Properties

Label 5445.2.a.bv.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} +O(q^{10})\) \(q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} -1.39026 q^{10} +1.41837 q^{13} +1.77618 q^{14} -3.86111 q^{16} +5.00000 q^{17} -0.158146 q^{19} -0.0671858 q^{20} +5.00829 q^{23} +1.00000 q^{25} -1.97189 q^{26} +0.0858360 q^{28} +6.27515 q^{29} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -4.69991 q^{37} +0.219863 q^{38} +2.87392 q^{40} +7.58597 q^{41} -5.41324 q^{43} -6.96281 q^{46} +8.23211 q^{47} -5.36776 q^{49} -1.39026 q^{50} -0.0952940 q^{52} -9.36648 q^{53} -3.67169 q^{56} -8.72406 q^{58} -8.09846 q^{59} +14.2917 q^{61} -4.24002 q^{62} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} +1.77618 q^{70} +6.77618 q^{71} -8.66107 q^{73} +6.53409 q^{74} +0.0106252 q^{76} +2.54572 q^{79} -3.86111 q^{80} -10.5464 q^{82} +10.1221 q^{83} +5.00000 q^{85} +7.52580 q^{86} -11.0447 q^{89} -1.81209 q^{91} -0.336486 q^{92} -11.4447 q^{94} -0.158146 q^{95} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39026 −0.983060 −0.491530 0.870861i \(-0.663562\pi\)
−0.491530 + 0.870861i \(0.663562\pi\)
\(3\) 0 0
\(4\) −0.0671858 −0.0335929
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.27759 −0.482884 −0.241442 0.970415i \(-0.577620\pi\)
−0.241442 + 0.970415i \(0.577620\pi\)
\(8\) 2.87392 1.01608
\(9\) 0 0
\(10\) −1.39026 −0.439638
\(11\) 0 0
\(12\) 0 0
\(13\) 1.41837 0.393384 0.196692 0.980465i \(-0.436980\pi\)
0.196692 + 0.980465i \(0.436980\pi\)
\(14\) 1.77618 0.474704
\(15\) 0 0
\(16\) −3.86111 −0.965279
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −0.158146 −0.0362811 −0.0181406 0.999835i \(-0.505775\pi\)
−0.0181406 + 0.999835i \(0.505775\pi\)
\(20\) −0.0671858 −0.0150232
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00829 1.04430 0.522150 0.852853i \(-0.325130\pi\)
0.522150 + 0.852853i \(0.325130\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.97189 −0.386720
\(27\) 0 0
\(28\) 0.0858360 0.0162215
\(29\) 6.27515 1.16527 0.582633 0.812736i \(-0.302023\pi\)
0.582633 + 0.812736i \(0.302023\pi\)
\(30\) 0 0
\(31\) 3.04981 0.547763 0.273881 0.961763i \(-0.411692\pi\)
0.273881 + 0.961763i \(0.411692\pi\)
\(32\) −0.379898 −0.0671570
\(33\) 0 0
\(34\) −6.95128 −1.19214
\(35\) −1.27759 −0.215952
\(36\) 0 0
\(37\) −4.69991 −0.772661 −0.386330 0.922360i \(-0.626258\pi\)
−0.386330 + 0.922360i \(0.626258\pi\)
\(38\) 0.219863 0.0356665
\(39\) 0 0
\(40\) 2.87392 0.454407
\(41\) 7.58597 1.18473 0.592365 0.805670i \(-0.298194\pi\)
0.592365 + 0.805670i \(0.298194\pi\)
\(42\) 0 0
\(43\) −5.41324 −0.825512 −0.412756 0.910842i \(-0.635434\pi\)
−0.412756 + 0.910842i \(0.635434\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.96281 −1.02661
\(47\) 8.23211 1.20078 0.600388 0.799709i \(-0.295013\pi\)
0.600388 + 0.799709i \(0.295013\pi\)
\(48\) 0 0
\(49\) −5.36776 −0.766823
\(50\) −1.39026 −0.196612
\(51\) 0 0
\(52\) −0.0952940 −0.0132149
\(53\) −9.36648 −1.28659 −0.643293 0.765620i \(-0.722432\pi\)
−0.643293 + 0.765620i \(0.722432\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.67169 −0.490651
\(57\) 0 0
\(58\) −8.72406 −1.14553
\(59\) −8.09846 −1.05433 −0.527165 0.849763i \(-0.676745\pi\)
−0.527165 + 0.849763i \(0.676745\pi\)
\(60\) 0 0
\(61\) 14.2917 1.82987 0.914934 0.403603i \(-0.132242\pi\)
0.914934 + 0.403603i \(0.132242\pi\)
\(62\) −4.24002 −0.538484
\(63\) 0 0
\(64\) 8.25038 1.03130
\(65\) 1.41837 0.175927
\(66\) 0 0
\(67\) −7.38362 −0.902053 −0.451026 0.892511i \(-0.648942\pi\)
−0.451026 + 0.892511i \(0.648942\pi\)
\(68\) −0.335929 −0.0407374
\(69\) 0 0
\(70\) 1.77618 0.212294
\(71\) 6.77618 0.804185 0.402092 0.915599i \(-0.368283\pi\)
0.402092 + 0.915599i \(0.368283\pi\)
\(72\) 0 0
\(73\) −8.66107 −1.01370 −0.506851 0.862034i \(-0.669190\pi\)
−0.506851 + 0.862034i \(0.669190\pi\)
\(74\) 6.53409 0.759572
\(75\) 0 0
\(76\) 0.0106252 0.00121879
\(77\) 0 0
\(78\) 0 0
\(79\) 2.54572 0.286416 0.143208 0.989693i \(-0.454258\pi\)
0.143208 + 0.989693i \(0.454258\pi\)
\(80\) −3.86111 −0.431686
\(81\) 0 0
\(82\) −10.5464 −1.16466
\(83\) 10.1221 1.11105 0.555524 0.831501i \(-0.312518\pi\)
0.555524 + 0.831501i \(0.312518\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 7.52580 0.811527
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0447 −1.17073 −0.585367 0.810768i \(-0.699050\pi\)
−0.585367 + 0.810768i \(0.699050\pi\)
\(90\) 0 0
\(91\) −1.81209 −0.189959
\(92\) −0.336486 −0.0350811
\(93\) 0 0
\(94\) −11.4447 −1.18044
\(95\) −0.158146 −0.0162254
\(96\) 0 0
\(97\) 6.41196 0.651036 0.325518 0.945536i \(-0.394461\pi\)
0.325518 + 0.945536i \(0.394461\pi\)
\(98\) 7.46257 0.753833
\(99\) 0 0
\(100\) −0.0671858 −0.00671858
\(101\) 8.77143 0.872790 0.436395 0.899755i \(-0.356255\pi\)
0.436395 + 0.899755i \(0.356255\pi\)
\(102\) 0 0
\(103\) 11.5666 1.13969 0.569847 0.821751i \(-0.307002\pi\)
0.569847 + 0.821751i \(0.307002\pi\)
\(104\) 4.07627 0.399711
\(105\) 0 0
\(106\) 13.0218 1.26479
\(107\) 15.3467 1.48362 0.741809 0.670611i \(-0.233968\pi\)
0.741809 + 0.670611i \(0.233968\pi\)
\(108\) 0 0
\(109\) −14.4004 −1.37931 −0.689656 0.724137i \(-0.742238\pi\)
−0.689656 + 0.724137i \(0.742238\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.93293 0.466118
\(113\) −19.3645 −1.82166 −0.910831 0.412780i \(-0.864558\pi\)
−0.910831 + 0.412780i \(0.864558\pi\)
\(114\) 0 0
\(115\) 5.00829 0.467026
\(116\) −0.421601 −0.0391446
\(117\) 0 0
\(118\) 11.2589 1.03647
\(119\) −6.38796 −0.585583
\(120\) 0 0
\(121\) 0 0
\(122\) −19.8692 −1.79887
\(123\) 0 0
\(124\) −0.204904 −0.0184009
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.02415 0.534557 0.267278 0.963619i \(-0.413876\pi\)
0.267278 + 0.963619i \(0.413876\pi\)
\(128\) −10.7104 −0.946671
\(129\) 0 0
\(130\) −1.97189 −0.172946
\(131\) −18.7278 −1.63626 −0.818130 0.575034i \(-0.804989\pi\)
−0.818130 + 0.575034i \(0.804989\pi\)
\(132\) 0 0
\(133\) 0.202046 0.0175196
\(134\) 10.2651 0.886772
\(135\) 0 0
\(136\) 14.3696 1.23218
\(137\) −3.03719 −0.259485 −0.129742 0.991548i \(-0.541415\pi\)
−0.129742 + 0.991548i \(0.541415\pi\)
\(138\) 0 0
\(139\) 2.21192 0.187612 0.0938062 0.995590i \(-0.470097\pi\)
0.0938062 + 0.995590i \(0.470097\pi\)
\(140\) 0.0858360 0.00725446
\(141\) 0 0
\(142\) −9.42063 −0.790562
\(143\) 0 0
\(144\) 0 0
\(145\) 6.27515 0.521122
\(146\) 12.0411 0.996529
\(147\) 0 0
\(148\) 0.315767 0.0259559
\(149\) 1.48122 0.121346 0.0606730 0.998158i \(-0.480675\pi\)
0.0606730 + 0.998158i \(0.480675\pi\)
\(150\) 0 0
\(151\) 9.03395 0.735173 0.367586 0.929989i \(-0.380184\pi\)
0.367586 + 0.929989i \(0.380184\pi\)
\(152\) −0.454498 −0.0368647
\(153\) 0 0
\(154\) 0 0
\(155\) 3.04981 0.244967
\(156\) 0 0
\(157\) −17.6377 −1.40764 −0.703820 0.710379i \(-0.748524\pi\)
−0.703820 + 0.710379i \(0.748524\pi\)
\(158\) −3.53921 −0.281564
\(159\) 0 0
\(160\) −0.379898 −0.0300335
\(161\) −6.39855 −0.504276
\(162\) 0 0
\(163\) 23.5040 1.84098 0.920489 0.390770i \(-0.127791\pi\)
0.920489 + 0.390770i \(0.127791\pi\)
\(164\) −0.509669 −0.0397985
\(165\) 0 0
\(166\) −14.0724 −1.09223
\(167\) 0.602731 0.0466407 0.0233203 0.999728i \(-0.492576\pi\)
0.0233203 + 0.999728i \(0.492576\pi\)
\(168\) 0 0
\(169\) −10.9882 −0.845249
\(170\) −6.95128 −0.533139
\(171\) 0 0
\(172\) 0.363693 0.0277313
\(173\) 9.66389 0.734732 0.367366 0.930076i \(-0.380260\pi\)
0.367366 + 0.930076i \(0.380260\pi\)
\(174\) 0 0
\(175\) −1.27759 −0.0965768
\(176\) 0 0
\(177\) 0 0
\(178\) 15.3550 1.15090
\(179\) 1.19571 0.0893717 0.0446859 0.999001i \(-0.485771\pi\)
0.0446859 + 0.999001i \(0.485771\pi\)
\(180\) 0 0
\(181\) 15.5741 1.15761 0.578806 0.815466i \(-0.303519\pi\)
0.578806 + 0.815466i \(0.303519\pi\)
\(182\) 2.51927 0.186741
\(183\) 0 0
\(184\) 14.3934 1.06110
\(185\) −4.69991 −0.345544
\(186\) 0 0
\(187\) 0 0
\(188\) −0.553081 −0.0403376
\(189\) 0 0
\(190\) 0.219863 0.0159506
\(191\) −18.4016 −1.33149 −0.665747 0.746178i \(-0.731887\pi\)
−0.665747 + 0.746178i \(0.731887\pi\)
\(192\) 0 0
\(193\) −1.56799 −0.112866 −0.0564331 0.998406i \(-0.517973\pi\)
−0.0564331 + 0.998406i \(0.517973\pi\)
\(194\) −8.91428 −0.640008
\(195\) 0 0
\(196\) 0.360637 0.0257598
\(197\) 7.97000 0.567839 0.283920 0.958848i \(-0.408365\pi\)
0.283920 + 0.958848i \(0.408365\pi\)
\(198\) 0 0
\(199\) 3.53141 0.250335 0.125167 0.992136i \(-0.460053\pi\)
0.125167 + 0.992136i \(0.460053\pi\)
\(200\) 2.87392 0.203217
\(201\) 0 0
\(202\) −12.1945 −0.858005
\(203\) −8.01707 −0.562688
\(204\) 0 0
\(205\) 7.58597 0.529827
\(206\) −16.0806 −1.12039
\(207\) 0 0
\(208\) −5.47647 −0.379725
\(209\) 0 0
\(210\) 0 0
\(211\) −20.2550 −1.39441 −0.697207 0.716870i \(-0.745574\pi\)
−0.697207 + 0.716870i \(0.745574\pi\)
\(212\) 0.629295 0.0432201
\(213\) 0 0
\(214\) −21.3358 −1.45849
\(215\) −5.41324 −0.369180
\(216\) 0 0
\(217\) −3.89642 −0.264506
\(218\) 20.0203 1.35595
\(219\) 0 0
\(220\) 0 0
\(221\) 7.09183 0.477048
\(222\) 0 0
\(223\) 22.5500 1.51006 0.755029 0.655691i \(-0.227623\pi\)
0.755029 + 0.655691i \(0.227623\pi\)
\(224\) 0.485354 0.0324291
\(225\) 0 0
\(226\) 26.9217 1.79080
\(227\) 18.8062 1.24821 0.624105 0.781341i \(-0.285464\pi\)
0.624105 + 0.781341i \(0.285464\pi\)
\(228\) 0 0
\(229\) −23.8320 −1.57486 −0.787431 0.616403i \(-0.788589\pi\)
−0.787431 + 0.616403i \(0.788589\pi\)
\(230\) −6.96281 −0.459114
\(231\) 0 0
\(232\) 18.0343 1.18401
\(233\) −10.0918 −0.661137 −0.330569 0.943782i \(-0.607241\pi\)
−0.330569 + 0.943782i \(0.607241\pi\)
\(234\) 0 0
\(235\) 8.23211 0.537004
\(236\) 0.544102 0.0354180
\(237\) 0 0
\(238\) 8.88090 0.575663
\(239\) 0.167227 0.0108170 0.00540850 0.999985i \(-0.498278\pi\)
0.00540850 + 0.999985i \(0.498278\pi\)
\(240\) 0 0
\(241\) −0.965256 −0.0621776 −0.0310888 0.999517i \(-0.509897\pi\)
−0.0310888 + 0.999517i \(0.509897\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −0.960201 −0.0614706
\(245\) −5.36776 −0.342934
\(246\) 0 0
\(247\) −0.224308 −0.0142724
\(248\) 8.76492 0.556573
\(249\) 0 0
\(250\) −1.39026 −0.0879276
\(251\) 23.5102 1.48395 0.741975 0.670428i \(-0.233889\pi\)
0.741975 + 0.670428i \(0.233889\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.37512 −0.525502
\(255\) 0 0
\(256\) −1.61062 −0.100664
\(257\) 6.58273 0.410620 0.205310 0.978697i \(-0.434180\pi\)
0.205310 + 0.978697i \(0.434180\pi\)
\(258\) 0 0
\(259\) 6.00457 0.373106
\(260\) −0.0952940 −0.00590988
\(261\) 0 0
\(262\) 26.0365 1.60854
\(263\) 12.4538 0.767936 0.383968 0.923346i \(-0.374557\pi\)
0.383968 + 0.923346i \(0.374557\pi\)
\(264\) 0 0
\(265\) −9.36648 −0.575378
\(266\) −0.280895 −0.0172228
\(267\) 0 0
\(268\) 0.496074 0.0303026
\(269\) −20.7729 −1.26655 −0.633274 0.773927i \(-0.718290\pi\)
−0.633274 + 0.773927i \(0.718290\pi\)
\(270\) 0 0
\(271\) −14.3903 −0.874146 −0.437073 0.899426i \(-0.643985\pi\)
−0.437073 + 0.899426i \(0.643985\pi\)
\(272\) −19.3056 −1.17057
\(273\) 0 0
\(274\) 4.22247 0.255089
\(275\) 0 0
\(276\) 0 0
\(277\) 17.4872 1.05070 0.525352 0.850885i \(-0.323934\pi\)
0.525352 + 0.850885i \(0.323934\pi\)
\(278\) −3.07513 −0.184434
\(279\) 0 0
\(280\) −3.67169 −0.219426
\(281\) −18.8860 −1.12664 −0.563322 0.826238i \(-0.690477\pi\)
−0.563322 + 0.826238i \(0.690477\pi\)
\(282\) 0 0
\(283\) 13.5813 0.807326 0.403663 0.914908i \(-0.367737\pi\)
0.403663 + 0.914908i \(0.367737\pi\)
\(284\) −0.455263 −0.0270149
\(285\) 0 0
\(286\) 0 0
\(287\) −9.69177 −0.572087
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) −8.72406 −0.512295
\(291\) 0 0
\(292\) 0.581901 0.0340532
\(293\) 19.9023 1.16271 0.581353 0.813651i \(-0.302523\pi\)
0.581353 + 0.813651i \(0.302523\pi\)
\(294\) 0 0
\(295\) −8.09846 −0.471511
\(296\) −13.5072 −0.785088
\(297\) 0 0
\(298\) −2.05927 −0.119290
\(299\) 7.10358 0.410811
\(300\) 0 0
\(301\) 6.91591 0.398626
\(302\) −12.5595 −0.722719
\(303\) 0 0
\(304\) 0.610619 0.0350214
\(305\) 14.2917 0.818342
\(306\) 0 0
\(307\) 14.9354 0.852410 0.426205 0.904627i \(-0.359850\pi\)
0.426205 + 0.904627i \(0.359850\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.24002 −0.240817
\(311\) 4.99700 0.283354 0.141677 0.989913i \(-0.454751\pi\)
0.141677 + 0.989913i \(0.454751\pi\)
\(312\) 0 0
\(313\) 9.39648 0.531120 0.265560 0.964094i \(-0.414443\pi\)
0.265560 + 0.964094i \(0.414443\pi\)
\(314\) 24.5209 1.38379
\(315\) 0 0
\(316\) −0.171037 −0.00962155
\(317\) −2.15253 −0.120898 −0.0604492 0.998171i \(-0.519253\pi\)
−0.0604492 + 0.998171i \(0.519253\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.25038 0.461210
\(321\) 0 0
\(322\) 8.89563 0.495734
\(323\) −0.790729 −0.0439973
\(324\) 0 0
\(325\) 1.41837 0.0786767
\(326\) −32.6766 −1.80979
\(327\) 0 0
\(328\) 21.8015 1.20378
\(329\) −10.5173 −0.579836
\(330\) 0 0
\(331\) −19.5116 −1.07245 −0.536227 0.844074i \(-0.680151\pi\)
−0.536227 + 0.844074i \(0.680151\pi\)
\(332\) −0.680063 −0.0373233
\(333\) 0 0
\(334\) −0.837950 −0.0458506
\(335\) −7.38362 −0.403410
\(336\) 0 0
\(337\) −31.6047 −1.72162 −0.860809 0.508929i \(-0.830042\pi\)
−0.860809 + 0.508929i \(0.830042\pi\)
\(338\) 15.2765 0.830931
\(339\) 0 0
\(340\) −0.335929 −0.0182183
\(341\) 0 0
\(342\) 0 0
\(343\) 15.8009 0.853171
\(344\) −15.5572 −0.838789
\(345\) 0 0
\(346\) −13.4353 −0.722286
\(347\) 9.38313 0.503713 0.251856 0.967765i \(-0.418959\pi\)
0.251856 + 0.967765i \(0.418959\pi\)
\(348\) 0 0
\(349\) 24.8131 1.32822 0.664108 0.747637i \(-0.268812\pi\)
0.664108 + 0.747637i \(0.268812\pi\)
\(350\) 1.77618 0.0949408
\(351\) 0 0
\(352\) 0 0
\(353\) 19.4788 1.03675 0.518375 0.855153i \(-0.326537\pi\)
0.518375 + 0.855153i \(0.326537\pi\)
\(354\) 0 0
\(355\) 6.77618 0.359642
\(356\) 0.742046 0.0393284
\(357\) 0 0
\(358\) −1.66235 −0.0878578
\(359\) 29.7701 1.57121 0.785603 0.618732i \(-0.212353\pi\)
0.785603 + 0.618732i \(0.212353\pi\)
\(360\) 0 0
\(361\) −18.9750 −0.998684
\(362\) −21.6520 −1.13800
\(363\) 0 0
\(364\) 0.121747 0.00638126
\(365\) −8.66107 −0.453341
\(366\) 0 0
\(367\) −6.43597 −0.335955 −0.167977 0.985791i \(-0.553724\pi\)
−0.167977 + 0.985791i \(0.553724\pi\)
\(368\) −19.3376 −1.00804
\(369\) 0 0
\(370\) 6.53409 0.339691
\(371\) 11.9665 0.621272
\(372\) 0 0
\(373\) 20.8924 1.08177 0.540883 0.841098i \(-0.318090\pi\)
0.540883 + 0.841098i \(0.318090\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 23.6584 1.22009
\(377\) 8.90045 0.458396
\(378\) 0 0
\(379\) −14.1991 −0.729359 −0.364680 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(380\) 0.0106252 0.000545059 0
\(381\) 0 0
\(382\) 25.5830 1.30894
\(383\) 26.4783 1.35298 0.676489 0.736453i \(-0.263501\pi\)
0.676489 + 0.736453i \(0.263501\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.17990 0.110954
\(387\) 0 0
\(388\) −0.430793 −0.0218702
\(389\) 36.4965 1.85044 0.925222 0.379427i \(-0.123879\pi\)
0.925222 + 0.379427i \(0.123879\pi\)
\(390\) 0 0
\(391\) 25.0415 1.26640
\(392\) −15.4265 −0.779157
\(393\) 0 0
\(394\) −11.0804 −0.558220
\(395\) 2.54572 0.128089
\(396\) 0 0
\(397\) −8.03969 −0.403500 −0.201750 0.979437i \(-0.564663\pi\)
−0.201750 + 0.979437i \(0.564663\pi\)
\(398\) −4.90956 −0.246094
\(399\) 0 0
\(400\) −3.86111 −0.193056
\(401\) 28.0902 1.40276 0.701379 0.712788i \(-0.252568\pi\)
0.701379 + 0.712788i \(0.252568\pi\)
\(402\) 0 0
\(403\) 4.32575 0.215481
\(404\) −0.589316 −0.0293196
\(405\) 0 0
\(406\) 11.1458 0.553156
\(407\) 0 0
\(408\) 0 0
\(409\) 5.97950 0.295667 0.147834 0.989012i \(-0.452770\pi\)
0.147834 + 0.989012i \(0.452770\pi\)
\(410\) −10.5464 −0.520852
\(411\) 0 0
\(412\) −0.777114 −0.0382857
\(413\) 10.3465 0.509119
\(414\) 0 0
\(415\) 10.1221 0.496876
\(416\) −0.538833 −0.0264185
\(417\) 0 0
\(418\) 0 0
\(419\) 15.4707 0.755795 0.377897 0.925847i \(-0.376647\pi\)
0.377897 + 0.925847i \(0.376647\pi\)
\(420\) 0 0
\(421\) 34.5746 1.68506 0.842532 0.538647i \(-0.181064\pi\)
0.842532 + 0.538647i \(0.181064\pi\)
\(422\) 28.1597 1.37079
\(423\) 0 0
\(424\) −26.9185 −1.30728
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) −18.2590 −0.883614
\(428\) −1.03108 −0.0498390
\(429\) 0 0
\(430\) 7.52580 0.362926
\(431\) 3.36052 0.161870 0.0809352 0.996719i \(-0.474209\pi\)
0.0809352 + 0.996719i \(0.474209\pi\)
\(432\) 0 0
\(433\) 3.67779 0.176744 0.0883718 0.996088i \(-0.471834\pi\)
0.0883718 + 0.996088i \(0.471834\pi\)
\(434\) 5.41702 0.260025
\(435\) 0 0
\(436\) 0.967505 0.0463351
\(437\) −0.792040 −0.0378884
\(438\) 0 0
\(439\) 1.55432 0.0741835 0.0370917 0.999312i \(-0.488191\pi\)
0.0370917 + 0.999312i \(0.488191\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9.85946 −0.468967
\(443\) −21.2059 −1.00752 −0.503761 0.863843i \(-0.668051\pi\)
−0.503761 + 0.863843i \(0.668051\pi\)
\(444\) 0 0
\(445\) −11.0447 −0.523569
\(446\) −31.3503 −1.48448
\(447\) 0 0
\(448\) −10.5406 −0.497997
\(449\) −15.1106 −0.713113 −0.356557 0.934274i \(-0.616049\pi\)
−0.356557 + 0.934274i \(0.616049\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.30102 0.0611949
\(453\) 0 0
\(454\) −26.1454 −1.22707
\(455\) −1.81209 −0.0849521
\(456\) 0 0
\(457\) −41.3042 −1.93213 −0.966064 0.258303i \(-0.916837\pi\)
−0.966064 + 0.258303i \(0.916837\pi\)
\(458\) 33.1326 1.54818
\(459\) 0 0
\(460\) −0.336486 −0.0156887
\(461\) −16.3158 −0.759901 −0.379951 0.925007i \(-0.624059\pi\)
−0.379951 + 0.925007i \(0.624059\pi\)
\(462\) 0 0
\(463\) 8.47904 0.394054 0.197027 0.980398i \(-0.436871\pi\)
0.197027 + 0.980398i \(0.436871\pi\)
\(464\) −24.2291 −1.12481
\(465\) 0 0
\(466\) 14.0302 0.649938
\(467\) 36.5669 1.69212 0.846058 0.533090i \(-0.178969\pi\)
0.846058 + 0.533090i \(0.178969\pi\)
\(468\) 0 0
\(469\) 9.43325 0.435587
\(470\) −11.4447 −0.527907
\(471\) 0 0
\(472\) −23.2743 −1.07129
\(473\) 0 0
\(474\) 0 0
\(475\) −0.158146 −0.00725623
\(476\) 0.429180 0.0196714
\(477\) 0 0
\(478\) −0.232488 −0.0106338
\(479\) −38.9513 −1.77973 −0.889866 0.456222i \(-0.849202\pi\)
−0.889866 + 0.456222i \(0.849202\pi\)
\(480\) 0 0
\(481\) −6.66619 −0.303952
\(482\) 1.34195 0.0611243
\(483\) 0 0
\(484\) 0 0
\(485\) 6.41196 0.291152
\(486\) 0 0
\(487\) 8.27209 0.374844 0.187422 0.982279i \(-0.439987\pi\)
0.187422 + 0.982279i \(0.439987\pi\)
\(488\) 41.0733 1.85930
\(489\) 0 0
\(490\) 7.46257 0.337124
\(491\) −12.1322 −0.547518 −0.273759 0.961798i \(-0.588267\pi\)
−0.273759 + 0.961798i \(0.588267\pi\)
\(492\) 0 0
\(493\) 31.3757 1.41309
\(494\) 0.311846 0.0140306
\(495\) 0 0
\(496\) −11.7757 −0.528744
\(497\) −8.65719 −0.388328
\(498\) 0 0
\(499\) −28.3581 −1.26948 −0.634740 0.772725i \(-0.718893\pi\)
−0.634740 + 0.772725i \(0.718893\pi\)
\(500\) −0.0671858 −0.00300464
\(501\) 0 0
\(502\) −32.6852 −1.45881
\(503\) −15.2800 −0.681300 −0.340650 0.940190i \(-0.610647\pi\)
−0.340650 + 0.940190i \(0.610647\pi\)
\(504\) 0 0
\(505\) 8.77143 0.390324
\(506\) 0 0
\(507\) 0 0
\(508\) −0.404737 −0.0179573
\(509\) 22.0494 0.977321 0.488661 0.872474i \(-0.337486\pi\)
0.488661 + 0.872474i \(0.337486\pi\)
\(510\) 0 0
\(511\) 11.0653 0.489500
\(512\) 23.6599 1.04563
\(513\) 0 0
\(514\) −9.15169 −0.403664
\(515\) 11.5666 0.509687
\(516\) 0 0
\(517\) 0 0
\(518\) −8.34789 −0.366785
\(519\) 0 0
\(520\) 4.07627 0.178756
\(521\) 23.9380 1.04874 0.524371 0.851490i \(-0.324300\pi\)
0.524371 + 0.851490i \(0.324300\pi\)
\(522\) 0 0
\(523\) 25.3421 1.10813 0.554066 0.832473i \(-0.313075\pi\)
0.554066 + 0.832473i \(0.313075\pi\)
\(524\) 1.25824 0.0549667
\(525\) 0 0
\(526\) −17.3140 −0.754927
\(527\) 15.2491 0.664260
\(528\) 0 0
\(529\) 2.08298 0.0905642
\(530\) 13.0218 0.565632
\(531\) 0 0
\(532\) −0.0135746 −0.000588534 0
\(533\) 10.7597 0.466053
\(534\) 0 0
\(535\) 15.3467 0.663494
\(536\) −21.2199 −0.916561
\(537\) 0 0
\(538\) 28.8797 1.24509
\(539\) 0 0
\(540\) 0 0
\(541\) 35.0634 1.50749 0.753747 0.657165i \(-0.228245\pi\)
0.753747 + 0.657165i \(0.228245\pi\)
\(542\) 20.0062 0.859338
\(543\) 0 0
\(544\) −1.89949 −0.0814399
\(545\) −14.4004 −0.616847
\(546\) 0 0
\(547\) 1.10787 0.0473689 0.0236845 0.999719i \(-0.492460\pi\)
0.0236845 + 0.999719i \(0.492460\pi\)
\(548\) 0.204056 0.00871684
\(549\) 0 0
\(550\) 0 0
\(551\) −0.992388 −0.0422771
\(552\) 0 0
\(553\) −3.25239 −0.138306
\(554\) −24.3117 −1.03291
\(555\) 0 0
\(556\) −0.148609 −0.00630244
\(557\) −5.41988 −0.229648 −0.114824 0.993386i \(-0.536630\pi\)
−0.114824 + 0.993386i \(0.536630\pi\)
\(558\) 0 0
\(559\) −7.67795 −0.324743
\(560\) 4.93293 0.208454
\(561\) 0 0
\(562\) 26.2564 1.10756
\(563\) 32.7448 1.38003 0.690015 0.723795i \(-0.257604\pi\)
0.690015 + 0.723795i \(0.257604\pi\)
\(564\) 0 0
\(565\) −19.3645 −0.814672
\(566\) −18.8815 −0.793650
\(567\) 0 0
\(568\) 19.4742 0.817119
\(569\) −3.51736 −0.147455 −0.0737277 0.997278i \(-0.523490\pi\)
−0.0737277 + 0.997278i \(0.523490\pi\)
\(570\) 0 0
\(571\) 36.0252 1.50761 0.753804 0.657099i \(-0.228217\pi\)
0.753804 + 0.657099i \(0.228217\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 13.4740 0.562396
\(575\) 5.00829 0.208860
\(576\) 0 0
\(577\) 15.5501 0.647357 0.323679 0.946167i \(-0.395080\pi\)
0.323679 + 0.946167i \(0.395080\pi\)
\(578\) −11.1221 −0.462617
\(579\) 0 0
\(580\) −0.421601 −0.0175060
\(581\) −12.9319 −0.536507
\(582\) 0 0
\(583\) 0 0
\(584\) −24.8912 −1.03001
\(585\) 0 0
\(586\) −27.6694 −1.14301
\(587\) 24.2604 1.00133 0.500667 0.865640i \(-0.333088\pi\)
0.500667 + 0.865640i \(0.333088\pi\)
\(588\) 0 0
\(589\) −0.482315 −0.0198735
\(590\) 11.2589 0.463523
\(591\) 0 0
\(592\) 18.1469 0.745833
\(593\) 27.4019 1.12526 0.562630 0.826709i \(-0.309790\pi\)
0.562630 + 0.826709i \(0.309790\pi\)
\(594\) 0 0
\(595\) −6.38796 −0.261881
\(596\) −0.0995167 −0.00407636
\(597\) 0 0
\(598\) −9.87581 −0.403852
\(599\) −10.6773 −0.436262 −0.218131 0.975920i \(-0.569996\pi\)
−0.218131 + 0.975920i \(0.569996\pi\)
\(600\) 0 0
\(601\) −11.0663 −0.451403 −0.225701 0.974197i \(-0.572467\pi\)
−0.225701 + 0.974197i \(0.572467\pi\)
\(602\) −9.61489 −0.391874
\(603\) 0 0
\(604\) −0.606953 −0.0246966
\(605\) 0 0
\(606\) 0 0
\(607\) 0.514518 0.0208836 0.0104418 0.999945i \(-0.496676\pi\)
0.0104418 + 0.999945i \(0.496676\pi\)
\(608\) 0.0600792 0.00243653
\(609\) 0 0
\(610\) −19.8692 −0.804479
\(611\) 11.6761 0.472366
\(612\) 0 0
\(613\) −19.0977 −0.771348 −0.385674 0.922635i \(-0.626031\pi\)
−0.385674 + 0.922635i \(0.626031\pi\)
\(614\) −20.7641 −0.837970
\(615\) 0 0
\(616\) 0 0
\(617\) 4.70745 0.189515 0.0947573 0.995500i \(-0.469792\pi\)
0.0947573 + 0.995500i \(0.469792\pi\)
\(618\) 0 0
\(619\) 37.3691 1.50199 0.750996 0.660307i \(-0.229574\pi\)
0.750996 + 0.660307i \(0.229574\pi\)
\(620\) −0.204904 −0.00822915
\(621\) 0 0
\(622\) −6.94711 −0.278554
\(623\) 14.1106 0.565329
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.0635 −0.522123
\(627\) 0 0
\(628\) 1.18500 0.0472867
\(629\) −23.4996 −0.936989
\(630\) 0 0
\(631\) 35.5669 1.41590 0.707949 0.706264i \(-0.249621\pi\)
0.707949 + 0.706264i \(0.249621\pi\)
\(632\) 7.31621 0.291023
\(633\) 0 0
\(634\) 2.99257 0.118850
\(635\) 6.02415 0.239061
\(636\) 0 0
\(637\) −7.61344 −0.301656
\(638\) 0 0
\(639\) 0 0
\(640\) −10.7104 −0.423364
\(641\) −12.2098 −0.482260 −0.241130 0.970493i \(-0.577518\pi\)
−0.241130 + 0.970493i \(0.577518\pi\)
\(642\) 0 0
\(643\) −20.9596 −0.826565 −0.413282 0.910603i \(-0.635618\pi\)
−0.413282 + 0.910603i \(0.635618\pi\)
\(644\) 0.429892 0.0169401
\(645\) 0 0
\(646\) 1.09932 0.0432520
\(647\) −14.8766 −0.584859 −0.292430 0.956287i \(-0.594464\pi\)
−0.292430 + 0.956287i \(0.594464\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.97189 −0.0773440
\(651\) 0 0
\(652\) −1.57914 −0.0618438
\(653\) 12.1308 0.474713 0.237357 0.971423i \(-0.423719\pi\)
0.237357 + 0.971423i \(0.423719\pi\)
\(654\) 0 0
\(655\) −18.7278 −0.731757
\(656\) −29.2903 −1.14359
\(657\) 0 0
\(658\) 14.6217 0.570014
\(659\) 7.49994 0.292156 0.146078 0.989273i \(-0.453335\pi\)
0.146078 + 0.989273i \(0.453335\pi\)
\(660\) 0 0
\(661\) 9.65248 0.375438 0.187719 0.982223i \(-0.439891\pi\)
0.187719 + 0.982223i \(0.439891\pi\)
\(662\) 27.1261 1.05429
\(663\) 0 0
\(664\) 29.0902 1.12892
\(665\) 0.202046 0.00783500
\(666\) 0 0
\(667\) 31.4278 1.21689
\(668\) −0.0404949 −0.00156680
\(669\) 0 0
\(670\) 10.2651 0.396577
\(671\) 0 0
\(672\) 0 0
\(673\) −24.4496 −0.942463 −0.471231 0.882010i \(-0.656190\pi\)
−0.471231 + 0.882010i \(0.656190\pi\)
\(674\) 43.9386 1.69245
\(675\) 0 0
\(676\) 0.738254 0.0283944
\(677\) 46.7394 1.79634 0.898171 0.439646i \(-0.144896\pi\)
0.898171 + 0.439646i \(0.144896\pi\)
\(678\) 0 0
\(679\) −8.19187 −0.314375
\(680\) 14.3696 0.551049
\(681\) 0 0
\(682\) 0 0
\(683\) 28.1941 1.07882 0.539408 0.842045i \(-0.318648\pi\)
0.539408 + 0.842045i \(0.318648\pi\)
\(684\) 0 0
\(685\) −3.03719 −0.116045
\(686\) −21.9674 −0.838718
\(687\) 0 0
\(688\) 20.9011 0.796849
\(689\) −13.2851 −0.506122
\(690\) 0 0
\(691\) 11.5971 0.441175 0.220587 0.975367i \(-0.429203\pi\)
0.220587 + 0.975367i \(0.429203\pi\)
\(692\) −0.649276 −0.0246818
\(693\) 0 0
\(694\) −13.0450 −0.495180
\(695\) 2.21192 0.0839028
\(696\) 0 0
\(697\) 37.9298 1.43670
\(698\) −34.4966 −1.30572
\(699\) 0 0
\(700\) 0.0858360 0.00324429
\(701\) −36.7019 −1.38621 −0.693107 0.720835i \(-0.743759\pi\)
−0.693107 + 0.720835i \(0.743759\pi\)
\(702\) 0 0
\(703\) 0.743272 0.0280330
\(704\) 0 0
\(705\) 0 0
\(706\) −27.0805 −1.01919
\(707\) −11.2063 −0.421456
\(708\) 0 0
\(709\) 36.4905 1.37043 0.685214 0.728341i \(-0.259708\pi\)
0.685214 + 0.728341i \(0.259708\pi\)
\(710\) −9.42063 −0.353550
\(711\) 0 0
\(712\) −31.7415 −1.18956
\(713\) 15.2744 0.572029
\(714\) 0 0
\(715\) 0 0
\(716\) −0.0803349 −0.00300225
\(717\) 0 0
\(718\) −41.3881 −1.54459
\(719\) 9.32888 0.347909 0.173954 0.984754i \(-0.444346\pi\)
0.173954 + 0.984754i \(0.444346\pi\)
\(720\) 0 0
\(721\) −14.7774 −0.550340
\(722\) 26.3801 0.981766
\(723\) 0 0
\(724\) −1.04636 −0.0388875
\(725\) 6.27515 0.233053
\(726\) 0 0
\(727\) −8.46883 −0.314091 −0.157046 0.987591i \(-0.550197\pi\)
−0.157046 + 0.987591i \(0.550197\pi\)
\(728\) −5.20780 −0.193014
\(729\) 0 0
\(730\) 12.0411 0.445661
\(731\) −27.0662 −1.00108
\(732\) 0 0
\(733\) −29.0470 −1.07288 −0.536438 0.843940i \(-0.680230\pi\)
−0.536438 + 0.843940i \(0.680230\pi\)
\(734\) 8.94765 0.330264
\(735\) 0 0
\(736\) −1.90264 −0.0701322
\(737\) 0 0
\(738\) 0 0
\(739\) 13.7551 0.505990 0.252995 0.967468i \(-0.418584\pi\)
0.252995 + 0.967468i \(0.418584\pi\)
\(740\) 0.315767 0.0116078
\(741\) 0 0
\(742\) −16.6366 −0.610747
\(743\) −34.9135 −1.28085 −0.640427 0.768019i \(-0.721242\pi\)
−0.640427 + 0.768019i \(0.721242\pi\)
\(744\) 0 0
\(745\) 1.48122 0.0542676
\(746\) −29.0458 −1.06344
\(747\) 0 0
\(748\) 0 0
\(749\) −19.6068 −0.716416
\(750\) 0 0
\(751\) 22.0992 0.806412 0.403206 0.915109i \(-0.367896\pi\)
0.403206 + 0.915109i \(0.367896\pi\)
\(752\) −31.7851 −1.15908
\(753\) 0 0
\(754\) −12.3739 −0.450631
\(755\) 9.03395 0.328779
\(756\) 0 0
\(757\) −45.0300 −1.63664 −0.818321 0.574761i \(-0.805095\pi\)
−0.818321 + 0.574761i \(0.805095\pi\)
\(758\) 19.7404 0.717004
\(759\) 0 0
\(760\) −0.454498 −0.0164864
\(761\) 33.6001 1.21800 0.609001 0.793169i \(-0.291570\pi\)
0.609001 + 0.793169i \(0.291570\pi\)
\(762\) 0 0
\(763\) 18.3979 0.666048
\(764\) 1.23633 0.0447287
\(765\) 0 0
\(766\) −36.8116 −1.33006
\(767\) −11.4866 −0.414756
\(768\) 0 0
\(769\) −24.6086 −0.887408 −0.443704 0.896173i \(-0.646336\pi\)
−0.443704 + 0.896173i \(0.646336\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.105346 0.00379150
\(773\) 23.6816 0.851768 0.425884 0.904778i \(-0.359963\pi\)
0.425884 + 0.904778i \(0.359963\pi\)
\(774\) 0 0
\(775\) 3.04981 0.109553
\(776\) 18.4275 0.661507
\(777\) 0 0
\(778\) −50.7394 −1.81910
\(779\) −1.19969 −0.0429833
\(780\) 0 0
\(781\) 0 0
\(782\) −34.8141 −1.24495
\(783\) 0 0
\(784\) 20.7255 0.740198
\(785\) −17.6377 −0.629515
\(786\) 0 0
\(787\) 18.4210 0.656639 0.328320 0.944567i \(-0.393518\pi\)
0.328320 + 0.944567i \(0.393518\pi\)
\(788\) −0.535471 −0.0190754
\(789\) 0 0
\(790\) −3.53921 −0.125919
\(791\) 24.7399 0.879651
\(792\) 0 0
\(793\) 20.2709 0.719840
\(794\) 11.1772 0.396665
\(795\) 0 0
\(796\) −0.237260 −0.00840947
\(797\) 6.50665 0.230477 0.115239 0.993338i \(-0.463237\pi\)
0.115239 + 0.993338i \(0.463237\pi\)
\(798\) 0 0
\(799\) 41.1606 1.45616
\(800\) −0.379898 −0.0134314
\(801\) 0 0
\(802\) −39.0526 −1.37900
\(803\) 0 0
\(804\) 0 0
\(805\) −6.39855 −0.225519
\(806\) −6.01390 −0.211831
\(807\) 0 0
\(808\) 25.2084 0.886828
\(809\) 11.8505 0.416642 0.208321 0.978060i \(-0.433200\pi\)
0.208321 + 0.978060i \(0.433200\pi\)
\(810\) 0 0
\(811\) −7.51391 −0.263849 −0.131925 0.991260i \(-0.542116\pi\)
−0.131925 + 0.991260i \(0.542116\pi\)
\(812\) 0.538633 0.0189023
\(813\) 0 0
\(814\) 0 0
\(815\) 23.5040 0.823310
\(816\) 0 0
\(817\) 0.856081 0.0299505
\(818\) −8.31305 −0.290659
\(819\) 0 0
\(820\) −0.509669 −0.0177984
\(821\) −37.3365 −1.30305 −0.651527 0.758625i \(-0.725871\pi\)
−0.651527 + 0.758625i \(0.725871\pi\)
\(822\) 0 0
\(823\) −43.0498 −1.50062 −0.750311 0.661085i \(-0.770096\pi\)
−0.750311 + 0.661085i \(0.770096\pi\)
\(824\) 33.2416 1.15803
\(825\) 0 0
\(826\) −14.3843 −0.500495
\(827\) −20.0304 −0.696524 −0.348262 0.937397i \(-0.613228\pi\)
−0.348262 + 0.937397i \(0.613228\pi\)
\(828\) 0 0
\(829\) −47.9207 −1.66435 −0.832177 0.554510i \(-0.812906\pi\)
−0.832177 + 0.554510i \(0.812906\pi\)
\(830\) −14.0724 −0.488458
\(831\) 0 0
\(832\) 11.7021 0.405696
\(833\) −26.8388 −0.929909
\(834\) 0 0
\(835\) 0.602731 0.0208584
\(836\) 0 0
\(837\) 0 0
\(838\) −21.5083 −0.742992
\(839\) −2.22093 −0.0766750 −0.0383375 0.999265i \(-0.512206\pi\)
−0.0383375 + 0.999265i \(0.512206\pi\)
\(840\) 0 0
\(841\) 10.3775 0.357843
\(842\) −48.0676 −1.65652
\(843\) 0 0
\(844\) 1.36085 0.0468424
\(845\) −10.9882 −0.378007
\(846\) 0 0
\(847\) 0 0
\(848\) 36.1651 1.24191
\(849\) 0 0
\(850\) −6.95128 −0.238427
\(851\) −23.5385 −0.806890
\(852\) 0 0
\(853\) 6.37161 0.218160 0.109080 0.994033i \(-0.465210\pi\)
0.109080 + 0.994033i \(0.465210\pi\)
\(854\) 25.3847 0.868646
\(855\) 0 0
\(856\) 44.1051 1.50748
\(857\) 35.9060 1.22653 0.613263 0.789878i \(-0.289856\pi\)
0.613263 + 0.789878i \(0.289856\pi\)
\(858\) 0 0
\(859\) 56.4697 1.92672 0.963360 0.268212i \(-0.0864328\pi\)
0.963360 + 0.268212i \(0.0864328\pi\)
\(860\) 0.363693 0.0124018
\(861\) 0 0
\(862\) −4.67198 −0.159128
\(863\) 31.0742 1.05778 0.528888 0.848691i \(-0.322609\pi\)
0.528888 + 0.848691i \(0.322609\pi\)
\(864\) 0 0
\(865\) 9.66389 0.328582
\(866\) −5.11308 −0.173749
\(867\) 0 0
\(868\) 0.261784 0.00888552
\(869\) 0 0
\(870\) 0 0
\(871\) −10.4727 −0.354853
\(872\) −41.3857 −1.40150
\(873\) 0 0
\(874\) 1.10114 0.0372466
\(875\) −1.27759 −0.0431905
\(876\) 0 0
\(877\) −16.8224 −0.568053 −0.284026 0.958816i \(-0.591670\pi\)
−0.284026 + 0.958816i \(0.591670\pi\)
\(878\) −2.16090 −0.0729268
\(879\) 0 0
\(880\) 0 0
\(881\) −26.5633 −0.894940 −0.447470 0.894299i \(-0.647675\pi\)
−0.447470 + 0.894299i \(0.647675\pi\)
\(882\) 0 0
\(883\) −54.3742 −1.82984 −0.914919 0.403638i \(-0.867746\pi\)
−0.914919 + 0.403638i \(0.867746\pi\)
\(884\) −0.476470 −0.0160254
\(885\) 0 0
\(886\) 29.4816 0.990455
\(887\) −41.2262 −1.38424 −0.692120 0.721783i \(-0.743323\pi\)
−0.692120 + 0.721783i \(0.743323\pi\)
\(888\) 0 0
\(889\) −7.69640 −0.258129
\(890\) 15.3550 0.514699
\(891\) 0 0
\(892\) −1.51504 −0.0507273
\(893\) −1.30187 −0.0435655
\(894\) 0 0
\(895\) 1.19571 0.0399682
\(896\) 13.6835 0.457132
\(897\) 0 0
\(898\) 21.0076 0.701033
\(899\) 19.1380 0.638289
\(900\) 0 0
\(901\) −46.8324 −1.56021
\(902\) 0 0
\(903\) 0 0
\(904\) −55.6521 −1.85096
\(905\) 15.5741 0.517699
\(906\) 0 0
\(907\) 42.4046 1.40802 0.704010 0.710190i \(-0.251391\pi\)
0.704010 + 0.710190i \(0.251391\pi\)
\(908\) −1.26351 −0.0419310
\(909\) 0 0
\(910\) 2.51927 0.0835130
\(911\) 8.01199 0.265449 0.132725 0.991153i \(-0.457627\pi\)
0.132725 + 0.991153i \(0.457627\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 57.4234 1.89940
\(915\) 0 0
\(916\) 1.60117 0.0529042
\(917\) 23.9265 0.790123
\(918\) 0 0
\(919\) −41.2336 −1.36017 −0.680086 0.733132i \(-0.738058\pi\)
−0.680086 + 0.733132i \(0.738058\pi\)
\(920\) 14.3934 0.474537
\(921\) 0 0
\(922\) 22.6831 0.747028
\(923\) 9.61110 0.316353
\(924\) 0 0
\(925\) −4.69991 −0.154532
\(926\) −11.7880 −0.387379
\(927\) 0 0
\(928\) −2.38391 −0.0782558
\(929\) 15.1939 0.498495 0.249247 0.968440i \(-0.419817\pi\)
0.249247 + 0.968440i \(0.419817\pi\)
\(930\) 0 0
\(931\) 0.848889 0.0278212
\(932\) 0.678027 0.0222095
\(933\) 0 0
\(934\) −50.8375 −1.66345
\(935\) 0 0
\(936\) 0 0
\(937\) −25.5213 −0.833746 −0.416873 0.908965i \(-0.636874\pi\)
−0.416873 + 0.908965i \(0.636874\pi\)
\(938\) −13.1146 −0.428208
\(939\) 0 0
\(940\) −0.553081 −0.0180395
\(941\) 32.3673 1.05514 0.527572 0.849511i \(-0.323103\pi\)
0.527572 + 0.849511i \(0.323103\pi\)
\(942\) 0 0
\(943\) 37.9927 1.23721
\(944\) 31.2691 1.01772
\(945\) 0 0
\(946\) 0 0
\(947\) 40.1516 1.30475 0.652375 0.757896i \(-0.273773\pi\)
0.652375 + 0.757896i \(0.273773\pi\)
\(948\) 0 0
\(949\) −12.2846 −0.398774
\(950\) 0.219863 0.00713331
\(951\) 0 0
\(952\) −18.3585 −0.595001
\(953\) 35.6513 1.15486 0.577429 0.816441i \(-0.304056\pi\)
0.577429 + 0.816441i \(0.304056\pi\)
\(954\) 0 0
\(955\) −18.4016 −0.595462
\(956\) −0.0112353 −0.000363374 0
\(957\) 0 0
\(958\) 54.1524 1.74958
\(959\) 3.88029 0.125301
\(960\) 0 0
\(961\) −21.6986 −0.699956
\(962\) 9.26772 0.298803
\(963\) 0 0
\(964\) 0.0648515 0.00208873
\(965\) −1.56799 −0.0504753
\(966\) 0 0
\(967\) 20.0622 0.645156 0.322578 0.946543i \(-0.395451\pi\)
0.322578 + 0.946543i \(0.395451\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −8.91428 −0.286220
\(971\) 41.6307 1.33599 0.667996 0.744165i \(-0.267152\pi\)
0.667996 + 0.744165i \(0.267152\pi\)
\(972\) 0 0
\(973\) −2.82593 −0.0905950
\(974\) −11.5003 −0.368494
\(975\) 0 0
\(976\) −55.1820 −1.76633
\(977\) −19.0722 −0.610173 −0.305087 0.952325i \(-0.598685\pi\)
−0.305087 + 0.952325i \(0.598685\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.360637 0.0115201
\(981\) 0 0
\(982\) 16.8668 0.538243
\(983\) −26.5563 −0.847015 −0.423507 0.905893i \(-0.639201\pi\)
−0.423507 + 0.905893i \(0.639201\pi\)
\(984\) 0 0
\(985\) 7.97000 0.253945
\(986\) −43.6203 −1.38915
\(987\) 0 0
\(988\) 0.0150703 0.000479452 0
\(989\) −27.1111 −0.862082
\(990\) 0 0
\(991\) −24.2494 −0.770307 −0.385153 0.922853i \(-0.625851\pi\)
−0.385153 + 0.922853i \(0.625851\pi\)
\(992\) −1.15862 −0.0367861
\(993\) 0 0
\(994\) 12.0357 0.381750
\(995\) 3.53141 0.111953
\(996\) 0 0
\(997\) −16.0616 −0.508675 −0.254338 0.967116i \(-0.581857\pi\)
−0.254338 + 0.967116i \(0.581857\pi\)
\(998\) 39.4250 1.24798
\(999\) 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 5445.2.a.bv.1.1 4
3.2 odd 2 1815.2.a.o.1.4 4
11.2 odd 10 495.2.n.d.136.1 8
11.6 odd 10 495.2.n.d.91.1 8
11.10 odd 2 5445.2.a.be.1.4 4
15.14 odd 2 9075.2.a.dj.1.1 4
33.2 even 10 165.2.m.a.136.2 yes 8
33.17 even 10 165.2.m.a.91.2 8
33.32 even 2 1815.2.a.x.1.1 4
165.2 odd 20 825.2.bx.h.499.2 16
165.17 odd 20 825.2.bx.h.124.3 16
165.68 odd 20 825.2.bx.h.499.3 16
165.83 odd 20 825.2.bx.h.124.2 16
165.134 even 10 825.2.n.k.301.1 8
165.149 even 10 825.2.n.k.751.1 8
165.164 even 2 9075.2.a.cl.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.91.2 8 33.17 even 10
165.2.m.a.136.2 yes 8 33.2 even 10
495.2.n.d.91.1 8 11.6 odd 10
495.2.n.d.136.1 8 11.2 odd 10
825.2.n.k.301.1 8 165.134 even 10
825.2.n.k.751.1 8 165.149 even 10
825.2.bx.h.124.2 16 165.83 odd 20
825.2.bx.h.124.3 16 165.17 odd 20
825.2.bx.h.499.2 16 165.2 odd 20
825.2.bx.h.499.3 16 165.68 odd 20
1815.2.a.o.1.4 4 3.2 odd 2
1815.2.a.x.1.1 4 33.32 even 2
5445.2.a.be.1.4 4 11.10 odd 2
5445.2.a.bv.1.1 4 1.1 even 1 trivial
9075.2.a.cl.1.4 4 165.164 even 2
9075.2.a.dj.1.1 4 15.14 odd 2