Properties

Label 3549.2.a.h.1.3
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Defining polynomial: \(x^{3} - x^{2} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.65109\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.65109 q^{2} -1.00000 q^{3} +0.726109 q^{4} +2.92498 q^{5} -1.65109 q^{6} -1.00000 q^{7} -2.10331 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.65109 q^{2} -1.00000 q^{3} +0.726109 q^{4} +2.92498 q^{5} -1.65109 q^{6} -1.00000 q^{7} -2.10331 q^{8} +1.00000 q^{9} +4.82942 q^{10} -4.37720 q^{11} -0.726109 q^{12} -1.65109 q^{14} -2.92498 q^{15} -4.92498 q^{16} +3.65109 q^{17} +1.65109 q^{18} -5.75441 q^{19} +2.12386 q^{20} +1.00000 q^{21} -7.22717 q^{22} +7.02830 q^{23} +2.10331 q^{24} +3.55553 q^{25} -1.00000 q^{27} -0.726109 q^{28} +1.19887 q^{29} -4.82942 q^{30} -6.47277 q^{31} -3.92498 q^{32} +4.37720 q^{33} +6.02830 q^{34} -2.92498 q^{35} +0.726109 q^{36} -2.92498 q^{37} -9.50106 q^{38} -6.15215 q^{40} -8.60437 q^{41} +1.65109 q^{42} -5.72611 q^{43} -3.17833 q^{44} +2.92498 q^{45} +11.6044 q^{46} -9.58383 q^{47} +4.92498 q^{48} +1.00000 q^{49} +5.87051 q^{50} -3.65109 q^{51} -0.302187 q^{53} -1.65109 q^{54} -12.8032 q^{55} +2.10331 q^{56} +5.75441 q^{57} +1.97945 q^{58} +3.02830 q^{59} -2.12386 q^{60} -0.302187 q^{61} -10.6871 q^{62} -1.00000 q^{63} +3.36945 q^{64} +7.22717 q^{66} -8.70769 q^{67} +2.65109 q^{68} -7.02830 q^{69} -4.82942 q^{70} +13.5371 q^{71} -2.10331 q^{72} +0.932734 q^{73} -4.82942 q^{74} -3.55553 q^{75} -4.17833 q^{76} +4.37720 q^{77} -16.9426 q^{79} -14.4055 q^{80} +1.00000 q^{81} -14.2066 q^{82} -1.26614 q^{83} +0.726109 q^{84} +10.6794 q^{85} -9.45434 q^{86} -1.19887 q^{87} +9.20662 q^{88} -13.3977 q^{89} +4.82942 q^{90} +5.10331 q^{92} +6.47277 q^{93} -15.8238 q^{94} -16.8315 q^{95} +3.92498 q^{96} +11.9250 q^{97} +1.65109 q^{98} -4.37720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + 13q^{10} - 8q^{11} - 4q^{12} + 2q^{14} - 6q^{16} + 4q^{17} - 2q^{18} - 7q^{19} - 13q^{20} + 3q^{21} + q^{22} + 9q^{23} + 3q^{24} + 11q^{25} - 3q^{27} - 4q^{28} - 7q^{29} - 13q^{30} - 7q^{31} - 3q^{32} + 8q^{33} + 6q^{34} + 4q^{36} - 4q^{38} + 13q^{40} + 2q^{41} - 2q^{42} - 19q^{43} - 15q^{44} + 7q^{46} - 17q^{47} + 6q^{48} + 3q^{49} - 16q^{50} - 4q^{51} + 13q^{53} + 2q^{54} + 3q^{56} + 7q^{57} + 22q^{58} - 3q^{59} + 13q^{60} + 13q^{61} - 17q^{62} - 3q^{63} + q^{64} - q^{66} + 5q^{67} + q^{68} - 9q^{69} - 13q^{70} + 8q^{71} - 3q^{72} - 2q^{73} - 13q^{74} - 11q^{75} - 18q^{76} + 8q^{77} - q^{79} - 26q^{80} + 3q^{81} - 36q^{82} + 2q^{83} + 4q^{84} + 13q^{85} + 17q^{86} + 7q^{87} + 21q^{88} - 19q^{89} + 13q^{90} + 12q^{92} + 7q^{93} + 7q^{94} + 3q^{96} + 27q^{97} - 2q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.65109 1.16750 0.583750 0.811934i \(-0.301585\pi\)
0.583750 + 0.811934i \(0.301585\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.726109 0.363055
\(5\) 2.92498 1.30809 0.654046 0.756455i \(-0.273070\pi\)
0.654046 + 0.756455i \(0.273070\pi\)
\(6\) −1.65109 −0.674056
\(7\) −1.00000 −0.377964
\(8\) −2.10331 −0.743633
\(9\) 1.00000 0.333333
\(10\) 4.82942 1.52720
\(11\) −4.37720 −1.31978 −0.659888 0.751364i \(-0.729396\pi\)
−0.659888 + 0.751364i \(0.729396\pi\)
\(12\) −0.726109 −0.209610
\(13\) 0 0
\(14\) −1.65109 −0.441273
\(15\) −2.92498 −0.755228
\(16\) −4.92498 −1.23125
\(17\) 3.65109 0.885520 0.442760 0.896640i \(-0.353999\pi\)
0.442760 + 0.896640i \(0.353999\pi\)
\(18\) 1.65109 0.389166
\(19\) −5.75441 −1.32015 −0.660076 0.751199i \(-0.729476\pi\)
−0.660076 + 0.751199i \(0.729476\pi\)
\(20\) 2.12386 0.474909
\(21\) 1.00000 0.218218
\(22\) −7.22717 −1.54084
\(23\) 7.02830 1.46550 0.732751 0.680497i \(-0.238236\pi\)
0.732751 + 0.680497i \(0.238236\pi\)
\(24\) 2.10331 0.429337
\(25\) 3.55553 0.711106
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −0.726109 −0.137222
\(29\) 1.19887 0.222625 0.111313 0.993785i \(-0.464494\pi\)
0.111313 + 0.993785i \(0.464494\pi\)
\(30\) −4.82942 −0.881728
\(31\) −6.47277 −1.16254 −0.581271 0.813710i \(-0.697445\pi\)
−0.581271 + 0.813710i \(0.697445\pi\)
\(32\) −3.92498 −0.693846
\(33\) 4.37720 0.761973
\(34\) 6.02830 1.03384
\(35\) −2.92498 −0.494413
\(36\) 0.726109 0.121018
\(37\) −2.92498 −0.480864 −0.240432 0.970666i \(-0.577289\pi\)
−0.240432 + 0.970666i \(0.577289\pi\)
\(38\) −9.50106 −1.54128
\(39\) 0 0
\(40\) −6.15215 −0.972741
\(41\) −8.60437 −1.34378 −0.671889 0.740652i \(-0.734517\pi\)
−0.671889 + 0.740652i \(0.734517\pi\)
\(42\) 1.65109 0.254769
\(43\) −5.72611 −0.873224 −0.436612 0.899650i \(-0.643822\pi\)
−0.436612 + 0.899650i \(0.643822\pi\)
\(44\) −3.17833 −0.479151
\(45\) 2.92498 0.436031
\(46\) 11.6044 1.71097
\(47\) −9.58383 −1.39794 −0.698972 0.715149i \(-0.746359\pi\)
−0.698972 + 0.715149i \(0.746359\pi\)
\(48\) 4.92498 0.710860
\(49\) 1.00000 0.142857
\(50\) 5.87051 0.830216
\(51\) −3.65109 −0.511255
\(52\) 0 0
\(53\) −0.302187 −0.0415086 −0.0207543 0.999785i \(-0.506607\pi\)
−0.0207543 + 0.999785i \(0.506607\pi\)
\(54\) −1.65109 −0.224685
\(55\) −12.8032 −1.72639
\(56\) 2.10331 0.281067
\(57\) 5.75441 0.762190
\(58\) 1.97945 0.259915
\(59\) 3.02830 0.394251 0.197125 0.980378i \(-0.436839\pi\)
0.197125 + 0.980378i \(0.436839\pi\)
\(60\) −2.12386 −0.274189
\(61\) −0.302187 −0.0386911 −0.0193455 0.999813i \(-0.506158\pi\)
−0.0193455 + 0.999813i \(0.506158\pi\)
\(62\) −10.6871 −1.35727
\(63\) −1.00000 −0.125988
\(64\) 3.36945 0.421182
\(65\) 0 0
\(66\) 7.22717 0.889603
\(67\) −8.70769 −1.06381 −0.531907 0.846803i \(-0.678524\pi\)
−0.531907 + 0.846803i \(0.678524\pi\)
\(68\) 2.65109 0.321492
\(69\) −7.02830 −0.846107
\(70\) −4.82942 −0.577226
\(71\) 13.5371 1.60656 0.803280 0.595602i \(-0.203087\pi\)
0.803280 + 0.595602i \(0.203087\pi\)
\(72\) −2.10331 −0.247878
\(73\) 0.932734 0.109168 0.0545841 0.998509i \(-0.482617\pi\)
0.0545841 + 0.998509i \(0.482617\pi\)
\(74\) −4.82942 −0.561409
\(75\) −3.55553 −0.410557
\(76\) −4.17833 −0.479287
\(77\) 4.37720 0.498829
\(78\) 0 0
\(79\) −16.9426 −1.90619 −0.953096 0.302668i \(-0.902123\pi\)
−0.953096 + 0.302668i \(0.902123\pi\)
\(80\) −14.4055 −1.61058
\(81\) 1.00000 0.111111
\(82\) −14.2066 −1.56886
\(83\) −1.26614 −0.138977 −0.0694885 0.997583i \(-0.522137\pi\)
−0.0694885 + 0.997583i \(0.522137\pi\)
\(84\) 0.726109 0.0792250
\(85\) 10.6794 1.15834
\(86\) −9.45434 −1.01949
\(87\) −1.19887 −0.128533
\(88\) 9.20662 0.981429
\(89\) −13.3977 −1.42016 −0.710079 0.704122i \(-0.751341\pi\)
−0.710079 + 0.704122i \(0.751341\pi\)
\(90\) 4.82942 0.509066
\(91\) 0 0
\(92\) 5.10331 0.532057
\(93\) 6.47277 0.671194
\(94\) −15.8238 −1.63210
\(95\) −16.8315 −1.72688
\(96\) 3.92498 0.400592
\(97\) 11.9250 1.21080 0.605399 0.795922i \(-0.293013\pi\)
0.605399 + 0.795922i \(0.293013\pi\)
\(98\) 1.65109 0.166786
\(99\) −4.37720 −0.439925
\(100\) 2.58170 0.258170
\(101\) 11.0283 1.09736 0.548678 0.836034i \(-0.315131\pi\)
0.548678 + 0.836034i \(0.315131\pi\)
\(102\) −6.02830 −0.596890
\(103\) −19.4543 −1.91689 −0.958447 0.285272i \(-0.907916\pi\)
−0.958447 + 0.285272i \(0.907916\pi\)
\(104\) 0 0
\(105\) 2.92498 0.285449
\(106\) −0.498939 −0.0484612
\(107\) −1.42392 −0.137656 −0.0688279 0.997629i \(-0.521926\pi\)
−0.0688279 + 0.997629i \(0.521926\pi\)
\(108\) −0.726109 −0.0698699
\(109\) −0.0467198 −0.00447494 −0.00223747 0.999997i \(-0.500712\pi\)
−0.00223747 + 0.999997i \(0.500712\pi\)
\(110\) −21.1394 −2.01556
\(111\) 2.92498 0.277627
\(112\) 4.92498 0.465367
\(113\) −10.3382 −0.972539 −0.486270 0.873809i \(-0.661643\pi\)
−0.486270 + 0.873809i \(0.661643\pi\)
\(114\) 9.50106 0.889856
\(115\) 20.5577 1.91701
\(116\) 0.870514 0.0808252
\(117\) 0 0
\(118\) 5.00000 0.460287
\(119\) −3.65109 −0.334695
\(120\) 6.15215 0.561612
\(121\) 8.15990 0.741810
\(122\) −0.498939 −0.0451718
\(123\) 8.60437 0.775830
\(124\) −4.69994 −0.422067
\(125\) −4.22505 −0.377900
\(126\) −1.65109 −0.147091
\(127\) −5.47277 −0.485629 −0.242815 0.970073i \(-0.578071\pi\)
−0.242815 + 0.970073i \(0.578071\pi\)
\(128\) 13.4132 1.18557
\(129\) 5.72611 0.504156
\(130\) 0 0
\(131\) 16.1706 1.41283 0.706415 0.707798i \(-0.250311\pi\)
0.706415 + 0.707798i \(0.250311\pi\)
\(132\) 3.17833 0.276638
\(133\) 5.75441 0.498970
\(134\) −14.3772 −1.24200
\(135\) −2.92498 −0.251743
\(136\) −7.67939 −0.658502
\(137\) −12.0176 −1.02673 −0.513367 0.858169i \(-0.671602\pi\)
−0.513367 + 0.858169i \(0.671602\pi\)
\(138\) −11.6044 −0.987830
\(139\) 1.34891 0.114413 0.0572064 0.998362i \(-0.481781\pi\)
0.0572064 + 0.998362i \(0.481781\pi\)
\(140\) −2.12386 −0.179499
\(141\) 9.58383 0.807104
\(142\) 22.3510 1.87566
\(143\) 0 0
\(144\) −4.92498 −0.410415
\(145\) 3.50669 0.291215
\(146\) 1.54003 0.127454
\(147\) −1.00000 −0.0824786
\(148\) −2.12386 −0.174580
\(149\) −3.07502 −0.251915 −0.125958 0.992036i \(-0.540200\pi\)
−0.125958 + 0.992036i \(0.540200\pi\)
\(150\) −5.87051 −0.479325
\(151\) 16.6610 1.35585 0.677925 0.735131i \(-0.262879\pi\)
0.677925 + 0.735131i \(0.262879\pi\)
\(152\) 12.1033 0.981708
\(153\) 3.65109 0.295173
\(154\) 7.22717 0.582382
\(155\) −18.9327 −1.52071
\(156\) 0 0
\(157\) 12.8294 1.02390 0.511950 0.859015i \(-0.328923\pi\)
0.511950 + 0.859015i \(0.328923\pi\)
\(158\) −27.9738 −2.22548
\(159\) 0.302187 0.0239650
\(160\) −11.4805 −0.907614
\(161\) −7.02830 −0.553907
\(162\) 1.65109 0.129722
\(163\) −21.6044 −1.69219 −0.846093 0.533036i \(-0.821051\pi\)
−0.846093 + 0.533036i \(0.821051\pi\)
\(164\) −6.24772 −0.487865
\(165\) 12.8032 0.996732
\(166\) −2.09052 −0.162256
\(167\) 21.3198 1.64978 0.824888 0.565296i \(-0.191238\pi\)
0.824888 + 0.565296i \(0.191238\pi\)
\(168\) −2.10331 −0.162274
\(169\) 0 0
\(170\) 17.6327 1.35236
\(171\) −5.75441 −0.440050
\(172\) −4.15778 −0.317028
\(173\) 3.12174 0.237341 0.118671 0.992934i \(-0.462137\pi\)
0.118671 + 0.992934i \(0.462137\pi\)
\(174\) −1.97945 −0.150062
\(175\) −3.55553 −0.268773
\(176\) 21.5577 1.62497
\(177\) −3.02830 −0.227621
\(178\) −22.1209 −1.65803
\(179\) 18.9709 1.41795 0.708976 0.705233i \(-0.249157\pi\)
0.708976 + 0.705233i \(0.249157\pi\)
\(180\) 2.12386 0.158303
\(181\) −1.93273 −0.143659 −0.0718295 0.997417i \(-0.522884\pi\)
−0.0718295 + 0.997417i \(0.522884\pi\)
\(182\) 0 0
\(183\) 0.302187 0.0223383
\(184\) −14.7827 −1.08980
\(185\) −8.55553 −0.629015
\(186\) 10.6871 0.783619
\(187\) −15.9816 −1.16869
\(188\) −6.95891 −0.507531
\(189\) 1.00000 0.0727393
\(190\) −27.7905 −2.01613
\(191\) −2.27389 −0.164533 −0.0822665 0.996610i \(-0.526216\pi\)
−0.0822665 + 0.996610i \(0.526216\pi\)
\(192\) −3.36945 −0.243169
\(193\) −4.38708 −0.315789 −0.157894 0.987456i \(-0.550471\pi\)
−0.157894 + 0.987456i \(0.550471\pi\)
\(194\) 19.6893 1.41361
\(195\) 0 0
\(196\) 0.726109 0.0518650
\(197\) −5.25839 −0.374645 −0.187322 0.982298i \(-0.559981\pi\)
−0.187322 + 0.982298i \(0.559981\pi\)
\(198\) −7.22717 −0.513613
\(199\) 13.1960 0.935436 0.467718 0.883878i \(-0.345076\pi\)
0.467718 + 0.883878i \(0.345076\pi\)
\(200\) −7.47839 −0.528802
\(201\) 8.70769 0.614193
\(202\) 18.2087 1.28116
\(203\) −1.19887 −0.0841445
\(204\) −2.65109 −0.185614
\(205\) −25.1677 −1.75779
\(206\) −32.1209 −2.23797
\(207\) 7.02830 0.488500
\(208\) 0 0
\(209\) 25.1882 1.74230
\(210\) 4.82942 0.333262
\(211\) −17.4415 −1.20073 −0.600363 0.799728i \(-0.704977\pi\)
−0.600363 + 0.799728i \(0.704977\pi\)
\(212\) −0.219421 −0.0150699
\(213\) −13.5371 −0.927547
\(214\) −2.35103 −0.160713
\(215\) −16.7488 −1.14226
\(216\) 2.10331 0.143112
\(217\) 6.47277 0.439400
\(218\) −0.0771387 −0.00522449
\(219\) −0.932734 −0.0630283
\(220\) −9.29656 −0.626774
\(221\) 0 0
\(222\) 4.82942 0.324130
\(223\) 8.00212 0.535862 0.267931 0.963438i \(-0.413660\pi\)
0.267931 + 0.963438i \(0.413660\pi\)
\(224\) 3.92498 0.262249
\(225\) 3.55553 0.237035
\(226\) −17.0694 −1.13544
\(227\) −8.05952 −0.534929 −0.267464 0.963568i \(-0.586186\pi\)
−0.267464 + 0.963568i \(0.586186\pi\)
\(228\) 4.17833 0.276717
\(229\) −8.48052 −0.560408 −0.280204 0.959940i \(-0.590402\pi\)
−0.280204 + 0.959940i \(0.590402\pi\)
\(230\) 33.9426 2.23811
\(231\) −4.37720 −0.287999
\(232\) −2.52161 −0.165552
\(233\) −10.2456 −0.671211 −0.335606 0.942003i \(-0.608941\pi\)
−0.335606 + 0.942003i \(0.608941\pi\)
\(234\) 0 0
\(235\) −28.0325 −1.82864
\(236\) 2.19887 0.143135
\(237\) 16.9426 1.10054
\(238\) −6.02830 −0.390756
\(239\) −11.3022 −0.731078 −0.365539 0.930796i \(-0.619115\pi\)
−0.365539 + 0.930796i \(0.619115\pi\)
\(240\) 14.4055 0.929871
\(241\) 1.03605 0.0667376 0.0333688 0.999443i \(-0.489376\pi\)
0.0333688 + 0.999443i \(0.489376\pi\)
\(242\) 13.4728 0.866062
\(243\) −1.00000 −0.0641500
\(244\) −0.219421 −0.0140470
\(245\) 2.92498 0.186870
\(246\) 14.2066 0.905781
\(247\) 0 0
\(248\) 13.6142 0.864506
\(249\) 1.26614 0.0802384
\(250\) −6.97595 −0.441198
\(251\) 13.2838 0.838464 0.419232 0.907879i \(-0.362299\pi\)
0.419232 + 0.907879i \(0.362299\pi\)
\(252\) −0.726109 −0.0457406
\(253\) −30.7643 −1.93413
\(254\) −9.03605 −0.566972
\(255\) −10.6794 −0.668769
\(256\) 15.4076 0.962976
\(257\) 12.4415 0.776082 0.388041 0.921642i \(-0.373152\pi\)
0.388041 + 0.921642i \(0.373152\pi\)
\(258\) 9.45434 0.588602
\(259\) 2.92498 0.181750
\(260\) 0 0
\(261\) 1.19887 0.0742085
\(262\) 26.6991 1.64948
\(263\) 6.59158 0.406454 0.203227 0.979132i \(-0.434857\pi\)
0.203227 + 0.979132i \(0.434857\pi\)
\(264\) −9.20662 −0.566629
\(265\) −0.883892 −0.0542970
\(266\) 9.50106 0.582547
\(267\) 13.3977 0.819929
\(268\) −6.32273 −0.386222
\(269\) −2.71624 −0.165612 −0.0828059 0.996566i \(-0.526388\pi\)
−0.0828059 + 0.996566i \(0.526388\pi\)
\(270\) −4.82942 −0.293909
\(271\) −30.7253 −1.86643 −0.933215 0.359319i \(-0.883009\pi\)
−0.933215 + 0.359319i \(0.883009\pi\)
\(272\) −17.9816 −1.09029
\(273\) 0 0
\(274\) −19.8422 −1.19871
\(275\) −15.5633 −0.938501
\(276\) −5.10331 −0.307183
\(277\) −21.5654 −1.29574 −0.647870 0.761751i \(-0.724340\pi\)
−0.647870 + 0.761751i \(0.724340\pi\)
\(278\) 2.22717 0.133577
\(279\) −6.47277 −0.387514
\(280\) 6.15215 0.367662
\(281\) 33.2058 1.98089 0.990447 0.137896i \(-0.0440340\pi\)
0.990447 + 0.137896i \(0.0440340\pi\)
\(282\) 15.8238 0.942293
\(283\) 1.48344 0.0881813 0.0440906 0.999028i \(-0.485961\pi\)
0.0440906 + 0.999028i \(0.485961\pi\)
\(284\) 9.82942 0.583269
\(285\) 16.8315 0.997015
\(286\) 0 0
\(287\) 8.60437 0.507900
\(288\) −3.92498 −0.231282
\(289\) −3.66952 −0.215854
\(290\) 5.78987 0.339993
\(291\) −11.9250 −0.699055
\(292\) 0.677267 0.0396341
\(293\) 8.83154 0.515944 0.257972 0.966152i \(-0.416946\pi\)
0.257972 + 0.966152i \(0.416946\pi\)
\(294\) −1.65109 −0.0962937
\(295\) 8.85772 0.515716
\(296\) 6.15215 0.357587
\(297\) 4.37720 0.253991
\(298\) −5.07714 −0.294111
\(299\) 0 0
\(300\) −2.58170 −0.149055
\(301\) 5.72611 0.330047
\(302\) 27.5088 1.58295
\(303\) −11.0283 −0.633559
\(304\) 28.3404 1.62543
\(305\) −0.883892 −0.0506115
\(306\) 6.02830 0.344615
\(307\) 11.6532 0.665084 0.332542 0.943088i \(-0.392094\pi\)
0.332542 + 0.943088i \(0.392094\pi\)
\(308\) 3.17833 0.181102
\(309\) 19.4543 1.10672
\(310\) −31.2597 −1.77543
\(311\) −25.0099 −1.41818 −0.709090 0.705118i \(-0.750894\pi\)
−0.709090 + 0.705118i \(0.750894\pi\)
\(312\) 0 0
\(313\) −0.186078 −0.0105178 −0.00525889 0.999986i \(-0.501674\pi\)
−0.00525889 + 0.999986i \(0.501674\pi\)
\(314\) 21.1826 1.19540
\(315\) −2.92498 −0.164804
\(316\) −12.3022 −0.692052
\(317\) −0.540031 −0.0303312 −0.0151656 0.999885i \(-0.504828\pi\)
−0.0151656 + 0.999885i \(0.504828\pi\)
\(318\) 0.498939 0.0279791
\(319\) −5.24772 −0.293816
\(320\) 9.85560 0.550945
\(321\) 1.42392 0.0794756
\(322\) −11.6044 −0.646686
\(323\) −21.0099 −1.16902
\(324\) 0.726109 0.0403394
\(325\) 0 0
\(326\) −35.6708 −1.97563
\(327\) 0.0467198 0.00258361
\(328\) 18.0977 0.999277
\(329\) 9.58383 0.528374
\(330\) 21.1394 1.16368
\(331\) 1.54003 0.0846478 0.0423239 0.999104i \(-0.486524\pi\)
0.0423239 + 0.999104i \(0.486524\pi\)
\(332\) −0.919357 −0.0504562
\(333\) −2.92498 −0.160288
\(334\) 35.2010 1.92611
\(335\) −25.4698 −1.39157
\(336\) −4.92498 −0.268680
\(337\) 2.82942 0.154128 0.0770642 0.997026i \(-0.475445\pi\)
0.0770642 + 0.997026i \(0.475445\pi\)
\(338\) 0 0
\(339\) 10.3382 0.561496
\(340\) 7.75441 0.420542
\(341\) 28.3326 1.53430
\(342\) −9.50106 −0.513759
\(343\) −1.00000 −0.0539949
\(344\) 12.0438 0.649358
\(345\) −20.5577 −1.10679
\(346\) 5.15428 0.277096
\(347\) −25.3900 −1.36301 −0.681503 0.731815i \(-0.738673\pi\)
−0.681503 + 0.731815i \(0.738673\pi\)
\(348\) −0.870514 −0.0466645
\(349\) −1.08569 −0.0581156 −0.0290578 0.999578i \(-0.509251\pi\)
−0.0290578 + 0.999578i \(0.509251\pi\)
\(350\) −5.87051 −0.313792
\(351\) 0 0
\(352\) 17.1805 0.915721
\(353\) 10.5294 0.560421 0.280211 0.959939i \(-0.409596\pi\)
0.280211 + 0.959939i \(0.409596\pi\)
\(354\) −5.00000 −0.265747
\(355\) 39.5958 2.10153
\(356\) −9.72823 −0.515595
\(357\) 3.65109 0.193236
\(358\) 31.3227 1.65546
\(359\) 27.2087 1.43602 0.718011 0.696031i \(-0.245053\pi\)
0.718011 + 0.696031i \(0.245053\pi\)
\(360\) −6.15215 −0.324247
\(361\) 14.1132 0.742799
\(362\) −3.19112 −0.167722
\(363\) −8.15990 −0.428284
\(364\) 0 0
\(365\) 2.72823 0.142802
\(366\) 0.498939 0.0260799
\(367\) −6.25839 −0.326685 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(368\) −34.6142 −1.80439
\(369\) −8.60437 −0.447926
\(370\) −14.1260 −0.734375
\(371\) 0.302187 0.0156888
\(372\) 4.69994 0.243680
\(373\) −1.17833 −0.0610115 −0.0305058 0.999535i \(-0.509712\pi\)
−0.0305058 + 0.999535i \(0.509712\pi\)
\(374\) −26.3871 −1.36444
\(375\) 4.22505 0.218181
\(376\) 20.1578 1.03956
\(377\) 0 0
\(378\) 1.65109 0.0849231
\(379\) −24.2066 −1.24341 −0.621705 0.783251i \(-0.713560\pi\)
−0.621705 + 0.783251i \(0.713560\pi\)
\(380\) −12.2215 −0.626952
\(381\) 5.47277 0.280378
\(382\) −3.75441 −0.192092
\(383\) −2.49119 −0.127294 −0.0636469 0.997972i \(-0.520273\pi\)
−0.0636469 + 0.997972i \(0.520273\pi\)
\(384\) −13.4132 −0.684492
\(385\) 12.8032 0.652514
\(386\) −7.24347 −0.368683
\(387\) −5.72611 −0.291075
\(388\) 8.65884 0.439586
\(389\) 23.6065 1.19690 0.598448 0.801161i \(-0.295784\pi\)
0.598448 + 0.801161i \(0.295784\pi\)
\(390\) 0 0
\(391\) 25.6610 1.29773
\(392\) −2.10331 −0.106233
\(393\) −16.1706 −0.815698
\(394\) −8.68209 −0.437398
\(395\) −49.5569 −2.49348
\(396\) −3.17833 −0.159717
\(397\) −8.84704 −0.444020 −0.222010 0.975044i \(-0.571262\pi\)
−0.222010 + 0.975044i \(0.571262\pi\)
\(398\) 21.7877 1.09212
\(399\) −5.75441 −0.288081
\(400\) −17.5109 −0.875547
\(401\) −18.9143 −0.944536 −0.472268 0.881455i \(-0.656564\pi\)
−0.472268 + 0.881455i \(0.656564\pi\)
\(402\) 14.3772 0.717070
\(403\) 0 0
\(404\) 8.00775 0.398400
\(405\) 2.92498 0.145344
\(406\) −1.97945 −0.0982386
\(407\) 12.8032 0.634633
\(408\) 7.67939 0.380186
\(409\) 3.49894 0.173011 0.0865057 0.996251i \(-0.472430\pi\)
0.0865057 + 0.996251i \(0.472430\pi\)
\(410\) −41.5541 −2.05221
\(411\) 12.0176 0.592786
\(412\) −14.1260 −0.695937
\(413\) −3.02830 −0.149013
\(414\) 11.6044 0.570324
\(415\) −3.70344 −0.181795
\(416\) 0 0
\(417\) −1.34891 −0.0660562
\(418\) 41.5881 2.03414
\(419\) 28.8315 1.40851 0.704257 0.709946i \(-0.251280\pi\)
0.704257 + 0.709946i \(0.251280\pi\)
\(420\) 2.12386 0.103634
\(421\) 34.3227 1.67279 0.836394 0.548129i \(-0.184660\pi\)
0.836394 + 0.548129i \(0.184660\pi\)
\(422\) −28.7976 −1.40185
\(423\) −9.58383 −0.465982
\(424\) 0.635593 0.0308671
\(425\) 12.9816 0.629699
\(426\) −22.3510 −1.08291
\(427\) 0.302187 0.0146238
\(428\) −1.03392 −0.0499766
\(429\) 0 0
\(430\) −27.6538 −1.33358
\(431\) 29.3559 1.41402 0.707011 0.707203i \(-0.250043\pi\)
0.707011 + 0.707203i \(0.250043\pi\)
\(432\) 4.92498 0.236953
\(433\) 28.6065 1.37474 0.687370 0.726307i \(-0.258765\pi\)
0.687370 + 0.726307i \(0.258765\pi\)
\(434\) 10.6871 0.512999
\(435\) −3.50669 −0.168133
\(436\) −0.0339237 −0.00162465
\(437\) −40.4437 −1.93468
\(438\) −1.54003 −0.0735855
\(439\) 34.4797 1.64563 0.822813 0.568311i \(-0.192403\pi\)
0.822813 + 0.568311i \(0.192403\pi\)
\(440\) 26.9292 1.28380
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −12.9066 −0.613209 −0.306605 0.951837i \(-0.599193\pi\)
−0.306605 + 0.951837i \(0.599193\pi\)
\(444\) 2.12386 0.100794
\(445\) −39.1882 −1.85770
\(446\) 13.2123 0.625618
\(447\) 3.07502 0.145443
\(448\) −3.36945 −0.159192
\(449\) −5.63830 −0.266088 −0.133044 0.991110i \(-0.542475\pi\)
−0.133044 + 0.991110i \(0.542475\pi\)
\(450\) 5.87051 0.276739
\(451\) 37.6631 1.77349
\(452\) −7.50669 −0.353085
\(453\) −16.6610 −0.782800
\(454\) −13.3070 −0.624529
\(455\) 0 0
\(456\) −12.1033 −0.566790
\(457\) 37.9893 1.77707 0.888533 0.458813i \(-0.151725\pi\)
0.888533 + 0.458813i \(0.151725\pi\)
\(458\) −14.0021 −0.654276
\(459\) −3.65109 −0.170418
\(460\) 14.9271 0.695980
\(461\) −6.60730 −0.307733 −0.153866 0.988092i \(-0.549172\pi\)
−0.153866 + 0.988092i \(0.549172\pi\)
\(462\) −7.22717 −0.336238
\(463\) 9.98933 0.464243 0.232122 0.972687i \(-0.425433\pi\)
0.232122 + 0.972687i \(0.425433\pi\)
\(464\) −5.90444 −0.274107
\(465\) 18.9327 0.877985
\(466\) −16.9164 −0.783639
\(467\) 3.42100 0.158305 0.0791525 0.996863i \(-0.474779\pi\)
0.0791525 + 0.996863i \(0.474779\pi\)
\(468\) 0 0
\(469\) 8.70769 0.402084
\(470\) −46.2843 −2.13494
\(471\) −12.8294 −0.591149
\(472\) −6.36945 −0.293178
\(473\) 25.0643 1.15246
\(474\) 27.9738 1.28488
\(475\) −20.4600 −0.938768
\(476\) −2.65109 −0.121513
\(477\) −0.302187 −0.0138362
\(478\) −18.6610 −0.853533
\(479\) 19.8414 0.906577 0.453289 0.891364i \(-0.350251\pi\)
0.453289 + 0.891364i \(0.350251\pi\)
\(480\) 11.4805 0.524011
\(481\) 0 0
\(482\) 1.71061 0.0779161
\(483\) 7.02830 0.319799
\(484\) 5.92498 0.269317
\(485\) 34.8804 1.58384
\(486\) −1.65109 −0.0748951
\(487\) −6.10331 −0.276567 −0.138284 0.990393i \(-0.544159\pi\)
−0.138284 + 0.990393i \(0.544159\pi\)
\(488\) 0.635593 0.0287720
\(489\) 21.6044 0.976984
\(490\) 4.82942 0.218171
\(491\) −2.36653 −0.106800 −0.0534000 0.998573i \(-0.517006\pi\)
−0.0534000 + 0.998573i \(0.517006\pi\)
\(492\) 6.24772 0.281669
\(493\) 4.37720 0.197139
\(494\) 0 0
\(495\) −12.8032 −0.575463
\(496\) 31.8783 1.43138
\(497\) −13.5371 −0.607222
\(498\) 2.09052 0.0936783
\(499\) −4.29231 −0.192150 −0.0960752 0.995374i \(-0.530629\pi\)
−0.0960752 + 0.995374i \(0.530629\pi\)
\(500\) −3.06785 −0.137198
\(501\) −21.3198 −0.952499
\(502\) 21.9327 0.978906
\(503\) −2.11106 −0.0941276 −0.0470638 0.998892i \(-0.514986\pi\)
−0.0470638 + 0.998892i \(0.514986\pi\)
\(504\) 2.10331 0.0936890
\(505\) 32.2576 1.43544
\(506\) −50.7947 −2.25810
\(507\) 0 0
\(508\) −3.97383 −0.176310
\(509\) −24.3764 −1.08047 −0.540233 0.841516i \(-0.681664\pi\)
−0.540233 + 0.841516i \(0.681664\pi\)
\(510\) −17.6327 −0.780788
\(511\) −0.932734 −0.0412617
\(512\) −1.38708 −0.0613007
\(513\) 5.75441 0.254063
\(514\) 20.5422 0.906076
\(515\) −56.9036 −2.50747
\(516\) 4.15778 0.183036
\(517\) 41.9504 1.84497
\(518\) 4.82942 0.212193
\(519\) −3.12174 −0.137029
\(520\) 0 0
\(521\) −17.8401 −0.781589 −0.390794 0.920478i \(-0.627800\pi\)
−0.390794 + 0.920478i \(0.627800\pi\)
\(522\) 1.97945 0.0866383
\(523\) −31.0558 −1.35797 −0.678987 0.734150i \(-0.737581\pi\)
−0.678987 + 0.734150i \(0.737581\pi\)
\(524\) 11.7416 0.512935
\(525\) 3.55553 0.155176
\(526\) 10.8833 0.474535
\(527\) −23.6327 −1.02946
\(528\) −21.5577 −0.938176
\(529\) 26.3969 1.14769
\(530\) −1.45939 −0.0633917
\(531\) 3.02830 0.131417
\(532\) 4.17833 0.181154
\(533\) 0 0
\(534\) 22.1209 0.957266
\(535\) −4.16495 −0.180067
\(536\) 18.3150 0.791087
\(537\) −18.9709 −0.818655
\(538\) −4.48476 −0.193352
\(539\) −4.37720 −0.188539
\(540\) −2.12386 −0.0913963
\(541\) −33.3900 −1.43555 −0.717774 0.696276i \(-0.754839\pi\)
−0.717774 + 0.696276i \(0.754839\pi\)
\(542\) −50.7304 −2.17906
\(543\) 1.93273 0.0829416
\(544\) −14.3305 −0.614414
\(545\) −0.136655 −0.00585364
\(546\) 0 0
\(547\) 9.12306 0.390074 0.195037 0.980796i \(-0.437517\pi\)
0.195037 + 0.980796i \(0.437517\pi\)
\(548\) −8.72611 −0.372761
\(549\) −0.302187 −0.0128970
\(550\) −25.6964 −1.09570
\(551\) −6.89881 −0.293899
\(552\) 14.7827 0.629194
\(553\) 16.9426 0.720473
\(554\) −35.6065 −1.51278
\(555\) 8.55553 0.363162
\(556\) 0.979454 0.0415381
\(557\) −38.0296 −1.61137 −0.805683 0.592347i \(-0.798202\pi\)
−0.805683 + 0.592347i \(0.798202\pi\)
\(558\) −10.6871 −0.452423
\(559\) 0 0
\(560\) 14.4055 0.608743
\(561\) 15.9816 0.674743
\(562\) 54.8259 2.31269
\(563\) −3.36653 −0.141882 −0.0709411 0.997481i \(-0.522600\pi\)
−0.0709411 + 0.997481i \(0.522600\pi\)
\(564\) 6.95891 0.293023
\(565\) −30.2392 −1.27217
\(566\) 2.44930 0.102952
\(567\) −1.00000 −0.0419961
\(568\) −28.4728 −1.19469
\(569\) −17.9327 −0.751779 −0.375890 0.926664i \(-0.622663\pi\)
−0.375890 + 0.926664i \(0.622663\pi\)
\(570\) 27.7905 1.16401
\(571\) 39.4274 1.64998 0.824992 0.565144i \(-0.191180\pi\)
0.824992 + 0.565144i \(0.191180\pi\)
\(572\) 0 0
\(573\) 2.27389 0.0949931
\(574\) 14.2066 0.592973
\(575\) 24.9893 1.04213
\(576\) 3.36945 0.140394
\(577\) 16.4239 0.683737 0.341868 0.939748i \(-0.388940\pi\)
0.341868 + 0.939748i \(0.388940\pi\)
\(578\) −6.05872 −0.252009
\(579\) 4.38708 0.182321
\(580\) 2.54624 0.105727
\(581\) 1.26614 0.0525284
\(582\) −19.6893 −0.816146
\(583\) 1.32273 0.0547820
\(584\) −1.96183 −0.0811811
\(585\) 0 0
\(586\) 14.5817 0.602365
\(587\) −14.9271 −0.616108 −0.308054 0.951369i \(-0.599678\pi\)
−0.308054 + 0.951369i \(0.599678\pi\)
\(588\) −0.726109 −0.0299442
\(589\) 37.2469 1.53473
\(590\) 14.6249 0.602098
\(591\) 5.25839 0.216301
\(592\) 14.4055 0.592062
\(593\) 29.9164 1.22852 0.614260 0.789103i \(-0.289454\pi\)
0.614260 + 0.789103i \(0.289454\pi\)
\(594\) 7.22717 0.296534
\(595\) −10.6794 −0.437812
\(596\) −2.23280 −0.0914590
\(597\) −13.1960 −0.540074
\(598\) 0 0
\(599\) 35.1911 1.43787 0.718935 0.695077i \(-0.244630\pi\)
0.718935 + 0.695077i \(0.244630\pi\)
\(600\) 7.47839 0.305304
\(601\) 17.1316 0.698813 0.349406 0.936971i \(-0.386383\pi\)
0.349406 + 0.936971i \(0.386383\pi\)
\(602\) 9.45434 0.385330
\(603\) −8.70769 −0.354604
\(604\) 12.0977 0.492248
\(605\) 23.8676 0.970356
\(606\) −18.2087 −0.739680
\(607\) 22.4466 0.911079 0.455540 0.890216i \(-0.349446\pi\)
0.455540 + 0.890216i \(0.349446\pi\)
\(608\) 22.5860 0.915981
\(609\) 1.19887 0.0485808
\(610\) −1.45939 −0.0590889
\(611\) 0 0
\(612\) 2.65109 0.107164
\(613\) −20.2058 −0.816106 −0.408053 0.912958i \(-0.633792\pi\)
−0.408053 + 0.912958i \(0.633792\pi\)
\(614\) 19.2405 0.776485
\(615\) 25.1677 1.01486
\(616\) −9.20662 −0.370945
\(617\) 38.6815 1.55726 0.778630 0.627484i \(-0.215915\pi\)
0.778630 + 0.627484i \(0.215915\pi\)
\(618\) 32.1209 1.29209
\(619\) 32.5109 1.30672 0.653362 0.757045i \(-0.273358\pi\)
0.653362 + 0.757045i \(0.273358\pi\)
\(620\) −13.7472 −0.552102
\(621\) −7.02830 −0.282036
\(622\) −41.2936 −1.65572
\(623\) 13.3977 0.536769
\(624\) 0 0
\(625\) −30.1359 −1.20543
\(626\) −0.307233 −0.0122795
\(627\) −25.1882 −1.00592
\(628\) 9.31556 0.371731
\(629\) −10.6794 −0.425815
\(630\) −4.82942 −0.192409
\(631\) −37.8620 −1.50726 −0.753630 0.657298i \(-0.771699\pi\)
−0.753630 + 0.657298i \(0.771699\pi\)
\(632\) 35.6356 1.41751
\(633\) 17.4415 0.693239
\(634\) −0.891642 −0.0354116
\(635\) −16.0078 −0.635248
\(636\) 0.219421 0.00870060
\(637\) 0 0
\(638\) −8.66447 −0.343030
\(639\) 13.5371 0.535520
\(640\) 39.2335 1.55084
\(641\) 22.8628 0.903025 0.451512 0.892265i \(-0.350885\pi\)
0.451512 + 0.892265i \(0.350885\pi\)
\(642\) 2.35103 0.0927877
\(643\) 43.5830 1.71875 0.859373 0.511349i \(-0.170854\pi\)
0.859373 + 0.511349i \(0.170854\pi\)
\(644\) −5.10331 −0.201099
\(645\) 16.7488 0.659483
\(646\) −34.6893 −1.36483
\(647\) −3.00775 −0.118247 −0.0591234 0.998251i \(-0.518831\pi\)
−0.0591234 + 0.998251i \(0.518831\pi\)
\(648\) −2.10331 −0.0826259
\(649\) −13.2555 −0.520323
\(650\) 0 0
\(651\) −6.47277 −0.253688
\(652\) −15.6871 −0.614356
\(653\) −33.9242 −1.32756 −0.663778 0.747930i \(-0.731048\pi\)
−0.663778 + 0.747930i \(0.731048\pi\)
\(654\) 0.0771387 0.00301636
\(655\) 47.2987 1.84811
\(656\) 42.3764 1.65452
\(657\) 0.932734 0.0363894
\(658\) 15.8238 0.616876
\(659\) −43.4594 −1.69294 −0.846469 0.532438i \(-0.821276\pi\)
−0.846469 + 0.532438i \(0.821276\pi\)
\(660\) 9.29656 0.361868
\(661\) 38.2859 1.48915 0.744574 0.667540i \(-0.232653\pi\)
0.744574 + 0.667540i \(0.232653\pi\)
\(662\) 2.54274 0.0988262
\(663\) 0 0
\(664\) 2.66309 0.103348
\(665\) 16.8315 0.652699
\(666\) −4.82942 −0.187136
\(667\) 8.42605 0.326258
\(668\) 15.4805 0.598959
\(669\) −8.00212 −0.309380
\(670\) −42.0531 −1.62465
\(671\) 1.32273 0.0510635
\(672\) −3.92498 −0.151410
\(673\) 28.1805 1.08628 0.543138 0.839643i \(-0.317236\pi\)
0.543138 + 0.839643i \(0.317236\pi\)
\(674\) 4.67164 0.179945
\(675\) −3.55553 −0.136852
\(676\) 0 0
\(677\) −31.7253 −1.21930 −0.609651 0.792670i \(-0.708691\pi\)
−0.609651 + 0.792670i \(0.708691\pi\)
\(678\) 17.0694 0.655546
\(679\) −11.9250 −0.457639
\(680\) −22.4621 −0.861382
\(681\) 8.05952 0.308841
\(682\) 46.7798 1.79129
\(683\) 8.57395 0.328073 0.164037 0.986454i \(-0.447548\pi\)
0.164037 + 0.986454i \(0.447548\pi\)
\(684\) −4.17833 −0.159762
\(685\) −35.1514 −1.34306
\(686\) −1.65109 −0.0630390
\(687\) 8.48052 0.323552
\(688\) 28.2010 1.07515
\(689\) 0 0
\(690\) −33.9426 −1.29217
\(691\) −6.95036 −0.264404 −0.132202 0.991223i \(-0.542205\pi\)
−0.132202 + 0.991223i \(0.542205\pi\)
\(692\) 2.26672 0.0861678
\(693\) 4.37720 0.166276
\(694\) −41.9213 −1.59131
\(695\) 3.94553 0.149662
\(696\) 2.52161 0.0955813
\(697\) −31.4154 −1.18994
\(698\) −1.79257 −0.0678500
\(699\) 10.2456 0.387524
\(700\) −2.58170 −0.0975793
\(701\) −21.8443 −0.825049 −0.412525 0.910946i \(-0.635353\pi\)
−0.412525 + 0.910946i \(0.635353\pi\)
\(702\) 0 0
\(703\) 16.8315 0.634814
\(704\) −14.7488 −0.555865
\(705\) 28.0325 1.05577
\(706\) 17.3850 0.654291
\(707\) −11.0283 −0.414762
\(708\) −2.19887 −0.0826388
\(709\) 9.74373 0.365934 0.182967 0.983119i \(-0.441430\pi\)
0.182967 + 0.983119i \(0.441430\pi\)
\(710\) 65.3764 2.45353
\(711\) −16.9426 −0.635397
\(712\) 28.1797 1.05608
\(713\) −45.4925 −1.70371
\(714\) 6.02830 0.225603
\(715\) 0 0
\(716\) 13.7750 0.514794
\(717\) 11.3022 0.422088
\(718\) 44.9242 1.67656
\(719\) −5.04592 −0.188181 −0.0940905 0.995564i \(-0.529994\pi\)
−0.0940905 + 0.995564i \(0.529994\pi\)
\(720\) −14.4055 −0.536861
\(721\) 19.4543 0.724518
\(722\) 23.3022 0.867218
\(723\) −1.03605 −0.0385310
\(724\) −1.40338 −0.0521561
\(725\) 4.26264 0.158310
\(726\) −13.4728 −0.500021
\(727\) 6.66659 0.247250 0.123625 0.992329i \(-0.460548\pi\)
0.123625 + 0.992329i \(0.460548\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.50457 0.166721
\(731\) −20.9066 −0.773257
\(732\) 0.219421 0.00811002
\(733\) 47.4076 1.75104 0.875520 0.483181i \(-0.160519\pi\)
0.875520 + 0.483181i \(0.160519\pi\)
\(734\) −10.3332 −0.381405
\(735\) −2.92498 −0.107890
\(736\) −27.5860 −1.01683
\(737\) 38.1153 1.40400
\(738\) −14.2066 −0.522953
\(739\) 6.18045 0.227352 0.113676 0.993518i \(-0.463737\pi\)
0.113676 + 0.993518i \(0.463737\pi\)
\(740\) −6.21225 −0.228367
\(741\) 0 0
\(742\) 0.498939 0.0183166
\(743\) −23.3334 −0.856020 −0.428010 0.903774i \(-0.640785\pi\)
−0.428010 + 0.903774i \(0.640785\pi\)
\(744\) −13.6142 −0.499122
\(745\) −8.99437 −0.329528
\(746\) −1.94553 −0.0712309
\(747\) −1.26614 −0.0463257
\(748\) −11.6044 −0.424298
\(749\) 1.42392 0.0520290
\(750\) 6.97595 0.254726
\(751\) −0.976750 −0.0356421 −0.0178211 0.999841i \(-0.505673\pi\)
−0.0178211 + 0.999841i \(0.505673\pi\)
\(752\) 47.2002 1.72121
\(753\) −13.2838 −0.484087
\(754\) 0 0
\(755\) 48.7331 1.77358
\(756\) 0.726109 0.0264083
\(757\) −8.76508 −0.318572 −0.159286 0.987232i \(-0.550919\pi\)
−0.159286 + 0.987232i \(0.550919\pi\)
\(758\) −39.9674 −1.45168
\(759\) 30.7643 1.11667
\(760\) 35.4020 1.28417
\(761\) −21.4551 −0.777748 −0.388874 0.921291i \(-0.627136\pi\)
−0.388874 + 0.921291i \(0.627136\pi\)
\(762\) 9.03605 0.327341
\(763\) 0.0467198 0.00169137
\(764\) −1.65109 −0.0597345
\(765\) 10.6794 0.386114
\(766\) −4.11319 −0.148615
\(767\) 0 0
\(768\) −15.4076 −0.555975
\(769\) 22.1500 0.798751 0.399375 0.916788i \(-0.369227\pi\)
0.399375 + 0.916788i \(0.369227\pi\)
\(770\) 21.1394 0.761810
\(771\) −12.4415 −0.448071
\(772\) −3.18550 −0.114649
\(773\) −19.1599 −0.689134 −0.344567 0.938762i \(-0.611974\pi\)
−0.344567 + 0.938762i \(0.611974\pi\)
\(774\) −9.45434 −0.339829
\(775\) −23.0141 −0.826692
\(776\) −25.0820 −0.900390
\(777\) −2.92498 −0.104933
\(778\) 38.9765 1.39738
\(779\) 49.5131 1.77399
\(780\) 0 0
\(781\) −59.2547 −2.12030
\(782\) 42.3687 1.51510
\(783\) −1.19887 −0.0428443
\(784\) −4.92498 −0.175892
\(785\) 37.5259 1.33936
\(786\) −26.6991 −0.952327
\(787\) 44.1797 1.57483 0.787417 0.616420i \(-0.211418\pi\)
0.787417 + 0.616420i \(0.211418\pi\)
\(788\) −3.81817 −0.136017
\(789\) −6.59158 −0.234666
\(790\) −81.8230 −2.91113
\(791\) 10.3382 0.367585
\(792\) 9.20662 0.327143
\(793\) 0 0
\(794\) −14.6073 −0.518394
\(795\) 0.883892 0.0313484
\(796\) 9.58170 0.339615
\(797\) 13.6561 0.483725 0.241863 0.970310i \(-0.422242\pi\)
0.241863 + 0.970310i \(0.422242\pi\)
\(798\) −9.50106 −0.336334
\(799\) −34.9914 −1.23791
\(800\) −13.9554 −0.493398
\(801\) −13.3977 −0.473386
\(802\) −31.2293 −1.10274
\(803\) −4.08277 −0.144078
\(804\) 6.32273 0.222986
\(805\) −20.5577 −0.724562
\(806\) 0 0
\(807\) 2.71624 0.0956161
\(808\) −23.1960 −0.816031
\(809\) 34.0078 1.19565 0.597824 0.801627i \(-0.296032\pi\)
0.597824 + 0.801627i \(0.296032\pi\)
\(810\) 4.82942 0.169689
\(811\) −21.4084 −0.751751 −0.375876 0.926670i \(-0.622658\pi\)
−0.375876 + 0.926670i \(0.622658\pi\)
\(812\) −0.870514 −0.0305491
\(813\) 30.7253 1.07758
\(814\) 21.1394 0.740934
\(815\) −63.1924 −2.21353
\(816\) 17.9816 0.629481
\(817\) 32.9504 1.15279
\(818\) 5.77707 0.201991
\(819\) 0 0
\(820\) −18.2745 −0.638172
\(821\) 27.1367 0.947076 0.473538 0.880773i \(-0.342977\pi\)
0.473538 + 0.880773i \(0.342977\pi\)
\(822\) 19.8422 0.692077
\(823\) −46.6417 −1.62583 −0.812914 0.582383i \(-0.802120\pi\)
−0.812914 + 0.582383i \(0.802120\pi\)
\(824\) 40.9186 1.42547
\(825\) 15.5633 0.541844
\(826\) −5.00000 −0.173972
\(827\) −27.2789 −0.948582 −0.474291 0.880368i \(-0.657295\pi\)
−0.474291 + 0.880368i \(0.657295\pi\)
\(828\) 5.10331 0.177352
\(829\) −25.3219 −0.879467 −0.439734 0.898128i \(-0.644927\pi\)
−0.439734 + 0.898128i \(0.644927\pi\)
\(830\) −6.11473 −0.212245
\(831\) 21.5654 0.748096
\(832\) 0 0
\(833\) 3.65109 0.126503
\(834\) −2.22717 −0.0771206
\(835\) 62.3601 2.15806
\(836\) 18.2894 0.632552
\(837\) 6.47277 0.223731
\(838\) 47.6036 1.64444
\(839\) 0.366529 0.0126540 0.00632700 0.999980i \(-0.497986\pi\)
0.00632700 + 0.999980i \(0.497986\pi\)
\(840\) −6.15215 −0.212270
\(841\) −27.5627 −0.950438
\(842\) 56.6700 1.95298
\(843\) −33.2058 −1.14367
\(844\) −12.6645 −0.435929
\(845\) 0 0
\(846\) −15.8238 −0.544033
\(847\) −8.15990 −0.280378
\(848\) 1.48827 0.0511072
\(849\) −1.48344 −0.0509115
\(850\) 21.4338 0.735173
\(851\) −20.5577 −0.704707
\(852\) −9.82942 −0.336750
\(853\) −49.3134 −1.68846 −0.844229 0.535983i \(-0.819941\pi\)
−0.844229 + 0.535983i \(0.819941\pi\)
\(854\) 0.498939 0.0170733
\(855\) −16.8315 −0.575627
\(856\) 2.99495 0.102365
\(857\) 26.0878 0.891143 0.445571 0.895246i \(-0.353001\pi\)
0.445571 + 0.895246i \(0.353001\pi\)
\(858\) 0 0
\(859\) 29.6815 1.01272 0.506360 0.862322i \(-0.330991\pi\)
0.506360 + 0.862322i \(0.330991\pi\)
\(860\) −12.1614 −0.414702
\(861\) −8.60437 −0.293236
\(862\) 48.4693 1.65087
\(863\) 17.8804 0.608655 0.304328 0.952567i \(-0.401568\pi\)
0.304328 + 0.952567i \(0.401568\pi\)
\(864\) 3.92498 0.133531
\(865\) 9.13103 0.310464
\(866\) 47.2320 1.60501
\(867\) 3.66952 0.124623
\(868\) 4.69994 0.159526
\(869\) 74.1612 2.51575
\(870\) −5.78987 −0.196295
\(871\) 0 0
\(872\) 0.0982663 0.00332772
\(873\) 11.9250 0.403600
\(874\) −66.7763 −2.25874
\(875\) 4.22505 0.142833
\(876\) −0.677267 −0.0228827
\(877\) 25.6121 0.864860 0.432430 0.901668i \(-0.357656\pi\)
0.432430 + 0.901668i \(0.357656\pi\)
\(878\) 56.9292 1.92127
\(879\) −8.83154 −0.297881
\(880\) 63.0558 2.12561
\(881\) −42.1642 −1.42055 −0.710273 0.703926i \(-0.751429\pi\)
−0.710273 + 0.703926i \(0.751429\pi\)
\(882\) 1.65109 0.0555952
\(883\) −32.9992 −1.11051 −0.555256 0.831680i \(-0.687380\pi\)
−0.555256 + 0.831680i \(0.687380\pi\)
\(884\) 0 0
\(885\) −8.85772 −0.297749
\(886\) −21.3099 −0.715921
\(887\) 9.28669 0.311816 0.155908 0.987772i \(-0.450170\pi\)
0.155908 + 0.987772i \(0.450170\pi\)
\(888\) −6.15215 −0.206453
\(889\) 5.47277 0.183551
\(890\) −64.7034 −2.16886
\(891\) −4.37720 −0.146642
\(892\) 5.81042 0.194547
\(893\) 55.1492 1.84550
\(894\) 5.07714 0.169805
\(895\) 55.4896 1.85481
\(896\) −13.4132 −0.448105
\(897\) 0 0
\(898\) −9.30936 −0.310657
\(899\) −7.76003 −0.258812
\(900\) 2.58170 0.0860568
\(901\) −1.10331 −0.0367567
\(902\) 62.1853 2.07054
\(903\) −5.72611 −0.190553
\(904\) 21.7445 0.723212
\(905\) −5.65322 −0.187919
\(906\) −27.5088 −0.913919
\(907\) 21.2448 0.705422 0.352711 0.935732i \(-0.385260\pi\)
0.352711 + 0.935732i \(0.385260\pi\)
\(908\) −5.85209 −0.194208
\(909\) 11.0283 0.365785
\(910\) 0 0
\(911\) −54.0785 −1.79170 −0.895850 0.444357i \(-0.853432\pi\)
−0.895850 + 0.444357i \(0.853432\pi\)
\(912\) −28.3404 −0.938443
\(913\) 5.54215 0.183419
\(914\) 62.7239 2.07472
\(915\) 0.883892 0.0292206
\(916\) −6.15778 −0.203459
\(917\) −16.1706 −0.534000
\(918\) −6.02830 −0.198963
\(919\) −18.9426 −0.624859 −0.312429 0.949941i \(-0.601143\pi\)
−0.312429 + 0.949941i \(0.601143\pi\)
\(920\) −43.2392 −1.42555
\(921\) −11.6532 −0.383987
\(922\) −10.9093 −0.359277
\(923\) 0 0
\(924\) −3.17833 −0.104559
\(925\) −10.3999 −0.341946
\(926\) 16.4933 0.542004
\(927\) −19.4543 −0.638964
\(928\) −4.70556 −0.154468
\(929\) −45.3347 −1.48738 −0.743692 0.668522i \(-0.766927\pi\)
−0.743692 + 0.668522i \(0.766927\pi\)
\(930\) 31.2597 1.02505
\(931\) −5.75441 −0.188593
\(932\) −7.43942 −0.243686
\(933\) 25.0099 0.818786
\(934\) 5.64839 0.184821
\(935\) −46.7459 −1.52875
\(936\) 0 0
\(937\) 3.71544 0.121378 0.0606890 0.998157i \(-0.480670\pi\)
0.0606890 + 0.998157i \(0.480670\pi\)
\(938\) 14.3772 0.469432
\(939\) 0.186078 0.00607244
\(940\) −20.3547 −0.663897
\(941\) −4.28376 −0.139647 −0.0698233 0.997559i \(-0.522244\pi\)
−0.0698233 + 0.997559i \(0.522244\pi\)
\(942\) −21.1826 −0.690166
\(943\) −60.4741 −1.96931
\(944\) −14.9143 −0.485419
\(945\) 2.92498 0.0951497
\(946\) 41.3836 1.34550
\(947\) −25.0205 −0.813059 −0.406529 0.913638i \(-0.633261\pi\)
−0.406529 + 0.913638i \(0.633261\pi\)
\(948\) 12.3022 0.399556
\(949\) 0 0
\(950\) −33.7813 −1.09601
\(951\) 0.540031 0.0175117
\(952\) 7.67939 0.248890
\(953\) −2.84434 −0.0921372 −0.0460686 0.998938i \(-0.514669\pi\)
−0.0460686 + 0.998938i \(0.514669\pi\)
\(954\) −0.498939 −0.0161537
\(955\) −6.65109 −0.215224
\(956\) −8.20662 −0.265421
\(957\) 5.24772 0.169635
\(958\) 32.7600 1.05843
\(959\) 12.0176 0.388069
\(960\) −9.85560 −0.318088
\(961\) 10.8967 0.351506
\(962\) 0 0
\(963\) −1.42392 −0.0458853
\(964\) 0.752283 0.0242294
\(965\) −12.8321 −0.413081
\(966\) 11.6044 0.373365
\(967\) −6.68444 −0.214957 −0.107478 0.994207i \(-0.534278\pi\)
−0.107478 + 0.994207i \(0.534278\pi\)
\(968\) −17.1628 −0.551634
\(969\) 21.0099 0.674934
\(970\) 57.5908 1.84913
\(971\) −26.6278 −0.854528 −0.427264 0.904127i \(-0.640523\pi\)
−0.427264 + 0.904127i \(0.640523\pi\)
\(972\) −0.726109 −0.0232900
\(973\) −1.34891 −0.0432440
\(974\) −10.0771 −0.322892
\(975\) 0 0
\(976\) 1.48827 0.0476382
\(977\) −27.8179 −0.889975 −0.444987 0.895537i \(-0.646792\pi\)
−0.444987 + 0.895537i \(0.646792\pi\)
\(978\) 35.6708 1.14063
\(979\) 58.6447 1.87429
\(980\) 2.12386 0.0678442
\(981\) −0.0467198 −0.00149165
\(982\) −3.90736 −0.124689
\(983\) 51.3454 1.63766 0.818832 0.574033i \(-0.194622\pi\)
0.818832 + 0.574033i \(0.194622\pi\)
\(984\) −18.0977 −0.576933
\(985\) −15.3807 −0.490070
\(986\) 7.22717 0.230160
\(987\) −9.58383 −0.305057
\(988\) 0 0
\(989\) −40.2448 −1.27971
\(990\) −21.1394 −0.671853
\(991\) −23.3745 −0.742515 −0.371258 0.928530i \(-0.621073\pi\)
−0.371258 + 0.928530i \(0.621073\pi\)
\(992\) 25.4055 0.806625
\(993\) −1.54003 −0.0488714
\(994\) −22.3510 −0.708932
\(995\) 38.5979 1.22364
\(996\) 0.919357 0.0291309
\(997\) −28.7955 −0.911963 −0.455981 0.889989i \(-0.650712\pi\)
−0.455981 + 0.889989i \(0.650712\pi\)
\(998\) −7.08701 −0.224335
\(999\) 2.92498 0.0925424
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 3549.2.a.h.1.3 3
13.3 even 3 273.2.k.d.22.1 6
13.9 even 3 273.2.k.d.211.1 yes 6
13.12 even 2 3549.2.a.s.1.1 3
39.29 odd 6 819.2.o.d.568.3 6
39.35 odd 6 819.2.o.d.757.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.k.d.22.1 6 13.3 even 3
273.2.k.d.211.1 yes 6 13.9 even 3
819.2.o.d.568.3 6 39.29 odd 6
819.2.o.d.757.3 6 39.35 odd 6
3549.2.a.h.1.3 3 1.1 even 1 trivial
3549.2.a.s.1.1 3 13.12 even 2