Properties

Label 825.4.c.j.199.3
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.j.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155i q^{2} -3.00000i q^{3} +5.56155 q^{4} +4.68466 q^{6} -10.2462i q^{7} +21.1771i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+1.56155i q^{2} -3.00000i q^{3} +5.56155 q^{4} +4.68466 q^{6} -10.2462i q^{7} +21.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -16.6847i q^{12} +40.8769i q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083i q^{17} -14.0540i q^{18} +39.6458 q^{19} -30.7386 q^{21} -17.1771i q^{22} -61.6932i q^{23} +63.5312 q^{24} -63.8314 q^{26} +27.0000i q^{27} -56.9848i q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255i q^{32} +33.0000i q^{33} +154.138 q^{34} -50.0540 q^{36} +44.8939i q^{37} +61.9091i q^{38} +122.631 q^{39} +336.479 q^{41} -48.0000i q^{42} +2.36745i q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295i q^{47} -34.2699i q^{48} +238.015 q^{49} -296.125 q^{51} +227.339i q^{52} -640.064i q^{53} -42.1619 q^{54} +216.985 q^{56} -118.938i q^{57} +232.816i q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773i q^{62} +92.2159i q^{63} -201.022 q^{64} -51.5312 q^{66} -404.985i q^{67} -548.972i q^{68} -185.080 q^{69} +939.292 q^{71} -190.594i q^{72} +362.570i q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708i q^{77} +191.494i q^{78} -951.835 q^{79} +81.0000 q^{81} +525.430i q^{82} -735.221i q^{83} -170.955 q^{84} -3.69690 q^{86} -447.278i q^{87} -232.948i q^{88} -385.879 q^{89} +418.833 q^{91} -343.110i q^{92} -164.216i q^{93} +520.458 q^{94} +561.764 q^{96} -966.345i q^{97} +371.673i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} + O(q^{10}) \) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} - 44q^{11} + 64q^{14} - 78q^{16} + 340q^{19} - 24q^{21} - 18q^{24} + 124q^{26} + 316q^{29} + 120q^{31} + 732q^{34} - 126q^{36} + 540q^{39} + 76q^{41} - 154q^{44} + 1144q^{46} + 1084q^{49} - 96q^{51} + 54q^{54} + 736q^{56} - 496q^{59} + 144q^{61} - 1538q^{64} + 66q^{66} + 744q^{69} + 4120q^{71} - 2276q^{74} + 816q^{76} - 1284q^{79} + 324q^{81} - 288q^{84} + 2624q^{86} - 488q^{89} + 224q^{91} + 3896q^{94} + 738q^{96} + 396q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155i 0.552092i 0.961144 + 0.276046i \(0.0890243\pi\)
−0.961144 + 0.276046i \(0.910976\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 5.56155 0.695194
\(5\) 0 0
\(6\) 4.68466 0.318751
\(7\) − 10.2462i − 0.553243i −0.960979 0.276622i \(-0.910785\pi\)
0.960979 0.276622i \(-0.0892149\pi\)
\(8\) 21.1771i 0.935904i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) − 16.6847i − 0.401371i
\(13\) 40.8769i 0.872093i 0.899924 + 0.436047i \(0.143622\pi\)
−0.899924 + 0.436047i \(0.856378\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) − 98.7083i − 1.40825i −0.710075 0.704126i \(-0.751339\pi\)
0.710075 0.704126i \(-0.248661\pi\)
\(18\) − 14.0540i − 0.184031i
\(19\) 39.6458 0.478704 0.239352 0.970933i \(-0.423065\pi\)
0.239352 + 0.970933i \(0.423065\pi\)
\(20\) 0 0
\(21\) −30.7386 −0.319415
\(22\) − 17.1771i − 0.166462i
\(23\) − 61.6932i − 0.559301i −0.960102 0.279650i \(-0.909781\pi\)
0.960102 0.279650i \(-0.0902185\pi\)
\(24\) 63.5312 0.540344
\(25\) 0 0
\(26\) −63.8314 −0.481476
\(27\) 27.0000i 0.192450i
\(28\) − 56.9848i − 0.384612i
\(29\) 149.093 0.954684 0.477342 0.878718i \(-0.341600\pi\)
0.477342 + 0.878718i \(0.341600\pi\)
\(30\) 0 0
\(31\) 54.7386 0.317140 0.158570 0.987348i \(-0.449312\pi\)
0.158570 + 0.987348i \(0.449312\pi\)
\(32\) 187.255i 1.03445i
\(33\) 33.0000i 0.174078i
\(34\) 154.138 0.777485
\(35\) 0 0
\(36\) −50.0540 −0.231731
\(37\) 44.8939i 0.199473i 0.995014 + 0.0997367i \(0.0318000\pi\)
−0.995014 + 0.0997367i \(0.968200\pi\)
\(38\) 61.9091i 0.264289i
\(39\) 122.631 0.503503
\(40\) 0 0
\(41\) 336.479 1.28169 0.640844 0.767671i \(-0.278585\pi\)
0.640844 + 0.767671i \(0.278585\pi\)
\(42\) − 48.0000i − 0.176347i
\(43\) 2.36745i 0.00839611i 0.999991 + 0.00419806i \(0.00133629\pi\)
−0.999991 + 0.00419806i \(0.998664\pi\)
\(44\) −61.1771 −0.209609
\(45\) 0 0
\(46\) 96.3371 0.308786
\(47\) − 333.295i − 1.03439i −0.855869 0.517193i \(-0.826977\pi\)
0.855869 0.517193i \(-0.173023\pi\)
\(48\) − 34.2699i − 0.103051i
\(49\) 238.015 0.693922
\(50\) 0 0
\(51\) −296.125 −0.813055
\(52\) 227.339i 0.606274i
\(53\) − 640.064i − 1.65886i −0.558610 0.829430i \(-0.688665\pi\)
0.558610 0.829430i \(-0.311335\pi\)
\(54\) −42.1619 −0.106250
\(55\) 0 0
\(56\) 216.985 0.517782
\(57\) − 118.938i − 0.276380i
\(58\) 232.816i 0.527074i
\(59\) 370.773 0.818144 0.409072 0.912502i \(-0.365853\pi\)
0.409072 + 0.912502i \(0.365853\pi\)
\(60\) 0 0
\(61\) −714.405 −1.49951 −0.749756 0.661715i \(-0.769829\pi\)
−0.749756 + 0.661715i \(0.769829\pi\)
\(62\) 85.4773i 0.175091i
\(63\) 92.2159i 0.184414i
\(64\) −201.022 −0.392621
\(65\) 0 0
\(66\) −51.5312 −0.0961069
\(67\) − 404.985i − 0.738459i −0.929338 0.369230i \(-0.879622\pi\)
0.929338 0.369230i \(-0.120378\pi\)
\(68\) − 548.972i − 0.979009i
\(69\) −185.080 −0.322912
\(70\) 0 0
\(71\) 939.292 1.57005 0.785024 0.619465i \(-0.212651\pi\)
0.785024 + 0.619465i \(0.212651\pi\)
\(72\) − 190.594i − 0.311968i
\(73\) 362.570i 0.581310i 0.956828 + 0.290655i \(0.0938732\pi\)
−0.956828 + 0.290655i \(0.906127\pi\)
\(74\) −70.1042 −0.110128
\(75\) 0 0
\(76\) 220.492 0.332792
\(77\) 112.708i 0.166809i
\(78\) 191.494i 0.277980i
\(79\) −951.835 −1.35557 −0.677784 0.735261i \(-0.737059\pi\)
−0.677784 + 0.735261i \(0.737059\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 525.430i 0.707610i
\(83\) − 735.221i − 0.972302i −0.873875 0.486151i \(-0.838401\pi\)
0.873875 0.486151i \(-0.161599\pi\)
\(84\) −170.955 −0.222056
\(85\) 0 0
\(86\) −3.69690 −0.00463543
\(87\) − 447.278i − 0.551187i
\(88\) − 232.948i − 0.282186i
\(89\) −385.879 −0.459585 −0.229793 0.973240i \(-0.573805\pi\)
−0.229793 + 0.973240i \(0.573805\pi\)
\(90\) 0 0
\(91\) 418.833 0.482480
\(92\) − 343.110i − 0.388823i
\(93\) − 164.216i − 0.183101i
\(94\) 520.458 0.571076
\(95\) 0 0
\(96\) 561.764 0.597238
\(97\) − 966.345i − 1.01152i −0.862674 0.505760i \(-0.831212\pi\)
0.862674 0.505760i \(-0.168788\pi\)
\(98\) 371.673i 0.383109i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 348.600 0.343436 0.171718 0.985146i \(-0.445068\pi\)
0.171718 + 0.985146i \(0.445068\pi\)
\(102\) − 462.415i − 0.448881i
\(103\) 1536.38i 1.46975i 0.678204 + 0.734873i \(0.262758\pi\)
−0.678204 + 0.734873i \(0.737242\pi\)
\(104\) −865.653 −0.816195
\(105\) 0 0
\(106\) 999.494 0.915844
\(107\) − 779.180i − 0.703983i −0.936003 0.351991i \(-0.885505\pi\)
0.936003 0.351991i \(-0.114495\pi\)
\(108\) 150.162i 0.133790i
\(109\) 1501.79 1.31968 0.659842 0.751404i \(-0.270623\pi\)
0.659842 + 0.751404i \(0.270623\pi\)
\(110\) 0 0
\(111\) 134.682 0.115166
\(112\) − 117.045i − 0.0987478i
\(113\) − 170.000i − 0.141524i −0.997493 0.0707622i \(-0.977457\pi\)
0.997493 0.0707622i \(-0.0225431\pi\)
\(114\) 185.727 0.152587
\(115\) 0 0
\(116\) 829.187 0.663691
\(117\) − 367.892i − 0.290698i
\(118\) 578.981i 0.451691i
\(119\) −1011.39 −0.779106
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 1115.58i − 0.827869i
\(123\) − 1009.44i − 0.739983i
\(124\) 304.432 0.220474
\(125\) 0 0
\(126\) −144.000 −0.101814
\(127\) − 1739.82i − 1.21562i −0.794082 0.607811i \(-0.792048\pi\)
0.794082 0.607811i \(-0.207952\pi\)
\(128\) 1184.13i 0.817683i
\(129\) 7.10235 0.00484750
\(130\) 0 0
\(131\) 312.837 0.208647 0.104323 0.994543i \(-0.466732\pi\)
0.104323 + 0.994543i \(0.466732\pi\)
\(132\) 183.531i 0.121018i
\(133\) − 406.220i − 0.264840i
\(134\) 632.405 0.407698
\(135\) 0 0
\(136\) 2090.35 1.31799
\(137\) − 716.928i − 0.447090i −0.974694 0.223545i \(-0.928237\pi\)
0.974694 0.223545i \(-0.0717629\pi\)
\(138\) − 289.011i − 0.178277i
\(139\) 876.483 0.534837 0.267418 0.963581i \(-0.413829\pi\)
0.267418 + 0.963581i \(0.413829\pi\)
\(140\) 0 0
\(141\) −999.886 −0.597203
\(142\) 1466.75i 0.866811i
\(143\) − 449.646i − 0.262946i
\(144\) −102.810 −0.0594963
\(145\) 0 0
\(146\) −566.172 −0.320937
\(147\) − 714.045i − 0.400636i
\(148\) 249.680i 0.138673i
\(149\) 2376.36 1.30657 0.653285 0.757112i \(-0.273390\pi\)
0.653285 + 0.757112i \(0.273390\pi\)
\(150\) 0 0
\(151\) −92.8466 −0.0500381 −0.0250190 0.999687i \(-0.507965\pi\)
−0.0250190 + 0.999687i \(0.507965\pi\)
\(152\) 839.583i 0.448021i
\(153\) 888.375i 0.469417i
\(154\) −176.000 −0.0920941
\(155\) 0 0
\(156\) 682.017 0.350032
\(157\) − 1881.24i − 0.956301i −0.878278 0.478150i \(-0.841307\pi\)
0.878278 0.478150i \(-0.158693\pi\)
\(158\) − 1486.34i − 0.748398i
\(159\) −1920.19 −0.957743
\(160\) 0 0
\(161\) −632.121 −0.309429
\(162\) 126.486i 0.0613436i
\(163\) 2465.49i 1.18474i 0.805667 + 0.592369i \(0.201807\pi\)
−0.805667 + 0.592369i \(0.798193\pi\)
\(164\) 1871.35 0.891022
\(165\) 0 0
\(166\) 1148.09 0.536800
\(167\) − 1254.30i − 0.581200i −0.956845 0.290600i \(-0.906145\pi\)
0.956845 0.290600i \(-0.0938549\pi\)
\(168\) − 650.955i − 0.298942i
\(169\) 526.080 0.239454
\(170\) 0 0
\(171\) −356.813 −0.159568
\(172\) 13.1667i 0.00583693i
\(173\) 1206.71i 0.530314i 0.964205 + 0.265157i \(0.0854238\pi\)
−0.964205 + 0.265157i \(0.914576\pi\)
\(174\) 698.449 0.304306
\(175\) 0 0
\(176\) −125.656 −0.0538164
\(177\) − 1112.32i − 0.472356i
\(178\) − 602.570i − 0.253733i
\(179\) 1442.29 0.602244 0.301122 0.953586i \(-0.402639\pi\)
0.301122 + 0.953586i \(0.402639\pi\)
\(180\) 0 0
\(181\) 4261.81 1.75015 0.875076 0.483985i \(-0.160811\pi\)
0.875076 + 0.483985i \(0.160811\pi\)
\(182\) 654.030i 0.266373i
\(183\) 2143.22i 0.865744i
\(184\) 1306.48 0.523451
\(185\) 0 0
\(186\) 256.432 0.101089
\(187\) 1085.79i 0.424604i
\(188\) − 1853.64i − 0.719099i
\(189\) 276.648 0.106472
\(190\) 0 0
\(191\) −852.223 −0.322852 −0.161426 0.986885i \(-0.551609\pi\)
−0.161426 + 0.986885i \(0.551609\pi\)
\(192\) 603.065i 0.226680i
\(193\) 2459.95i 0.917468i 0.888574 + 0.458734i \(0.151697\pi\)
−0.888574 + 0.458734i \(0.848303\pi\)
\(194\) 1509.00 0.558452
\(195\) 0 0
\(196\) 1323.73 0.482410
\(197\) 3477.06i 1.25751i 0.777602 + 0.628756i \(0.216436\pi\)
−0.777602 + 0.628756i \(0.783564\pi\)
\(198\) 154.594i 0.0554874i
\(199\) −3995.04 −1.42312 −0.711560 0.702626i \(-0.752011\pi\)
−0.711560 + 0.702626i \(0.752011\pi\)
\(200\) 0 0
\(201\) −1214.95 −0.426350
\(202\) 544.358i 0.189608i
\(203\) − 1527.64i − 0.528173i
\(204\) −1646.91 −0.565231
\(205\) 0 0
\(206\) −2399.14 −0.811436
\(207\) 555.239i 0.186434i
\(208\) 466.949i 0.155659i
\(209\) −436.104 −0.144335
\(210\) 0 0
\(211\) 1046.13 0.341319 0.170660 0.985330i \(-0.445410\pi\)
0.170660 + 0.985330i \(0.445410\pi\)
\(212\) − 3559.75i − 1.15323i
\(213\) − 2817.88i − 0.906468i
\(214\) 1216.73 0.388664
\(215\) 0 0
\(216\) −571.781 −0.180115
\(217\) − 560.864i − 0.175456i
\(218\) 2345.13i 0.728587i
\(219\) 1087.71 0.335619
\(220\) 0 0
\(221\) 4034.89 1.22813
\(222\) 210.313i 0.0635823i
\(223\) 506.265i 0.152027i 0.997107 + 0.0760135i \(0.0242192\pi\)
−0.997107 + 0.0760135i \(0.975781\pi\)
\(224\) 1918.65 0.572300
\(225\) 0 0
\(226\) 265.464 0.0781345
\(227\) − 4286.29i − 1.25326i −0.779315 0.626632i \(-0.784433\pi\)
0.779315 0.626632i \(-0.215567\pi\)
\(228\) − 661.477i − 0.192138i
\(229\) −5709.37 −1.64754 −0.823769 0.566926i \(-0.808133\pi\)
−0.823769 + 0.566926i \(0.808133\pi\)
\(230\) 0 0
\(231\) 338.125 0.0963073
\(232\) 3157.35i 0.893492i
\(233\) − 2946.09i − 0.828348i −0.910198 0.414174i \(-0.864071\pi\)
0.910198 0.414174i \(-0.135929\pi\)
\(234\) 574.483 0.160492
\(235\) 0 0
\(236\) 2062.07 0.568769
\(237\) 2855.51i 0.782637i
\(238\) − 1579.33i − 0.430139i
\(239\) 2078.89 0.562646 0.281323 0.959613i \(-0.409227\pi\)
0.281323 + 0.959613i \(0.409227\pi\)
\(240\) 0 0
\(241\) 1853.37 0.495378 0.247689 0.968840i \(-0.420329\pi\)
0.247689 + 0.968840i \(0.420329\pi\)
\(242\) 188.948i 0.0501902i
\(243\) − 243.000i − 0.0641500i
\(244\) −3973.20 −1.04245
\(245\) 0 0
\(246\) 1576.29 0.408539
\(247\) 1620.60i 0.417475i
\(248\) 1159.20i 0.296813i
\(249\) −2205.66 −0.561359
\(250\) 0 0
\(251\) −2358.39 −0.593068 −0.296534 0.955022i \(-0.595831\pi\)
−0.296534 + 0.955022i \(0.595831\pi\)
\(252\) 512.864i 0.128204i
\(253\) 678.625i 0.168635i
\(254\) 2716.82 0.671135
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) 5519.25i 1.33962i 0.742534 + 0.669809i \(0.233624\pi\)
−0.742534 + 0.669809i \(0.766376\pi\)
\(258\) 11.0907i 0.00267627i
\(259\) 459.993 0.110357
\(260\) 0 0
\(261\) −1341.84 −0.318228
\(262\) 488.512i 0.115192i
\(263\) − 2259.65i − 0.529795i −0.964277 0.264898i \(-0.914662\pi\)
0.964277 0.264898i \(-0.0853382\pi\)
\(264\) −698.844 −0.162920
\(265\) 0 0
\(266\) 634.333 0.146216
\(267\) 1157.64i 0.265342i
\(268\) − 2252.34i − 0.513373i
\(269\) −7039.53 −1.59557 −0.797783 0.602944i \(-0.793994\pi\)
−0.797783 + 0.602944i \(0.793994\pi\)
\(270\) 0 0
\(271\) 5155.08 1.15553 0.577765 0.816203i \(-0.303925\pi\)
0.577765 + 0.816203i \(0.303925\pi\)
\(272\) − 1127.57i − 0.251357i
\(273\) − 1256.50i − 0.278560i
\(274\) 1119.52 0.246835
\(275\) 0 0
\(276\) −1029.33 −0.224487
\(277\) 9074.52i 1.96836i 0.177175 + 0.984179i \(0.443304\pi\)
−0.177175 + 0.984179i \(0.556696\pi\)
\(278\) 1368.67i 0.295279i
\(279\) −492.648 −0.105713
\(280\) 0 0
\(281\) −3407.79 −0.723459 −0.361729 0.932283i \(-0.617814\pi\)
−0.361729 + 0.932283i \(0.617814\pi\)
\(282\) − 1561.38i − 0.329711i
\(283\) 8827.73i 1.85425i 0.374746 + 0.927127i \(0.377730\pi\)
−0.374746 + 0.927127i \(0.622270\pi\)
\(284\) 5223.92 1.09149
\(285\) 0 0
\(286\) 702.146 0.145170
\(287\) − 3447.64i − 0.709085i
\(288\) − 1685.29i − 0.344815i
\(289\) −4830.33 −0.983174
\(290\) 0 0
\(291\) −2899.03 −0.584001
\(292\) 2016.45i 0.404123i
\(293\) 4528.29i 0.902886i 0.892300 + 0.451443i \(0.149091\pi\)
−0.892300 + 0.451443i \(0.850909\pi\)
\(294\) 1115.02 0.221188
\(295\) 0 0
\(296\) −950.722 −0.186688
\(297\) − 297.000i − 0.0580259i
\(298\) 3710.81i 0.721347i
\(299\) 2521.83 0.487762
\(300\) 0 0
\(301\) 24.2574 0.00464509
\(302\) − 144.985i − 0.0276256i
\(303\) − 1045.80i − 0.198283i
\(304\) 452.886 0.0854434
\(305\) 0 0
\(306\) −1387.24 −0.259162
\(307\) 568.106i 0.105614i 0.998605 + 0.0528071i \(0.0168168\pi\)
−0.998605 + 0.0528071i \(0.983183\pi\)
\(308\) 626.833i 0.115965i
\(309\) 4609.14 0.848559
\(310\) 0 0
\(311\) −6853.59 −1.24962 −0.624809 0.780778i \(-0.714823\pi\)
−0.624809 + 0.780778i \(0.714823\pi\)
\(312\) 2596.96i 0.471230i
\(313\) 1138.92i 0.205673i 0.994698 + 0.102837i \(0.0327919\pi\)
−0.994698 + 0.102837i \(0.967208\pi\)
\(314\) 2937.65 0.527966
\(315\) 0 0
\(316\) −5293.68 −0.942382
\(317\) 3207.48i 0.568297i 0.958780 + 0.284148i \(0.0917109\pi\)
−0.958780 + 0.284148i \(0.908289\pi\)
\(318\) − 2998.48i − 0.528763i
\(319\) −1640.02 −0.287848
\(320\) 0 0
\(321\) −2337.54 −0.406445
\(322\) − 987.091i − 0.170834i
\(323\) − 3913.37i − 0.674136i
\(324\) 450.486 0.0772438
\(325\) 0 0
\(326\) −3850.00 −0.654085
\(327\) − 4505.37i − 0.761920i
\(328\) 7125.65i 1.19954i
\(329\) −3415.02 −0.572267
\(330\) 0 0
\(331\) −9135.12 −1.51695 −0.758477 0.651700i \(-0.774056\pi\)
−0.758477 + 0.651700i \(0.774056\pi\)
\(332\) − 4088.97i − 0.675938i
\(333\) − 404.045i − 0.0664911i
\(334\) 1958.65 0.320876
\(335\) 0 0
\(336\) −351.136 −0.0570121
\(337\) − 3470.05i − 0.560907i −0.959868 0.280453i \(-0.909515\pi\)
0.959868 0.280453i \(-0.0904848\pi\)
\(338\) 821.501i 0.132200i
\(339\) −510.000 −0.0817091
\(340\) 0 0
\(341\) −602.125 −0.0956214
\(342\) − 557.182i − 0.0880963i
\(343\) − 5953.20i − 0.937151i
\(344\) −50.1357 −0.00785795
\(345\) 0 0
\(346\) −1884.34 −0.292782
\(347\) − 89.3315i − 0.0138201i −0.999976 0.00691004i \(-0.997800\pi\)
0.999976 0.00691004i \(-0.00219955\pi\)
\(348\) − 2487.56i − 0.383182i
\(349\) 149.375 0.0229107 0.0114554 0.999934i \(-0.496354\pi\)
0.0114554 + 0.999934i \(0.496354\pi\)
\(350\) 0 0
\(351\) −1103.68 −0.167834
\(352\) − 2059.80i − 0.311897i
\(353\) − 7867.64i − 1.18627i −0.805104 0.593133i \(-0.797891\pi\)
0.805104 0.593133i \(-0.202109\pi\)
\(354\) 1736.94 0.260784
\(355\) 0 0
\(356\) −2146.09 −0.319501
\(357\) 3034.16i 0.449817i
\(358\) 2252.21i 0.332494i
\(359\) −4974.22 −0.731279 −0.365639 0.930757i \(-0.619150\pi\)
−0.365639 + 0.930757i \(0.619150\pi\)
\(360\) 0 0
\(361\) −5287.21 −0.770842
\(362\) 6655.04i 0.966246i
\(363\) − 363.000i − 0.0524864i
\(364\) 2329.36 0.335417
\(365\) 0 0
\(366\) −3346.74 −0.477970
\(367\) − 13266.7i − 1.88696i −0.331433 0.943479i \(-0.607532\pi\)
0.331433 0.943479i \(-0.392468\pi\)
\(368\) − 704.739i − 0.0998290i
\(369\) −3028.31 −0.427229
\(370\) 0 0
\(371\) −6558.23 −0.917754
\(372\) − 913.295i − 0.127291i
\(373\) 4632.77i 0.643099i 0.946893 + 0.321549i \(0.104204\pi\)
−0.946893 + 0.321549i \(0.895796\pi\)
\(374\) −1695.52 −0.234421
\(375\) 0 0
\(376\) 7058.22 0.968085
\(377\) 6094.45i 0.832573i
\(378\) 432.000i 0.0587822i
\(379\) −6503.31 −0.881406 −0.440703 0.897653i \(-0.645271\pi\)
−0.440703 + 0.897653i \(0.645271\pi\)
\(380\) 0 0
\(381\) −5219.45 −0.701839
\(382\) − 1330.79i − 0.178244i
\(383\) − 12734.5i − 1.69897i −0.527614 0.849484i \(-0.676913\pi\)
0.527614 0.849484i \(-0.323087\pi\)
\(384\) 3552.39 0.472090
\(385\) 0 0
\(386\) −3841.35 −0.506527
\(387\) − 21.3071i − 0.00279870i
\(388\) − 5374.38i − 0.703203i
\(389\) −12024.6 −1.56728 −0.783639 0.621216i \(-0.786639\pi\)
−0.783639 + 0.621216i \(0.786639\pi\)
\(390\) 0 0
\(391\) −6089.63 −0.787636
\(392\) 5040.47i 0.649444i
\(393\) − 938.511i − 0.120462i
\(394\) −5429.61 −0.694263
\(395\) 0 0
\(396\) 550.594 0.0698696
\(397\) − 5223.65i − 0.660371i −0.943916 0.330186i \(-0.892889\pi\)
0.943916 0.330186i \(-0.107111\pi\)
\(398\) − 6238.46i − 0.785693i
\(399\) −1218.66 −0.152905
\(400\) 0 0
\(401\) 9648.18 1.20151 0.600757 0.799432i \(-0.294866\pi\)
0.600757 + 0.799432i \(0.294866\pi\)
\(402\) − 1897.22i − 0.235384i
\(403\) 2237.55i 0.276576i
\(404\) 1938.76 0.238755
\(405\) 0 0
\(406\) 2385.48 0.291600
\(407\) − 493.833i − 0.0601435i
\(408\) − 6271.06i − 0.760941i
\(409\) 2010.47 0.243060 0.121530 0.992588i \(-0.461220\pi\)
0.121530 + 0.992588i \(0.461220\pi\)
\(410\) 0 0
\(411\) −2150.78 −0.258127
\(412\) 8544.65i 1.02176i
\(413\) − 3799.02i − 0.452633i
\(414\) −867.034 −0.102929
\(415\) 0 0
\(416\) −7654.39 −0.902133
\(417\) − 2629.45i − 0.308788i
\(418\) − 681.000i − 0.0796861i
\(419\) −4435.27 −0.517129 −0.258565 0.965994i \(-0.583250\pi\)
−0.258565 + 0.965994i \(0.583250\pi\)
\(420\) 0 0
\(421\) 15217.9 1.76170 0.880852 0.473392i \(-0.156971\pi\)
0.880852 + 0.473392i \(0.156971\pi\)
\(422\) 1633.58i 0.188440i
\(423\) 2999.66i 0.344795i
\(424\) 13554.7 1.55253
\(425\) 0 0
\(426\) 4400.26 0.500454
\(427\) 7319.95i 0.829595i
\(428\) − 4333.45i − 0.489405i
\(429\) −1348.94 −0.151812
\(430\) 0 0
\(431\) −5622.11 −0.628324 −0.314162 0.949369i \(-0.601723\pi\)
−0.314162 + 0.949369i \(0.601723\pi\)
\(432\) 308.429i 0.0343502i
\(433\) 14306.3i 1.58780i 0.608051 + 0.793898i \(0.291951\pi\)
−0.608051 + 0.793898i \(0.708049\pi\)
\(434\) 875.818 0.0968678
\(435\) 0 0
\(436\) 8352.29 0.917436
\(437\) − 2445.88i − 0.267740i
\(438\) 1698.52i 0.185293i
\(439\) −4384.20 −0.476643 −0.238322 0.971186i \(-0.576597\pi\)
−0.238322 + 0.971186i \(0.576597\pi\)
\(440\) 0 0
\(441\) −2142.14 −0.231307
\(442\) 6300.69i 0.678039i
\(443\) 10090.0i 1.08214i 0.840977 + 0.541071i \(0.181981\pi\)
−0.840977 + 0.541071i \(0.818019\pi\)
\(444\) 749.040 0.0800627
\(445\) 0 0
\(446\) −790.560 −0.0839330
\(447\) − 7129.07i − 0.754348i
\(448\) 2059.71i 0.217215i
\(449\) −9582.52 −1.00719 −0.503594 0.863941i \(-0.667989\pi\)
−0.503594 + 0.863941i \(0.667989\pi\)
\(450\) 0 0
\(451\) −3701.27 −0.386444
\(452\) − 945.464i − 0.0983869i
\(453\) 278.540i 0.0288895i
\(454\) 6693.27 0.691918
\(455\) 0 0
\(456\) 2518.75 0.258665
\(457\) 9999.34i 1.02352i 0.859128 + 0.511761i \(0.171007\pi\)
−0.859128 + 0.511761i \(0.828993\pi\)
\(458\) − 8915.49i − 0.909593i
\(459\) 2665.12 0.271018
\(460\) 0 0
\(461\) −11115.8 −1.12302 −0.561512 0.827468i \(-0.689780\pi\)
−0.561512 + 0.827468i \(0.689780\pi\)
\(462\) 528.000i 0.0531705i
\(463\) 1567.16i 0.157305i 0.996902 + 0.0786524i \(0.0250617\pi\)
−0.996902 + 0.0786524i \(0.974938\pi\)
\(464\) 1703.13 0.170401
\(465\) 0 0
\(466\) 4600.48 0.457325
\(467\) 12648.8i 1.25335i 0.779281 + 0.626675i \(0.215585\pi\)
−0.779281 + 0.626675i \(0.784415\pi\)
\(468\) − 2046.05i − 0.202091i
\(469\) −4149.56 −0.408548
\(470\) 0 0
\(471\) −5643.72 −0.552120
\(472\) 7851.88i 0.765704i
\(473\) − 26.0420i − 0.00253152i
\(474\) −4459.02 −0.432088
\(475\) 0 0
\(476\) −5624.88 −0.541630
\(477\) 5760.58i 0.552953i
\(478\) 3246.30i 0.310633i
\(479\) 10719.2 1.02249 0.511247 0.859434i \(-0.329184\pi\)
0.511247 + 0.859434i \(0.329184\pi\)
\(480\) 0 0
\(481\) −1835.12 −0.173959
\(482\) 2894.14i 0.273494i
\(483\) 1896.36i 0.178649i
\(484\) 672.948 0.0631995
\(485\) 0 0
\(486\) 379.457 0.0354167
\(487\) 7161.20i 0.666335i 0.942868 + 0.333167i \(0.108117\pi\)
−0.942868 + 0.333167i \(0.891883\pi\)
\(488\) − 15129.0i − 1.40340i
\(489\) 7396.48 0.684009
\(490\) 0 0
\(491\) 14567.3 1.33893 0.669463 0.742845i \(-0.266524\pi\)
0.669463 + 0.742845i \(0.266524\pi\)
\(492\) − 5614.04i − 0.514432i
\(493\) − 14716.7i − 1.34444i
\(494\) −2530.65 −0.230485
\(495\) 0 0
\(496\) 625.295 0.0566060
\(497\) − 9624.18i − 0.868619i
\(498\) − 3444.26i − 0.309922i
\(499\) 4638.99 0.416172 0.208086 0.978111i \(-0.433277\pi\)
0.208086 + 0.978111i \(0.433277\pi\)
\(500\) 0 0
\(501\) −3762.89 −0.335556
\(502\) − 3682.75i − 0.327428i
\(503\) 12206.3i 1.08201i 0.841019 + 0.541006i \(0.181956\pi\)
−0.841019 + 0.541006i \(0.818044\pi\)
\(504\) −1952.86 −0.172594
\(505\) 0 0
\(506\) −1059.71 −0.0931024
\(507\) − 1578.24i − 0.138249i
\(508\) − 9676.09i − 0.845093i
\(509\) −10018.6 −0.872427 −0.436214 0.899843i \(-0.643681\pi\)
−0.436214 + 0.899843i \(0.643681\pi\)
\(510\) 0 0
\(511\) 3714.97 0.321606
\(512\) 4074.36i 0.351686i
\(513\) 1070.44i 0.0921267i
\(514\) −8618.61 −0.739592
\(515\) 0 0
\(516\) 39.5001 0.00336995
\(517\) 3666.25i 0.311879i
\(518\) 718.303i 0.0609274i
\(519\) 3620.12 0.306177
\(520\) 0 0
\(521\) 1054.72 0.0886916 0.0443458 0.999016i \(-0.485880\pi\)
0.0443458 + 0.999016i \(0.485880\pi\)
\(522\) − 2095.35i − 0.175691i
\(523\) 16234.2i 1.35730i 0.734460 + 0.678652i \(0.237436\pi\)
−0.734460 + 0.678652i \(0.762564\pi\)
\(524\) 1739.86 0.145050
\(525\) 0 0
\(526\) 3528.57 0.292496
\(527\) − 5403.16i − 0.446613i
\(528\) 376.969i 0.0310709i
\(529\) 8360.95 0.687183
\(530\) 0 0
\(531\) −3336.95 −0.272715
\(532\) − 2259.21i − 0.184115i
\(533\) 13754.2i 1.11775i
\(534\) −1807.71 −0.146493
\(535\) 0 0
\(536\) 8576.40 0.691127
\(537\) − 4326.86i − 0.347706i
\(538\) − 10992.6i − 0.880900i
\(539\) −2618.17 −0.209225
\(540\) 0 0
\(541\) 675.936 0.0537167 0.0268584 0.999639i \(-0.491450\pi\)
0.0268584 + 0.999639i \(0.491450\pi\)
\(542\) 8049.93i 0.637960i
\(543\) − 12785.4i − 1.01045i
\(544\) 18483.6 1.45676
\(545\) 0 0
\(546\) 1962.09 0.153791
\(547\) 13058.2i 1.02071i 0.859964 + 0.510355i \(0.170486\pi\)
−0.859964 + 0.510355i \(0.829514\pi\)
\(548\) − 3987.23i − 0.310814i
\(549\) 6429.65 0.499837
\(550\) 0 0
\(551\) 5910.91 0.457011
\(552\) − 3919.44i − 0.302215i
\(553\) 9752.70i 0.749959i
\(554\) −14170.3 −1.08672
\(555\) 0 0
\(556\) 4874.61 0.371815
\(557\) − 6710.48i − 0.510471i −0.966879 0.255236i \(-0.917847\pi\)
0.966879 0.255236i \(-0.0821530\pi\)
\(558\) − 769.295i − 0.0583636i
\(559\) −96.7741 −0.00732219
\(560\) 0 0
\(561\) 3257.37 0.245145
\(562\) − 5321.45i − 0.399416i
\(563\) 20820.5i 1.55858i 0.626666 + 0.779288i \(0.284419\pi\)
−0.626666 + 0.779288i \(0.715581\pi\)
\(564\) −5560.92 −0.415172
\(565\) 0 0
\(566\) −13785.0 −1.02372
\(567\) − 829.943i − 0.0614715i
\(568\) 19891.5i 1.46941i
\(569\) −3251.08 −0.239530 −0.119765 0.992802i \(-0.538214\pi\)
−0.119765 + 0.992802i \(0.538214\pi\)
\(570\) 0 0
\(571\) −4637.50 −0.339883 −0.169941 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(572\) − 2500.73i − 0.182798i
\(573\) 2556.67i 0.186399i
\(574\) 5383.67 0.391481
\(575\) 0 0
\(576\) 1809.20 0.130874
\(577\) 14462.4i 1.04346i 0.853111 + 0.521730i \(0.174713\pi\)
−0.853111 + 0.521730i \(0.825287\pi\)
\(578\) − 7542.82i − 0.542803i
\(579\) 7379.86 0.529700
\(580\) 0 0
\(581\) −7533.23 −0.537920
\(582\) − 4526.99i − 0.322423i
\(583\) 7040.71i 0.500165i
\(584\) −7678.18 −0.544050
\(585\) 0 0
\(586\) −7071.17 −0.498476
\(587\) − 22759.7i − 1.60033i −0.599779 0.800166i \(-0.704745\pi\)
0.599779 0.800166i \(-0.295255\pi\)
\(588\) − 3971.20i − 0.278520i
\(589\) 2170.16 0.151816
\(590\) 0 0
\(591\) 10431.2 0.726025
\(592\) 512.836i 0.0356038i
\(593\) 14956.4i 1.03573i 0.855463 + 0.517864i \(0.173273\pi\)
−0.855463 + 0.517864i \(0.826727\pi\)
\(594\) 463.781 0.0320356
\(595\) 0 0
\(596\) 13216.2 0.908319
\(597\) 11985.1i 0.821638i
\(598\) 3937.96i 0.269290i
\(599\) −2150.77 −0.146708 −0.0733539 0.997306i \(-0.523370\pi\)
−0.0733539 + 0.997306i \(0.523370\pi\)
\(600\) 0 0
\(601\) 27759.8 1.88410 0.942050 0.335472i \(-0.108896\pi\)
0.942050 + 0.335472i \(0.108896\pi\)
\(602\) 37.8792i 0.00256452i
\(603\) 3644.86i 0.246153i
\(604\) −516.371 −0.0347862
\(605\) 0 0
\(606\) 1633.07 0.109470
\(607\) − 10991.5i − 0.734974i −0.930029 0.367487i \(-0.880218\pi\)
0.930029 0.367487i \(-0.119782\pi\)
\(608\) 7423.87i 0.495194i
\(609\) −4582.91 −0.304941
\(610\) 0 0
\(611\) 13624.1 0.902081
\(612\) 4940.74i 0.326336i
\(613\) − 10646.1i − 0.701457i −0.936477 0.350728i \(-0.885934\pi\)
0.936477 0.350728i \(-0.114066\pi\)
\(614\) −887.128 −0.0583088
\(615\) 0 0
\(616\) −2386.83 −0.156117
\(617\) 7199.92i 0.469786i 0.972021 + 0.234893i \(0.0754739\pi\)
−0.972021 + 0.234893i \(0.924526\pi\)
\(618\) 7197.41i 0.468483i
\(619\) −12186.9 −0.791332 −0.395666 0.918395i \(-0.629486\pi\)
−0.395666 + 0.918395i \(0.629486\pi\)
\(620\) 0 0
\(621\) 1665.72 0.107637
\(622\) − 10702.2i − 0.689905i
\(623\) 3953.80i 0.254262i
\(624\) 1400.85 0.0898698
\(625\) 0 0
\(626\) −1778.49 −0.113551
\(627\) 1308.31i 0.0833317i
\(628\) − 10462.6i − 0.664814i
\(629\) 4431.40 0.280909
\(630\) 0 0
\(631\) 7370.64 0.465009 0.232505 0.972595i \(-0.425308\pi\)
0.232505 + 0.972595i \(0.425308\pi\)
\(632\) − 20157.1i − 1.26868i
\(633\) − 3138.38i − 0.197061i
\(634\) −5008.65 −0.313752
\(635\) 0 0
\(636\) −10679.3 −0.665818
\(637\) 9729.32i 0.605164i
\(638\) − 2560.98i − 0.158919i
\(639\) −8453.63 −0.523349
\(640\) 0 0
\(641\) −25014.9 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(642\) − 3650.19i − 0.224395i
\(643\) 21668.2i 1.32894i 0.747313 + 0.664472i \(0.231343\pi\)
−0.747313 + 0.664472i \(0.768657\pi\)
\(644\) −3515.58 −0.215113
\(645\) 0 0
\(646\) 6110.94 0.372185
\(647\) − 27625.3i − 1.67861i −0.543661 0.839305i \(-0.682962\pi\)
0.543661 0.839305i \(-0.317038\pi\)
\(648\) 1715.34i 0.103989i
\(649\) −4078.50 −0.246680
\(650\) 0 0
\(651\) −1682.59 −0.101299
\(652\) 13712.0i 0.823623i
\(653\) 14314.0i 0.857810i 0.903350 + 0.428905i \(0.141101\pi\)
−0.903350 + 0.428905i \(0.858899\pi\)
\(654\) 7035.38 0.420650
\(655\) 0 0
\(656\) 3843.70 0.228767
\(657\) − 3263.13i − 0.193770i
\(658\) − 5332.73i − 0.315944i
\(659\) 28327.8 1.67450 0.837249 0.546822i \(-0.184163\pi\)
0.837249 + 0.546822i \(0.184163\pi\)
\(660\) 0 0
\(661\) −32190.9 −1.89422 −0.947112 0.320905i \(-0.896013\pi\)
−0.947112 + 0.320905i \(0.896013\pi\)
\(662\) − 14265.0i − 0.837498i
\(663\) − 12104.7i − 0.709059i
\(664\) 15569.8 0.909981
\(665\) 0 0
\(666\) 630.938 0.0367092
\(667\) − 9198.01i − 0.533955i
\(668\) − 6975.84i − 0.404047i
\(669\) 1518.80 0.0877729
\(670\) 0 0
\(671\) 7858.46 0.452120
\(672\) − 5755.95i − 0.330418i
\(673\) 6207.38i 0.355538i 0.984072 + 0.177769i \(0.0568879\pi\)
−0.984072 + 0.177769i \(0.943112\pi\)
\(674\) 5418.66 0.309672
\(675\) 0 0
\(676\) 2925.82 0.166467
\(677\) 28831.1i 1.63674i 0.574695 + 0.818368i \(0.305121\pi\)
−0.574695 + 0.818368i \(0.694879\pi\)
\(678\) − 796.392i − 0.0451110i
\(679\) −9901.37 −0.559617
\(680\) 0 0
\(681\) −12858.9 −0.723573
\(682\) − 940.250i − 0.0527918i
\(683\) − 3193.10i − 0.178888i −0.995992 0.0894441i \(-0.971491\pi\)
0.995992 0.0894441i \(-0.0285090\pi\)
\(684\) −1984.43 −0.110931
\(685\) 0 0
\(686\) 9296.24 0.517394
\(687\) 17128.1i 0.951206i
\(688\) 27.0441i 0.00149861i
\(689\) 26163.8 1.44668
\(690\) 0 0
\(691\) 7682.49 0.422946 0.211473 0.977384i \(-0.432174\pi\)
0.211473 + 0.977384i \(0.432174\pi\)
\(692\) 6711.17i 0.368671i
\(693\) − 1014.37i − 0.0556031i
\(694\) 139.496 0.00762996
\(695\) 0 0
\(696\) 9472.05 0.515858
\(697\) − 33213.3i − 1.80494i
\(698\) 233.256i 0.0126488i
\(699\) −8838.28 −0.478247
\(700\) 0 0
\(701\) 26551.6 1.43058 0.715292 0.698825i \(-0.246293\pi\)
0.715292 + 0.698825i \(0.246293\pi\)
\(702\) − 1723.45i − 0.0926601i
\(703\) 1779.86i 0.0954887i
\(704\) 2211.24 0.118380
\(705\) 0 0
\(706\) 12285.7 0.654928
\(707\) − 3571.83i − 0.190004i
\(708\) − 6186.22i − 0.328379i
\(709\) 16304.6 0.863655 0.431828 0.901956i \(-0.357869\pi\)
0.431828 + 0.901956i \(0.357869\pi\)
\(710\) 0 0
\(711\) 8566.52 0.451856
\(712\) − 8171.79i − 0.430127i
\(713\) − 3377.00i − 0.177377i
\(714\) −4738.00 −0.248341
\(715\) 0 0
\(716\) 8021.36 0.418676
\(717\) − 6236.68i − 0.324844i
\(718\) − 7767.50i − 0.403733i
\(719\) 3973.62 0.206107 0.103053 0.994676i \(-0.467139\pi\)
0.103053 + 0.994676i \(0.467139\pi\)
\(720\) 0 0
\(721\) 15742.1 0.813128
\(722\) − 8256.25i − 0.425576i
\(723\) − 5560.11i − 0.286007i
\(724\) 23702.3 1.21670
\(725\) 0 0
\(726\) 566.844 0.0289773
\(727\) − 10780.4i − 0.549961i −0.961450 0.274980i \(-0.911329\pi\)
0.961450 0.274980i \(-0.0886714\pi\)
\(728\) 8869.67i 0.451555i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 233.687 0.0118238
\(732\) 11919.6i 0.601860i
\(733\) 9211.46i 0.464165i 0.972696 + 0.232083i \(0.0745540\pi\)
−0.972696 + 0.232083i \(0.925446\pi\)
\(734\) 20716.6 1.04177
\(735\) 0 0
\(736\) 11552.3 0.578566
\(737\) 4454.83i 0.222654i
\(738\) − 4728.87i − 0.235870i
\(739\) −11084.7 −0.551768 −0.275884 0.961191i \(-0.588971\pi\)
−0.275884 + 0.961191i \(0.588971\pi\)
\(740\) 0 0
\(741\) 4861.80 0.241029
\(742\) − 10241.0i − 0.506685i
\(743\) 27420.4i 1.35391i 0.736024 + 0.676955i \(0.236701\pi\)
−0.736024 + 0.676955i \(0.763299\pi\)
\(744\) 3477.61 0.171365
\(745\) 0 0
\(746\) −7234.32 −0.355050
\(747\) 6616.99i 0.324101i
\(748\) 6038.69i 0.295182i
\(749\) −7983.64 −0.389474
\(750\) 0 0
\(751\) −11290.8 −0.548614 −0.274307 0.961642i \(-0.588448\pi\)
−0.274307 + 0.961642i \(0.588448\pi\)
\(752\) − 3807.33i − 0.184626i
\(753\) 7075.16i 0.342408i
\(754\) −9516.81 −0.459657
\(755\) 0 0
\(756\) 1538.59 0.0740185
\(757\) 3739.19i 0.179528i 0.995963 + 0.0897642i \(0.0286113\pi\)
−0.995963 + 0.0897642i \(0.971389\pi\)
\(758\) − 10155.3i − 0.486617i
\(759\) 2035.87 0.0973617
\(760\) 0 0
\(761\) −15621.1 −0.744107 −0.372053 0.928211i \(-0.621346\pi\)
−0.372053 + 0.928211i \(0.621346\pi\)
\(762\) − 8150.45i − 0.387480i
\(763\) − 15387.7i − 0.730106i
\(764\) −4739.69 −0.224445
\(765\) 0 0
\(766\) 19885.7 0.937987
\(767\) 15156.0i 0.713498i
\(768\) 10371.8i 0.487317i
\(769\) −40241.7 −1.88706 −0.943531 0.331284i \(-0.892518\pi\)
−0.943531 + 0.331284i \(0.892518\pi\)
\(770\) 0 0
\(771\) 16557.8 0.773428
\(772\) 13681.2i 0.637818i
\(773\) − 22821.4i − 1.06187i −0.847412 0.530936i \(-0.821840\pi\)
0.847412 0.530936i \(-0.178160\pi\)
\(774\) 33.2721 0.00154514
\(775\) 0 0
\(776\) 20464.4 0.946685
\(777\) − 1379.98i − 0.0637148i
\(778\) − 18777.0i − 0.865282i
\(779\) 13340.0 0.613549
\(780\) 0 0
\(781\) −10332.2 −0.473387
\(782\) − 9509.28i − 0.434848i
\(783\) 4025.51i 0.183729i
\(784\) 2718.92 0.123857
\(785\) 0 0
\(786\) 1465.53 0.0665062
\(787\) − 29454.3i − 1.33410i −0.745015 0.667048i \(-0.767558\pi\)
0.745015 0.667048i \(-0.232442\pi\)
\(788\) 19337.8i 0.874216i
\(789\) −6778.96 −0.305878
\(790\) 0 0
\(791\) −1741.86 −0.0782974
\(792\) 2096.53i 0.0940619i
\(793\) − 29202.7i − 1.30771i
\(794\) 8157.00 0.364586
\(795\) 0 0
\(796\) −22218.6 −0.989344
\(797\) − 27440.3i − 1.21955i −0.792573 0.609777i \(-0.791259\pi\)
0.792573 0.609777i \(-0.208741\pi\)
\(798\) − 1903.00i − 0.0844179i
\(799\) −32899.0 −1.45668
\(800\) 0 0
\(801\) 3472.91 0.153195
\(802\) 15066.1i 0.663346i
\(803\) − 3988.27i − 0.175272i
\(804\) −6757.03 −0.296396
\(805\) 0 0
\(806\) −3494.05 −0.152695
\(807\) 21118.6i 0.921201i
\(808\) 7382.34i 0.321423i
\(809\) 5060.18 0.219909 0.109954 0.993937i \(-0.464930\pi\)
0.109954 + 0.993937i \(0.464930\pi\)
\(810\) 0 0
\(811\) 30480.1 1.31973 0.659865 0.751384i \(-0.270613\pi\)
0.659865 + 0.751384i \(0.270613\pi\)
\(812\) − 8496.03i − 0.367183i
\(813\) − 15465.2i − 0.667146i
\(814\) 771.146 0.0332048
\(815\) 0 0
\(816\) −3382.72 −0.145121
\(817\) 93.8596i 0.00401925i
\(818\) 3139.46i 0.134192i
\(819\) −3769.50 −0.160827
\(820\) 0 0
\(821\) 37909.0 1.61149 0.805745 0.592263i \(-0.201765\pi\)
0.805745 + 0.592263i \(0.201765\pi\)
\(822\) − 3358.56i − 0.142510i
\(823\) − 23636.0i − 1.00109i −0.865710 0.500546i \(-0.833133\pi\)
0.865710 0.500546i \(-0.166867\pi\)
\(824\) −32536.0 −1.37554
\(825\) 0 0
\(826\) 5932.36 0.249895
\(827\) − 42634.3i − 1.79267i −0.443376 0.896336i \(-0.646219\pi\)
0.443376 0.896336i \(-0.353781\pi\)
\(828\) 3087.99i 0.129608i
\(829\) 45152.5 1.89169 0.945845 0.324619i \(-0.105236\pi\)
0.945845 + 0.324619i \(0.105236\pi\)
\(830\) 0 0
\(831\) 27223.6 1.13643
\(832\) − 8217.15i − 0.342402i
\(833\) − 23494.1i − 0.977217i
\(834\) 4106.02 0.170480
\(835\) 0 0
\(836\) −2425.42 −0.100341
\(837\) 1477.94i 0.0610337i
\(838\) − 6925.91i − 0.285503i
\(839\) −30431.5 −1.25222 −0.626110 0.779734i \(-0.715354\pi\)
−0.626110 + 0.779734i \(0.715354\pi\)
\(840\) 0 0
\(841\) −2160.34 −0.0885784
\(842\) 23763.6i 0.972623i
\(843\) 10223.4i 0.417689i
\(844\) 5818.09 0.237283
\(845\) 0 0
\(846\) −4684.13 −0.190359
\(847\) − 1239.79i − 0.0502949i
\(848\) − 7311.64i − 0.296088i
\(849\) 26483.2 1.07055
\(850\) 0 0
\(851\) 2769.65 0.111566
\(852\) − 15671.8i − 0.630171i
\(853\) − 10367.2i − 0.416139i −0.978114 0.208070i \(-0.933282\pi\)
0.978114 0.208070i \(-0.0667180\pi\)
\(854\) −11430.5 −0.458013
\(855\) 0 0
\(856\) 16500.8 0.658860
\(857\) 12947.1i 0.516063i 0.966136 + 0.258032i \(0.0830739\pi\)
−0.966136 + 0.258032i \(0.916926\pi\)
\(858\) − 2106.44i − 0.0838142i
\(859\) 20383.5 0.809636 0.404818 0.914397i \(-0.367335\pi\)
0.404818 + 0.914397i \(0.367335\pi\)
\(860\) 0 0
\(861\) −10342.9 −0.409391
\(862\) − 8779.22i − 0.346893i
\(863\) − 9056.42i − 0.357224i −0.983920 0.178612i \(-0.942839\pi\)
0.983920 0.178612i \(-0.0571606\pi\)
\(864\) −5055.88 −0.199079
\(865\) 0 0
\(866\) −22340.0 −0.876610
\(867\) 14491.0i 0.567636i
\(868\) − 3119.27i − 0.121976i
\(869\) 10470.2 0.408719
\(870\) 0 0
\(871\) 16554.5 0.644005
\(872\) 31803.6i 1.23510i
\(873\) 8697.10i 0.337173i
\(874\) 3819.37 0.147817
\(875\) 0 0
\(876\) 6049.36 0.233321
\(877\) − 2867.88i − 0.110424i −0.998475 0.0552118i \(-0.982417\pi\)
0.998475 0.0552118i \(-0.0175834\pi\)
\(878\) − 6846.16i − 0.263151i
\(879\) 13584.9 0.521281
\(880\) 0 0
\(881\) −11862.5 −0.453640 −0.226820 0.973937i \(-0.572833\pi\)
−0.226820 + 0.973937i \(0.572833\pi\)
\(882\) − 3345.06i − 0.127703i
\(883\) − 33463.8i − 1.27537i −0.770299 0.637683i \(-0.779893\pi\)
0.770299 0.637683i \(-0.220107\pi\)
\(884\) 22440.3 0.853787
\(885\) 0 0
\(886\) −15756.0 −0.597442
\(887\) 2420.75i 0.0916357i 0.998950 + 0.0458178i \(0.0145894\pi\)
−0.998950 + 0.0458178i \(0.985411\pi\)
\(888\) 2852.17i 0.107784i
\(889\) −17826.5 −0.672534
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 2815.62i 0.105688i
\(893\) − 13213.8i − 0.495165i
\(894\) 11132.4 0.416470
\(895\) 0 0
\(896\) 12132.9 0.452378
\(897\) − 7565.48i − 0.281610i
\(898\) − 14963.6i − 0.556060i
\(899\) 8161.14 0.302769
\(900\) 0 0
\(901\) −63179.7 −2.33609
\(902\) − 5779.73i − 0.213352i
\(903\) − 72.7722i − 0.00268185i
\(904\) 3600.10 0.132453
\(905\) 0 0
\(906\) −434.955 −0.0159497
\(907\) − 38154.1i − 1.39679i −0.715714 0.698393i \(-0.753899\pi\)
0.715714 0.698393i \(-0.246101\pi\)
\(908\) − 23838.4i − 0.871262i
\(909\) −3137.40 −0.114479
\(910\) 0 0
\(911\) −35758.0 −1.30045 −0.650227 0.759740i \(-0.725326\pi\)
−0.650227 + 0.759740i \(0.725326\pi\)
\(912\) − 1358.66i − 0.0493308i
\(913\) 8087.44i 0.293160i
\(914\) −15614.5 −0.565079
\(915\) 0 0
\(916\) −31753.0 −1.14536
\(917\) − 3205.39i − 0.115432i
\(918\) 4161.73i 0.149627i
\(919\) −17387.5 −0.624115 −0.312058 0.950063i \(-0.601018\pi\)
−0.312058 + 0.950063i \(0.601018\pi\)
\(920\) 0 0
\(921\) 1704.32 0.0609764
\(922\) − 17357.9i − 0.620013i
\(923\) 38395.3i 1.36923i
\(924\) 1880.50 0.0669523
\(925\) 0 0
\(926\) −2447.20 −0.0868467
\(927\) − 13827.4i − 0.489915i
\(928\) 27918.3i 0.987569i
\(929\) −6955.93 −0.245658 −0.122829 0.992428i \(-0.539197\pi\)
−0.122829 + 0.992428i \(0.539197\pi\)
\(930\) 0 0
\(931\) 9436.31 0.332183
\(932\) − 16384.9i − 0.575863i
\(933\) 20560.8i 0.721467i
\(934\) −19751.7 −0.691965
\(935\) 0 0
\(936\) 7790.88 0.272065
\(937\) − 16074.5i − 0.560438i −0.959936 0.280219i \(-0.909593\pi\)
0.959936 0.280219i \(-0.0904070\pi\)
\(938\) − 6479.76i − 0.225556i
\(939\) 3416.77 0.118746
\(940\) 0 0
\(941\) −687.126 −0.0238041 −0.0119021 0.999929i \(-0.503789\pi\)
−0.0119021 + 0.999929i \(0.503789\pi\)
\(942\) − 8812.96i − 0.304821i
\(943\) − 20758.5i − 0.716849i
\(944\) 4235.44 0.146030
\(945\) 0 0
\(946\) 40.6659 0.00139763
\(947\) 35352.0i 1.21308i 0.795053 + 0.606540i \(0.207443\pi\)
−0.795053 + 0.606540i \(0.792557\pi\)
\(948\) 15881.0i 0.544085i
\(949\) −14820.7 −0.506956
\(950\) 0 0
\(951\) 9622.44 0.328106
\(952\) − 21418.2i − 0.729168i
\(953\) 19390.7i 0.659103i 0.944138 + 0.329552i \(0.106898\pi\)
−0.944138 + 0.329552i \(0.893102\pi\)
\(954\) −8995.45 −0.305281
\(955\) 0 0
\(956\) 11561.9 0.391148
\(957\) 4920.06i 0.166189i
\(958\) 16738.7i 0.564511i
\(959\) −7345.80 −0.247349
\(960\) 0 0
\(961\) −26794.7 −0.899422
\(962\) − 2865.64i − 0.0960416i
\(963\) 7012.62i 0.234661i
\(964\) 10307.6 0.344384
\(965\) 0 0
\(966\) −2961.27 −0.0986308
\(967\) 28643.6i 0.952551i 0.879296 + 0.476275i \(0.158013\pi\)
−0.879296 + 0.476275i \(0.841987\pi\)
\(968\) 2562.43i 0.0850821i
\(969\) −11740.1 −0.389213
\(970\) 0 0
\(971\) −19574.8 −0.646946 −0.323473 0.946237i \(-0.604851\pi\)
−0.323473 + 0.946237i \(0.604851\pi\)
\(972\) − 1351.46i − 0.0445967i
\(973\) − 8980.63i − 0.295895i
\(974\) −11182.6 −0.367878
\(975\) 0 0
\(976\) −8160.86 −0.267646
\(977\) 50095.5i 1.64043i 0.572058 + 0.820213i \(0.306145\pi\)
−0.572058 + 0.820213i \(0.693855\pi\)
\(978\) 11550.0i 0.377636i
\(979\) 4244.67 0.138570
\(980\) 0 0
\(981\) −13516.1 −0.439895
\(982\) 22747.6i 0.739211i
\(983\) − 14445.4i − 0.468706i −0.972152 0.234353i \(-0.924703\pi\)
0.972152 0.234353i \(-0.0752971\pi\)
\(984\) 21376.9 0.692553
\(985\) 0 0
\(986\) 22980.9 0.742253
\(987\) 10245.0i 0.330399i
\(988\) 9013.05i 0.290226i
\(989\) 146.056 0.00469595
\(990\) 0 0
\(991\) 29120.1 0.933430 0.466715 0.884408i \(-0.345437\pi\)
0.466715 + 0.884408i \(0.345437\pi\)
\(992\) 10250.1i 0.328064i
\(993\) 27405.4i 0.875814i
\(994\) 15028.7 0.479558
\(995\) 0 0
\(996\) −12266.9 −0.390253
\(997\) − 9137.45i − 0.290257i −0.989413 0.145128i \(-0.953640\pi\)
0.989413 0.145128i \(-0.0463595\pi\)
\(998\) 7244.03i 0.229765i
\(999\) −1212.14 −0.0383887
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 825.4.c.j.199.3 4
5.2 odd 4 825.4.a.m.1.1 2
5.3 odd 4 165.4.a.c.1.2 2
5.4 even 2 inner 825.4.c.j.199.2 4
15.2 even 4 2475.4.a.n.1.2 2
15.8 even 4 495.4.a.d.1.1 2
55.43 even 4 1815.4.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 5.3 odd 4
495.4.a.d.1.1 2 15.8 even 4
825.4.a.m.1.1 2 5.2 odd 4
825.4.c.j.199.2 4 5.4 even 2 inner
825.4.c.j.199.3 4 1.1 even 1 trivial
1815.4.a.n.1.1 2 55.43 even 4
2475.4.a.n.1.2 2 15.2 even 4