Properties

Label 18.6.a.b.1.1
Level $18$
Weight $6$
Character 18.1
Self dual yes
Analytic conductor $2.887$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +66.0000 q^{5} +176.000 q^{7} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} +66.0000 q^{5} +176.000 q^{7} -64.0000 q^{8} -264.000 q^{10} +60.0000 q^{11} -658.000 q^{13} -704.000 q^{14} +256.000 q^{16} +414.000 q^{17} +956.000 q^{19} +1056.00 q^{20} -240.000 q^{22} -600.000 q^{23} +1231.00 q^{25} +2632.00 q^{26} +2816.00 q^{28} -5574.00 q^{29} -3592.00 q^{31} -1024.00 q^{32} -1656.00 q^{34} +11616.0 q^{35} -8458.00 q^{37} -3824.00 q^{38} -4224.00 q^{40} -19194.0 q^{41} +13316.0 q^{43} +960.000 q^{44} +2400.00 q^{46} +19680.0 q^{47} +14169.0 q^{49} -4924.00 q^{50} -10528.0 q^{52} +31266.0 q^{53} +3960.00 q^{55} -11264.0 q^{56} +22296.0 q^{58} -26340.0 q^{59} -31090.0 q^{61} +14368.0 q^{62} +4096.00 q^{64} -43428.0 q^{65} -16804.0 q^{67} +6624.00 q^{68} -46464.0 q^{70} -6120.00 q^{71} -25558.0 q^{73} +33832.0 q^{74} +15296.0 q^{76} +10560.0 q^{77} +74408.0 q^{79} +16896.0 q^{80} +76776.0 q^{82} +6468.00 q^{83} +27324.0 q^{85} -53264.0 q^{86} -3840.00 q^{88} +32742.0 q^{89} -115808. q^{91} -9600.00 q^{92} -78720.0 q^{94} +63096.0 q^{95} +166082. q^{97} -56676.0 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 66.0000 1.18064 0.590322 0.807168i \(-0.299001\pi\)
0.590322 + 0.807168i \(0.299001\pi\)
\(6\) 0 0
\(7\) 176.000 1.35759 0.678793 0.734329i \(-0.262503\pi\)
0.678793 + 0.734329i \(0.262503\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) −264.000 −0.834841
\(11\) 60.0000 0.149510 0.0747549 0.997202i \(-0.476183\pi\)
0.0747549 + 0.997202i \(0.476183\pi\)
\(12\) 0 0
\(13\) −658.000 −1.07986 −0.539930 0.841710i \(-0.681549\pi\)
−0.539930 + 0.841710i \(0.681549\pi\)
\(14\) −704.000 −0.959959
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 414.000 0.347439 0.173719 0.984795i \(-0.444421\pi\)
0.173719 + 0.984795i \(0.444421\pi\)
\(18\) 0 0
\(19\) 956.000 0.607539 0.303769 0.952746i \(-0.401755\pi\)
0.303769 + 0.952746i \(0.401755\pi\)
\(20\) 1056.00 0.590322
\(21\) 0 0
\(22\) −240.000 −0.105719
\(23\) −600.000 −0.236500 −0.118250 0.992984i \(-0.537728\pi\)
−0.118250 + 0.992984i \(0.537728\pi\)
\(24\) 0 0
\(25\) 1231.00 0.393920
\(26\) 2632.00 0.763576
\(27\) 0 0
\(28\) 2816.00 0.678793
\(29\) −5574.00 −1.23076 −0.615378 0.788232i \(-0.710997\pi\)
−0.615378 + 0.788232i \(0.710997\pi\)
\(30\) 0 0
\(31\) −3592.00 −0.671324 −0.335662 0.941983i \(-0.608960\pi\)
−0.335662 + 0.941983i \(0.608960\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) −1656.00 −0.245676
\(35\) 11616.0 1.60283
\(36\) 0 0
\(37\) −8458.00 −1.01570 −0.507848 0.861447i \(-0.669559\pi\)
−0.507848 + 0.861447i \(0.669559\pi\)
\(38\) −3824.00 −0.429595
\(39\) 0 0
\(40\) −4224.00 −0.417421
\(41\) −19194.0 −1.78322 −0.891612 0.452800i \(-0.850425\pi\)
−0.891612 + 0.452800i \(0.850425\pi\)
\(42\) 0 0
\(43\) 13316.0 1.09825 0.549127 0.835739i \(-0.314960\pi\)
0.549127 + 0.835739i \(0.314960\pi\)
\(44\) 960.000 0.0747549
\(45\) 0 0
\(46\) 2400.00 0.167231
\(47\) 19680.0 1.29951 0.649756 0.760143i \(-0.274871\pi\)
0.649756 + 0.760143i \(0.274871\pi\)
\(48\) 0 0
\(49\) 14169.0 0.843042
\(50\) −4924.00 −0.278544
\(51\) 0 0
\(52\) −10528.0 −0.539930
\(53\) 31266.0 1.52891 0.764456 0.644676i \(-0.223008\pi\)
0.764456 + 0.644676i \(0.223008\pi\)
\(54\) 0 0
\(55\) 3960.00 0.176518
\(56\) −11264.0 −0.479979
\(57\) 0 0
\(58\) 22296.0 0.870276
\(59\) −26340.0 −0.985112 −0.492556 0.870281i \(-0.663937\pi\)
−0.492556 + 0.870281i \(0.663937\pi\)
\(60\) 0 0
\(61\) −31090.0 −1.06978 −0.534892 0.844920i \(-0.679648\pi\)
−0.534892 + 0.844920i \(0.679648\pi\)
\(62\) 14368.0 0.474698
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −43428.0 −1.27493
\(66\) 0 0
\(67\) −16804.0 −0.457326 −0.228663 0.973506i \(-0.573435\pi\)
−0.228663 + 0.973506i \(0.573435\pi\)
\(68\) 6624.00 0.173719
\(69\) 0 0
\(70\) −46464.0 −1.13337
\(71\) −6120.00 −0.144081 −0.0720403 0.997402i \(-0.522951\pi\)
−0.0720403 + 0.997402i \(0.522951\pi\)
\(72\) 0 0
\(73\) −25558.0 −0.561332 −0.280666 0.959806i \(-0.590555\pi\)
−0.280666 + 0.959806i \(0.590555\pi\)
\(74\) 33832.0 0.718205
\(75\) 0 0
\(76\) 15296.0 0.303769
\(77\) 10560.0 0.202972
\(78\) 0 0
\(79\) 74408.0 1.34138 0.670690 0.741738i \(-0.265998\pi\)
0.670690 + 0.741738i \(0.265998\pi\)
\(80\) 16896.0 0.295161
\(81\) 0 0
\(82\) 76776.0 1.26093
\(83\) 6468.00 0.103056 0.0515282 0.998672i \(-0.483591\pi\)
0.0515282 + 0.998672i \(0.483591\pi\)
\(84\) 0 0
\(85\) 27324.0 0.410201
\(86\) −53264.0 −0.776583
\(87\) 0 0
\(88\) −3840.00 −0.0528597
\(89\) 32742.0 0.438157 0.219079 0.975707i \(-0.429695\pi\)
0.219079 + 0.975707i \(0.429695\pi\)
\(90\) 0 0
\(91\) −115808. −1.46600
\(92\) −9600.00 −0.118250
\(93\) 0 0
\(94\) −78720.0 −0.918894
\(95\) 63096.0 0.717287
\(96\) 0 0
\(97\) 166082. 1.79223 0.896114 0.443824i \(-0.146378\pi\)
0.896114 + 0.443824i \(0.146378\pi\)
\(98\) −56676.0 −0.596120
\(99\) 0 0
\(100\) 19696.0 0.196960
\(101\) 22002.0 0.214614 0.107307 0.994226i \(-0.465777\pi\)
0.107307 + 0.994226i \(0.465777\pi\)
\(102\) 0 0
\(103\) −79264.0 −0.736178 −0.368089 0.929791i \(-0.619988\pi\)
−0.368089 + 0.929791i \(0.619988\pi\)
\(104\) 42112.0 0.381788
\(105\) 0 0
\(106\) −125064. −1.08110
\(107\) −227988. −1.92510 −0.962548 0.271110i \(-0.912609\pi\)
−0.962548 + 0.271110i \(0.912609\pi\)
\(108\) 0 0
\(109\) −8530.00 −0.0687674 −0.0343837 0.999409i \(-0.510947\pi\)
−0.0343837 + 0.999409i \(0.510947\pi\)
\(110\) −15840.0 −0.124817
\(111\) 0 0
\(112\) 45056.0 0.339397
\(113\) 195438. 1.43984 0.719918 0.694059i \(-0.244179\pi\)
0.719918 + 0.694059i \(0.244179\pi\)
\(114\) 0 0
\(115\) −39600.0 −0.279223
\(116\) −89184.0 −0.615378
\(117\) 0 0
\(118\) 105360. 0.696580
\(119\) 72864.0 0.471678
\(120\) 0 0
\(121\) −157451. −0.977647
\(122\) 124360. 0.756452
\(123\) 0 0
\(124\) −57472.0 −0.335662
\(125\) −125004. −0.715565
\(126\) 0 0
\(127\) 173000. 0.951780 0.475890 0.879505i \(-0.342126\pi\)
0.475890 + 0.879505i \(0.342126\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) 173712. 0.901512
\(131\) −151260. −0.770098 −0.385049 0.922896i \(-0.625815\pi\)
−0.385049 + 0.922896i \(0.625815\pi\)
\(132\) 0 0
\(133\) 168256. 0.824786
\(134\) 67216.0 0.323378
\(135\) 0 0
\(136\) −26496.0 −0.122838
\(137\) 128454. 0.584718 0.292359 0.956309i \(-0.405560\pi\)
0.292359 + 0.956309i \(0.405560\pi\)
\(138\) 0 0
\(139\) 154196. 0.676918 0.338459 0.940981i \(-0.390094\pi\)
0.338459 + 0.940981i \(0.390094\pi\)
\(140\) 185856. 0.801413
\(141\) 0 0
\(142\) 24480.0 0.101880
\(143\) −39480.0 −0.161450
\(144\) 0 0
\(145\) −367884. −1.45308
\(146\) 102232. 0.396922
\(147\) 0 0
\(148\) −135328. −0.507848
\(149\) −29454.0 −0.108687 −0.0543436 0.998522i \(-0.517307\pi\)
−0.0543436 + 0.998522i \(0.517307\pi\)
\(150\) 0 0
\(151\) −203872. −0.727638 −0.363819 0.931470i \(-0.618527\pi\)
−0.363819 + 0.931470i \(0.618527\pi\)
\(152\) −61184.0 −0.214797
\(153\) 0 0
\(154\) −42240.0 −0.143523
\(155\) −237072. −0.792594
\(156\) 0 0
\(157\) 136142. 0.440801 0.220401 0.975409i \(-0.429263\pi\)
0.220401 + 0.975409i \(0.429263\pi\)
\(158\) −297632. −0.948499
\(159\) 0 0
\(160\) −67584.0 −0.208710
\(161\) −105600. −0.321070
\(162\) 0 0
\(163\) −171124. −0.504478 −0.252239 0.967665i \(-0.581167\pi\)
−0.252239 + 0.967665i \(0.581167\pi\)
\(164\) −307104. −0.891612
\(165\) 0 0
\(166\) −25872.0 −0.0728718
\(167\) 676200. 1.87622 0.938110 0.346336i \(-0.112574\pi\)
0.938110 + 0.346336i \(0.112574\pi\)
\(168\) 0 0
\(169\) 61671.0 0.166098
\(170\) −109296. −0.290056
\(171\) 0 0
\(172\) 213056. 0.549127
\(173\) −133158. −0.338261 −0.169131 0.985594i \(-0.554096\pi\)
−0.169131 + 0.985594i \(0.554096\pi\)
\(174\) 0 0
\(175\) 216656. 0.534781
\(176\) 15360.0 0.0373774
\(177\) 0 0
\(178\) −130968. −0.309824
\(179\) 693396. 1.61752 0.808758 0.588141i \(-0.200140\pi\)
0.808758 + 0.588141i \(0.200140\pi\)
\(180\) 0 0
\(181\) 377174. 0.855747 0.427873 0.903839i \(-0.359263\pi\)
0.427873 + 0.903839i \(0.359263\pi\)
\(182\) 463232. 1.03662
\(183\) 0 0
\(184\) 38400.0 0.0836155
\(185\) −558228. −1.19917
\(186\) 0 0
\(187\) 24840.0 0.0519455
\(188\) 314880. 0.649756
\(189\) 0 0
\(190\) −252384. −0.507198
\(191\) 265344. 0.526291 0.263145 0.964756i \(-0.415240\pi\)
0.263145 + 0.964756i \(0.415240\pi\)
\(192\) 0 0
\(193\) 295298. 0.570647 0.285323 0.958431i \(-0.407899\pi\)
0.285323 + 0.958431i \(0.407899\pi\)
\(194\) −664328. −1.26730
\(195\) 0 0
\(196\) 226704. 0.421521
\(197\) −201294. −0.369543 −0.184772 0.982781i \(-0.559155\pi\)
−0.184772 + 0.982781i \(0.559155\pi\)
\(198\) 0 0
\(199\) 652448. 1.16792 0.583960 0.811782i \(-0.301502\pi\)
0.583960 + 0.811782i \(0.301502\pi\)
\(200\) −78784.0 −0.139272
\(201\) 0 0
\(202\) −88008.0 −0.151755
\(203\) −981024. −1.67086
\(204\) 0 0
\(205\) −1.26680e6 −2.10535
\(206\) 317056. 0.520557
\(207\) 0 0
\(208\) −168448. −0.269965
\(209\) 57360.0 0.0908330
\(210\) 0 0
\(211\) −1.14706e6 −1.77370 −0.886850 0.462058i \(-0.847111\pi\)
−0.886850 + 0.462058i \(0.847111\pi\)
\(212\) 500256. 0.764456
\(213\) 0 0
\(214\) 911952. 1.36125
\(215\) 878856. 1.29665
\(216\) 0 0
\(217\) −632192. −0.911380
\(218\) 34120.0 0.0486259
\(219\) 0 0
\(220\) 63360.0 0.0882589
\(221\) −272412. −0.375185
\(222\) 0 0
\(223\) 701960. 0.945258 0.472629 0.881262i \(-0.343305\pi\)
0.472629 + 0.881262i \(0.343305\pi\)
\(224\) −180224. −0.239990
\(225\) 0 0
\(226\) −781752. −1.01812
\(227\) −1.23611e6 −1.59218 −0.796089 0.605179i \(-0.793101\pi\)
−0.796089 + 0.605179i \(0.793101\pi\)
\(228\) 0 0
\(229\) 105830. 0.133358 0.0666792 0.997774i \(-0.478760\pi\)
0.0666792 + 0.997774i \(0.478760\pi\)
\(230\) 158400. 0.197440
\(231\) 0 0
\(232\) 356736. 0.435138
\(233\) 438678. 0.529366 0.264683 0.964335i \(-0.414733\pi\)
0.264683 + 0.964335i \(0.414733\pi\)
\(234\) 0 0
\(235\) 1.29888e6 1.53426
\(236\) −421440. −0.492556
\(237\) 0 0
\(238\) −291456. −0.333527
\(239\) −28464.0 −0.0322330 −0.0161165 0.999870i \(-0.505130\pi\)
−0.0161165 + 0.999870i \(0.505130\pi\)
\(240\) 0 0
\(241\) 892562. 0.989910 0.494955 0.868919i \(-0.335185\pi\)
0.494955 + 0.868919i \(0.335185\pi\)
\(242\) 629804. 0.691301
\(243\) 0 0
\(244\) −497440. −0.534892
\(245\) 935154. 0.995332
\(246\) 0 0
\(247\) −629048. −0.656057
\(248\) 229888. 0.237349
\(249\) 0 0
\(250\) 500016. 0.505981
\(251\) 110124. 0.110331 0.0551655 0.998477i \(-0.482431\pi\)
0.0551655 + 0.998477i \(0.482431\pi\)
\(252\) 0 0
\(253\) −36000.0 −0.0353591
\(254\) −692000. −0.673010
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −140802. −0.132977 −0.0664884 0.997787i \(-0.521180\pi\)
−0.0664884 + 0.997787i \(0.521180\pi\)
\(258\) 0 0
\(259\) −1.48861e6 −1.37889
\(260\) −694848. −0.637465
\(261\) 0 0
\(262\) 605040. 0.544541
\(263\) 938760. 0.836884 0.418442 0.908244i \(-0.362576\pi\)
0.418442 + 0.908244i \(0.362576\pi\)
\(264\) 0 0
\(265\) 2.06356e6 1.80510
\(266\) −673024. −0.583212
\(267\) 0 0
\(268\) −268864. −0.228663
\(269\) 1.11451e6 0.939078 0.469539 0.882912i \(-0.344420\pi\)
0.469539 + 0.882912i \(0.344420\pi\)
\(270\) 0 0
\(271\) 567704. 0.469568 0.234784 0.972048i \(-0.424562\pi\)
0.234784 + 0.972048i \(0.424562\pi\)
\(272\) 105984. 0.0868596
\(273\) 0 0
\(274\) −513816. −0.413458
\(275\) 73860.0 0.0588949
\(276\) 0 0
\(277\) −1.21326e6 −0.950066 −0.475033 0.879968i \(-0.657564\pi\)
−0.475033 + 0.879968i \(0.657564\pi\)
\(278\) −616784. −0.478653
\(279\) 0 0
\(280\) −743424. −0.566685
\(281\) −687738. −0.519586 −0.259793 0.965664i \(-0.583654\pi\)
−0.259793 + 0.965664i \(0.583654\pi\)
\(282\) 0 0
\(283\) −830908. −0.616718 −0.308359 0.951270i \(-0.599780\pi\)
−0.308359 + 0.951270i \(0.599780\pi\)
\(284\) −97920.0 −0.0720403
\(285\) 0 0
\(286\) 157920. 0.114162
\(287\) −3.37814e6 −2.42088
\(288\) 0 0
\(289\) −1.24846e6 −0.879286
\(290\) 1.47154e6 1.02749
\(291\) 0 0
\(292\) −408928. −0.280666
\(293\) 1.31263e6 0.893248 0.446624 0.894722i \(-0.352626\pi\)
0.446624 + 0.894722i \(0.352626\pi\)
\(294\) 0 0
\(295\) −1.73844e6 −1.16307
\(296\) 541312. 0.359102
\(297\) 0 0
\(298\) 117816. 0.0768535
\(299\) 394800. 0.255387
\(300\) 0 0
\(301\) 2.34362e6 1.49097
\(302\) 815488. 0.514518
\(303\) 0 0
\(304\) 244736. 0.151885
\(305\) −2.05194e6 −1.26303
\(306\) 0 0
\(307\) 1.69022e6 1.02352 0.511761 0.859128i \(-0.328993\pi\)
0.511761 + 0.859128i \(0.328993\pi\)
\(308\) 168960. 0.101486
\(309\) 0 0
\(310\) 948288. 0.560449
\(311\) 1.50204e6 0.880604 0.440302 0.897850i \(-0.354871\pi\)
0.440302 + 0.897850i \(0.354871\pi\)
\(312\) 0 0
\(313\) 810842. 0.467816 0.233908 0.972259i \(-0.424848\pi\)
0.233908 + 0.972259i \(0.424848\pi\)
\(314\) −544568. −0.311694
\(315\) 0 0
\(316\) 1.19053e6 0.670690
\(317\) −903558. −0.505019 −0.252510 0.967594i \(-0.581256\pi\)
−0.252510 + 0.967594i \(0.581256\pi\)
\(318\) 0 0
\(319\) −334440. −0.184010
\(320\) 270336. 0.147580
\(321\) 0 0
\(322\) 422400. 0.227031
\(323\) 395784. 0.211082
\(324\) 0 0
\(325\) −809998. −0.425379
\(326\) 684496. 0.356720
\(327\) 0 0
\(328\) 1.22842e6 0.630465
\(329\) 3.46368e6 1.76420
\(330\) 0 0
\(331\) 1.12197e6 0.562875 0.281438 0.959580i \(-0.409189\pi\)
0.281438 + 0.959580i \(0.409189\pi\)
\(332\) 103488. 0.0515282
\(333\) 0 0
\(334\) −2.70480e6 −1.32669
\(335\) −1.10906e6 −0.539939
\(336\) 0 0
\(337\) −2.75217e6 −1.32008 −0.660041 0.751229i \(-0.729461\pi\)
−0.660041 + 0.751229i \(0.729461\pi\)
\(338\) −246684. −0.117449
\(339\) 0 0
\(340\) 437184. 0.205101
\(341\) −215520. −0.100369
\(342\) 0 0
\(343\) −464288. −0.213085
\(344\) −852224. −0.388291
\(345\) 0 0
\(346\) 532632. 0.239187
\(347\) −1.91749e6 −0.854889 −0.427445 0.904042i \(-0.640586\pi\)
−0.427445 + 0.904042i \(0.640586\pi\)
\(348\) 0 0
\(349\) 1.83659e6 0.807140 0.403570 0.914949i \(-0.367769\pi\)
0.403570 + 0.914949i \(0.367769\pi\)
\(350\) −866624. −0.378147
\(351\) 0 0
\(352\) −61440.0 −0.0264298
\(353\) 622014. 0.265683 0.132841 0.991137i \(-0.457590\pi\)
0.132841 + 0.991137i \(0.457590\pi\)
\(354\) 0 0
\(355\) −403920. −0.170108
\(356\) 523872. 0.219079
\(357\) 0 0
\(358\) −2.77358e6 −1.14376
\(359\) −3.74062e6 −1.53182 −0.765909 0.642949i \(-0.777711\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(360\) 0 0
\(361\) −1.56216e6 −0.630897
\(362\) −1.50870e6 −0.605104
\(363\) 0 0
\(364\) −1.85293e6 −0.733002
\(365\) −1.68683e6 −0.662733
\(366\) 0 0
\(367\) 16232.0 0.00629081 0.00314541 0.999995i \(-0.498999\pi\)
0.00314541 + 0.999995i \(0.498999\pi\)
\(368\) −153600. −0.0591251
\(369\) 0 0
\(370\) 2.23291e6 0.847944
\(371\) 5.50282e6 2.07563
\(372\) 0 0
\(373\) 293606. 0.109268 0.0546340 0.998506i \(-0.482601\pi\)
0.0546340 + 0.998506i \(0.482601\pi\)
\(374\) −99360.0 −0.0367310
\(375\) 0 0
\(376\) −1.25952e6 −0.459447
\(377\) 3.66769e6 1.32904
\(378\) 0 0
\(379\) 3.18012e6 1.13722 0.568611 0.822607i \(-0.307481\pi\)
0.568611 + 0.822607i \(0.307481\pi\)
\(380\) 1.00954e6 0.358643
\(381\) 0 0
\(382\) −1.06138e6 −0.372144
\(383\) 2.97984e6 1.03800 0.518998 0.854775i \(-0.326305\pi\)
0.518998 + 0.854775i \(0.326305\pi\)
\(384\) 0 0
\(385\) 696960. 0.239638
\(386\) −1.18119e6 −0.403508
\(387\) 0 0
\(388\) 2.65731e6 0.896114
\(389\) −3.45977e6 −1.15924 −0.579620 0.814887i \(-0.696799\pi\)
−0.579620 + 0.814887i \(0.696799\pi\)
\(390\) 0 0
\(391\) −248400. −0.0821693
\(392\) −906816. −0.298060
\(393\) 0 0
\(394\) 805176. 0.261307
\(395\) 4.91093e6 1.58369
\(396\) 0 0
\(397\) −3.90416e6 −1.24323 −0.621615 0.783323i \(-0.713523\pi\)
−0.621615 + 0.783323i \(0.713523\pi\)
\(398\) −2.60979e6 −0.825844
\(399\) 0 0
\(400\) 315136. 0.0984800
\(401\) −5.44115e6 −1.68978 −0.844890 0.534940i \(-0.820334\pi\)
−0.844890 + 0.534940i \(0.820334\pi\)
\(402\) 0 0
\(403\) 2.36354e6 0.724936
\(404\) 352032. 0.107307
\(405\) 0 0
\(406\) 3.92410e6 1.18148
\(407\) −507480. −0.151856
\(408\) 0 0
\(409\) 1.96995e6 0.582299 0.291150 0.956678i \(-0.405962\pi\)
0.291150 + 0.956678i \(0.405962\pi\)
\(410\) 5.06722e6 1.48871
\(411\) 0 0
\(412\) −1.26822e6 −0.368089
\(413\) −4.63584e6 −1.33738
\(414\) 0 0
\(415\) 426888. 0.121673
\(416\) 673792. 0.190894
\(417\) 0 0
\(418\) −229440. −0.0642286
\(419\) −139020. −0.0386850 −0.0193425 0.999813i \(-0.506157\pi\)
−0.0193425 + 0.999813i \(0.506157\pi\)
\(420\) 0 0
\(421\) 4.32743e6 1.18994 0.594970 0.803748i \(-0.297164\pi\)
0.594970 + 0.803748i \(0.297164\pi\)
\(422\) 4.58824e6 1.25419
\(423\) 0 0
\(424\) −2.00102e6 −0.540552
\(425\) 509634. 0.136863
\(426\) 0 0
\(427\) −5.47184e6 −1.45232
\(428\) −3.64781e6 −0.962548
\(429\) 0 0
\(430\) −3.51542e6 −0.916867
\(431\) 2.79936e6 0.725881 0.362941 0.931812i \(-0.381773\pi\)
0.362941 + 0.931812i \(0.381773\pi\)
\(432\) 0 0
\(433\) −5.90241e6 −1.51290 −0.756449 0.654052i \(-0.773068\pi\)
−0.756449 + 0.654052i \(0.773068\pi\)
\(434\) 2.52877e6 0.644443
\(435\) 0 0
\(436\) −136480. −0.0343837
\(437\) −573600. −0.143683
\(438\) 0 0
\(439\) −446512. −0.110579 −0.0552894 0.998470i \(-0.517608\pi\)
−0.0552894 + 0.998470i \(0.517608\pi\)
\(440\) −253440. −0.0624085
\(441\) 0 0
\(442\) 1.08965e6 0.265296
\(443\) −3.49525e6 −0.846193 −0.423096 0.906085i \(-0.639057\pi\)
−0.423096 + 0.906085i \(0.639057\pi\)
\(444\) 0 0
\(445\) 2.16097e6 0.517308
\(446\) −2.80784e6 −0.668398
\(447\) 0 0
\(448\) 720896. 0.169698
\(449\) 1.20613e6 0.282343 0.141171 0.989985i \(-0.454913\pi\)
0.141171 + 0.989985i \(0.454913\pi\)
\(450\) 0 0
\(451\) −1.15164e6 −0.266609
\(452\) 3.12701e6 0.719918
\(453\) 0 0
\(454\) 4.94443e6 1.12584
\(455\) −7.64333e6 −1.73083
\(456\) 0 0
\(457\) 233546. 0.0523097 0.0261548 0.999658i \(-0.491674\pi\)
0.0261548 + 0.999658i \(0.491674\pi\)
\(458\) −423320. −0.0942986
\(459\) 0 0
\(460\) −633600. −0.139611
\(461\) 1.74489e6 0.382398 0.191199 0.981551i \(-0.438762\pi\)
0.191199 + 0.981551i \(0.438762\pi\)
\(462\) 0 0
\(463\) −2.91786e6 −0.632576 −0.316288 0.948663i \(-0.602437\pi\)
−0.316288 + 0.948663i \(0.602437\pi\)
\(464\) −1.42694e6 −0.307689
\(465\) 0 0
\(466\) −1.75471e6 −0.374318
\(467\) 5.31076e6 1.12684 0.563422 0.826169i \(-0.309484\pi\)
0.563422 + 0.826169i \(0.309484\pi\)
\(468\) 0 0
\(469\) −2.95750e6 −0.620859
\(470\) −5.19552e6 −1.08489
\(471\) 0 0
\(472\) 1.68576e6 0.348290
\(473\) 798960. 0.164200
\(474\) 0 0
\(475\) 1.17684e6 0.239322
\(476\) 1.16582e6 0.235839
\(477\) 0 0
\(478\) 113856. 0.0227922
\(479\) −2.34466e6 −0.466918 −0.233459 0.972367i \(-0.575004\pi\)
−0.233459 + 0.972367i \(0.575004\pi\)
\(480\) 0 0
\(481\) 5.56536e6 1.09681
\(482\) −3.57025e6 −0.699972
\(483\) 0 0
\(484\) −2.51922e6 −0.488823
\(485\) 1.09614e7 2.11598
\(486\) 0 0
\(487\) 9.81531e6 1.87535 0.937674 0.347517i \(-0.112975\pi\)
0.937674 + 0.347517i \(0.112975\pi\)
\(488\) 1.98976e6 0.378226
\(489\) 0 0
\(490\) −3.74062e6 −0.703806
\(491\) 5.94520e6 1.11292 0.556458 0.830876i \(-0.312160\pi\)
0.556458 + 0.830876i \(0.312160\pi\)
\(492\) 0 0
\(493\) −2.30764e6 −0.427612
\(494\) 2.51619e6 0.463902
\(495\) 0 0
\(496\) −919552. −0.167831
\(497\) −1.07712e6 −0.195602
\(498\) 0 0
\(499\) 6.47832e6 1.16469 0.582346 0.812941i \(-0.302135\pi\)
0.582346 + 0.812941i \(0.302135\pi\)
\(500\) −2.00006e6 −0.357782
\(501\) 0 0
\(502\) −440496. −0.0780158
\(503\) −4.71794e6 −0.831444 −0.415722 0.909492i \(-0.636471\pi\)
−0.415722 + 0.909492i \(0.636471\pi\)
\(504\) 0 0
\(505\) 1.45213e6 0.253383
\(506\) 144000. 0.0250027
\(507\) 0 0
\(508\) 2.76800e6 0.475890
\(509\) 1.90771e6 0.326375 0.163188 0.986595i \(-0.447822\pi\)
0.163188 + 0.986595i \(0.447822\pi\)
\(510\) 0 0
\(511\) −4.49821e6 −0.762057
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) 563208. 0.0940288
\(515\) −5.23142e6 −0.869164
\(516\) 0 0
\(517\) 1.18080e6 0.194290
\(518\) 5.95443e6 0.975025
\(519\) 0 0
\(520\) 2.77939e6 0.450756
\(521\) −8.01974e6 −1.29439 −0.647196 0.762324i \(-0.724059\pi\)
−0.647196 + 0.762324i \(0.724059\pi\)
\(522\) 0 0
\(523\) 1.91162e6 0.305596 0.152798 0.988257i \(-0.451172\pi\)
0.152798 + 0.988257i \(0.451172\pi\)
\(524\) −2.42016e6 −0.385049
\(525\) 0 0
\(526\) −3.75504e6 −0.591766
\(527\) −1.48709e6 −0.233244
\(528\) 0 0
\(529\) −6.07634e6 −0.944068
\(530\) −8.25422e6 −1.27640
\(531\) 0 0
\(532\) 2.69210e6 0.412393
\(533\) 1.26297e7 1.92563
\(534\) 0 0
\(535\) −1.50472e7 −2.27285
\(536\) 1.07546e6 0.161689
\(537\) 0 0
\(538\) −4.45802e6 −0.664028
\(539\) 850140. 0.126043
\(540\) 0 0
\(541\) −1.19900e7 −1.76128 −0.880639 0.473788i \(-0.842886\pi\)
−0.880639 + 0.473788i \(0.842886\pi\)
\(542\) −2.27082e6 −0.332035
\(543\) 0 0
\(544\) −423936. −0.0614190
\(545\) −562980. −0.0811898
\(546\) 0 0
\(547\) 4.45809e6 0.637061 0.318530 0.947913i \(-0.396811\pi\)
0.318530 + 0.947913i \(0.396811\pi\)
\(548\) 2.05526e6 0.292359
\(549\) 0 0
\(550\) −295440. −0.0416450
\(551\) −5.32874e6 −0.747732
\(552\) 0 0
\(553\) 1.30958e7 1.82104
\(554\) 4.85303e6 0.671798
\(555\) 0 0
\(556\) 2.46714e6 0.338459
\(557\) −9.02612e6 −1.23272 −0.616358 0.787466i \(-0.711393\pi\)
−0.616358 + 0.787466i \(0.711393\pi\)
\(558\) 0 0
\(559\) −8.76193e6 −1.18596
\(560\) 2.97370e6 0.400707
\(561\) 0 0
\(562\) 2.75095e6 0.367403
\(563\) −6.84899e6 −0.910658 −0.455329 0.890323i \(-0.650478\pi\)
−0.455329 + 0.890323i \(0.650478\pi\)
\(564\) 0 0
\(565\) 1.28989e7 1.69993
\(566\) 3.32363e6 0.436086
\(567\) 0 0
\(568\) 391680. 0.0509402
\(569\) 5.46322e6 0.707405 0.353703 0.935358i \(-0.384923\pi\)
0.353703 + 0.935358i \(0.384923\pi\)
\(570\) 0 0
\(571\) −1.02324e7 −1.31337 −0.656684 0.754166i \(-0.728041\pi\)
−0.656684 + 0.754166i \(0.728041\pi\)
\(572\) −631680. −0.0807248
\(573\) 0 0
\(574\) 1.35126e7 1.71182
\(575\) −738600. −0.0931622
\(576\) 0 0
\(577\) 1.59437e7 1.99365 0.996825 0.0796186i \(-0.0253702\pi\)
0.996825 + 0.0796186i \(0.0253702\pi\)
\(578\) 4.99384e6 0.621749
\(579\) 0 0
\(580\) −5.88614e6 −0.726542
\(581\) 1.13837e6 0.139908
\(582\) 0 0
\(583\) 1.87596e6 0.228587
\(584\) 1.63571e6 0.198461
\(585\) 0 0
\(586\) −5.25050e6 −0.631622
\(587\) 9.47713e6 1.13522 0.567612 0.823296i \(-0.307867\pi\)
0.567612 + 0.823296i \(0.307867\pi\)
\(588\) 0 0
\(589\) −3.43395e6 −0.407855
\(590\) 6.95376e6 0.822412
\(591\) 0 0
\(592\) −2.16525e6 −0.253924
\(593\) −2.45349e6 −0.286515 −0.143258 0.989685i \(-0.545758\pi\)
−0.143258 + 0.989685i \(0.545758\pi\)
\(594\) 0 0
\(595\) 4.80902e6 0.556884
\(596\) −471264. −0.0543436
\(597\) 0 0
\(598\) −1.57920e6 −0.180586
\(599\) 9.29978e6 1.05902 0.529512 0.848302i \(-0.322375\pi\)
0.529512 + 0.848302i \(0.322375\pi\)
\(600\) 0 0
\(601\) −1.14617e7 −1.29438 −0.647192 0.762327i \(-0.724057\pi\)
−0.647192 + 0.762327i \(0.724057\pi\)
\(602\) −9.37446e6 −1.05428
\(603\) 0 0
\(604\) −3.26195e6 −0.363819
\(605\) −1.03918e7 −1.15425
\(606\) 0 0
\(607\) 1.12784e7 1.24244 0.621219 0.783637i \(-0.286638\pi\)
0.621219 + 0.783637i \(0.286638\pi\)
\(608\) −978944. −0.107399
\(609\) 0 0
\(610\) 8.20776e6 0.893100
\(611\) −1.29494e7 −1.40329
\(612\) 0 0
\(613\) 93782.0 0.0100802 0.00504009 0.999987i \(-0.498396\pi\)
0.00504009 + 0.999987i \(0.498396\pi\)
\(614\) −6.76088e6 −0.723740
\(615\) 0 0
\(616\) −675840. −0.0717616
\(617\) 1.49642e7 1.58248 0.791242 0.611504i \(-0.209435\pi\)
0.791242 + 0.611504i \(0.209435\pi\)
\(618\) 0 0
\(619\) −5.06888e6 −0.531723 −0.265861 0.964011i \(-0.585656\pi\)
−0.265861 + 0.964011i \(0.585656\pi\)
\(620\) −3.79315e6 −0.396297
\(621\) 0 0
\(622\) −6.00816e6 −0.622681
\(623\) 5.76259e6 0.594837
\(624\) 0 0
\(625\) −1.20971e7 −1.23875
\(626\) −3.24337e6 −0.330796
\(627\) 0 0
\(628\) 2.17827e6 0.220401
\(629\) −3.50161e6 −0.352892
\(630\) 0 0
\(631\) 1.55919e7 1.55892 0.779462 0.626450i \(-0.215493\pi\)
0.779462 + 0.626450i \(0.215493\pi\)
\(632\) −4.76211e6 −0.474250
\(633\) 0 0
\(634\) 3.61423e6 0.357102
\(635\) 1.14180e7 1.12371
\(636\) 0 0
\(637\) −9.32320e6 −0.910367
\(638\) 1.33776e6 0.130115
\(639\) 0 0
\(640\) −1.08134e6 −0.104355
\(641\) −1.09701e7 −1.05455 −0.527274 0.849695i \(-0.676786\pi\)
−0.527274 + 0.849695i \(0.676786\pi\)
\(642\) 0 0
\(643\) −2.83704e6 −0.270607 −0.135303 0.990804i \(-0.543201\pi\)
−0.135303 + 0.990804i \(0.543201\pi\)
\(644\) −1.68960e6 −0.160535
\(645\) 0 0
\(646\) −1.58314e6 −0.149258
\(647\) 6.05686e6 0.568835 0.284418 0.958700i \(-0.408200\pi\)
0.284418 + 0.958700i \(0.408200\pi\)
\(648\) 0 0
\(649\) −1.58040e6 −0.147284
\(650\) 3.23999e6 0.300788
\(651\) 0 0
\(652\) −2.73798e6 −0.252239
\(653\) 1.08892e6 0.0999341 0.0499671 0.998751i \(-0.484088\pi\)
0.0499671 + 0.998751i \(0.484088\pi\)
\(654\) 0 0
\(655\) −9.98316e6 −0.909211
\(656\) −4.91366e6 −0.445806
\(657\) 0 0
\(658\) −1.38547e7 −1.24748
\(659\) −7.41803e6 −0.665388 −0.332694 0.943035i \(-0.607958\pi\)
−0.332694 + 0.943035i \(0.607958\pi\)
\(660\) 0 0
\(661\) 767654. 0.0683379 0.0341690 0.999416i \(-0.489122\pi\)
0.0341690 + 0.999416i \(0.489122\pi\)
\(662\) −4.48789e6 −0.398013
\(663\) 0 0
\(664\) −413952. −0.0364359
\(665\) 1.11049e7 0.973779
\(666\) 0 0
\(667\) 3.34440e6 0.291074
\(668\) 1.08192e7 0.938110
\(669\) 0 0
\(670\) 4.43626e6 0.381794
\(671\) −1.86540e6 −0.159943
\(672\) 0 0
\(673\) 1.42263e6 0.121075 0.0605373 0.998166i \(-0.480719\pi\)
0.0605373 + 0.998166i \(0.480719\pi\)
\(674\) 1.10087e7 0.933439
\(675\) 0 0
\(676\) 986736. 0.0830490
\(677\) 6.16231e6 0.516739 0.258370 0.966046i \(-0.416815\pi\)
0.258370 + 0.966046i \(0.416815\pi\)
\(678\) 0 0
\(679\) 2.92304e7 2.43310
\(680\) −1.74874e6 −0.145028
\(681\) 0 0
\(682\) 862080. 0.0709719
\(683\) −1.50621e7 −1.23548 −0.617739 0.786383i \(-0.711951\pi\)
−0.617739 + 0.786383i \(0.711951\pi\)
\(684\) 0 0
\(685\) 8.47796e6 0.690343
\(686\) 1.85715e6 0.150674
\(687\) 0 0
\(688\) 3.40890e6 0.274563
\(689\) −2.05730e7 −1.65101
\(690\) 0 0
\(691\) −5.87636e6 −0.468180 −0.234090 0.972215i \(-0.575211\pi\)
−0.234090 + 0.972215i \(0.575211\pi\)
\(692\) −2.13053e6 −0.169131
\(693\) 0 0
\(694\) 7.66997e6 0.604498
\(695\) 1.01769e7 0.799199
\(696\) 0 0
\(697\) −7.94632e6 −0.619561
\(698\) −7.34636e6 −0.570734
\(699\) 0 0
\(700\) 3.46650e6 0.267390
\(701\) −3.60077e6 −0.276758 −0.138379 0.990379i \(-0.544189\pi\)
−0.138379 + 0.990379i \(0.544189\pi\)
\(702\) 0 0
\(703\) −8.08585e6 −0.617074
\(704\) 245760. 0.0186887
\(705\) 0 0
\(706\) −2.48806e6 −0.187866
\(707\) 3.87235e6 0.291358
\(708\) 0 0
\(709\) 9.22516e6 0.689221 0.344610 0.938746i \(-0.388011\pi\)
0.344610 + 0.938746i \(0.388011\pi\)
\(710\) 1.61568e6 0.120284
\(711\) 0 0
\(712\) −2.09549e6 −0.154912
\(713\) 2.15520e6 0.158768
\(714\) 0 0
\(715\) −2.60568e6 −0.190615
\(716\) 1.10943e7 0.808758
\(717\) 0 0
\(718\) 1.49625e7 1.08316
\(719\) 2.63923e7 1.90395 0.951975 0.306177i \(-0.0990500\pi\)
0.951975 + 0.306177i \(0.0990500\pi\)
\(720\) 0 0
\(721\) −1.39505e7 −0.999426
\(722\) 6.24865e6 0.446111
\(723\) 0 0
\(724\) 6.03478e6 0.427873
\(725\) −6.86159e6 −0.484819
\(726\) 0 0
\(727\) −9.79485e6 −0.687324 −0.343662 0.939093i \(-0.611667\pi\)
−0.343662 + 0.939093i \(0.611667\pi\)
\(728\) 7.41171e6 0.518311
\(729\) 0 0
\(730\) 6.74731e6 0.468623
\(731\) 5.51282e6 0.381576
\(732\) 0 0
\(733\) 4.07584e6 0.280193 0.140096 0.990138i \(-0.455259\pi\)
0.140096 + 0.990138i \(0.455259\pi\)
\(734\) −64928.0 −0.00444828
\(735\) 0 0
\(736\) 614400. 0.0418077
\(737\) −1.00824e6 −0.0683747
\(738\) 0 0
\(739\) −1.65709e7 −1.11618 −0.558089 0.829781i \(-0.688465\pi\)
−0.558089 + 0.829781i \(0.688465\pi\)
\(740\) −8.93165e6 −0.599587
\(741\) 0 0
\(742\) −2.20113e7 −1.46769
\(743\) −1.44141e7 −0.957892 −0.478946 0.877844i \(-0.658981\pi\)
−0.478946 + 0.877844i \(0.658981\pi\)
\(744\) 0 0
\(745\) −1.94396e6 −0.128321
\(746\) −1.17442e6 −0.0772641
\(747\) 0 0
\(748\) 397440. 0.0259727
\(749\) −4.01259e7 −2.61349
\(750\) 0 0
\(751\) 1.67944e7 1.08659 0.543295 0.839542i \(-0.317177\pi\)
0.543295 + 0.839542i \(0.317177\pi\)
\(752\) 5.03808e6 0.324878
\(753\) 0 0
\(754\) −1.46708e7 −0.939776
\(755\) −1.34556e7 −0.859081
\(756\) 0 0
\(757\) 1.32943e7 0.843188 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(758\) −1.27205e7 −0.804137
\(759\) 0 0
\(760\) −4.03814e6 −0.253599
\(761\) 2.14786e6 0.134445 0.0672225 0.997738i \(-0.478586\pi\)
0.0672225 + 0.997738i \(0.478586\pi\)
\(762\) 0 0
\(763\) −1.50128e6 −0.0933577
\(764\) 4.24550e6 0.263145
\(765\) 0 0
\(766\) −1.19194e7 −0.733975
\(767\) 1.73317e7 1.06378
\(768\) 0 0
\(769\) −1.31059e7 −0.799193 −0.399596 0.916691i \(-0.630850\pi\)
−0.399596 + 0.916691i \(0.630850\pi\)
\(770\) −2.78784e6 −0.169450
\(771\) 0 0
\(772\) 4.72477e6 0.285323
\(773\) 2.37154e7 1.42752 0.713759 0.700392i \(-0.246991\pi\)
0.713759 + 0.700392i \(0.246991\pi\)
\(774\) 0 0
\(775\) −4.42175e6 −0.264448
\(776\) −1.06292e7 −0.633648
\(777\) 0 0
\(778\) 1.38391e7 0.819707
\(779\) −1.83495e7 −1.08338
\(780\) 0 0
\(781\) −367200. −0.0215415
\(782\) 993600. 0.0581025
\(783\) 0 0
\(784\) 3.62726e6 0.210760
\(785\) 8.98537e6 0.520430
\(786\) 0 0
\(787\) −8.40048e6 −0.483468 −0.241734 0.970343i \(-0.577716\pi\)
−0.241734 + 0.970343i \(0.577716\pi\)
\(788\) −3.22070e6 −0.184772
\(789\) 0 0
\(790\) −1.96437e7 −1.11984
\(791\) 3.43971e7 1.95470
\(792\) 0 0
\(793\) 2.04572e7 1.15522
\(794\) 1.56166e7 0.879097
\(795\) 0 0
\(796\) 1.04392e7 0.583960
\(797\) −5.41023e6 −0.301696 −0.150848 0.988557i \(-0.548200\pi\)
−0.150848 + 0.988557i \(0.548200\pi\)
\(798\) 0 0
\(799\) 8.14752e6 0.451501
\(800\) −1.26054e6 −0.0696359
\(801\) 0 0
\(802\) 2.17646e7 1.19485
\(803\) −1.53348e6 −0.0839246
\(804\) 0 0
\(805\) −6.96960e6 −0.379069
\(806\) −9.45414e6 −0.512607
\(807\) 0 0
\(808\) −1.40813e6 −0.0758776
\(809\) 2.60777e7 1.40087 0.700436 0.713715i \(-0.252989\pi\)
0.700436 + 0.713715i \(0.252989\pi\)
\(810\) 0 0
\(811\) 1.90021e7 1.01449 0.507247 0.861800i \(-0.330663\pi\)
0.507247 + 0.861800i \(0.330663\pi\)
\(812\) −1.56964e7 −0.835429
\(813\) 0 0
\(814\) 2.02992e6 0.107379
\(815\) −1.12942e7 −0.595608
\(816\) 0 0
\(817\) 1.27301e7 0.667231
\(818\) −7.87978e6 −0.411748
\(819\) 0 0
\(820\) −2.02689e7 −1.05268
\(821\) 3.10173e7 1.60600 0.803001 0.595978i \(-0.203236\pi\)
0.803001 + 0.595978i \(0.203236\pi\)
\(822\) 0 0
\(823\) −1.56290e7 −0.804323 −0.402162 0.915569i \(-0.631741\pi\)
−0.402162 + 0.915569i \(0.631741\pi\)
\(824\) 5.07290e6 0.260278
\(825\) 0 0
\(826\) 1.85434e7 0.945667
\(827\) −1.58421e7 −0.805467 −0.402733 0.915317i \(-0.631940\pi\)
−0.402733 + 0.915317i \(0.631940\pi\)
\(828\) 0 0
\(829\) 2.06176e6 0.104196 0.0520980 0.998642i \(-0.483409\pi\)
0.0520980 + 0.998642i \(0.483409\pi\)
\(830\) −1.70755e6 −0.0860357
\(831\) 0 0
\(832\) −2.69517e6 −0.134983
\(833\) 5.86597e6 0.292905
\(834\) 0 0
\(835\) 4.46292e7 2.21515
\(836\) 917760. 0.0454165
\(837\) 0 0
\(838\) 556080. 0.0273544
\(839\) −3.03900e7 −1.49048 −0.745240 0.666796i \(-0.767665\pi\)
−0.745240 + 0.666796i \(0.767665\pi\)
\(840\) 0 0
\(841\) 1.05583e7 0.514760
\(842\) −1.73097e7 −0.841414
\(843\) 0 0
\(844\) −1.83530e7 −0.886850
\(845\) 4.07029e6 0.196103
\(846\) 0 0
\(847\) −2.77114e7 −1.32724
\(848\) 8.00410e6 0.382228
\(849\) 0 0
\(850\) −2.03854e6 −0.0967768
\(851\) 5.07480e6 0.240212
\(852\) 0 0
\(853\) −2.97738e7 −1.40108 −0.700538 0.713615i \(-0.747056\pi\)
−0.700538 + 0.713615i \(0.747056\pi\)
\(854\) 2.18874e7 1.02695
\(855\) 0 0
\(856\) 1.45912e7 0.680624
\(857\) −8.64100e6 −0.401894 −0.200947 0.979602i \(-0.564402\pi\)
−0.200947 + 0.979602i \(0.564402\pi\)
\(858\) 0 0
\(859\) −3.35663e7 −1.55210 −0.776051 0.630670i \(-0.782780\pi\)
−0.776051 + 0.630670i \(0.782780\pi\)
\(860\) 1.40617e7 0.648323
\(861\) 0 0
\(862\) −1.11974e7 −0.513276
\(863\) −3.90191e7 −1.78341 −0.891703 0.452621i \(-0.850489\pi\)
−0.891703 + 0.452621i \(0.850489\pi\)
\(864\) 0 0
\(865\) −8.78843e6 −0.399366
\(866\) 2.36097e7 1.06978
\(867\) 0 0
\(868\) −1.01151e7 −0.455690
\(869\) 4.46448e6 0.200549
\(870\) 0 0
\(871\) 1.10570e7 0.493848
\(872\) 545920. 0.0243130
\(873\) 0 0
\(874\) 2.29440e6 0.101599
\(875\) −2.20007e7 −0.971441
\(876\) 0 0
\(877\) −1.81382e7 −0.796333 −0.398166 0.917313i \(-0.630353\pi\)
−0.398166 + 0.917313i \(0.630353\pi\)
\(878\) 1.78605e6 0.0781910
\(879\) 0 0
\(880\) 1.01376e6 0.0441294
\(881\) −3.05312e7 −1.32527 −0.662634 0.748943i \(-0.730562\pi\)
−0.662634 + 0.748943i \(0.730562\pi\)
\(882\) 0 0
\(883\) −4.35533e7 −1.87983 −0.939916 0.341405i \(-0.889097\pi\)
−0.939916 + 0.341405i \(0.889097\pi\)
\(884\) −4.35859e6 −0.187593
\(885\) 0 0
\(886\) 1.39810e7 0.598348
\(887\) 1.34152e7 0.572515 0.286257 0.958153i \(-0.407589\pi\)
0.286257 + 0.958153i \(0.407589\pi\)
\(888\) 0 0
\(889\) 3.04480e7 1.29212
\(890\) −8.64389e6 −0.365792
\(891\) 0 0
\(892\) 1.12314e7 0.472629
\(893\) 1.88141e7 0.789504
\(894\) 0 0
\(895\) 4.57641e7 1.90971
\(896\) −2.88358e6 −0.119995
\(897\) 0 0
\(898\) −4.82450e6 −0.199647
\(899\) 2.00218e7 0.826236
\(900\) 0 0
\(901\) 1.29441e7 0.531203
\(902\) 4.60656e6 0.188521
\(903\) 0 0
\(904\) −1.25080e7 −0.509059
\(905\) 2.48935e7 1.01033
\(906\) 0 0
\(907\) 3.10816e6 0.125454 0.0627272 0.998031i \(-0.480020\pi\)
0.0627272 + 0.998031i \(0.480020\pi\)
\(908\) −1.97777e7 −0.796089
\(909\) 0 0
\(910\) 3.05733e7 1.22388
\(911\) −1.19035e6 −0.0475203 −0.0237602 0.999718i \(-0.507564\pi\)
−0.0237602 + 0.999718i \(0.507564\pi\)
\(912\) 0 0
\(913\) 388080. 0.0154079
\(914\) −934184. −0.0369885
\(915\) 0 0
\(916\) 1.69328e6 0.0666792
\(917\) −2.66218e7 −1.04547
\(918\) 0 0
\(919\) −4.71996e7 −1.84353 −0.921764 0.387752i \(-0.873252\pi\)
−0.921764 + 0.387752i \(0.873252\pi\)
\(920\) 2.53440e6 0.0987201
\(921\) 0 0
\(922\) −6.97956e6 −0.270396
\(923\) 4.02696e6 0.155587
\(924\) 0 0
\(925\) −1.04118e7 −0.400103
\(926\) 1.16715e7 0.447299
\(927\) 0 0
\(928\) 5.70778e6 0.217569
\(929\) −1.33595e6 −0.0507870 −0.0253935 0.999678i \(-0.508084\pi\)
−0.0253935 + 0.999678i \(0.508084\pi\)
\(930\) 0 0
\(931\) 1.35456e7 0.512180
\(932\) 7.01885e6 0.264683
\(933\) 0 0
\(934\) −2.12430e7 −0.796800
\(935\) 1.63944e6 0.0613291
\(936\) 0 0
\(937\) 1.47238e7 0.547861 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(938\) 1.18300e7 0.439014
\(939\) 0 0
\(940\) 2.07821e7 0.767131
\(941\) 2.69196e7 0.991049 0.495525 0.868594i \(-0.334976\pi\)
0.495525 + 0.868594i \(0.334976\pi\)
\(942\) 0 0
\(943\) 1.15164e7 0.421733
\(944\) −6.74304e6 −0.246278
\(945\) 0 0
\(946\) −3.19584e6 −0.116107
\(947\) 3.73160e6 0.135214 0.0676068 0.997712i \(-0.478464\pi\)
0.0676068 + 0.997712i \(0.478464\pi\)
\(948\) 0 0
\(949\) 1.68172e7 0.606160
\(950\) −4.70734e6 −0.169226
\(951\) 0 0
\(952\) −4.66330e6 −0.166763
\(953\) −2.18735e7 −0.780166 −0.390083 0.920780i \(-0.627554\pi\)
−0.390083 + 0.920780i \(0.627554\pi\)
\(954\) 0 0
\(955\) 1.75127e7 0.621362
\(956\) −455424. −0.0161165
\(957\) 0 0
\(958\) 9.37862e6 0.330161
\(959\) 2.26079e7 0.793805
\(960\) 0 0
\(961\) −1.57267e7 −0.549324
\(962\) −2.22615e7 −0.775561
\(963\) 0 0
\(964\) 1.42810e7 0.494955
\(965\) 1.94897e7 0.673730
\(966\) 0 0
\(967\) 1.76025e7 0.605352 0.302676 0.953093i \(-0.402120\pi\)
0.302676 + 0.953093i \(0.402120\pi\)
\(968\) 1.00769e7 0.345650
\(969\) 0 0
\(970\) −4.38456e7 −1.49623
\(971\) −1.67317e7 −0.569497 −0.284749 0.958602i \(-0.591910\pi\)
−0.284749 + 0.958602i \(0.591910\pi\)
\(972\) 0 0
\(973\) 2.71385e7 0.918975
\(974\) −3.92612e7 −1.32607
\(975\) 0 0
\(976\) −7.95904e6 −0.267446
\(977\) −5.55382e7 −1.86147 −0.930733 0.365699i \(-0.880830\pi\)
−0.930733 + 0.365699i \(0.880830\pi\)
\(978\) 0 0
\(979\) 1.96452e6 0.0655088
\(980\) 1.49625e7 0.497666
\(981\) 0 0
\(982\) −2.37808e7 −0.786951
\(983\) 3.86784e7 1.27669 0.638344 0.769751i \(-0.279620\pi\)
0.638344 + 0.769751i \(0.279620\pi\)
\(984\) 0 0
\(985\) −1.32854e7 −0.436299
\(986\) 9.23054e6 0.302367
\(987\) 0 0
\(988\) −1.00648e7 −0.328028
\(989\) −7.98960e6 −0.259737
\(990\) 0 0
\(991\) 9.58498e6 0.310033 0.155016 0.987912i \(-0.450457\pi\)
0.155016 + 0.987912i \(0.450457\pi\)
\(992\) 3.67821e6 0.118674
\(993\) 0 0
\(994\) 4.30848e6 0.138311
\(995\) 4.30616e7 1.37890
\(996\) 0 0
\(997\) −1.03650e7 −0.330242 −0.165121 0.986273i \(-0.552802\pi\)
−0.165121 + 0.986273i \(0.552802\pi\)
\(998\) −2.59133e7 −0.823561
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 18.6.a.b.1.1 1
3.2 odd 2 6.6.a.a.1.1 1
4.3 odd 2 144.6.a.j.1.1 1
5.2 odd 4 450.6.c.j.199.1 2
5.3 odd 4 450.6.c.j.199.2 2
5.4 even 2 450.6.a.m.1.1 1
7.6 odd 2 882.6.a.a.1.1 1
8.3 odd 2 576.6.a.i.1.1 1
8.5 even 2 576.6.a.j.1.1 1
9.2 odd 6 162.6.c.e.109.1 2
9.4 even 3 162.6.c.h.55.1 2
9.5 odd 6 162.6.c.e.55.1 2
9.7 even 3 162.6.c.h.109.1 2
12.11 even 2 48.6.a.c.1.1 1
15.2 even 4 150.6.c.b.49.2 2
15.8 even 4 150.6.c.b.49.1 2
15.14 odd 2 150.6.a.d.1.1 1
21.2 odd 6 294.6.e.g.67.1 2
21.5 even 6 294.6.e.a.67.1 2
21.11 odd 6 294.6.e.g.79.1 2
21.17 even 6 294.6.e.a.79.1 2
21.20 even 2 294.6.a.m.1.1 1
24.5 odd 2 192.6.a.o.1.1 1
24.11 even 2 192.6.a.g.1.1 1
33.32 even 2 726.6.a.a.1.1 1
39.38 odd 2 1014.6.a.c.1.1 1
48.5 odd 4 768.6.d.c.385.2 2
48.11 even 4 768.6.d.p.385.1 2
48.29 odd 4 768.6.d.c.385.1 2
48.35 even 4 768.6.d.p.385.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.6.a.a.1.1 1 3.2 odd 2
18.6.a.b.1.1 1 1.1 even 1 trivial
48.6.a.c.1.1 1 12.11 even 2
144.6.a.j.1.1 1 4.3 odd 2
150.6.a.d.1.1 1 15.14 odd 2
150.6.c.b.49.1 2 15.8 even 4
150.6.c.b.49.2 2 15.2 even 4
162.6.c.e.55.1 2 9.5 odd 6
162.6.c.e.109.1 2 9.2 odd 6
162.6.c.h.55.1 2 9.4 even 3
162.6.c.h.109.1 2 9.7 even 3
192.6.a.g.1.1 1 24.11 even 2
192.6.a.o.1.1 1 24.5 odd 2
294.6.a.m.1.1 1 21.20 even 2
294.6.e.a.67.1 2 21.5 even 6
294.6.e.a.79.1 2 21.17 even 6
294.6.e.g.67.1 2 21.2 odd 6
294.6.e.g.79.1 2 21.11 odd 6
450.6.a.m.1.1 1 5.4 even 2
450.6.c.j.199.1 2 5.2 odd 4
450.6.c.j.199.2 2 5.3 odd 4
576.6.a.i.1.1 1 8.3 odd 2
576.6.a.j.1.1 1 8.5 even 2
726.6.a.a.1.1 1 33.32 even 2
768.6.d.c.385.1 2 48.29 odd 4
768.6.d.c.385.2 2 48.5 odd 4
768.6.d.p.385.1 2 48.11 even 4
768.6.d.p.385.2 2 48.35 even 4
882.6.a.a.1.1 1 7.6 odd 2
1014.6.a.c.1.1 1 39.38 odd 2