Properties

Label 43.2.a.a.1.1
Level 43
Weight 2
Character 43.1
Self dual yes
Analytic conductor 0.343
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.343356728692\)
Analytic rank: \(1\)
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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} +4.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} +4.00000 q^{6} +1.00000 q^{9} +8.00000 q^{10} +3.00000 q^{11} -4.00000 q^{12} -5.00000 q^{13} +8.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -2.00000 q^{19} -8.00000 q^{20} -6.00000 q^{22} -1.00000 q^{23} +11.0000 q^{25} +10.0000 q^{26} +4.00000 q^{27} -6.00000 q^{29} -16.0000 q^{30} -1.00000 q^{31} +8.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +2.00000 q^{36} +4.00000 q^{38} +10.0000 q^{39} +5.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} -4.00000 q^{45} +2.00000 q^{46} +4.00000 q^{47} +8.00000 q^{48} -7.00000 q^{49} -22.0000 q^{50} +6.00000 q^{51} -10.0000 q^{52} -5.00000 q^{53} -8.00000 q^{54} -12.0000 q^{55} +4.00000 q^{57} +12.0000 q^{58} -12.0000 q^{59} +16.0000 q^{60} +2.00000 q^{61} +2.00000 q^{62} -8.00000 q^{64} +20.0000 q^{65} +12.0000 q^{66} -3.00000 q^{67} -6.00000 q^{68} +2.00000 q^{69} +2.00000 q^{71} +2.00000 q^{73} -22.0000 q^{75} -4.00000 q^{76} -20.0000 q^{78} -8.00000 q^{79} +16.0000 q^{80} -11.0000 q^{81} -10.0000 q^{82} +15.0000 q^{83} +12.0000 q^{85} +2.00000 q^{86} +12.0000 q^{87} -4.00000 q^{89} +8.00000 q^{90} -2.00000 q^{92} +2.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -16.0000 q^{96} +7.00000 q^{97} +14.0000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 2.00000 1.00000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 4.00000 1.63299
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 8.00000 2.52982
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −4.00000 −1.15470
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −2.00000 −0.471405
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −8.00000 −1.78885
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 10.0000 1.96116
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −16.0000 −2.92119
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 8.00000 1.41421
\(33\) −6.00000 −1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 4.00000 0.648886
\(39\) 10.0000 1.60128
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 6.00000 0.904534
\(45\) −4.00000 −0.596285
\(46\) 2.00000 0.294884
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 8.00000 1.15470
\(49\) −7.00000 −1.00000
\(50\) −22.0000 −3.11127
\(51\) 6.00000 0.840168
\(52\) −10.0000 −1.38675
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) −8.00000 −1.08866
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 12.0000 1.57568
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 16.0000 2.06559
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 20.0000 2.48069
\(66\) 12.0000 1.47710
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −6.00000 −0.727607
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −22.0000 −2.54034
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −20.0000 −2.26455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 16.0000 1.78885
\(81\) −11.0000 −1.22222
\(82\) −10.0000 −1.10432
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 2.00000 0.215666
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 8.00000 0.843274
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 2.00000 0.207390
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) −16.0000 −1.63299
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 14.0000 1.41421
\(99\) 3.00000 0.301511
\(100\) 22.0000 2.20000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) −12.0000 −1.18818
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 8.00000 0.769800
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 24.0000 2.28831
\(111\) 0 0
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) −8.00000 −0.749269
\(115\) 4.00000 0.373002
\(116\) −12.0000 −1.11417
\(117\) −5.00000 −0.462250
\(118\) 24.0000 2.20938
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −4.00000 −0.362143
\(123\) −10.0000 −0.901670
\(124\) −2.00000 −0.179605
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) −40.0000 −3.50823
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −12.0000 −1.04447
\(133\) 0 0
\(134\) 6.00000 0.518321
\(135\) −16.0000 −1.37706
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −4.00000 −0.340503
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −4.00000 −0.335673
\(143\) −15.0000 −1.25436
\(144\) −4.00000 −0.333333
\(145\) 24.0000 1.99309
\(146\) −4.00000 −0.331042
\(147\) 14.0000 1.15470
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 44.0000 3.59258
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 20.0000 1.60128
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 16.0000 1.27289
\(159\) 10.0000 0.793052
\(160\) −32.0000 −2.52982
\(161\) 0 0
\(162\) 22.0000 1.72848
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 10.0000 0.780869
\(165\) 24.0000 1.86840
\(166\) −30.0000 −2.32845
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) −24.0000 −1.84072
\(171\) −2.00000 −0.152944
\(172\) −2.00000 −0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −24.0000 −1.81944
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 24.0000 1.80395
\(178\) 8.00000 0.599625
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −8.00000 −0.596285
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) −9.00000 −0.658145
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −16.0000 −1.16076
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 16.0000 1.15470
\(193\) 3.00000 0.215945 0.107972 0.994154i \(-0.465564\pi\)
0.107972 + 0.994154i \(0.465564\pi\)
\(194\) −14.0000 −1.00514
\(195\) −40.0000 −2.86446
\(196\) −14.0000 −1.00000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −6.00000 −0.426401
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 18.0000 1.26648
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) −20.0000 −1.39686
\(206\) −2.00000 −0.139347
\(207\) −1.00000 −0.0695048
\(208\) 20.0000 1.38675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −10.0000 −0.686803
\(213\) −4.00000 −0.274075
\(214\) 24.0000 1.64061
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −4.00000 −0.270295
\(220\) −24.0000 −1.61808
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 40.0000 2.66076
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 8.00000 0.529813
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 10.0000 0.653720
\(235\) −16.0000 −1.04372
\(236\) −24.0000 −1.56227
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −32.0000 −2.06559
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 4.00000 0.257130
\(243\) 10.0000 0.641500
\(244\) 4.00000 0.256074
\(245\) 28.0000 1.78885
\(246\) 20.0000 1.27515
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) −30.0000 −1.90117
\(250\) 48.0000 3.03579
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −2.00000 −0.125491
\(255\) −24.0000 −1.50294
\(256\) 16.0000 1.00000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 40.0000 2.48069
\(261\) −6.00000 −0.371391
\(262\) −16.0000 −0.988483
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −6.00000 −0.366508
\(269\) −25.0000 −1.52428 −0.762138 0.647414i \(-0.775850\pi\)
−0.762138 + 0.647414i \(0.775850\pi\)
\(270\) 32.0000 1.94746
\(271\) 23.0000 1.39715 0.698575 0.715537i \(-0.253818\pi\)
0.698575 + 0.715537i \(0.253818\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 33.0000 1.98997
\(276\) 4.00000 0.240772
\(277\) −32.0000 −1.92269 −0.961347 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) −38.0000 −2.27909
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 16.0000 0.952786
\(283\) 21.0000 1.24832 0.624160 0.781296i \(-0.285441\pi\)
0.624160 + 0.781296i \(0.285441\pi\)
\(284\) 4.00000 0.237356
\(285\) −16.0000 −0.947758
\(286\) 30.0000 1.77394
\(287\) 0 0
\(288\) 8.00000 0.471405
\(289\) −8.00000 −0.470588
\(290\) −48.0000 −2.81866
\(291\) −14.0000 −0.820695
\(292\) 4.00000 0.234082
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −28.0000 −1.63299
\(295\) 48.0000 2.79467
\(296\) 0 0
\(297\) 12.0000 0.696311
\(298\) −24.0000 −1.39028
\(299\) 5.00000 0.289157
\(300\) −44.0000 −2.54034
\(301\) 0 0
\(302\) 40.0000 2.30174
\(303\) 18.0000 1.03407
\(304\) 8.00000 0.458831
\(305\) −8.00000 −0.458079
\(306\) 6.00000 0.342997
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) −8.00000 −0.454369
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) −20.0000 −1.12154
\(319\) −18.0000 −1.00781
\(320\) 32.0000 1.78885
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) −22.0000 −1.22222
\(325\) −55.0000 −3.05085
\(326\) −28.0000 −1.55078
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) −48.0000 −2.64231
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) 30.0000 1.64646
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −3.00000 −0.163420 −0.0817102 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) −24.0000 −1.30543
\(339\) 40.0000 2.17250
\(340\) 24.0000 1.30158
\(341\) −3.00000 −0.162459
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) −12.0000 −0.645124
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 24.0000 1.28654
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 24.0000 1.27920
\(353\) −31.0000 −1.64996 −0.824982 0.565159i \(-0.808815\pi\)
−0.824982 + 0.565159i \(0.808815\pi\)
\(354\) −48.0000 −2.55117
\(355\) −8.00000 −0.424596
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) −40.0000 −2.11407
\(359\) 19.0000 1.00278 0.501391 0.865221i \(-0.332822\pi\)
0.501391 + 0.865221i \(0.332822\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −20.0000 −1.05118
\(363\) 4.00000 0.209946
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 8.00000 0.418167
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 4.00000 0.208514
\(369\) 5.00000 0.260290
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 18.0000 0.930758
\(375\) 48.0000 2.47871
\(376\) 0 0
\(377\) 30.0000 1.54508
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 16.0000 0.820783
\(381\) −2.00000 −0.102463
\(382\) 32.0000 1.63726
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −1.00000 −0.0508329
\(388\) 14.0000 0.710742
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 80.0000 4.05096
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) −4.00000 −0.201517
\(395\) 32.0000 1.61009
\(396\) 6.00000 0.301511
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −28.0000 −1.40351
\(399\) 0 0
\(400\) −44.0000 −2.20000
\(401\) 5.00000 0.249688 0.124844 0.992176i \(-0.460157\pi\)
0.124844 + 0.992176i \(0.460157\pi\)
\(402\) −12.0000 −0.598506
\(403\) 5.00000 0.249068
\(404\) −18.0000 −0.895533
\(405\) 44.0000 2.18638
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 40.0000 1.97546
\(411\) −12.0000 −0.591916
\(412\) 2.00000 0.0985329
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) −60.0000 −2.94528
\(416\) −40.0000 −1.96116
\(417\) −38.0000 −1.86087
\(418\) 12.0000 0.586939
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) −33.0000 −1.60074
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −24.0000 −1.16008
\(429\) 30.0000 1.44841
\(430\) −8.00000 −0.385794
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) −16.0000 −0.769800
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 0 0
\(435\) −48.0000 −2.30142
\(436\) 14.0000 0.670478
\(437\) 2.00000 0.0956730
\(438\) 8.00000 0.382255
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −30.0000 −1.42695
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 56.0000 2.65168
\(447\) −24.0000 −1.13516
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −22.0000 −1.03709
\(451\) 15.0000 0.706322
\(452\) −40.0000 −1.88144
\(453\) 40.0000 1.87936
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 30.0000 1.40181
\(459\) −12.0000 −0.560112
\(460\) 8.00000 0.373002
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 24.0000 1.11417
\(465\) −8.00000 −0.370991
\(466\) −12.0000 −0.555889
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −10.0000 −0.462250
\(469\) 0 0
\(470\) 32.0000 1.47605
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) −32.0000 −1.46981
\(475\) −22.0000 −1.00943
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) −32.0000 −1.46365
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 64.0000 2.92119
\(481\) 0 0
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) −28.0000 −1.27141
\(486\) −20.0000 −0.907218
\(487\) 36.0000 1.63132 0.815658 0.578535i \(-0.196375\pi\)
0.815658 + 0.578535i \(0.196375\pi\)
\(488\) 0 0
\(489\) −28.0000 −1.26620
\(490\) −56.0000 −2.52982
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) −20.0000 −0.901670
\(493\) 18.0000 0.810679
\(494\) −20.0000 −0.899843
\(495\) −12.0000 −0.539360
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 60.0000 2.68866
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) −48.0000 −2.14663
\(501\) 18.0000 0.804181
\(502\) 46.0000 2.05308
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 6.00000 0.266733
\(507\) −24.0000 −1.06588
\(508\) 2.00000 0.0887357
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 48.0000 2.12548
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) −8.00000 −0.353209
\(514\) 48.0000 2.11719
\(515\) −4.00000 −0.176261
\(516\) 4.00000 0.176090
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 12.0000 0.525226
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 36.0000 1.56967
\(527\) 3.00000 0.130682
\(528\) 24.0000 1.04447
\(529\) −22.0000 −0.956522
\(530\) −40.0000 −1.73749
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −25.0000 −1.08287
\(534\) −16.0000 −0.692388
\(535\) 48.0000 2.07522
\(536\) 0 0
\(537\) −40.0000 −1.72613
\(538\) 50.0000 2.15565
\(539\) −21.0000 −0.904534
\(540\) −32.0000 −1.37706
\(541\) 1.00000 0.0429934 0.0214967 0.999769i \(-0.493157\pi\)
0.0214967 + 0.999769i \(0.493157\pi\)
\(542\) −46.0000 −1.97587
\(543\) −20.0000 −0.858282
\(544\) −24.0000 −1.02899
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −29.0000 −1.23995 −0.619975 0.784621i \(-0.712857\pi\)
−0.619975 + 0.784621i \(0.712857\pi\)
\(548\) 12.0000 0.512615
\(549\) 2.00000 0.0853579
\(550\) −66.0000 −2.81425
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 64.0000 2.71910
\(555\) 0 0
\(556\) 38.0000 1.61156
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 2.00000 0.0846668
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) −38.0000 −1.60293
\(563\) 37.0000 1.55936 0.779682 0.626176i \(-0.215381\pi\)
0.779682 + 0.626176i \(0.215381\pi\)
\(564\) −16.0000 −0.673722
\(565\) 80.0000 3.36563
\(566\) −42.0000 −1.76539
\(567\) 0 0
\(568\) 0 0
\(569\) 7.00000 0.293455 0.146728 0.989177i \(-0.453126\pi\)
0.146728 + 0.989177i \(0.453126\pi\)
\(570\) 32.0000 1.34033
\(571\) −14.0000 −0.585882 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) −30.0000 −1.25436
\(573\) 32.0000 1.33682
\(574\) 0 0
\(575\) −11.0000 −0.458732
\(576\) −8.00000 −0.333333
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 16.0000 0.665512
\(579\) −6.00000 −0.249351
\(580\) 48.0000 1.99309
\(581\) 0 0
\(582\) 28.0000 1.16064
\(583\) −15.0000 −0.621237
\(584\) 0 0
\(585\) 20.0000 0.826898
\(586\) 52.0000 2.14810
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 28.0000 1.15470
\(589\) 2.00000 0.0824086
\(590\) −96.0000 −3.95226
\(591\) −4.00000 −0.164538
\(592\) 0 0
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) −24.0000 −0.984732
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) −28.0000 −1.14596
\(598\) −10.0000 −0.408930
\(599\) −1.00000 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) −40.0000 −1.62758
\(605\) 8.00000 0.325246
\(606\) −36.0000 −1.46240
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −16.0000 −0.648886
\(609\) 0 0
\(610\) 16.0000 0.647821
\(611\) −20.0000 −0.809113
\(612\) −6.00000 −0.242536
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 14.0000 0.564994
\(615\) 40.0000 1.61296
\(616\) 0 0
\(617\) −21.0000 −0.845428 −0.422714 0.906263i \(-0.638923\pi\)
−0.422714 + 0.906263i \(0.638923\pi\)
\(618\) 4.00000 0.160904
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 8.00000 0.321288
\(621\) −4.00000 −0.160514
\(622\) −30.0000 −1.20289
\(623\) 0 0
\(624\) −40.0000 −1.60128
\(625\) 41.0000 1.64000
\(626\) −44.0000 −1.75859
\(627\) 12.0000 0.479234
\(628\) −20.0000 −0.798087
\(629\) 0 0
\(630\) 0 0
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) −4.00000 −0.158735
\(636\) 20.0000 0.793052
\(637\) 35.0000 1.38675
\(638\) 36.0000 1.42525
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) −48.0000 −1.89441
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −12.0000 −0.472134
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 110.000 4.31455
\(651\) 0 0
\(652\) 28.0000 1.09656
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 28.0000 1.09489
\(655\) −32.0000 −1.25034
\(656\) −20.0000 −0.780869
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 48.0000 1.86840
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 52.0000 2.02104
\(663\) −30.0000 −1.16510
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) −18.0000 −0.696441
\(669\) 56.0000 2.16509
\(670\) −24.0000 −0.927201
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 6.00000 0.231111
\(675\) 44.0000 1.69356
\(676\) 24.0000 0.923077
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −80.0000 −3.07238
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 6.00000 0.229752
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) −4.00000 −0.152944
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 30.0000 1.14457
\(688\) 4.00000 0.152499
\(689\) 25.0000 0.952424
\(690\) 16.0000 0.609110
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −56.0000 −2.12573
\(695\) −76.0000 −2.88284
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) −28.0000 −1.05982
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 40.0000 1.50970
\(703\) 0 0
\(704\) −24.0000 −0.904534
\(705\) 32.0000 1.20519
\(706\) 62.0000 2.33340
\(707\) 0 0
\(708\) 48.0000 1.80395
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 16.0000 0.600469
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) 60.0000 2.24387
\(716\) 40.0000 1.49487
\(717\) −32.0000 −1.19506
\(718\) −38.0000 −1.41815
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 16.0000 0.596285
\(721\) 0 0
\(722\) 30.0000 1.11648
\(723\) 24.0000 0.892570
\(724\) 20.0000 0.743294
\(725\) −66.0000 −2.45118
\(726\) −8.00000 −0.296908
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 16.0000 0.592187
\(731\) 3.00000 0.110959
\(732\) −8.00000 −0.295689
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 64.0000 2.36228
\(735\) −56.0000 −2.06559
\(736\) −8.00000 −0.294884
\(737\) −9.00000 −0.331519
\(738\) −10.0000 −0.368105
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −48.0000 −1.75858
\(746\) −64.0000 −2.34321
\(747\) 15.0000 0.548821
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) −96.0000 −3.50542
\(751\) −6.00000 −0.218943 −0.109472 0.993990i \(-0.534916\pi\)
−0.109472 + 0.993990i \(0.534916\pi\)
\(752\) −16.0000 −0.583460
\(753\) 46.0000 1.67633
\(754\) −60.0000 −2.18507
\(755\) 80.0000 2.91150
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) −22.0000 −0.799076
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) −32.0000 −1.15772
\(765\) 12.0000 0.433861
\(766\) −64.0000 −2.31241
\(767\) 60.0000 2.16647
\(768\) −32.0000 −1.15470
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 6.00000 0.215945
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 2.00000 0.0718885
\(775\) −11.0000 −0.395132
\(776\) 0 0
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) −10.0000 −0.358287
\(780\) −80.0000 −2.86446
\(781\) 6.00000 0.214697
\(782\) −6.00000 −0.214560
\(783\) −24.0000 −0.857690
\(784\) 28.0000 1.00000
\(785\) 40.0000 1.42766
\(786\) 32.0000 1.14140
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 4.00000 0.142494
\(789\) 36.0000 1.28163
\(790\) −64.0000 −2.27702
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 12.0000 0.425864
\(795\) −40.0000 −1.41865
\(796\) 28.0000 0.992434
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 88.0000 3.11127
\(801\) −4.00000 −0.141333
\(802\) −10.0000 −0.353112
\(803\) 6.00000 0.211735
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 50.0000 1.76008
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −88.0000 −3.09200
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) −46.0000 −1.61329
\(814\) 0 0
\(815\) −56.0000 −1.96159
\(816\) −24.0000 −0.840168
\(817\) 2.00000 0.0699711
\(818\) 48.0000 1.67828
\(819\) 0 0
\(820\) −40.0000 −1.39686
\(821\) 49.0000 1.71011 0.855056 0.518536i \(-0.173523\pi\)
0.855056 + 0.518536i \(0.173523\pi\)
\(822\) 24.0000 0.837096
\(823\) −1.00000 −0.0348578 −0.0174289 0.999848i \(-0.505548\pi\)
−0.0174289 + 0.999848i \(0.505548\pi\)
\(824\) 0 0
\(825\) −66.0000 −2.29783
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 44.0000 1.52818 0.764092 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 120.000 4.16526
\(831\) 64.0000 2.22014
\(832\) 40.0000 1.38675
\(833\) 21.0000 0.727607
\(834\) 76.0000 2.63166
\(835\) 36.0000 1.24583
\(836\) −12.0000 −0.415029
\(837\) −4.00000 −0.138260
\(838\) 56.0000 1.93449
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000 0.689246
\(843\) −38.0000 −1.30879
\(844\) 4.00000 0.137686
\(845\) −48.0000 −1.65125
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 20.0000 0.686803
\(849\) −42.0000 −1.44144
\(850\) 66.0000 2.26378
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) −29.0000 −0.992941 −0.496471 0.868054i \(-0.665371\pi\)
−0.496471 + 0.868054i \(0.665371\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) −60.0000 −2.04837
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 42.0000 1.43053
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 32.0000 1.08866
\(865\) −24.0000 −0.816024
\(866\) 24.0000 0.815553
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 96.0000 3.25470
\(871\) 15.0000 0.508256
\(872\) 0 0
\(873\) 7.00000 0.236914
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 41.0000 1.38447 0.692236 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) −34.0000 −1.14744
\(879\) 52.0000 1.75392
\(880\) 48.0000 1.61808
\(881\) 37.0000 1.24656 0.623281 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 14.0000 0.471405
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 30.0000 1.00901
\(885\) −96.0000 −3.22700
\(886\) 8.00000 0.268765
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −32.0000 −1.07264
\(891\) −33.0000 −1.10554
\(892\) −56.0000 −1.87502
\(893\) −8.00000 −0.267710
\(894\) 48.0000 1.60536
\(895\) −80.0000 −2.67411
\(896\) 0 0
\(897\) −10.0000 −0.333890
\(898\) −60.0000 −2.00223
\(899\) 6.00000 0.200111
\(900\) 22.0000 0.733333
\(901\) 15.0000 0.499722
\(902\) −30.0000 −0.998891
\(903\) 0 0
\(904\) 0 0
\(905\) −40.0000 −1.32964
\(906\) −80.0000 −2.65782
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) −8.00000 −0.265489
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) −22.0000 −0.728893 −0.364446 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) −16.0000 −0.529813
\(913\) 45.0000 1.48928
\(914\) 36.0000 1.19077
\(915\) 16.0000 0.528944
\(916\) −30.0000 −0.991228
\(917\) 0 0
\(918\) 24.0000 0.792118
\(919\) −49.0000 −1.61636 −0.808180 0.588935i \(-0.799547\pi\)
−0.808180 + 0.588935i \(0.799547\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) −60.0000 −1.97599
\(923\) −10.0000 −0.329154
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 1.00000 0.0328443
\(928\) −48.0000 −1.57568
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 16.0000 0.524661
\(931\) 14.0000 0.458831
\(932\) 12.0000 0.393073
\(933\) −30.0000 −0.982156
\(934\) −12.0000 −0.392652
\(935\) 36.0000 1.17733
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 0 0
\(939\) −44.0000 −1.43589
\(940\) −32.0000 −1.04372
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) −40.0000 −1.30327
\(943\) −5.00000 −0.162822
\(944\) 48.0000 1.56227
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 32.0000 1.03931
\(949\) −10.0000 −0.324614
\(950\) 44.0000 1.42755
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 10.0000 0.323762
\(955\) 64.0000 2.07099
\(956\) 32.0000 1.03495
\(957\) 36.0000 1.16371
\(958\) −42.0000 −1.35696
\(959\) 0 0
\(960\) −64.0000 −2.06559
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −24.0000 −0.772988
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 37.0000 1.18984 0.594920 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 56.0000 1.79805
\(971\) −13.0000 −0.417190 −0.208595 0.978002i \(-0.566889\pi\)
−0.208595 + 0.978002i \(0.566889\pi\)
\(972\) 20.0000 0.641500
\(973\) 0 0
\(974\) −72.0000 −2.30703
\(975\) 110.000 3.52282
\(976\) −8.00000 −0.256074
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 56.0000 1.79068
\(979\) −12.0000 −0.383522
\(980\) 56.0000 1.78885
\(981\) 7.00000 0.223493
\(982\) 12.0000 0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 20.0000 0.636285
\(989\) 1.00000 0.0317982
\(990\) 24.0000 0.762770
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −8.00000 −0.254000
\(993\) 52.0000 1.65017
\(994\) 0 0
\(995\) −56.0000 −1.77532
\(996\) −60.0000 −1.90117
\(997\) 4.00000 0.126681 0.0633406 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) 16.0000 0.506471
\(999\) 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 43.2.a.a.1.1 1
3.2 odd 2 387.2.a.e.1.1 1
4.3 odd 2 688.2.a.b.1.1 1
5.2 odd 4 1075.2.b.b.474.1 2
5.3 odd 4 1075.2.b.b.474.2 2
5.4 even 2 1075.2.a.h.1.1 1
7.6 odd 2 2107.2.a.a.1.1 1
8.3 odd 2 2752.2.a.b.1.1 1
8.5 even 2 2752.2.a.f.1.1 1
11.10 odd 2 5203.2.a.a.1.1 1
12.11 even 2 6192.2.a.ba.1.1 1
13.12 even 2 7267.2.a.a.1.1 1
15.14 odd 2 9675.2.a.b.1.1 1
43.42 odd 2 1849.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.a.a.1.1 1 1.1 even 1 trivial
387.2.a.e.1.1 1 3.2 odd 2
688.2.a.b.1.1 1 4.3 odd 2
1075.2.a.h.1.1 1 5.4 even 2
1075.2.b.b.474.1 2 5.2 odd 4
1075.2.b.b.474.2 2 5.3 odd 4
1849.2.a.d.1.1 1 43.42 odd 2
2107.2.a.a.1.1 1 7.6 odd 2
2752.2.a.b.1.1 1 8.3 odd 2
2752.2.a.f.1.1 1 8.5 even 2
5203.2.a.a.1.1 1 11.10 odd 2
6192.2.a.ba.1.1 1 12.11 even 2
7267.2.a.a.1.1 1 13.12 even 2
9675.2.a.b.1.1 1 15.14 odd 2