Properties

Label 3025.2.a.y.1.3
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.517638 q^{2} -1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} -3.34607 q^{7} -1.93185 q^{8} +0.732051 q^{9} +O(q^{10})\) \(q+0.517638 q^{2} -1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} -3.34607 q^{7} -1.93185 q^{8} +0.732051 q^{9} +3.34607 q^{12} +4.24264 q^{13} -1.73205 q^{14} +2.46410 q^{16} +3.86370 q^{17} +0.378937 q^{18} -4.19615 q^{19} +6.46410 q^{21} +3.48477 q^{23} +3.73205 q^{24} +2.19615 q^{26} +4.38134 q^{27} +5.79555 q^{28} +6.92820 q^{29} -8.73205 q^{31} +5.13922 q^{32} +2.00000 q^{34} -1.26795 q^{36} +1.79315 q^{37} -2.17209 q^{38} -8.19615 q^{39} +1.73205 q^{41} +3.34607 q^{42} -6.45189 q^{43} +1.80385 q^{46} +11.4524 q^{47} -4.76028 q^{48} +4.19615 q^{49} -7.46410 q^{51} -7.34847 q^{52} +2.17209 q^{53} +2.26795 q^{54} +6.46410 q^{56} +8.10634 q^{57} +3.58630 q^{58} -1.26795 q^{59} -11.7321 q^{61} -4.52004 q^{62} -2.44949 q^{63} -2.26795 q^{64} +2.20925 q^{67} -6.69213 q^{68} -6.73205 q^{69} -8.19615 q^{71} -1.41421 q^{72} -4.89898 q^{73} +0.928203 q^{74} +7.26795 q^{76} -4.24264 q^{78} +1.46410 q^{79} -10.6603 q^{81} +0.896575 q^{82} -9.89949 q^{83} -11.1962 q^{84} -3.33975 q^{86} -13.3843 q^{87} +0.464102 q^{89} -14.1962 q^{91} -6.03579 q^{92} +16.8690 q^{93} +5.92820 q^{94} -9.92820 q^{96} -9.14162 q^{97} +2.17209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638 0.366025 0.183013 0.983111i \(-0.441415\pi\)
0.183013 + 0.983111i \(0.441415\pi\)
\(3\) −1.93185 −1.11536 −0.557678 0.830058i \(-0.688307\pi\)
−0.557678 + 0.830058i \(0.688307\pi\)
\(4\) −1.73205 −0.866025
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −3.34607 −1.26469 −0.632347 0.774685i \(-0.717908\pi\)
−0.632347 + 0.774685i \(0.717908\pi\)
\(8\) −1.93185 −0.683013
\(9\) 0.732051 0.244017
\(10\) 0 0
\(11\) 0 0
\(12\) 3.34607 0.965926
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) −1.73205 −0.462910
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 3.86370 0.937086 0.468543 0.883441i \(-0.344779\pi\)
0.468543 + 0.883441i \(0.344779\pi\)
\(18\) 0.378937 0.0893164
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) 0 0
\(21\) 6.46410 1.41058
\(22\) 0 0
\(23\) 3.48477 0.726624 0.363312 0.931668i \(-0.381646\pi\)
0.363312 + 0.931668i \(0.381646\pi\)
\(24\) 3.73205 0.761802
\(25\) 0 0
\(26\) 2.19615 0.430701
\(27\) 4.38134 0.843190
\(28\) 5.79555 1.09526
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) −8.73205 −1.56832 −0.784161 0.620557i \(-0.786907\pi\)
−0.784161 + 0.620557i \(0.786907\pi\)
\(32\) 5.13922 0.908494
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.26795 −0.211325
\(37\) 1.79315 0.294792 0.147396 0.989078i \(-0.452911\pi\)
0.147396 + 0.989078i \(0.452911\pi\)
\(38\) −2.17209 −0.352359
\(39\) −8.19615 −1.31243
\(40\) 0 0
\(41\) 1.73205 0.270501 0.135250 0.990811i \(-0.456816\pi\)
0.135250 + 0.990811i \(0.456816\pi\)
\(42\) 3.34607 0.516309
\(43\) −6.45189 −0.983905 −0.491952 0.870622i \(-0.663717\pi\)
−0.491952 + 0.870622i \(0.663717\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.80385 0.265963
\(47\) 11.4524 1.67051 0.835253 0.549866i \(-0.185321\pi\)
0.835253 + 0.549866i \(0.185321\pi\)
\(48\) −4.76028 −0.687087
\(49\) 4.19615 0.599450
\(50\) 0 0
\(51\) −7.46410 −1.04518
\(52\) −7.34847 −1.01905
\(53\) 2.17209 0.298359 0.149180 0.988810i \(-0.452337\pi\)
0.149180 + 0.988810i \(0.452337\pi\)
\(54\) 2.26795 0.308629
\(55\) 0 0
\(56\) 6.46410 0.863802
\(57\) 8.10634 1.07371
\(58\) 3.58630 0.470905
\(59\) −1.26795 −0.165073 −0.0825365 0.996588i \(-0.526302\pi\)
−0.0825365 + 0.996588i \(0.526302\pi\)
\(60\) 0 0
\(61\) −11.7321 −1.50214 −0.751068 0.660225i \(-0.770461\pi\)
−0.751068 + 0.660225i \(0.770461\pi\)
\(62\) −4.52004 −0.574046
\(63\) −2.44949 −0.308607
\(64\) −2.26795 −0.283494
\(65\) 0 0
\(66\) 0 0
\(67\) 2.20925 0.269903 0.134952 0.990852i \(-0.456912\pi\)
0.134952 + 0.990852i \(0.456912\pi\)
\(68\) −6.69213 −0.811540
\(69\) −6.73205 −0.810444
\(70\) 0 0
\(71\) −8.19615 −0.972704 −0.486352 0.873763i \(-0.661673\pi\)
−0.486352 + 0.873763i \(0.661673\pi\)
\(72\) −1.41421 −0.166667
\(73\) −4.89898 −0.573382 −0.286691 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0.928203 0.107901
\(75\) 0 0
\(76\) 7.26795 0.833691
\(77\) 0 0
\(78\) −4.24264 −0.480384
\(79\) 1.46410 0.164724 0.0823622 0.996602i \(-0.473754\pi\)
0.0823622 + 0.996602i \(0.473754\pi\)
\(80\) 0 0
\(81\) −10.6603 −1.18447
\(82\) 0.896575 0.0990102
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) −11.1962 −1.22160
\(85\) 0 0
\(86\) −3.33975 −0.360134
\(87\) −13.3843 −1.43494
\(88\) 0 0
\(89\) 0.464102 0.0491947 0.0245973 0.999697i \(-0.492170\pi\)
0.0245973 + 0.999697i \(0.492170\pi\)
\(90\) 0 0
\(91\) −14.1962 −1.48816
\(92\) −6.03579 −0.629275
\(93\) 16.8690 1.74924
\(94\) 5.92820 0.611447
\(95\) 0 0
\(96\) −9.92820 −1.01329
\(97\) −9.14162 −0.928191 −0.464095 0.885785i \(-0.653621\pi\)
−0.464095 + 0.885785i \(0.653621\pi\)
\(98\) 2.17209 0.219414
\(99\) 0 0
\(100\) 0 0
\(101\) 19.3923 1.92961 0.964803 0.262973i \(-0.0847030\pi\)
0.964803 + 0.262973i \(0.0847030\pi\)
\(102\) −3.86370 −0.382564
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) −8.19615 −0.803699
\(105\) 0 0
\(106\) 1.12436 0.109207
\(107\) 2.96713 0.286843 0.143422 0.989662i \(-0.454190\pi\)
0.143422 + 0.989662i \(0.454190\pi\)
\(108\) −7.58871 −0.730224
\(109\) 9.19615 0.880832 0.440416 0.897794i \(-0.354831\pi\)
0.440416 + 0.897794i \(0.354831\pi\)
\(110\) 0 0
\(111\) −3.46410 −0.328798
\(112\) −8.24504 −0.779083
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 4.19615 0.393006
\(115\) 0 0
\(116\) −12.0000 −1.11417
\(117\) 3.10583 0.287134
\(118\) −0.656339 −0.0604209
\(119\) −12.9282 −1.18513
\(120\) 0 0
\(121\) 0 0
\(122\) −6.07296 −0.549820
\(123\) −3.34607 −0.301705
\(124\) 15.1244 1.35821
\(125\) 0 0
\(126\) −1.26795 −0.112958
\(127\) −0.896575 −0.0795582 −0.0397791 0.999208i \(-0.512665\pi\)
−0.0397791 + 0.999208i \(0.512665\pi\)
\(128\) −11.4524 −1.01226
\(129\) 12.4641 1.09740
\(130\) 0 0
\(131\) −3.12436 −0.272976 −0.136488 0.990642i \(-0.543582\pi\)
−0.136488 + 0.990642i \(0.543582\pi\)
\(132\) 0 0
\(133\) 14.0406 1.21747
\(134\) 1.14359 0.0987914
\(135\) 0 0
\(136\) −7.46410 −0.640041
\(137\) −8.86422 −0.757321 −0.378661 0.925536i \(-0.623615\pi\)
−0.378661 + 0.925536i \(0.623615\pi\)
\(138\) −3.48477 −0.296643
\(139\) −14.5885 −1.23738 −0.618688 0.785636i \(-0.712336\pi\)
−0.618688 + 0.785636i \(0.712336\pi\)
\(140\) 0 0
\(141\) −22.1244 −1.86321
\(142\) −4.24264 −0.356034
\(143\) 0 0
\(144\) 1.80385 0.150321
\(145\) 0 0
\(146\) −2.53590 −0.209872
\(147\) −8.10634 −0.668600
\(148\) −3.10583 −0.255298
\(149\) −10.2679 −0.841183 −0.420592 0.907250i \(-0.638177\pi\)
−0.420592 + 0.907250i \(0.638177\pi\)
\(150\) 0 0
\(151\) 4.19615 0.341478 0.170739 0.985316i \(-0.445384\pi\)
0.170739 + 0.985316i \(0.445384\pi\)
\(152\) 8.10634 0.657511
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 14.1962 1.13660
\(157\) −21.8695 −1.74538 −0.872690 0.488275i \(-0.837626\pi\)
−0.872690 + 0.488275i \(0.837626\pi\)
\(158\) 0.757875 0.0602933
\(159\) −4.19615 −0.332777
\(160\) 0 0
\(161\) −11.6603 −0.918957
\(162\) −5.51815 −0.433547
\(163\) −5.13922 −0.402534 −0.201267 0.979536i \(-0.564506\pi\)
−0.201267 + 0.979536i \(0.564506\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −5.12436 −0.397727
\(167\) 0.517638 0.0400560 0.0200280 0.999799i \(-0.493624\pi\)
0.0200280 + 0.999799i \(0.493624\pi\)
\(168\) −12.4877 −0.963446
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −3.07180 −0.234906
\(172\) 11.1750 0.852086
\(173\) 21.7680 1.65499 0.827495 0.561472i \(-0.189765\pi\)
0.827495 + 0.561472i \(0.189765\pi\)
\(174\) −6.92820 −0.525226
\(175\) 0 0
\(176\) 0 0
\(177\) 2.44949 0.184115
\(178\) 0.240237 0.0180065
\(179\) −15.4641 −1.15584 −0.577921 0.816093i \(-0.696136\pi\)
−0.577921 + 0.816093i \(0.696136\pi\)
\(180\) 0 0
\(181\) −17.3923 −1.29276 −0.646380 0.763016i \(-0.723718\pi\)
−0.646380 + 0.763016i \(0.723718\pi\)
\(182\) −7.34847 −0.544705
\(183\) 22.6646 1.67541
\(184\) −6.73205 −0.496293
\(185\) 0 0
\(186\) 8.73205 0.640265
\(187\) 0 0
\(188\) −19.8362 −1.44670
\(189\) −14.6603 −1.06638
\(190\) 0 0
\(191\) −7.26795 −0.525890 −0.262945 0.964811i \(-0.584694\pi\)
−0.262945 + 0.964811i \(0.584694\pi\)
\(192\) 4.38134 0.316196
\(193\) 15.1774 1.09249 0.546247 0.837624i \(-0.316056\pi\)
0.546247 + 0.837624i \(0.316056\pi\)
\(194\) −4.73205 −0.339741
\(195\) 0 0
\(196\) −7.26795 −0.519139
\(197\) 17.2480 1.22887 0.614433 0.788969i \(-0.289385\pi\)
0.614433 + 0.788969i \(0.289385\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) −4.26795 −0.301038
\(202\) 10.0382 0.706285
\(203\) −23.1822 −1.62707
\(204\) 12.9282 0.905155
\(205\) 0 0
\(206\) 2.19615 0.153013
\(207\) 2.55103 0.177309
\(208\) 10.4543 0.724875
\(209\) 0 0
\(210\) 0 0
\(211\) −13.1244 −0.903518 −0.451759 0.892140i \(-0.649203\pi\)
−0.451759 + 0.892140i \(0.649203\pi\)
\(212\) −3.76217 −0.258387
\(213\) 15.8338 1.08491
\(214\) 1.53590 0.104992
\(215\) 0 0
\(216\) −8.46410 −0.575909
\(217\) 29.2180 1.98345
\(218\) 4.76028 0.322407
\(219\) 9.46410 0.639525
\(220\) 0 0
\(221\) 16.3923 1.10267
\(222\) −1.79315 −0.120348
\(223\) 1.55291 0.103991 0.0519954 0.998647i \(-0.483442\pi\)
0.0519954 + 0.998647i \(0.483442\pi\)
\(224\) −17.1962 −1.14897
\(225\) 0 0
\(226\) 1.46410 0.0973906
\(227\) 20.1136 1.33498 0.667492 0.744617i \(-0.267368\pi\)
0.667492 + 0.744617i \(0.267368\pi\)
\(228\) −14.0406 −0.929861
\(229\) −4.07180 −0.269072 −0.134536 0.990909i \(-0.542954\pi\)
−0.134536 + 0.990909i \(0.542954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −13.3843 −0.878720
\(233\) 1.69161 0.110821 0.0554107 0.998464i \(-0.482353\pi\)
0.0554107 + 0.998464i \(0.482353\pi\)
\(234\) 1.60770 0.105098
\(235\) 0 0
\(236\) 2.19615 0.142957
\(237\) −2.82843 −0.183726
\(238\) −6.69213 −0.433786
\(239\) 5.66025 0.366131 0.183066 0.983101i \(-0.441398\pi\)
0.183066 + 0.983101i \(0.441398\pi\)
\(240\) 0 0
\(241\) 14.1244 0.909830 0.454915 0.890535i \(-0.349670\pi\)
0.454915 + 0.890535i \(0.349670\pi\)
\(242\) 0 0
\(243\) 7.45001 0.477918
\(244\) 20.3205 1.30089
\(245\) 0 0
\(246\) −1.73205 −0.110432
\(247\) −17.8028 −1.13276
\(248\) 16.8690 1.07118
\(249\) 19.1244 1.21196
\(250\) 0 0
\(251\) −3.80385 −0.240097 −0.120048 0.992768i \(-0.538305\pi\)
−0.120048 + 0.992768i \(0.538305\pi\)
\(252\) 4.24264 0.267261
\(253\) 0 0
\(254\) −0.464102 −0.0291203
\(255\) 0 0
\(256\) −1.39230 −0.0870191
\(257\) 14.7985 0.923103 0.461552 0.887113i \(-0.347293\pi\)
0.461552 + 0.887113i \(0.347293\pi\)
\(258\) 6.45189 0.401677
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 5.07180 0.313936
\(262\) −1.61729 −0.0999162
\(263\) −6.79367 −0.418915 −0.209458 0.977818i \(-0.567170\pi\)
−0.209458 + 0.977818i \(0.567170\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.26795 0.445627
\(267\) −0.896575 −0.0548695
\(268\) −3.82654 −0.233743
\(269\) −16.2679 −0.991874 −0.495937 0.868358i \(-0.665175\pi\)
−0.495937 + 0.868358i \(0.665175\pi\)
\(270\) 0 0
\(271\) 11.4641 0.696395 0.348197 0.937421i \(-0.386794\pi\)
0.348197 + 0.937421i \(0.386794\pi\)
\(272\) 9.52056 0.577269
\(273\) 27.4249 1.65983
\(274\) −4.58846 −0.277199
\(275\) 0 0
\(276\) 11.6603 0.701865
\(277\) −31.1870 −1.87385 −0.936923 0.349535i \(-0.886340\pi\)
−0.936923 + 0.349535i \(0.886340\pi\)
\(278\) −7.55154 −0.452911
\(279\) −6.39230 −0.382697
\(280\) 0 0
\(281\) −17.3205 −1.03325 −0.516627 0.856210i \(-0.672813\pi\)
−0.516627 + 0.856210i \(0.672813\pi\)
\(282\) −11.4524 −0.681981
\(283\) −8.06918 −0.479663 −0.239831 0.970815i \(-0.577092\pi\)
−0.239831 + 0.970815i \(0.577092\pi\)
\(284\) 14.1962 0.842387
\(285\) 0 0
\(286\) 0 0
\(287\) −5.79555 −0.342101
\(288\) 3.76217 0.221688
\(289\) −2.07180 −0.121870
\(290\) 0 0
\(291\) 17.6603 1.03526
\(292\) 8.48528 0.496564
\(293\) 15.0759 0.880742 0.440371 0.897816i \(-0.354847\pi\)
0.440371 + 0.897816i \(0.354847\pi\)
\(294\) −4.19615 −0.244725
\(295\) 0 0
\(296\) −3.46410 −0.201347
\(297\) 0 0
\(298\) −5.31508 −0.307894
\(299\) 14.7846 0.855016
\(300\) 0 0
\(301\) 21.5885 1.24434
\(302\) 2.17209 0.124990
\(303\) −37.4631 −2.15220
\(304\) −10.3397 −0.593025
\(305\) 0 0
\(306\) 1.46410 0.0836971
\(307\) −7.82894 −0.446821 −0.223411 0.974724i \(-0.571719\pi\)
−0.223411 + 0.974724i \(0.571719\pi\)
\(308\) 0 0
\(309\) −8.19615 −0.466263
\(310\) 0 0
\(311\) 10.0526 0.570028 0.285014 0.958523i \(-0.408002\pi\)
0.285014 + 0.958523i \(0.408002\pi\)
\(312\) 15.8338 0.896410
\(313\) −17.1464 −0.969173 −0.484587 0.874743i \(-0.661030\pi\)
−0.484587 + 0.874743i \(0.661030\pi\)
\(314\) −11.3205 −0.638853
\(315\) 0 0
\(316\) −2.53590 −0.142655
\(317\) −12.6264 −0.709168 −0.354584 0.935024i \(-0.615378\pi\)
−0.354584 + 0.935024i \(0.615378\pi\)
\(318\) −2.17209 −0.121805
\(319\) 0 0
\(320\) 0 0
\(321\) −5.73205 −0.319932
\(322\) −6.03579 −0.336362
\(323\) −16.2127 −0.902098
\(324\) 18.4641 1.02578
\(325\) 0 0
\(326\) −2.66025 −0.147338
\(327\) −17.7656 −0.982440
\(328\) −3.34607 −0.184756
\(329\) −38.3205 −2.11268
\(330\) 0 0
\(331\) −18.1962 −1.00015 −0.500075 0.865982i \(-0.666694\pi\)
−0.500075 + 0.865982i \(0.666694\pi\)
\(332\) 17.1464 0.941033
\(333\) 1.31268 0.0719343
\(334\) 0.267949 0.0146615
\(335\) 0 0
\(336\) 15.9282 0.868955
\(337\) −17.6269 −0.960199 −0.480099 0.877214i \(-0.659399\pi\)
−0.480099 + 0.877214i \(0.659399\pi\)
\(338\) 2.58819 0.140779
\(339\) −5.46410 −0.296769
\(340\) 0 0
\(341\) 0 0
\(342\) −1.59008 −0.0859816
\(343\) 9.38186 0.506573
\(344\) 12.4641 0.672019
\(345\) 0 0
\(346\) 11.2679 0.605769
\(347\) −4.38134 −0.235203 −0.117601 0.993061i \(-0.537521\pi\)
−0.117601 + 0.993061i \(0.537521\pi\)
\(348\) 23.1822 1.24270
\(349\) −7.32051 −0.391858 −0.195929 0.980618i \(-0.562772\pi\)
−0.195929 + 0.980618i \(0.562772\pi\)
\(350\) 0 0
\(351\) 18.5885 0.992178
\(352\) 0 0
\(353\) −13.0053 −0.692204 −0.346102 0.938197i \(-0.612495\pi\)
−0.346102 + 0.938197i \(0.612495\pi\)
\(354\) 1.26795 0.0673907
\(355\) 0 0
\(356\) −0.803848 −0.0426038
\(357\) 24.9754 1.32184
\(358\) −8.00481 −0.423067
\(359\) 35.6603 1.88208 0.941038 0.338301i \(-0.109852\pi\)
0.941038 + 0.338301i \(0.109852\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) −9.00292 −0.473183
\(363\) 0 0
\(364\) 24.5885 1.28879
\(365\) 0 0
\(366\) 11.7321 0.613244
\(367\) −16.9062 −0.882496 −0.441248 0.897385i \(-0.645464\pi\)
−0.441248 + 0.897385i \(0.645464\pi\)
\(368\) 8.58682 0.447619
\(369\) 1.26795 0.0660068
\(370\) 0 0
\(371\) −7.26795 −0.377333
\(372\) −29.2180 −1.51488
\(373\) 0.175865 0.00910597 0.00455298 0.999990i \(-0.498551\pi\)
0.00455298 + 0.999990i \(0.498551\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −22.1244 −1.14098
\(377\) 29.3939 1.51386
\(378\) −7.58871 −0.390321
\(379\) 1.46410 0.0752058 0.0376029 0.999293i \(-0.488028\pi\)
0.0376029 + 0.999293i \(0.488028\pi\)
\(380\) 0 0
\(381\) 1.73205 0.0887357
\(382\) −3.76217 −0.192489
\(383\) −38.4612 −1.96527 −0.982637 0.185539i \(-0.940597\pi\)
−0.982637 + 0.185539i \(0.940597\pi\)
\(384\) 22.1244 1.12903
\(385\) 0 0
\(386\) 7.85641 0.399881
\(387\) −4.72311 −0.240089
\(388\) 15.8338 0.803837
\(389\) 6.12436 0.310517 0.155259 0.987874i \(-0.450379\pi\)
0.155259 + 0.987874i \(0.450379\pi\)
\(390\) 0 0
\(391\) 13.4641 0.680909
\(392\) −8.10634 −0.409432
\(393\) 6.03579 0.304465
\(394\) 8.92820 0.449796
\(395\) 0 0
\(396\) 0 0
\(397\) 32.9802 1.65523 0.827614 0.561298i \(-0.189698\pi\)
0.827614 + 0.561298i \(0.189698\pi\)
\(398\) −1.03528 −0.0518937
\(399\) −27.1244 −1.35792
\(400\) 0 0
\(401\) 2.32051 0.115881 0.0579403 0.998320i \(-0.481547\pi\)
0.0579403 + 0.998320i \(0.481547\pi\)
\(402\) −2.20925 −0.110188
\(403\) −37.0470 −1.84544
\(404\) −33.5885 −1.67109
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 14.4195 0.713873
\(409\) −26.1244 −1.29177 −0.645883 0.763436i \(-0.723511\pi\)
−0.645883 + 0.763436i \(0.723511\pi\)
\(410\) 0 0
\(411\) 17.1244 0.844682
\(412\) −7.34847 −0.362033
\(413\) 4.24264 0.208767
\(414\) 1.32051 0.0648994
\(415\) 0 0
\(416\) 21.8038 1.06902
\(417\) 28.1827 1.38011
\(418\) 0 0
\(419\) −12.3397 −0.602836 −0.301418 0.953492i \(-0.597460\pi\)
−0.301418 + 0.953492i \(0.597460\pi\)
\(420\) 0 0
\(421\) −17.0526 −0.831091 −0.415545 0.909572i \(-0.636409\pi\)
−0.415545 + 0.909572i \(0.636409\pi\)
\(422\) −6.79367 −0.330711
\(423\) 8.38375 0.407632
\(424\) −4.19615 −0.203783
\(425\) 0 0
\(426\) 8.19615 0.397105
\(427\) 39.2562 1.89974
\(428\) −5.13922 −0.248413
\(429\) 0 0
\(430\) 0 0
\(431\) −5.07180 −0.244300 −0.122150 0.992512i \(-0.538979\pi\)
−0.122150 + 0.992512i \(0.538979\pi\)
\(432\) 10.7961 0.519426
\(433\) −28.7375 −1.38104 −0.690519 0.723314i \(-0.742618\pi\)
−0.690519 + 0.723314i \(0.742618\pi\)
\(434\) 15.1244 0.725992
\(435\) 0 0
\(436\) −15.9282 −0.762823
\(437\) −14.6226 −0.699494
\(438\) 4.89898 0.234082
\(439\) 28.2487 1.34824 0.674119 0.738623i \(-0.264524\pi\)
0.674119 + 0.738623i \(0.264524\pi\)
\(440\) 0 0
\(441\) 3.07180 0.146276
\(442\) 8.48528 0.403604
\(443\) 27.5636 1.30958 0.654792 0.755809i \(-0.272756\pi\)
0.654792 + 0.755809i \(0.272756\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 0.803848 0.0380633
\(447\) 19.8362 0.938218
\(448\) 7.58871 0.358533
\(449\) 34.5167 1.62894 0.814471 0.580204i \(-0.197027\pi\)
0.814471 + 0.580204i \(0.197027\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4.89898 −0.230429
\(453\) −8.10634 −0.380869
\(454\) 10.4115 0.488638
\(455\) 0 0
\(456\) −15.6603 −0.733359
\(457\) −21.2132 −0.992312 −0.496156 0.868233i \(-0.665256\pi\)
−0.496156 + 0.868233i \(0.665256\pi\)
\(458\) −2.10772 −0.0984872
\(459\) 16.9282 0.790141
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) −20.3166 −0.944194 −0.472097 0.881547i \(-0.656503\pi\)
−0.472097 + 0.881547i \(0.656503\pi\)
\(464\) 17.0718 0.792538
\(465\) 0 0
\(466\) 0.875644 0.0405634
\(467\) −0.619174 −0.0286520 −0.0143260 0.999897i \(-0.504560\pi\)
−0.0143260 + 0.999897i \(0.504560\pi\)
\(468\) −5.37945 −0.248665
\(469\) −7.39230 −0.341345
\(470\) 0 0
\(471\) 42.2487 1.94672
\(472\) 2.44949 0.112747
\(473\) 0 0
\(474\) −1.46410 −0.0672484
\(475\) 0 0
\(476\) 22.3923 1.02635
\(477\) 1.59008 0.0728047
\(478\) 2.92996 0.134013
\(479\) 2.53590 0.115868 0.0579341 0.998320i \(-0.481549\pi\)
0.0579341 + 0.998320i \(0.481549\pi\)
\(480\) 0 0
\(481\) 7.60770 0.346881
\(482\) 7.31130 0.333021
\(483\) 22.5259 1.02496
\(484\) 0 0
\(485\) 0 0
\(486\) 3.85641 0.174930
\(487\) 12.7279 0.576757 0.288379 0.957516i \(-0.406884\pi\)
0.288379 + 0.957516i \(0.406884\pi\)
\(488\) 22.6646 1.02598
\(489\) 9.92820 0.448969
\(490\) 0 0
\(491\) 39.7128 1.79221 0.896107 0.443838i \(-0.146383\pi\)
0.896107 + 0.443838i \(0.146383\pi\)
\(492\) 5.79555 0.261284
\(493\) 26.7685 1.20559
\(494\) −9.21539 −0.414620
\(495\) 0 0
\(496\) −21.5167 −0.966127
\(497\) 27.4249 1.23017
\(498\) 9.89949 0.443607
\(499\) 17.6077 0.788229 0.394114 0.919061i \(-0.371051\pi\)
0.394114 + 0.919061i \(0.371051\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) −1.96902 −0.0878815
\(503\) −7.38563 −0.329309 −0.164655 0.986351i \(-0.552651\pi\)
−0.164655 + 0.986351i \(0.552651\pi\)
\(504\) 4.73205 0.210782
\(505\) 0 0
\(506\) 0 0
\(507\) −9.65926 −0.428983
\(508\) 1.55291 0.0688994
\(509\) −8.07180 −0.357776 −0.178888 0.983869i \(-0.557250\pi\)
−0.178888 + 0.983869i \(0.557250\pi\)
\(510\) 0 0
\(511\) 16.3923 0.725153
\(512\) 22.1841 0.980408
\(513\) −18.3848 −0.811708
\(514\) 7.66025 0.337879
\(515\) 0 0
\(516\) −21.5885 −0.950379
\(517\) 0 0
\(518\) −3.10583 −0.136462
\(519\) −42.0526 −1.84590
\(520\) 0 0
\(521\) −9.24871 −0.405193 −0.202597 0.979262i \(-0.564938\pi\)
−0.202597 + 0.979262i \(0.564938\pi\)
\(522\) 2.62536 0.114909
\(523\) 25.6317 1.12080 0.560398 0.828223i \(-0.310648\pi\)
0.560398 + 0.828223i \(0.310648\pi\)
\(524\) 5.41154 0.236404
\(525\) 0 0
\(526\) −3.51666 −0.153334
\(527\) −33.7381 −1.46965
\(528\) 0 0
\(529\) −10.8564 −0.472018
\(530\) 0 0
\(531\) −0.928203 −0.0402806
\(532\) −24.3190 −1.05436
\(533\) 7.34847 0.318298
\(534\) −0.464102 −0.0200836
\(535\) 0 0
\(536\) −4.26795 −0.184347
\(537\) 29.8744 1.28917
\(538\) −8.42091 −0.363051
\(539\) 0 0
\(540\) 0 0
\(541\) −2.60770 −0.112114 −0.0560568 0.998428i \(-0.517853\pi\)
−0.0560568 + 0.998428i \(0.517853\pi\)
\(542\) 5.93426 0.254898
\(543\) 33.5994 1.44189
\(544\) 19.8564 0.851336
\(545\) 0 0
\(546\) 14.1962 0.607539
\(547\) −28.7375 −1.22873 −0.614364 0.789023i \(-0.710587\pi\)
−0.614364 + 0.789023i \(0.710587\pi\)
\(548\) 15.3533 0.655859
\(549\) −8.58846 −0.366546
\(550\) 0 0
\(551\) −29.0718 −1.23850
\(552\) 13.0053 0.553543
\(553\) −4.89898 −0.208326
\(554\) −16.1436 −0.685876
\(555\) 0 0
\(556\) 25.2679 1.07160
\(557\) −37.6018 −1.59324 −0.796619 0.604482i \(-0.793380\pi\)
−0.796619 + 0.604482i \(0.793380\pi\)
\(558\) −3.30890 −0.140077
\(559\) −27.3731 −1.15776
\(560\) 0 0
\(561\) 0 0
\(562\) −8.96575 −0.378198
\(563\) 15.9725 0.673159 0.336579 0.941655i \(-0.390730\pi\)
0.336579 + 0.941655i \(0.390730\pi\)
\(564\) 38.3205 1.61358
\(565\) 0 0
\(566\) −4.17691 −0.175569
\(567\) 35.6699 1.49800
\(568\) 15.8338 0.664369
\(569\) 30.1244 1.26288 0.631439 0.775425i \(-0.282464\pi\)
0.631439 + 0.775425i \(0.282464\pi\)
\(570\) 0 0
\(571\) −19.7128 −0.824956 −0.412478 0.910968i \(-0.635337\pi\)
−0.412478 + 0.910968i \(0.635337\pi\)
\(572\) 0 0
\(573\) 14.0406 0.586554
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) −1.66025 −0.0691773
\(577\) −3.76217 −0.156621 −0.0783105 0.996929i \(-0.524953\pi\)
−0.0783105 + 0.996929i \(0.524953\pi\)
\(578\) −1.07244 −0.0446077
\(579\) −29.3205 −1.21852
\(580\) 0 0
\(581\) 33.1244 1.37423
\(582\) 9.14162 0.378932
\(583\) 0 0
\(584\) 9.46410 0.391627
\(585\) 0 0
\(586\) 7.80385 0.322374
\(587\) −20.3910 −0.841625 −0.420812 0.907148i \(-0.638255\pi\)
−0.420812 + 0.907148i \(0.638255\pi\)
\(588\) 14.0406 0.579025
\(589\) 36.6410 1.50977
\(590\) 0 0
\(591\) −33.3205 −1.37062
\(592\) 4.41851 0.181599
\(593\) 0.859411 0.0352918 0.0176459 0.999844i \(-0.494383\pi\)
0.0176459 + 0.999844i \(0.494383\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.7846 0.728486
\(597\) 3.86370 0.158131
\(598\) 7.65308 0.312958
\(599\) −38.4449 −1.57081 −0.785407 0.618979i \(-0.787546\pi\)
−0.785407 + 0.618979i \(0.787546\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 11.1750 0.455459
\(603\) 1.61729 0.0658610
\(604\) −7.26795 −0.295729
\(605\) 0 0
\(606\) −19.3923 −0.787759
\(607\) 36.3906 1.47705 0.738525 0.674226i \(-0.235523\pi\)
0.738525 + 0.674226i \(0.235523\pi\)
\(608\) −21.5649 −0.874574
\(609\) 44.7846 1.81476
\(610\) 0 0
\(611\) 48.5885 1.96568
\(612\) −4.89898 −0.198030
\(613\) 1.79315 0.0724247 0.0362123 0.999344i \(-0.488471\pi\)
0.0362123 + 0.999344i \(0.488471\pi\)
\(614\) −4.05256 −0.163548
\(615\) 0 0
\(616\) 0 0
\(617\) 13.0053 0.523575 0.261787 0.965126i \(-0.415688\pi\)
0.261787 + 0.965126i \(0.415688\pi\)
\(618\) −4.24264 −0.170664
\(619\) 12.7846 0.513857 0.256928 0.966430i \(-0.417290\pi\)
0.256928 + 0.966430i \(0.417290\pi\)
\(620\) 0 0
\(621\) 15.2679 0.612682
\(622\) 5.20359 0.208645
\(623\) −1.55291 −0.0622162
\(624\) −20.1962 −0.808493
\(625\) 0 0
\(626\) −8.87564 −0.354742
\(627\) 0 0
\(628\) 37.8792 1.51154
\(629\) 6.92820 0.276246
\(630\) 0 0
\(631\) −15.3205 −0.609900 −0.304950 0.952368i \(-0.598640\pi\)
−0.304950 + 0.952368i \(0.598640\pi\)
\(632\) −2.82843 −0.112509
\(633\) 25.3543 1.00774
\(634\) −6.53590 −0.259574
\(635\) 0 0
\(636\) 7.26795 0.288193
\(637\) 17.8028 0.705371
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −7.85641 −0.310309 −0.155155 0.987890i \(-0.549588\pi\)
−0.155155 + 0.987890i \(0.549588\pi\)
\(642\) −2.96713 −0.117103
\(643\) −10.5187 −0.414816 −0.207408 0.978255i \(-0.566503\pi\)
−0.207408 + 0.978255i \(0.566503\pi\)
\(644\) 20.1962 0.795840
\(645\) 0 0
\(646\) −8.39230 −0.330191
\(647\) −37.3615 −1.46883 −0.734416 0.678699i \(-0.762544\pi\)
−0.734416 + 0.678699i \(0.762544\pi\)
\(648\) 20.5940 0.809010
\(649\) 0 0
\(650\) 0 0
\(651\) −56.4449 −2.21225
\(652\) 8.90138 0.348605
\(653\) 12.1459 0.475306 0.237653 0.971350i \(-0.423622\pi\)
0.237653 + 0.971350i \(0.423622\pi\)
\(654\) −9.19615 −0.359598
\(655\) 0 0
\(656\) 4.26795 0.166635
\(657\) −3.58630 −0.139915
\(658\) −19.8362 −0.773294
\(659\) −23.3205 −0.908438 −0.454219 0.890890i \(-0.650082\pi\)
−0.454219 + 0.890890i \(0.650082\pi\)
\(660\) 0 0
\(661\) 29.5885 1.15086 0.575429 0.817852i \(-0.304835\pi\)
0.575429 + 0.817852i \(0.304835\pi\)
\(662\) −9.41902 −0.366081
\(663\) −31.6675 −1.22986
\(664\) 19.1244 0.742169
\(665\) 0 0
\(666\) 0.679492 0.0263298
\(667\) 24.1432 0.934827
\(668\) −0.896575 −0.0346895
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) 0 0
\(672\) 33.2204 1.28151
\(673\) −40.8091 −1.57308 −0.786538 0.617542i \(-0.788129\pi\)
−0.786538 + 0.617542i \(0.788129\pi\)
\(674\) −9.12436 −0.351457
\(675\) 0 0
\(676\) −8.66025 −0.333087
\(677\) 20.3538 0.782260 0.391130 0.920335i \(-0.372084\pi\)
0.391130 + 0.920335i \(0.372084\pi\)
\(678\) −2.82843 −0.108625
\(679\) 30.5885 1.17388
\(680\) 0 0
\(681\) −38.8564 −1.48898
\(682\) 0 0
\(683\) 29.8372 1.14169 0.570844 0.821058i \(-0.306616\pi\)
0.570844 + 0.821058i \(0.306616\pi\)
\(684\) 5.32051 0.203435
\(685\) 0 0
\(686\) 4.85641 0.185418
\(687\) 7.86611 0.300111
\(688\) −15.8981 −0.606110
\(689\) 9.21539 0.351078
\(690\) 0 0
\(691\) 37.9090 1.44213 0.721063 0.692870i \(-0.243654\pi\)
0.721063 + 0.692870i \(0.243654\pi\)
\(692\) −37.7033 −1.43326
\(693\) 0 0
\(694\) −2.26795 −0.0860902
\(695\) 0 0
\(696\) 25.8564 0.980085
\(697\) 6.69213 0.253483
\(698\) −3.78937 −0.143430
\(699\) −3.26795 −0.123605
\(700\) 0 0
\(701\) 37.8564 1.42982 0.714908 0.699218i \(-0.246468\pi\)
0.714908 + 0.699218i \(0.246468\pi\)
\(702\) 9.62209 0.363163
\(703\) −7.52433 −0.283786
\(704\) 0 0
\(705\) 0 0
\(706\) −6.73205 −0.253364
\(707\) −64.8879 −2.44036
\(708\) −4.24264 −0.159448
\(709\) 2.60770 0.0979340 0.0489670 0.998800i \(-0.484407\pi\)
0.0489670 + 0.998800i \(0.484407\pi\)
\(710\) 0 0
\(711\) 1.07180 0.0401955
\(712\) −0.896575 −0.0336006
\(713\) −30.4292 −1.13958
\(714\) 12.9282 0.483826
\(715\) 0 0
\(716\) 26.7846 1.00099
\(717\) −10.9348 −0.408367
\(718\) 18.4591 0.688888
\(719\) −50.1962 −1.87200 −0.936000 0.351999i \(-0.885502\pi\)
−0.936000 + 0.351999i \(0.885502\pi\)
\(720\) 0 0
\(721\) −14.1962 −0.528692
\(722\) −0.720710 −0.0268220
\(723\) −27.2862 −1.01478
\(724\) 30.1244 1.11956
\(725\) 0 0
\(726\) 0 0
\(727\) −8.24504 −0.305792 −0.152896 0.988242i \(-0.548860\pi\)
−0.152896 + 0.988242i \(0.548860\pi\)
\(728\) 27.4249 1.01643
\(729\) 17.5885 0.651424
\(730\) 0 0
\(731\) −24.9282 −0.922003
\(732\) −39.2562 −1.45095
\(733\) −9.31749 −0.344149 −0.172075 0.985084i \(-0.555047\pi\)
−0.172075 + 0.985084i \(0.555047\pi\)
\(734\) −8.75129 −0.323016
\(735\) 0 0
\(736\) 17.9090 0.660133
\(737\) 0 0
\(738\) 0.656339 0.0241602
\(739\) −38.9282 −1.43200 −0.715999 0.698102i \(-0.754028\pi\)
−0.715999 + 0.698102i \(0.754028\pi\)
\(740\) 0 0
\(741\) 34.3923 1.26343
\(742\) −3.76217 −0.138114
\(743\) 43.3973 1.59209 0.796046 0.605235i \(-0.206921\pi\)
0.796046 + 0.605235i \(0.206921\pi\)
\(744\) −32.5885 −1.19475
\(745\) 0 0
\(746\) 0.0910347 0.00333302
\(747\) −7.24693 −0.265151
\(748\) 0 0
\(749\) −9.92820 −0.362769
\(750\) 0 0
\(751\) 48.6410 1.77494 0.887468 0.460869i \(-0.152462\pi\)
0.887468 + 0.460869i \(0.152462\pi\)
\(752\) 28.2199 1.02907
\(753\) 7.34847 0.267793
\(754\) 15.2154 0.554112
\(755\) 0 0
\(756\) 25.3923 0.923509
\(757\) −49.7749 −1.80910 −0.904549 0.426369i \(-0.859793\pi\)
−0.904549 + 0.426369i \(0.859793\pi\)
\(758\) 0.757875 0.0275273
\(759\) 0 0
\(760\) 0 0
\(761\) −19.8564 −0.719794 −0.359897 0.932992i \(-0.617188\pi\)
−0.359897 + 0.932992i \(0.617188\pi\)
\(762\) 0.896575 0.0324795
\(763\) −30.7709 −1.11398
\(764\) 12.5885 0.455434
\(765\) 0 0
\(766\) −19.9090 −0.719340
\(767\) −5.37945 −0.194241
\(768\) 2.68973 0.0970571
\(769\) −9.85641 −0.355431 −0.177716 0.984082i \(-0.556871\pi\)
−0.177716 + 0.984082i \(0.556871\pi\)
\(770\) 0 0
\(771\) −28.5885 −1.02959
\(772\) −26.2880 −0.946128
\(773\) −12.0444 −0.433206 −0.216603 0.976260i \(-0.569498\pi\)
−0.216603 + 0.976260i \(0.569498\pi\)
\(774\) −2.44486 −0.0878788
\(775\) 0 0
\(776\) 17.6603 0.633966
\(777\) 11.5911 0.415829
\(778\) 3.17020 0.113657
\(779\) −7.26795 −0.260401
\(780\) 0 0
\(781\) 0 0
\(782\) 6.96953 0.249230
\(783\) 30.3548 1.08479
\(784\) 10.3397 0.369277
\(785\) 0 0
\(786\) 3.12436 0.111442
\(787\) −17.3867 −0.619768 −0.309884 0.950774i \(-0.600290\pi\)
−0.309884 + 0.950774i \(0.600290\pi\)
\(788\) −29.8744 −1.06423
\(789\) 13.1244 0.467239
\(790\) 0 0
\(791\) −9.46410 −0.336505
\(792\) 0 0
\(793\) −49.7749 −1.76756
\(794\) 17.0718 0.605855
\(795\) 0 0
\(796\) 3.46410 0.122782
\(797\) 36.1875 1.28183 0.640914 0.767612i \(-0.278555\pi\)
0.640914 + 0.767612i \(0.278555\pi\)
\(798\) −14.0406 −0.497032
\(799\) 44.2487 1.56541
\(800\) 0 0
\(801\) 0.339746 0.0120043
\(802\) 1.20118 0.0424153
\(803\) 0 0
\(804\) 7.39230 0.260706
\(805\) 0 0
\(806\) −19.1769 −0.675478
\(807\) 31.4273 1.10629
\(808\) −37.4631 −1.31795
\(809\) −14.5359 −0.511055 −0.255527 0.966802i \(-0.582249\pi\)
−0.255527 + 0.966802i \(0.582249\pi\)
\(810\) 0 0
\(811\) 18.9808 0.666505 0.333252 0.942838i \(-0.391854\pi\)
0.333252 + 0.942838i \(0.391854\pi\)
\(812\) 40.1528 1.40909
\(813\) −22.1469 −0.776727
\(814\) 0 0
\(815\) 0 0
\(816\) −18.3923 −0.643859
\(817\) 27.0731 0.947169
\(818\) −13.5230 −0.472819
\(819\) −10.3923 −0.363137
\(820\) 0 0
\(821\) 29.1962 1.01895 0.509476 0.860485i \(-0.329839\pi\)
0.509476 + 0.860485i \(0.329839\pi\)
\(822\) 8.86422 0.309175
\(823\) −6.75650 −0.235517 −0.117758 0.993042i \(-0.537571\pi\)
−0.117758 + 0.993042i \(0.537571\pi\)
\(824\) −8.19615 −0.285526
\(825\) 0 0
\(826\) 2.19615 0.0764139
\(827\) −34.0798 −1.18507 −0.592536 0.805544i \(-0.701873\pi\)
−0.592536 + 0.805544i \(0.701873\pi\)
\(828\) −4.41851 −0.153554
\(829\) −9.98076 −0.346646 −0.173323 0.984865i \(-0.555450\pi\)
−0.173323 + 0.984865i \(0.555450\pi\)
\(830\) 0 0
\(831\) 60.2487 2.09000
\(832\) −9.62209 −0.333586
\(833\) 16.2127 0.561736
\(834\) 14.5885 0.505157
\(835\) 0 0
\(836\) 0 0
\(837\) −38.2581 −1.32239
\(838\) −6.38752 −0.220653
\(839\) −38.7846 −1.33899 −0.669497 0.742815i \(-0.733490\pi\)
−0.669497 + 0.742815i \(0.733490\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) −8.82705 −0.304200
\(843\) 33.4607 1.15245
\(844\) 22.7321 0.782469
\(845\) 0 0
\(846\) 4.33975 0.149204
\(847\) 0 0
\(848\) 5.35225 0.183797
\(849\) 15.5885 0.534994
\(850\) 0 0
\(851\) 6.24871 0.214203
\(852\) −27.4249 −0.939560
\(853\) −46.3644 −1.58749 −0.793744 0.608252i \(-0.791871\pi\)
−0.793744 + 0.608252i \(0.791871\pi\)
\(854\) 20.3205 0.695353
\(855\) 0 0
\(856\) −5.73205 −0.195917
\(857\) 29.3195 1.00154 0.500768 0.865581i \(-0.333051\pi\)
0.500768 + 0.865581i \(0.333051\pi\)
\(858\) 0 0
\(859\) −22.1962 −0.757323 −0.378661 0.925535i \(-0.623616\pi\)
−0.378661 + 0.925535i \(0.623616\pi\)
\(860\) 0 0
\(861\) 11.1962 0.381564
\(862\) −2.62536 −0.0894200
\(863\) 17.1093 0.582406 0.291203 0.956661i \(-0.405944\pi\)
0.291203 + 0.956661i \(0.405944\pi\)
\(864\) 22.5167 0.766032
\(865\) 0 0
\(866\) −14.8756 −0.505495
\(867\) 4.00240 0.135929
\(868\) −50.6071 −1.71772
\(869\) 0 0
\(870\) 0 0
\(871\) 9.37307 0.317594
\(872\) −17.7656 −0.601619
\(873\) −6.69213 −0.226494
\(874\) −7.56922 −0.256033
\(875\) 0 0
\(876\) −16.3923 −0.553845
\(877\) 11.8957 0.401690 0.200845 0.979623i \(-0.435631\pi\)
0.200845 + 0.979623i \(0.435631\pi\)
\(878\) 14.6226 0.493489
\(879\) −29.1244 −0.982340
\(880\) 0 0
\(881\) −26.9090 −0.906586 −0.453293 0.891362i \(-0.649751\pi\)
−0.453293 + 0.891362i \(0.649751\pi\)
\(882\) 1.59008 0.0535407
\(883\) −21.5649 −0.725718 −0.362859 0.931844i \(-0.618199\pi\)
−0.362859 + 0.931844i \(0.618199\pi\)
\(884\) −28.3923 −0.954937
\(885\) 0 0
\(886\) 14.2679 0.479341
\(887\) 15.6950 0.526988 0.263494 0.964661i \(-0.415125\pi\)
0.263494 + 0.964661i \(0.415125\pi\)
\(888\) 6.69213 0.224573
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) −2.68973 −0.0900587
\(893\) −48.0561 −1.60813
\(894\) 10.2679 0.343412
\(895\) 0 0
\(896\) 38.3205 1.28020
\(897\) −28.5617 −0.953646
\(898\) 17.8671 0.596234
\(899\) −60.4974 −2.01770
\(900\) 0 0
\(901\) 8.39230 0.279588
\(902\) 0 0
\(903\) −41.7057 −1.38788
\(904\) −5.46410 −0.181733
\(905\) 0 0
\(906\) −4.19615 −0.139408
\(907\) −30.1146 −0.999938 −0.499969 0.866043i \(-0.666655\pi\)
−0.499969 + 0.866043i \(0.666655\pi\)
\(908\) −34.8377 −1.15613
\(909\) 14.1962 0.470857
\(910\) 0 0
\(911\) −30.2487 −1.00218 −0.501092 0.865394i \(-0.667068\pi\)
−0.501092 + 0.865394i \(0.667068\pi\)
\(912\) 19.9749 0.661434
\(913\) 0 0
\(914\) −10.9808 −0.363211
\(915\) 0 0
\(916\) 7.05256 0.233023
\(917\) 10.4543 0.345231
\(918\) 8.76268 0.289212
\(919\) −18.9808 −0.626118 −0.313059 0.949734i \(-0.601354\pi\)
−0.313059 + 0.949734i \(0.601354\pi\)
\(920\) 0 0
\(921\) 15.1244 0.498364
\(922\) −17.0821 −0.562568
\(923\) −34.7733 −1.14458
\(924\) 0 0
\(925\) 0 0
\(926\) −10.5167 −0.345599
\(927\) 3.10583 0.102009
\(928\) 35.6055 1.16881
\(929\) −43.8564 −1.43888 −0.719441 0.694554i \(-0.755602\pi\)
−0.719441 + 0.694554i \(0.755602\pi\)
\(930\) 0 0
\(931\) −17.6077 −0.577069
\(932\) −2.92996 −0.0959741
\(933\) −19.4201 −0.635784
\(934\) −0.320508 −0.0104873
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −31.1870 −1.01884 −0.509418 0.860519i \(-0.670139\pi\)
−0.509418 + 0.860519i \(0.670139\pi\)
\(938\) −3.82654 −0.124941
\(939\) 33.1244 1.08097
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 21.8695 0.712548
\(943\) 6.03579 0.196552
\(944\) −3.12436 −0.101689
\(945\) 0 0
\(946\) 0 0
\(947\) −19.9749 −0.649096 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(948\) 4.89898 0.159111
\(949\) −20.7846 −0.674697
\(950\) 0 0
\(951\) 24.3923 0.790975
\(952\) 24.9754 0.809456
\(953\) 16.6932 0.540745 0.270372 0.962756i \(-0.412853\pi\)
0.270372 + 0.962756i \(0.412853\pi\)
\(954\) 0.823085 0.0266484
\(955\) 0 0
\(956\) −9.80385 −0.317079
\(957\) 0 0
\(958\) 1.31268 0.0424107
\(959\) 29.6603 0.957780
\(960\) 0 0
\(961\) 45.2487 1.45964
\(962\) 3.93803 0.126967
\(963\) 2.17209 0.0699946
\(964\) −24.4641 −0.787936
\(965\) 0 0
\(966\) 11.6603 0.375163
\(967\) 4.24264 0.136434 0.0682171 0.997671i \(-0.478269\pi\)
0.0682171 + 0.997671i \(0.478269\pi\)
\(968\) 0 0
\(969\) 31.3205 1.00616
\(970\) 0 0
\(971\) −24.9282 −0.799984 −0.399992 0.916519i \(-0.630987\pi\)
−0.399992 + 0.916519i \(0.630987\pi\)
\(972\) −12.9038 −0.413889
\(973\) 48.8139 1.56490
\(974\) 6.58846 0.211108
\(975\) 0 0
\(976\) −28.9090 −0.925353
\(977\) −29.1165 −0.931519 −0.465759 0.884911i \(-0.654219\pi\)
−0.465759 + 0.884911i \(0.654219\pi\)
\(978\) 5.13922 0.164334
\(979\) 0 0
\(980\) 0 0
\(981\) 6.73205 0.214938
\(982\) 20.5569 0.655996
\(983\) −56.0237 −1.78688 −0.893439 0.449184i \(-0.851715\pi\)
−0.893439 + 0.449184i \(0.851715\pi\)
\(984\) 6.46410 0.206068
\(985\) 0 0
\(986\) 13.8564 0.441278
\(987\) 74.0295 2.35639
\(988\) 30.8353 0.981001
\(989\) −22.4833 −0.714929
\(990\) 0 0
\(991\) 19.9090 0.632429 0.316215 0.948688i \(-0.397588\pi\)
0.316215 + 0.948688i \(0.397588\pi\)
\(992\) −44.8759 −1.42481
\(993\) 35.1523 1.11552
\(994\) 14.1962 0.450275
\(995\) 0 0
\(996\) −33.1244 −1.04959
\(997\) −34.1170 −1.08050 −0.540248 0.841506i \(-0.681670\pi\)
−0.540248 + 0.841506i \(0.681670\pi\)
\(998\) 9.11441 0.288512
\(999\) 7.85641 0.248566
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 3025.2.a.y.1.3 4
5.2 odd 4 605.2.b.d.364.3 yes 4
5.3 odd 4 605.2.b.d.364.2 4
5.4 even 2 inner 3025.2.a.y.1.2 4
11.10 odd 2 3025.2.a.z.1.2 4
55.2 even 20 605.2.j.e.444.2 16
55.3 odd 20 605.2.j.f.9.2 16
55.7 even 20 605.2.j.e.269.3 16
55.8 even 20 605.2.j.e.9.3 16
55.13 even 20 605.2.j.e.444.3 16
55.17 even 20 605.2.j.e.124.3 16
55.18 even 20 605.2.j.e.269.2 16
55.27 odd 20 605.2.j.f.124.2 16
55.28 even 20 605.2.j.e.124.2 16
55.32 even 4 605.2.b.e.364.2 yes 4
55.37 odd 20 605.2.j.f.269.2 16
55.38 odd 20 605.2.j.f.124.3 16
55.42 odd 20 605.2.j.f.444.3 16
55.43 even 4 605.2.b.e.364.3 yes 4
55.47 odd 20 605.2.j.f.9.3 16
55.48 odd 20 605.2.j.f.269.3 16
55.52 even 20 605.2.j.e.9.2 16
55.53 odd 20 605.2.j.f.444.2 16
55.54 odd 2 3025.2.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.b.d.364.2 4 5.3 odd 4
605.2.b.d.364.3 yes 4 5.2 odd 4
605.2.b.e.364.2 yes 4 55.32 even 4
605.2.b.e.364.3 yes 4 55.43 even 4
605.2.j.e.9.2 16 55.52 even 20
605.2.j.e.9.3 16 55.8 even 20
605.2.j.e.124.2 16 55.28 even 20
605.2.j.e.124.3 16 55.17 even 20
605.2.j.e.269.2 16 55.18 even 20
605.2.j.e.269.3 16 55.7 even 20
605.2.j.e.444.2 16 55.2 even 20
605.2.j.e.444.3 16 55.13 even 20
605.2.j.f.9.2 16 55.3 odd 20
605.2.j.f.9.3 16 55.47 odd 20
605.2.j.f.124.2 16 55.27 odd 20
605.2.j.f.124.3 16 55.38 odd 20
605.2.j.f.269.2 16 55.37 odd 20
605.2.j.f.269.3 16 55.48 odd 20
605.2.j.f.444.2 16 55.53 odd 20
605.2.j.f.444.3 16 55.42 odd 20
3025.2.a.y.1.2 4 5.4 even 2 inner
3025.2.a.y.1.3 4 1.1 even 1 trivial
3025.2.a.z.1.2 4 11.10 odd 2
3025.2.a.z.1.3 4 55.54 odd 2