 978944. 0.107399
\(609\) −8.82922e6 −0.964670
\(610\) 8.20776e6 0.893100
\(611\) 1.29494e7 1.40329
\(612\) −536544. −0.0579064
\(613\) 93782.0 0.0100802 0.00504009 0.999987i \(-0.498396\pi\)
0.00504009 + 0.999987i \(0.498396\pi\)
\(614\) 6.76088e6 0.723740
\(615\) 1.14012e7 1.21553
\(616\) −675840. −0.0717616
\(617\) −1.49642e7 −1.58248 −0.791242 0.611504i \(-0.790565\pi\)
−0.791242 + 0.611504i \(0.790565\pi\)
\(618\) 2.85350e6 0.300543
\(619\) −5.06888e6 −0.531723 −0.265861 0.964011i \(-0.585656\pi\)
−0.265861 + 0.964011i \(0.585656\pi\)
\(620\) 3.79315e6 0.396297
\(621\) −437400. −0.0455145
\(622\) −6.00816e6 −0.622681
\(623\) −5.76259e6 −0.594837
\(624\) 1.51603e6 0.155864
\(625\) −1.20971e7 −1.23875
\(626\) 3.24337e6 0.330796
\(627\) 516240. 0.0524424
\(628\) 2.17827e6 0.220401
\(629\) 3.50161e6 0.352892
\(630\) −3.76358e6 −0.377790
\(631\) 1.55919e7 1.55892 0.779462 0.626450i \(-0.215493\pi\)
0.779462 + 0.626450i \(0.215493\pi\)
\(632\) 4.76211e6 0.474250
\(633\) 1.03235e7 1.02405
\(634\) 3.61423e6 0.357102
\(635\) −1.14180e7 −1.12371
\(636\) 4.50230e6 0.441359
\(637\) −9.32320e6 −0.910367
\(638\) −1.33776e6 −0.130115
\(639\) 495720. 0.0480269
\(640\) −1.08134e6 −0.104355
\(641\) 1.09701e7 1.05455 0.527274 0.849695i \(-0.323214\pi\)
0.527274 + 0.849695i \(0.323214\pi\)
\(642\) −8.20757e6 −0.785917
\(643\) −2.83704e6 −0.270607 −0.135303 0.990804i \(-0.543201\pi\)
−0.135303 + 0.990804i \(0.543201\pi\)
\(644\) 1.68960e6 0.160535
\(645\) 7.90970e6 0.748619
\(646\) −1.58314e6 −0.149258
\(647\) −6.05686e6 −0.568835 −0.284418 0.958700i \(-0.591800\pi\)
−0.284418 + 0.958700i \(0.591800\pi\)
\(648\) 419904. 0.0392837
\(649\) −1.58040e6 −0.147284
\(650\) −3.23999e6 −0.300788
\(651\) 5.68973e6 0.526186
\(652\) −2.73798e6 −0.252239
\(653\) −1.08892e6 −0.0999341 −0.0499671 0.998751i \(-0.515912\pi\)
−0.0499671 + 0.998751i \(0.515912\pi\)
\(654\) 307080. 0.0280742
\(655\) −9.98316e6 −0.909211
\(656\) 4.91366e6 0.445806
\(657\) −2.07020e6 −0.187111
\(658\) −1.38547e7 −1.24748
\(659\) 7.41803e6 0.665388 0.332694 0.943035i \(-0.392042\pi\)
0.332694 + 0.943035i \(0.392042\pi\)
\(660\) −570240. −0.0509563
\(661\) 767654. 0.0683379 0.0341690 0.999416i \(-0.489122\pi\)
0.0341690 + 0.999416i \(0.489122\pi\)
\(662\) 4.48789e6 0.398013
\(663\) −2.45171e6 −0.216613
\(664\) −413952. −0.0364359
\(665\) −1.11049e7 −0.973779
\(666\) −2.74039e6 −0.239402
\(667\) 3.34440e6 0.291074
\(668\) −1.08192e7 −0.938110
\(669\) −6.31764e6 −0.545745
\(670\) 4.43626e6 0.381794
\(671\) 1.86540e6 0.159943
\(672\) −1.62202e6 −0.138558
\(673\) 1.42263e6 0.121075 0.0605373 0.998166i \(-0.480719\pi\)
0.0605373 + 0.998166i \(0.480719\pi\)
\(674\) −1.10087e7 −0.933439
\(675\) −897399. −0.0758099
\(676\) 986736. 0.0830490
\(677\) −6.16231e6 −0.516739 −0.258370 0.966046i \(-0.583185\pi\)
−0.258370 + 0.966046i \(0.583185\pi\)
\(678\) 7.03577e6 0.587810
\(679\) 2.92304e7 2.43310
\(680\) 1.74874e6 0.145028
\(681\) −1.11250e7 −0.919245
\(682\) 862080. 0.0709719
\(683\) 1.50621e7 1.23548 0.617739 0.786383i \(-0.288049\pi\)
0.617739 + 0.786383i \(0.288049\pi\)
\(684\) 1.23898e6 0.101256
\(685\) 8.47796e6 0.690343
\(686\) −1.85715e6 −0.150674
\(687\) −952470. −0.0769945
\(688\) 3.40890e6 0.274563
\(689\) 2.05730e7 1.65101
\(690\) 1.42560e6 0.113992
\(691\) −5.87636e6 −0.468180 −0.234090 0.972215i \(-0.575211\pi\)
−0.234090 + 0.972215i \(0.575211\pi\)
\(692\) 2.13053e6 0.169131
\(693\) −855360. −0.0676575
\(694\) 7.66997e6 0.604498
\(695\) −1.01769e7 −0.799199
\(696\) −3.21062e6 −0.251227
\(697\) −7.94632e6 −0.619561
\(698\) 7.34636e6 0.570734
\(699\) 3.94810e6 0.305630
\(700\) 3.46650e6 0.267390
\(701\) 3.60077e6 0.276758 0.138379 0.990379i \(-0.455811\pi\)
0.138379 + 0.990379i \(0.455811\pi\)
\(702\) 1.91873e6 0.146950
\(703\) −8.08585e6 −0.617074
\(704\) −245760. −0.0186887
\(705\) −1.16899e7 −0.885806
\(706\) −2.48806e6 −0.187866
\(707\) −3.87235e6 −0.291358
\(708\) −3.79296e6 −0.284377
\(709\) 9.22516e6 0.689221 0.344610 0.938746i \(-0.388011\pi\)
0.344610 + 0.938746i \(0.388011\pi\)
\(710\) −1.61568e6 −0.120284
\(711\) 6.02705e6 0.447127
\(712\) −2.09549e6 −0.154912
\(713\) −2.15520e6 −0.158768
\(714\) 2.62310e6 0.192562
\(715\) −2.60568e6 −0.190615
\(716\) −1.10943e7 −0.808758
\(717\) −256176. −0.0186098
\(718\) 1.49625e7 1.08316
\(719\) −2.63923e7 −1.90395 −0.951975 0.306177i \(-0.900950\pi\)
−0.951975 + 0.306177i \(0.900950\pi\)
\(720\) −1.36858e6 −0.0983870
\(721\) −1.39505e7 −0.999426
\(722\) −6.24865e6 −0.446111
\(723\) −8.03306e6 −0.571525
\(724\) 6.03478e6 0.427873
\(725\) 6.86159e6 0.484819
\(726\) 5.66824e6 0.399123
\(727\) −9.79485e6 −0.687324 −0.343662 0.939093i \(-0.611667\pi\)
−0.343662 + 0.939093i \(0.611667\pi\)
\(728\) −7.41171e6 −0.518311
\(729\) 531441. 0.0370370
\(730\) 6.74731e6 0.468623
\(731\) −5.51282e6 −0.381576
\(732\) 4.47696e6 0.308820
\(733\) 4.07584e6 0.280193 0.140096 0.990138i \(-0.455259\pi\)
0.140096 + 0.990138i \(0.455259\pi\)
\(734\) 64928.0 0.00444828
\(735\) 8.41639e6 0.574655
\(736\) 614400. 0.0418077
\(737\) 1.00824e6 0.0683747
\(738\) 6.21886e6 0.420310
\(739\) −1.65709e7 −1.11618 −0.558089 0.829781i \(-0.688465\pi\)
−0.558089 + 0.829781i \(0.688465\pi\)
\(740\) 8.93165e6 0.599587
\(741\) 5.66143e6 0.378775
\(742\) −2.20113e7 −1.46769
\(743\) 1.44141e7 0.957892 0.478946 0.877844i \(-0.341019\pi\)
0.478946 + 0.877844i \(0.341019\pi\)
\(744\) 2.06899e6 0.137033
\(745\) −1.94396e6 −0.128321
\(746\) 1.17442e6 0.0772641
\(747\) −523908. −0.0343521
\(748\) 397440. 0.0259727
\(749\) 4.01259e7 2.61349
\(750\) −4.50014e6 −0.292128
\(751\) 1.67944e7 1.08659 0.543295 0.839542i \(-0.317177\pi\)
0.543295 + 0.839542i \(0.317177\pi\)
\(752\) −5.03808e6 −0.324878
\(753\) 991116. 0.0636997
\(754\) −1.46708e7 −0.939776
\(755\) 1.34556e7 0.859081
\(756\) −2.05286e6 −0.130634
\(757\) 1.32943e7 0.843188 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(758\) 1.27205e7 0.804137
\(759\) 324000. 0.0204146
\(760\) −4.03814e6 −0.253599
\(761\) −2.14786e6 −0.134445 −0.0672225 0.997738i \(-0.521414\pi\)
−0.0672225 + 0.997738i \(0.521414\pi\)
\(762\) −6.22800e6 −0.388563
\(763\) −1.50128e6 −0.0933577
\(764\) −4.24550e6 −0.263145
\(765\) 2.21324e6 0.136734
\(766\) −1.19194e7 −0.733975
\(767\) −1.73317e7 −1.06378
\(768\) −589824. −0.0360844
\(769\) −1.31059e7 −0.799193 −0.399596 0.916691i \(-0.630850\pi\)
−0.399596 + 0.916691i \(0.630850\pi\)
\(770\) 2.78784e6 0.169450
\(771\) −1.26722e6 −0.0767742
\(772\) 4.72477e6 0.285323
\(773\) −2.37154e7 −1.42752 −0.713759 0.700392i \(-0.753009\pi\)
−0.713759 + 0.700392i \(0.753009\pi\)
\(774\) 4.31438e6 0.258861
\(775\) −4.42175e6 −0.264448
\(776\) 1.06292e7 0.633648
\(777\) 1.33975e7 0.796105
\(778\) 1.38391e7 0.819707
\(779\) 1.83495e7 1.08338
\(780\) −6.25363e6 −0.368041
\(781\) −367200. −0.0215415
\(782\) −993600. −0.0581025
\(783\) −4.06345e6 −0.236859
\(784\) 3.62726e6 0.210760
\(785\) −8.98537e6 −0.520430
\(786\) −5.44536e6 −0.314391
\(787\) −8.40048e6 −0.483468 −0.241734 0.970343i \(-0.577716\pi\)
−0.241734 + 0.970343i \(0.577716\pi\)
\(788\) 3.22070e6 0.184772
\(789\) 8.44884e6 0.483175
\(790\) −1.96437e7 −1.11984
\(791\) −3.43971e7 −1.95470
\(792\) −311040. −0.0176199
\(793\) 2.04572e7 1.15522
\(794\) −1.56166e7 −0.879097
\(795\) −1.85720e7 −1.04218
\(796\) 1.04392e7 0.583960
\(797\) 5.41023e6 0.301696 0.150848 0.988557i \(-0.451800\pi\)
0.150848 + 0.988557i \(0.451800\pi\)
\(798\) −6.05722e6 −0.336718
\(799\) 8.14752e6 0.451501
\(800\) 1.26054e6 0.0696359
\(801\) −2.65210e6 −0.146052
\(802\) 2.17646e7 1.19485
\(803\) 1.53348e6 0.0839246
\(804\) 2.41978e6 0.132019
\(805\) −6.96960e6 −0.379069
\(806\) 9.45414e6 0.512607
\(807\) 1.00306e7 0.542177
\(808\) −1.40813e6 −0.0758776
\(809\) −2.60777e7 −1.40087 −0.700436 0.713715i \(-0.747011\pi\)
−0.700436 + 0.713715i \(0.747011\pi\)
\(810\) −1.73210e6 −0.0927601
\(811\) 1.90021e7 1.01449 0.507247 0.861800i \(-0.330663\pi\)
0.507247 + 0.861800i \(0.330663\pi\)
\(812\) 1.56964e7 0.835429
\(813\) −5.10934e6 −0.271105
\(814\) 2.02992e6 0.107379
\(815\) 1.12942e7 0.595608
\(816\) 953856. 0.0501484
\(817\) 1.27301e7 0.667231
\(818\) 7.87978e6 0.411748
\(819\) −9.38045e6 −0.488668
\(820\) −2.02689e7 −1.05268
\(821\) −3.10173e7 −1.60600 −0.803001 0.595978i \(-0.796764\pi\)
−0.803001 + 0.595978i \(0.796764\pi\)
\(822\) 4.62434e6 0.238710
\(823\) −1.56290e7 −0.804323 −0.402162 0.915569i \(-0.631741\pi\)
−0.402162 + 0.915569i \(0.631741\pi\)
\(824\) −5.07290e6 −0.260278
\(825\) 664740. 0.0340030
\(826\) 1.85434e7 0.945667
\(827\) 1.58421e7 0.805467 0.402733 0.915317i \(-0.368060\pi\)
0.402733 + 0.915317i \(0.368060\pi\)
\(828\) 777600. 0.0394167
\(829\) 2.06176e6 0.104196 0.0520980 0.998642i \(-0.483409\pi\)
0.0520980 + 0.998642i \(0.483409\pi\)
\(830\) 1.70755e6 0.0860357
\(831\) 1.09193e7 0.548521
\(832\) −2.69517e6 −0.134983
\(833\) −5.86597e6 −0.292905
\(834\) −5.55106e6 −0.276351
\(835\) 4.46292e7 2.21515
\(836\) −917760. −0.0454165
\(837\) 2.61857e6 0.129196
\(838\) 556080. 0.0273544
\(839\) 3.03900e7 1.49048 0.745240 0.666796i \(-0.232335\pi\)
0.745240 + 0.666796i \(0.232335\pi\)
\(840\) 6.69082e6 0.327176
\(841\) 1.05583e7 0.514760
\(842\) 1.73097e7 0.841414
\(843\) −6.18964e6 −0.299983
\(844\) −1.83530e7 −0.886850
\(845\) −4.07029e6 −0.196103
\(846\) −6.37632e6 −0.306298
\(847\) −2.77114e7 −1.32724
\(848\) −8.00410e6 −0.382228
\(849\) 7.47817e6 0.356062
\(850\) −2.03854e6 −0.0967768
\(851\) −5.07480e6 −0.240212
\(852\) −881280. −0.0415925
\(853\) −2.97738e7 −1.40108 −0.700538 0.713615i \(-0.747056\pi\)
−0.700538 + 0.713615i \(0.747056\pi\)
\(854\) −2.18874e7 −1.02695
\(855\) −5.11078e6 −0.239096
\(856\) 1.45912e7 0.680624
\(857\) 8.64100e6 0.401894 0.200947 0.979602i \(-0.435598\pi\)
0.200947 + 0.979602i \(0.435598\pi\)
\(858\) −1.42128e6 −0.0659115
\(859\) −3.35663e7 −1.55210 −0.776051 0.630670i \(-0.782780\pi\)
−0.776051 + 0.630670i \(0.782780\pi\)
\(860\) −1.40617e7 −0.648323
\(861\) −3.04033e7 −1.39770
\(862\) −1.11974e7 −0.513276
\(863\) 3.90191e7 1.78341 0.891703 0.452621i \(-0.149511\pi\)
0.891703 + 0.452621i \(0.149511\pi\)
\(864\) −746496. −0.0340207
\(865\) −8.78843e6 −0.399366
\(866\) −2.36097e7 −1.06978
\(867\) 1.12361e7 0.507656
\(868\) −1.01151e7 −0.455690
\(869\) −4.46448e6 −0.200549
\(870\) 1.32438e7 0.593219
\(871\) 1.10570e7 0.493848
\(872\) −545920. −0.0243130
\(873\) 1.34526e7 0.597409
\(874\) 2.29440e6 0.101599
\(875\) 2.20007e7 0.971441
\(876\) 3.68035e6 0.162043
\(877\) −1.81382e7 −0.796333 −0.398166 0.917313i \(-0.630353\pi\)
−0.398166 + 0.917313i \(0.630353\pi\)
\(878\) −1.78605e6 −0.0781910
\(879\) 1.18136e7 0.515717
\(880\) 1.01376e6 0.0441294
\(881\) 3.05312e7 1.32527 0.662634 0.748943i \(-0.269438\pi\)
0.662634 + 0.748943i \(0.269438\pi\)
\(882\) 4.59076e6 0.198707
\(883\) −4.35533e7 −1.87983 −0.939916 0.341405i \(-0.889097\pi\)
−0.939916 + 0.341405i \(0.889097\pi\)
\(884\) 4.35859e6 0.187593
\(885\) 1.56460e7 0.671497
\(886\) 1.39810e7 0.598348
\(887\) −1.34152e7 −0.572515 −0.286257 0.958153i \(-0.592411\pi\)
−0.286257 + 0.958153i \(0.592411\pi\)
\(888\) 4.87181e6 0.207328
\(889\) 3.04480e7 1.29212
\(890\) 8.64389e6 0.365792
\(891\) −393660. −0.0166122
\(892\) 1.12314e7 0.472629
\(893\) −1.88141e7 −0.789504
\(894\) −1.06034e6 −0.0443714
\(895\) 4.57641e7 1.90971
\(896\) 2.88358e6 0.119995
\(897\) 3.55320e6 0.147448
\(898\) −4.82450e6 −0.199647
\(899\) −2.00218e7 −0.826236
\(900\) 1.59538e6 0.0656533
\(901\) 1.29441e7 0.531203
\(902\) −4.60656e6 −0.188521
\(903\) −2.10925e7 −0.860815
\(904\) −1.25080e7 −0.509059
\(905\) −2.48935e7 −1.01033
\(906\) 7.33939e6 0.297057
\(907\) 3.10816e6 0.125454 0.0627272 0.998031i \(-0.480020\pi\)
0.0627272 + 0.998031i \(0.480020\pi\)
\(908\) 1.97777e7 0.796089
\(909\) −1.78216e6 −0.0715381
\(910\) 3.05733e7 1.22388
\(911\) 1.19035e6 0.0475203 0.0237602 0.999718i \(-0.492436\pi\)
0.0237602 + 0.999718i \(0.492436\pi\)
\(912\) −2.20262e6 −0.0876906
\(913\) 388080. 0.0154079
\(914\) 934184. 0.0369885
\(915\) −1.84675e7 −0.729213
\(916\) 1.69328e6 0.0666792
\(917\) 2.66218e7 1.04547
\(918\) 1.20722e6 0.0472804
\(919\) −4.71996e7 −1.84353 −0.921764 0.387752i \(-0.873252\pi\)
−0.921764 + 0.387752i \(0.873252\pi\)
\(920\) −2.53440e6 −0.0987201
\(921\) −1.52120e7 −0.590931
\(922\) −6.97956e6 −0.270396
\(923\) −4.02696e6 −0.155587
\(924\) 1.52064e6 0.0585931
\(925\) −1.04118e7 −0.400103
\(926\) −1.16715e7 −0.447299
\(927\) −6.42038e6 −0.245393
\(928\) 5.70778e6 0.217569
\(929\) 1.33595e6 0.0507870 0.0253935 0.999678i \(-0.491916\pi\)
0.0253935 + 0.999678i \(0.491916\pi\)
\(930\) −8.53459e6 −0.323575
\(931\) 1.35456e7 0.512180
\(932\) −7.01885e6 −0.264683
\(933\) 1.35184e7 0.508417
\(934\) −2.12430e7 −0.796800
\(935\) −1.63944e6 −0.0613291
\(936\) −3.41107e6 −0.127263
\(937\) 1.47238e7 0.547861 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(938\) −1.18300e7 −0.439014
\(939\) −7.29758e6 −0.270094
\(940\) 2.07821e7 0.767131
\(941\) −2.69196e7 −0.991049 −0.495525 0.868594i \(-0.665024\pi\)
−0.495525 + 0.868594i \(0.665024\pi\)
\(942\) −4.90111e6 −0.179956
\(943\) 1.15164e7 0.421733
\(944\) 6.74304e6 0.246278
\(945\) 8.46806e6 0.308464
\(946\) −3.19584e6 −0.116107
\(947\) −3.73160e6 −0.135214 −0.0676068 0.997712i \(-0.521536\pi\)
−0.0676068 + 0.997712i \(0.521536\pi\)
\(948\) −1.07148e7 −0.387223
\(949\) 1.68172e7 0.606160
\(950\) 4.70734e6 0.169226
\(951\) −8.13202e6 −0.291573
\(952\) −4.66330e6 −0.166763
\(953\) 2.18735e7 0.780166 0.390083 0.920780i \(-0.372446\pi\)
0.390083 + 0.920780i \(0.372446\pi\)
\(954\) −1.01302e7 −0.360368
\(955\) 1.75127e7 0.621362
\(956\) 455424. 0.0161165
\(957\) 3.00996e6 0.106238
\(958\) 9.37862e6 0.330161
\(959\) −2.26079e7 −0.793805
\(960\) 2.43302e6 0.0852056
\(961\) −1.57267e7 −0.549324
\(962\) 2.22615e7 0.775561
\(963\) 1.84670e7 0.641699
\(964\) 1.42810e7 0.494955
\(965\) −1.94897e7 −0.673730
\(966\) −3.80160e6 −0.131076
\(967\) 1.76025e7 0.605352 0.302676 0.953093i \(-0.402120\pi\)
0.302676 + 0.953093i \(0.402120\pi\)
\(968\) −1.00769e7 −0.345650
\(969\) 3.56206e6 0.121868
\(970\) −4.38456e7 −1.49623
\(971\) 1.67317e7 0.569497 0.284749 0.958602i \(-0.408090\pi\)
0.284749 + 0.958602i \(0.408090\pi\)
\(972\) −944784. −0.0320750
\(973\) 2.71385e7 0.918975
\(974\) 3.92612e7 1.32607
\(975\) 7.28998e6 0.245592
\(976\) −7.95904e6 −0.267446
\(977\) 5.55382e7 1.86147 0.930733 0.365699i \(-0.119170\pi\)
0.930733 + 0.365699i \(0.119170\pi\)
\(978\) 6.16046e6 0.205952
\(979\) 1.96452e6 0.0655088
\(980\) −1.49625e7 −0.497666
\(981\) −690930. −0.0229225
\(982\) −2.37808e7 −0.786951
\(983\) −3.86784e7 −1.27669 −0.638344 0.769751i \(-0.720380\pi\)
−0.638344 + 0.769751i \(0.720380\pi\)
\(984\) −1.10557e7 −0.363999
\(985\) −1.32854e7 −0.436299
\(986\) −9.23054e6 −0.302367
\(987\) 3.11731e7 1.01856
\(988\) −1.00648e7 −0.328028
\(989\) 7.98960e6 0.259737
\(990\) 1.28304e6 0.0416056
\(991\) 9.58498e6 0.310033 0.155016 0.987912i \(-0.450457\pi\)
0.155016 + 0.987912i \(0.450457\pi\)
\(992\) −3.67821e6 −0.118674
\(993\) −1.00977e7 −0.324976
\(994\) 4.30848e6 0.138311
\(995\) −4.30616e7 −1.37890
\(996\) 931392. 0.0297498
\(997\) −1.03650e7 −0.330242 −0.165121 0.986273i \(-0.552802\pi\)
−0.165121 + 0.986273i \(0.552802\pi\)
\(998\) 2.59133e7 0.823561
\(999\) 6.16588e6 0.195471
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 6.6.a.a.1.1 1
3.2 odd 2 18.6.a.b.1.1 1
4.3 odd 2 48.6.a.c.1.1 1
5.2 odd 4 150.6.c.b.49.2 2
5.3 odd 4 150.6.c.b.49.1 2
5.4 even 2 150.6.a.d.1.1 1
7.2 even 3 294.6.e.g.67.1 2
7.3 odd 6 294.6.e.a.79.1 2
7.4 even 3 294.6.e.g.79.1 2
7.5 odd 6 294.6.e.a.67.1 2
7.6 odd 2 294.6.a.m.1.1 1
8.3 odd 2 192.6.a.g.1.1 1
8.5 even 2 192.6.a.o.1.1 1
9.2 odd 6 162.6.c.h.109.1 2
9.4 even 3 162.6.c.e.55.1 2
9.5 odd 6 162.6.c.h.55.1 2
9.7 even 3 162.6.c.e.109.1 2
11.10 odd 2 726.6.a.a.1.1 1
12.11 even 2 144.6.a.j.1.1 1
13.12 even 2 1014.6.a.c.1.1 1
15.2 even 4 450.6.c.j.199.1 2
15.8 even 4 450.6.c.j.199.2 2
15.14 odd 2 450.6.a.m.1.1 1
16.3 odd 4 768.6.d.p.385.2 2
16.5 even 4 768.6.d.c.385.2 2
16.11 odd 4 768.6.d.p.385.1 2
16.13 even 4 768.6.d.c.385.1 2
21.20 even 2 882.6.a.a.1.1 1
24.5 odd 2 576.6.a.j.1.1 1
24.11 even 2 576.6.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.6.a.a.1.1 1 1.1 even 1 trivial
18.6.a.b.1.1 1 3.2 odd 2
48.6.a.c.1.1 1 4.3 odd 2
144.6.a.j.1.1 1 12.11 even 2
150.6.a.d.1.1 1 5.4 even 2
150.6.c.b.49.1 2 5.3 odd 4
150.6.c.b.49.2 2 5.2 odd 4
162.6.c.e.55.1 2 9.4 even 3
162.6.c.e.109.1 2 9.7 even 3
162.6.c.h.55.1 2 9.5 odd 6
162.6.c.h.109.1 2 9.2 odd 6
192.6.a.g.1.1 1 8.3 odd 2
192.6.a.o.1.1 1 8.5 even 2
294.6.a.m.1.1 1 7.6 odd 2
294.6.e.a.67.1 2 7.5 odd 6
294.6.e.a.79.1 2 7.3 odd 6
294.6.e.g.67.1 2 7.2 even 3
294.6.e.g.79.1 2 7.4 even 3
450.6.a.m.1.1 1 15.14 odd 2
450.6.c.j.199.1 2 15.2 even 4
450.6.c.j.199.2 2 15.8 even 4
576.6.a.i.1.1 1 24.11 even 2
576.6.a.j.1.1 1 24.5 odd 2
726.6.a.a.1.1 1 11.10 odd 2
768.6.d.c.385.1 2 16.13 even 4
768.6.d.c.385.2 2 16.5 even 4
768.6.d.p.385.1 2 16.11 odd 4
768.6.d.p.385.2 2 16.3 odd 4
882.6.a.a.1.1 1 21.20 even 2
1014.6.a.c.1.1 1 13.12 even 2