Properties

Label 4650.2.a.bg.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -2.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +12.0000 q^{37} -8.00000 q^{38} -4.00000 q^{39} +10.0000 q^{41} -2.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} +4.00000 q^{58} +2.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +6.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +12.0000 q^{74} -8.00000 q^{76} -8.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -4.00000 q^{87} -4.00000 q^{88} +8.00000 q^{91} +8.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +18.0000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −4.00000 −0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 12.0000 1.97279 0.986394 0.164399i \(-0.0525685\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) −8.00000 −1.29777
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000 1.05963
\(58\) 4.00000 0.525226
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 12.0000 1.39497
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −8.00000 −0.911685
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −4.00000 −0.428845
\(88\) −4.00000 −0.426401
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 8.00000 0.834058
\(93\) 1.00000 0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −3.00000 −0.303046
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 2.00000 0.198030
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 4.00000 0.369800
\(118\) 2.00000 0.184115
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 4.00000 0.348155
\(133\) −16.0000 −1.38738
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −8.00000 −0.681005
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 6.00000 0.503509
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) 12.0000 0.986394
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −8.00000 −0.648886
\(153\) −2.00000 −0.161690
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −8.00000 −0.609994
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −2.00000 −0.150329
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 8.00000 0.592999
\(183\) −10.0000 −0.739221
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 8.00000 0.585018
\(188\) 4.00000 0.291730
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) −18.0000 −1.26648
\(203\) 8.00000 0.561490
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 8.00000 0.556038
\(208\) 4.00000 0.277350
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −2.00000 −0.135769
\(218\) 18.0000 1.21911
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −12.0000 −0.805387
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 8.00000 0.529813
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 4.00000 0.262613
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 2.00000 0.130189
\(237\) 8.00000 0.519656
\(238\) −4.00000 −0.259281
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −32.0000 −2.03611
\(248\) −1.00000 −0.0635001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000 0.125988
\(253\) −32.0000 −2.01182
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 10.0000 0.617802
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) 6.00000 0.366508
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) −8.00000 −0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 4.00000 0.239904
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −4.00000 −0.238197
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 4.00000 0.234082
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 4.00000 0.232104
\(298\) −18.0000 −1.04271
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) 18.0000 1.03407
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −8.00000 −0.455842
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) −4.00000 −0.226455
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 16.0000 0.891645
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) −18.0000 −0.995402
\(328\) 10.0000 0.552158
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) 12.0000 0.657596
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −8.00000 −0.432590
\(343\) −20.0000 −1.07990
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −4.00000 −0.214423
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −4.00000 −0.213201
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) −2.00000 −0.106299
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) −20.0000 −1.05703
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 18.0000 0.946059
\(363\) −5.00000 −0.262432
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 8.00000 0.417029
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 1.00000 0.0518476
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 16.0000 0.824042
\(378\) −2.00000 −0.102869
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 18.0000 0.920960
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −8.00000 −0.406663
\(388\) 18.0000 0.913812
\(389\) −28.0000 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −3.00000 −0.151523
\(393\) −10.0000 −0.504433
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 16.0000 0.802008
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −6.00000 −0.299253
\(403\) −4.00000 −0.199254
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −48.0000 −2.37927
\(408\) 2.00000 0.0990148
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 14.0000 0.689730
\(413\) 4.00000 0.196827
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −4.00000 −0.195881
\(418\) 32.0000 1.56517
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 24.0000 1.16830
\(423\) 4.00000 0.194487
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 20.0000 0.967868
\(428\) −8.00000 −0.386695
\(429\) 16.0000 0.772487
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −64.0000 −3.06154
\(438\) −4.00000 −0.191127
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −8.00000 −0.380521
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −12.0000 −0.569495
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 18.0000 0.851371
\(448\) 2.00000 0.0944911
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 6.00000 0.282216
\(453\) 16.0000 0.751746
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −36.0000 −1.68401 −0.842004 0.539471i \(-0.818624\pi\)
−0.842004 + 0.539471i \(0.818624\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 8.00000 0.372194
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 4.00000 0.184900
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 2.00000 0.0920575
\(473\) 32.0000 1.47136
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) −6.00000 −0.274721
\(478\) −4.00000 −0.182956
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 10.0000 0.455488
\(483\) −16.0000 −0.728025
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 10.0000 0.452679
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −10.0000 −0.450835
\(493\) −8.00000 −0.360302
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 12.0000 0.538274
\(498\) 4.00000 0.179244
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 20.0000 0.892644
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) −3.00000 −0.133235
\(508\) −4.00000 −0.177471
\(509\) 16.0000 0.709188 0.354594 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −16.0000 −0.703679
\(518\) 24.0000 1.05450
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 4.00000 0.175075
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) −16.0000 −0.693688
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 20.0000 0.863064
\(538\) −12.0000 −0.517357
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 8.00000 0.343629
\(543\) −18.0000 −0.772454
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −6.00000 −0.256307
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) −8.00000 −0.340503
\(553\) −16.0000 −0.680389
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 6.00000 0.253095
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 2.00000 0.0839921
\(568\) 6.00000 0.251754
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −16.0000 −0.668994
\(573\) −18.0000 −0.751961
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) −13.0000 −0.540729
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −18.0000 −0.746124
\(583\) 24.0000 0.993978
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 3.00000 0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 12.0000 0.493197
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −16.0000 −0.654836
\(598\) 32.0000 1.30858
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −16.0000 −0.652111
\(603\) 6.00000 0.244339
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −8.00000 −0.324443
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) −2.00000 −0.0808452
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −14.0000 −0.563163
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −30.0000 −1.20289
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) −32.0000 −1.27796
\(628\) 22.0000 0.877896
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −8.00000 −0.318223
\(633\) −24.0000 −0.953914
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −12.0000 −0.475457
\(638\) −16.0000 −0.633446
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 8.00000 0.315735
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 6.00000 0.234978
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 4.00000 0.156055
\(658\) 8.00000 0.311872
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −12.0000 −0.466393
\(663\) 8.00000 0.310694
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 32.0000 1.23904
\(668\) 8.00000 0.309529
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) −2.00000 −0.0771517
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) 36.0000 1.38155
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 4.00000 0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −22.0000 −0.839352
\(688\) −8.00000 −0.304997
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −22.0000 −0.836315
\(693\) −8.00000 −0.303895
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) −20.0000 −0.757554
\(698\) −30.0000 −1.13552
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −4.00000 −0.150970
\(703\) −96.0000 −3.62071
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) −36.0000 −1.35392
\(708\) −2.00000 −0.0751646
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 4.00000 0.149383
\(718\) 10.0000 0.373197
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 45.0000 1.67473
\(723\) −10.0000 −0.371904
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −10.0000 −0.369611
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −24.0000 −0.884051
\(738\) 10.0000 0.368105
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 32.0000 1.17555
\(742\) −12.0000 −0.440534
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 4.00000 0.145865
\(753\) −20.0000 −0.728841
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −36.0000 −1.30844 −0.654221 0.756303i \(-0.727003\pi\)
−0.654221 + 0.756303i \(0.727003\pi\)
\(758\) 16.0000 0.581146
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −20.0000 −0.724999 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 4.00000 0.144905
\(763\) 36.0000 1.30329
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) −24.0000 −0.860995
\(778\) −28.0000 −1.00385
\(779\) −80.0000 −2.86630
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) −16.0000 −0.572159
\(783\) −4.00000 −0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −4.00000 −0.142134
\(793\) 40.0000 1.42044
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 16.0000 0.566394
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) −16.0000 −0.564628
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 12.0000 0.422420
\(808\) −18.0000 −0.633238
\(809\) 28.0000 0.984428 0.492214 0.870474i \(-0.336188\pi\)
0.492214 + 0.870474i \(0.336188\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 8.00000 0.280745
\(813\) −8.00000 −0.280572
\(814\) −48.0000 −1.68240
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 64.0000 2.23908
\(818\) 6.00000 0.209785
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) 6.00000 0.209274
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 8.00000 0.278019
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 4.00000 0.138675
\(833\) 6.00000 0.207888
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 32.0000 1.10674
\(837\) 1.00000 0.0345651
\(838\) −10.0000 −0.345444
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 2.00000 0.0689246
\(843\) −6.00000 −0.206651
\(844\) 24.0000 0.826114
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 10.0000 0.343604
\(848\) −6.00000 −0.206041
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) 96.0000 3.29084
\(852\) −6.00000 −0.205557
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 16.0000 0.546231
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) −30.0000 −1.02180
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −4.00000 −0.135926
\(867\) 13.0000 0.441503
\(868\) −2.00000 −0.0678844
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 18.0000 0.609557
\(873\) 18.0000 0.609208
\(874\) −64.0000 −2.16483
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −40.0000 −1.34993
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −3.00000 −0.101015
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) −12.0000 −0.402694
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000 0.267860
\(893\) −32.0000 −1.07084
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) −32.0000 −1.06845
\(898\) 28.0000 0.934372
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −40.0000 −1.33185
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −4.00000 −0.132745
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 8.00000 0.264906
\(913\) 16.0000 0.529523
\(914\) −36.0000 −1.19077
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 20.0000 0.660458
\(918\) 2.00000 0.0660098
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) −40.0000 −1.31733
\(923\) 24.0000 0.789970
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 14.0000 0.459820
\(928\) 4.00000 0.131306
\(929\) 32.0000 1.04989 0.524943 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 10.0000 0.327561
\(933\) 30.0000 0.982156
\(934\) 0 0
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 12.0000 0.391814
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) −22.0000 −0.716799
\(943\) 80.0000 2.60516
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 8.00000 0.259828
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) −4.00000 −0.129641
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 16.0000 0.517207
\(958\) −10.0000 −0.323085
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 48.0000 1.54758
\(963\) −8.00000 −0.257796
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 5.00000 0.160706
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) −2.00000 −0.0641831 −0.0320915 0.999485i \(-0.510217\pi\)
−0.0320915 + 0.999485i \(0.510217\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) −6.00000 −0.191859
\(979\) 0 0
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) −8.00000 −0.255290
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) −8.00000 −0.254643
\(988\) −32.0000 −1.01806
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 12.0000 0.380808
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −36.0000 −1.13956
\(999\) −12.0000 −0.379663
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 4650.2.a.bg.1.1 1
5.2 odd 4 4650.2.d.b.3349.2 2
5.3 odd 4 4650.2.d.b.3349.1 2
5.4 even 2 930.2.a.h.1.1 1
15.14 odd 2 2790.2.a.p.1.1 1
20.19 odd 2 7440.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 1 5.4 even 2
2790.2.a.p.1.1 1 15.14 odd 2
4650.2.a.bg.1.1 1 1.1 even 1 trivial
4650.2.d.b.3349.1 2 5.3 odd 4
4650.2.d.b.3349.2 2 5.2 odd 4
7440.2.a.m.1.1 1 20.19 odd 2