Properties

Label 1521.4.a.l.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.438447 q^{2} -7.80776 q^{4} -17.8078 q^{5} -5.43845 q^{7} +6.93087 q^{8} +O(q^{10})\) \(q-0.438447 q^{2} -7.80776 q^{4} -17.8078 q^{5} -5.43845 q^{7} +6.93087 q^{8} +7.80776 q^{10} -22.4233 q^{11} +2.38447 q^{14} +59.4233 q^{16} -67.9848 q^{17} +80.8078 q^{19} +139.039 q^{20} +9.83143 q^{22} -140.531 q^{23} +192.116 q^{25} +42.4621 q^{28} +106.693 q^{29} +276.155 q^{31} -81.5009 q^{32} +29.8078 q^{34} +96.8466 q^{35} +4.29168 q^{37} -35.4299 q^{38} -123.423 q^{40} +227.769 q^{41} +27.5294 q^{43} +175.076 q^{44} +61.6155 q^{46} +318.617 q^{47} -313.423 q^{49} -84.2329 q^{50} +67.6562 q^{53} +399.309 q^{55} -37.6932 q^{56} -46.7793 q^{58} -291.115 q^{59} +663.311 q^{61} -121.080 q^{62} -439.652 q^{64} +425.101 q^{67} +530.810 q^{68} -42.4621 q^{70} -152.963 q^{71} -117.268 q^{73} -1.88167 q^{74} -630.928 q^{76} +121.948 q^{77} +202.462 q^{79} -1058.20 q^{80} -99.8647 q^{82} +336.155 q^{83} +1210.66 q^{85} -12.0702 q^{86} -155.413 q^{88} +718.194 q^{89} +1097.23 q^{92} -139.697 q^{94} -1439.01 q^{95} -759.368 q^{97} +137.420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8} - 5 q^{10} + 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} + 141 q^{19} + 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} - 80 q^{28} - 34 q^{29} + 140 q^{31} + 105 q^{32} + 39 q^{34} + 70 q^{35} + 190 q^{37} - 310 q^{38} - 185 q^{40} + 538 q^{41} + 455 q^{43} + 680 q^{44} + 82 q^{46} + 60 q^{47} - 565 q^{49} + 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} + 595 q^{58} - 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} + 475 q^{67} + 505 q^{68} + 80 q^{70} + 127 q^{71} - 585 q^{73} - 849 q^{74} + 140 q^{76} - 255 q^{77} + 240 q^{79} - 1065 q^{80} - 1515 q^{82} + 260 q^{83} + 1205 q^{85} - 1962 q^{86} - 1020 q^{88} + 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.438447 −0.155014 −0.0775072 0.996992i \(-0.524696\pi\)
−0.0775072 + 0.996992i \(0.524696\pi\)
\(3\) 0 0
\(4\) −7.80776 −0.975971
\(5\) −17.8078 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) −5.43845 −0.293649 −0.146824 0.989163i \(-0.546905\pi\)
−0.146824 + 0.989163i \(0.546905\pi\)
\(8\) 6.93087 0.306304
\(9\) 0 0
\(10\) 7.80776 0.246903
\(11\) −22.4233 −0.614625 −0.307313 0.951609i \(-0.599430\pi\)
−0.307313 + 0.951609i \(0.599430\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 2.38447 0.0455198
\(15\) 0 0
\(16\) 59.4233 0.928489
\(17\) −67.9848 −0.969926 −0.484963 0.874535i \(-0.661167\pi\)
−0.484963 + 0.874535i \(0.661167\pi\)
\(18\) 0 0
\(19\) 80.8078 0.975714 0.487857 0.872923i \(-0.337779\pi\)
0.487857 + 0.872923i \(0.337779\pi\)
\(20\) 139.039 1.55450
\(21\) 0 0
\(22\) 9.83143 0.0952758
\(23\) −140.531 −1.27403 −0.637017 0.770850i \(-0.719832\pi\)
−0.637017 + 0.770850i \(0.719832\pi\)
\(24\) 0 0
\(25\) 192.116 1.53693
\(26\) 0 0
\(27\) 0 0
\(28\) 42.4621 0.286592
\(29\) 106.693 0.683187 0.341594 0.939848i \(-0.389033\pi\)
0.341594 + 0.939848i \(0.389033\pi\)
\(30\) 0 0
\(31\) 276.155 1.59997 0.799983 0.600023i \(-0.204842\pi\)
0.799983 + 0.600023i \(0.204842\pi\)
\(32\) −81.5009 −0.450233
\(33\) 0 0
\(34\) 29.8078 0.150353
\(35\) 96.8466 0.467716
\(36\) 0 0
\(37\) 4.29168 0.0190688 0.00953442 0.999955i \(-0.496965\pi\)
0.00953442 + 0.999955i \(0.496965\pi\)
\(38\) −35.4299 −0.151250
\(39\) 0 0
\(40\) −123.423 −0.487873
\(41\) 227.769 0.867598 0.433799 0.901010i \(-0.357173\pi\)
0.433799 + 0.901010i \(0.357173\pi\)
\(42\) 0 0
\(43\) 27.5294 0.0976323 0.0488162 0.998808i \(-0.484455\pi\)
0.0488162 + 0.998808i \(0.484455\pi\)
\(44\) 175.076 0.599856
\(45\) 0 0
\(46\) 61.6155 0.197494
\(47\) 318.617 0.988832 0.494416 0.869225i \(-0.335382\pi\)
0.494416 + 0.869225i \(0.335382\pi\)
\(48\) 0 0
\(49\) −313.423 −0.913771
\(50\) −84.2329 −0.238247
\(51\) 0 0
\(52\) 0 0
\(53\) 67.6562 0.175345 0.0876726 0.996149i \(-0.472057\pi\)
0.0876726 + 0.996149i \(0.472057\pi\)
\(54\) 0 0
\(55\) 399.309 0.978960
\(56\) −37.6932 −0.0899457
\(57\) 0 0
\(58\) −46.7793 −0.105904
\(59\) −291.115 −0.642371 −0.321186 0.947016i \(-0.604081\pi\)
−0.321186 + 0.947016i \(0.604081\pi\)
\(60\) 0 0
\(61\) 663.311 1.39227 0.696133 0.717913i \(-0.254902\pi\)
0.696133 + 0.717913i \(0.254902\pi\)
\(62\) −121.080 −0.248018
\(63\) 0 0
\(64\) −439.652 −0.858696
\(65\) 0 0
\(66\) 0 0
\(67\) 425.101 0.775140 0.387570 0.921840i \(-0.373315\pi\)
0.387570 + 0.921840i \(0.373315\pi\)
\(68\) 530.810 0.946619
\(69\) 0 0
\(70\) −42.4621 −0.0725028
\(71\) −152.963 −0.255681 −0.127841 0.991795i \(-0.540805\pi\)
−0.127841 + 0.991795i \(0.540805\pi\)
\(72\) 0 0
\(73\) −117.268 −0.188016 −0.0940081 0.995571i \(-0.529968\pi\)
−0.0940081 + 0.995571i \(0.529968\pi\)
\(74\) −1.88167 −0.00295595
\(75\) 0 0
\(76\) −630.928 −0.952268
\(77\) 121.948 0.180484
\(78\) 0 0
\(79\) 202.462 0.288339 0.144169 0.989553i \(-0.453949\pi\)
0.144169 + 0.989553i \(0.453949\pi\)
\(80\) −1058.20 −1.47887
\(81\) 0 0
\(82\) −99.8647 −0.134490
\(83\) 336.155 0.444552 0.222276 0.974984i \(-0.428651\pi\)
0.222276 + 0.974984i \(0.428651\pi\)
\(84\) 0 0
\(85\) 1210.66 1.54487
\(86\) −12.0702 −0.0151344
\(87\) 0 0
\(88\) −155.413 −0.188262
\(89\) 718.194 0.855376 0.427688 0.903927i \(-0.359328\pi\)
0.427688 + 0.903927i \(0.359328\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1097.23 1.24342
\(93\) 0 0
\(94\) −139.697 −0.153283
\(95\) −1439.01 −1.55409
\(96\) 0 0
\(97\) −759.368 −0.794868 −0.397434 0.917631i \(-0.630099\pi\)
−0.397434 + 0.917631i \(0.630099\pi\)
\(98\) 137.420 0.141648
\(99\) 0 0
\(100\) −1500.00 −1.50000
\(101\) 348.697 0.343531 0.171766 0.985138i \(-0.445053\pi\)
0.171766 + 0.985138i \(0.445053\pi\)
\(102\) 0 0
\(103\) −580.303 −0.555136 −0.277568 0.960706i \(-0.589528\pi\)
−0.277568 + 0.960706i \(0.589528\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −29.6637 −0.0271810
\(107\) −571.493 −0.516340 −0.258170 0.966100i \(-0.583119\pi\)
−0.258170 + 0.966100i \(0.583119\pi\)
\(108\) 0 0
\(109\) −176.004 −0.154661 −0.0773307 0.997005i \(-0.524640\pi\)
−0.0773307 + 0.997005i \(0.524640\pi\)
\(110\) −175.076 −0.151753
\(111\) 0 0
\(112\) −323.170 −0.272649
\(113\) −1264.88 −1.05301 −0.526505 0.850172i \(-0.676498\pi\)
−0.526505 + 0.850172i \(0.676498\pi\)
\(114\) 0 0
\(115\) 2502.55 2.02925
\(116\) −833.035 −0.666770
\(117\) 0 0
\(118\) 127.638 0.0995768
\(119\) 369.732 0.284817
\(120\) 0 0
\(121\) −828.196 −0.622236
\(122\) −290.827 −0.215821
\(123\) 0 0
\(124\) −2156.16 −1.56152
\(125\) −1195.19 −0.855211
\(126\) 0 0
\(127\) 2604.11 1.81950 0.909752 0.415151i \(-0.136271\pi\)
0.909752 + 0.415151i \(0.136271\pi\)
\(128\) 844.772 0.583344
\(129\) 0 0
\(130\) 0 0
\(131\) −2131.70 −1.42174 −0.710870 0.703324i \(-0.751698\pi\)
−0.710870 + 0.703324i \(0.751698\pi\)
\(132\) 0 0
\(133\) −439.469 −0.286517
\(134\) −186.384 −0.120158
\(135\) 0 0
\(136\) −471.194 −0.297092
\(137\) 687.985 0.429040 0.214520 0.976720i \(-0.431181\pi\)
0.214520 + 0.976720i \(0.431181\pi\)
\(138\) 0 0
\(139\) −679.580 −0.414685 −0.207343 0.978268i \(-0.566482\pi\)
−0.207343 + 0.978268i \(0.566482\pi\)
\(140\) −756.155 −0.456477
\(141\) 0 0
\(142\) 67.0662 0.0396343
\(143\) 0 0
\(144\) 0 0
\(145\) −1899.97 −1.08816
\(146\) 51.4158 0.0291452
\(147\) 0 0
\(148\) −33.5084 −0.0186106
\(149\) −1975.46 −1.08615 −0.543074 0.839685i \(-0.682740\pi\)
−0.543074 + 0.839685i \(0.682740\pi\)
\(150\) 0 0
\(151\) −1803.24 −0.971824 −0.485912 0.874008i \(-0.661513\pi\)
−0.485912 + 0.874008i \(0.661513\pi\)
\(152\) 560.068 0.298865
\(153\) 0 0
\(154\) −53.4677 −0.0279776
\(155\) −4917.71 −2.54839
\(156\) 0 0
\(157\) −397.168 −0.201894 −0.100947 0.994892i \(-0.532187\pi\)
−0.100947 + 0.994892i \(0.532187\pi\)
\(158\) −88.7689 −0.0446967
\(159\) 0 0
\(160\) 1451.35 0.717120
\(161\) 764.272 0.374118
\(162\) 0 0
\(163\) −941.393 −0.452365 −0.226183 0.974085i \(-0.572625\pi\)
−0.226183 + 0.974085i \(0.572625\pi\)
\(164\) −1778.37 −0.846750
\(165\) 0 0
\(166\) −147.386 −0.0689120
\(167\) 3680.43 1.70539 0.852696 0.522408i \(-0.174966\pi\)
0.852696 + 0.522408i \(0.174966\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −530.810 −0.239478
\(171\) 0 0
\(172\) −214.943 −0.0952863
\(173\) −1422.77 −0.625269 −0.312634 0.949874i \(-0.601211\pi\)
−0.312634 + 0.949874i \(0.601211\pi\)
\(174\) 0 0
\(175\) −1044.82 −0.451318
\(176\) −1332.47 −0.570673
\(177\) 0 0
\(178\) −314.890 −0.132596
\(179\) −1167.89 −0.487666 −0.243833 0.969817i \(-0.578405\pi\)
−0.243833 + 0.969817i \(0.578405\pi\)
\(180\) 0 0
\(181\) −1133.96 −0.465673 −0.232836 0.972516i \(-0.574801\pi\)
−0.232836 + 0.972516i \(0.574801\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −974.004 −0.390242
\(185\) −76.4252 −0.0303724
\(186\) 0 0
\(187\) 1524.44 0.596141
\(188\) −2487.69 −0.965071
\(189\) 0 0
\(190\) 630.928 0.240907
\(191\) −2682.12 −1.01608 −0.508040 0.861333i \(-0.669630\pi\)
−0.508040 + 0.861333i \(0.669630\pi\)
\(192\) 0 0
\(193\) 1970.67 0.734983 0.367491 0.930027i \(-0.380217\pi\)
0.367491 + 0.930027i \(0.380217\pi\)
\(194\) 332.943 0.123216
\(195\) 0 0
\(196\) 2447.14 0.891813
\(197\) −4016.05 −1.45244 −0.726222 0.687460i \(-0.758726\pi\)
−0.726222 + 0.687460i \(0.758726\pi\)
\(198\) 0 0
\(199\) −4226.06 −1.50541 −0.752707 0.658356i \(-0.771252\pi\)
−0.752707 + 0.658356i \(0.771252\pi\)
\(200\) 1331.53 0.470768
\(201\) 0 0
\(202\) −152.885 −0.0532523
\(203\) −580.245 −0.200617
\(204\) 0 0
\(205\) −4056.06 −1.38189
\(206\) 254.432 0.0860541
\(207\) 0 0
\(208\) 0 0
\(209\) −1811.98 −0.599699
\(210\) 0 0
\(211\) 1364.67 0.445249 0.222625 0.974904i \(-0.428538\pi\)
0.222625 + 0.974904i \(0.428538\pi\)
\(212\) −528.244 −0.171132
\(213\) 0 0
\(214\) 250.570 0.0800401
\(215\) −490.237 −0.155506
\(216\) 0 0
\(217\) −1501.86 −0.469828
\(218\) 77.1683 0.0239748
\(219\) 0 0
\(220\) −3117.71 −0.955436
\(221\) 0 0
\(222\) 0 0
\(223\) 1059.47 0.318149 0.159075 0.987267i \(-0.449149\pi\)
0.159075 + 0.987267i \(0.449149\pi\)
\(224\) 443.239 0.132210
\(225\) 0 0
\(226\) 554.584 0.163232
\(227\) 3464.19 1.01289 0.506446 0.862272i \(-0.330959\pi\)
0.506446 + 0.862272i \(0.330959\pi\)
\(228\) 0 0
\(229\) 2324.64 0.670815 0.335407 0.942073i \(-0.391126\pi\)
0.335407 + 0.942073i \(0.391126\pi\)
\(230\) −1097.23 −0.314563
\(231\) 0 0
\(232\) 739.476 0.209263
\(233\) 3731.01 1.04904 0.524521 0.851398i \(-0.324245\pi\)
0.524521 + 0.851398i \(0.324245\pi\)
\(234\) 0 0
\(235\) −5673.86 −1.57499
\(236\) 2272.95 0.626935
\(237\) 0 0
\(238\) −162.108 −0.0441508
\(239\) 6044.47 1.63592 0.817958 0.575278i \(-0.195106\pi\)
0.817958 + 0.575278i \(0.195106\pi\)
\(240\) 0 0
\(241\) 5173.96 1.38292 0.691461 0.722414i \(-0.256967\pi\)
0.691461 + 0.722414i \(0.256967\pi\)
\(242\) 363.120 0.0964556
\(243\) 0 0
\(244\) −5178.97 −1.35881
\(245\) 5581.37 1.45543
\(246\) 0 0
\(247\) 0 0
\(248\) 1914.00 0.490076
\(249\) 0 0
\(250\) 524.029 0.132570
\(251\) −5620.73 −1.41346 −0.706728 0.707486i \(-0.749829\pi\)
−0.706728 + 0.707486i \(0.749829\pi\)
\(252\) 0 0
\(253\) 3151.17 0.783054
\(254\) −1141.76 −0.282050
\(255\) 0 0
\(256\) 3146.83 0.768270
\(257\) 1674.14 0.406342 0.203171 0.979143i \(-0.434875\pi\)
0.203171 + 0.979143i \(0.434875\pi\)
\(258\) 0 0
\(259\) −23.3401 −0.00559954
\(260\) 0 0
\(261\) 0 0
\(262\) 934.640 0.220390
\(263\) 6309.18 1.47924 0.739622 0.673023i \(-0.235004\pi\)
0.739622 + 0.673023i \(0.235004\pi\)
\(264\) 0 0
\(265\) −1204.81 −0.279285
\(266\) 192.684 0.0444143
\(267\) 0 0
\(268\) −3319.09 −0.756514
\(269\) 2482.73 0.562731 0.281366 0.959601i \(-0.409213\pi\)
0.281366 + 0.959601i \(0.409213\pi\)
\(270\) 0 0
\(271\) −2835.72 −0.635638 −0.317819 0.948151i \(-0.602950\pi\)
−0.317819 + 0.948151i \(0.602950\pi\)
\(272\) −4039.88 −0.900566
\(273\) 0 0
\(274\) −301.645 −0.0665075
\(275\) −4307.88 −0.944637
\(276\) 0 0
\(277\) −3837.51 −0.832396 −0.416198 0.909274i \(-0.636638\pi\)
−0.416198 + 0.909274i \(0.636638\pi\)
\(278\) 297.960 0.0642822
\(279\) 0 0
\(280\) 671.231 0.143263
\(281\) −9122.13 −1.93659 −0.968293 0.249819i \(-0.919629\pi\)
−0.968293 + 0.249819i \(0.919629\pi\)
\(282\) 0 0
\(283\) 2127.85 0.446952 0.223476 0.974709i \(-0.428260\pi\)
0.223476 + 0.974709i \(0.428260\pi\)
\(284\) 1194.30 0.249537
\(285\) 0 0
\(286\) 0 0
\(287\) −1238.71 −0.254769
\(288\) 0 0
\(289\) −291.061 −0.0592430
\(290\) 833.035 0.168681
\(291\) 0 0
\(292\) 915.601 0.183498
\(293\) 8274.77 1.64989 0.824944 0.565215i \(-0.191207\pi\)
0.824944 + 0.565215i \(0.191207\pi\)
\(294\) 0 0
\(295\) 5184.10 1.02315
\(296\) 29.7450 0.00584086
\(297\) 0 0
\(298\) 866.136 0.168369
\(299\) 0 0
\(300\) 0 0
\(301\) −149.717 −0.0286696
\(302\) 790.625 0.150647
\(303\) 0 0
\(304\) 4801.86 0.905940
\(305\) −11812.1 −2.21757
\(306\) 0 0
\(307\) 3610.49 0.671211 0.335605 0.942003i \(-0.391059\pi\)
0.335605 + 0.942003i \(0.391059\pi\)
\(308\) −952.140 −0.176147
\(309\) 0 0
\(310\) 2156.16 0.395037
\(311\) −3331.06 −0.607354 −0.303677 0.952775i \(-0.598214\pi\)
−0.303677 + 0.952775i \(0.598214\pi\)
\(312\) 0 0
\(313\) −358.125 −0.0646724 −0.0323362 0.999477i \(-0.510295\pi\)
−0.0323362 + 0.999477i \(0.510295\pi\)
\(314\) 174.137 0.0312966
\(315\) 0 0
\(316\) −1580.78 −0.281410
\(317\) 3047.46 0.539944 0.269972 0.962868i \(-0.412986\pi\)
0.269972 + 0.962868i \(0.412986\pi\)
\(318\) 0 0
\(319\) −2392.41 −0.419904
\(320\) 7829.23 1.36771
\(321\) 0 0
\(322\) −335.093 −0.0579938
\(323\) −5493.70 −0.946371
\(324\) 0 0
\(325\) 0 0
\(326\) 412.751 0.0701232
\(327\) 0 0
\(328\) 1578.64 0.265749
\(329\) −1732.78 −0.290369
\(330\) 0 0
\(331\) −7694.77 −1.27777 −0.638887 0.769301i \(-0.720605\pi\)
−0.638887 + 0.769301i \(0.720605\pi\)
\(332\) −2624.62 −0.433870
\(333\) 0 0
\(334\) −1613.68 −0.264360
\(335\) −7570.10 −1.23462
\(336\) 0 0
\(337\) 4712.21 0.761693 0.380846 0.924638i \(-0.375633\pi\)
0.380846 + 0.924638i \(0.375633\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −9452.53 −1.50775
\(341\) −6192.31 −0.983380
\(342\) 0 0
\(343\) 3569.92 0.561976
\(344\) 190.803 0.0299052
\(345\) 0 0
\(346\) 623.811 0.0969257
\(347\) 5261.98 0.814058 0.407029 0.913415i \(-0.366565\pi\)
0.407029 + 0.913415i \(0.366565\pi\)
\(348\) 0 0
\(349\) −50.3345 −0.00772018 −0.00386009 0.999993i \(-0.501229\pi\)
−0.00386009 + 0.999993i \(0.501229\pi\)
\(350\) 458.096 0.0699608
\(351\) 0 0
\(352\) 1827.52 0.276725
\(353\) 9057.64 1.36569 0.682846 0.730562i \(-0.260742\pi\)
0.682846 + 0.730562i \(0.260742\pi\)
\(354\) 0 0
\(355\) 2723.93 0.407243
\(356\) −5607.49 −0.834821
\(357\) 0 0
\(358\) 512.059 0.0755953
\(359\) 7177.86 1.05525 0.527623 0.849479i \(-0.323083\pi\)
0.527623 + 0.849479i \(0.323083\pi\)
\(360\) 0 0
\(361\) −329.105 −0.0479815
\(362\) 497.183 0.0721861
\(363\) 0 0
\(364\) 0 0
\(365\) 2088.28 0.299467
\(366\) 0 0
\(367\) 4004.14 0.569522 0.284761 0.958599i \(-0.408086\pi\)
0.284761 + 0.958599i \(0.408086\pi\)
\(368\) −8350.83 −1.18293
\(369\) 0 0
\(370\) 33.5084 0.00470816
\(371\) −367.945 −0.0514899
\(372\) 0 0
\(373\) −10014.2 −1.39012 −0.695060 0.718952i \(-0.744622\pi\)
−0.695060 + 0.718952i \(0.744622\pi\)
\(374\) −668.388 −0.0924105
\(375\) 0 0
\(376\) 2208.30 0.302883
\(377\) 0 0
\(378\) 0 0
\(379\) 8169.12 1.10717 0.553587 0.832791i \(-0.313258\pi\)
0.553587 + 0.832791i \(0.313258\pi\)
\(380\) 11235.4 1.51675
\(381\) 0 0
\(382\) 1175.97 0.157507
\(383\) −7310.25 −0.975290 −0.487645 0.873042i \(-0.662144\pi\)
−0.487645 + 0.873042i \(0.662144\pi\)
\(384\) 0 0
\(385\) −2171.62 −0.287470
\(386\) −864.033 −0.113933
\(387\) 0 0
\(388\) 5928.97 0.775767
\(389\) −8785.47 −1.14509 −0.572546 0.819872i \(-0.694044\pi\)
−0.572546 + 0.819872i \(0.694044\pi\)
\(390\) 0 0
\(391\) 9553.99 1.23572
\(392\) −2172.30 −0.279892
\(393\) 0 0
\(394\) 1760.82 0.225150
\(395\) −3605.40 −0.459259
\(396\) 0 0
\(397\) −11266.8 −1.42434 −0.712171 0.702006i \(-0.752288\pi\)
−0.712171 + 0.702006i \(0.752288\pi\)
\(398\) 1852.90 0.233361
\(399\) 0 0
\(400\) 11416.2 1.42702
\(401\) 1576.23 0.196293 0.0981464 0.995172i \(-0.468709\pi\)
0.0981464 + 0.995172i \(0.468709\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −2722.54 −0.335276
\(405\) 0 0
\(406\) 254.407 0.0310985
\(407\) −96.2335 −0.0117202
\(408\) 0 0
\(409\) −6755.78 −0.816753 −0.408377 0.912814i \(-0.633905\pi\)
−0.408377 + 0.912814i \(0.633905\pi\)
\(410\) 1778.37 0.214213
\(411\) 0 0
\(412\) 4530.87 0.541796
\(413\) 1583.21 0.188631
\(414\) 0 0
\(415\) −5986.17 −0.708072
\(416\) 0 0
\(417\) 0 0
\(418\) 794.456 0.0929620
\(419\) −10756.2 −1.25411 −0.627057 0.778973i \(-0.715741\pi\)
−0.627057 + 0.778973i \(0.715741\pi\)
\(420\) 0 0
\(421\) −7886.03 −0.912925 −0.456463 0.889743i \(-0.650884\pi\)
−0.456463 + 0.889743i \(0.650884\pi\)
\(422\) −598.335 −0.0690201
\(423\) 0 0
\(424\) 468.916 0.0537089
\(425\) −13061.0 −1.49071
\(426\) 0 0
\(427\) −3607.38 −0.408837
\(428\) 4462.08 0.503932
\(429\) 0 0
\(430\) 214.943 0.0241057
\(431\) −14084.6 −1.57409 −0.787044 0.616897i \(-0.788390\pi\)
−0.787044 + 0.616897i \(0.788390\pi\)
\(432\) 0 0
\(433\) 1864.14 0.206894 0.103447 0.994635i \(-0.467013\pi\)
0.103447 + 0.994635i \(0.467013\pi\)
\(434\) 658.485 0.0728301
\(435\) 0 0
\(436\) 1374.20 0.150945
\(437\) −11356.0 −1.24309
\(438\) 0 0
\(439\) 6154.49 0.669106 0.334553 0.942377i \(-0.391415\pi\)
0.334553 + 0.942377i \(0.391415\pi\)
\(440\) 2767.56 0.299859
\(441\) 0 0
\(442\) 0 0
\(443\) 14539.3 1.55933 0.779663 0.626200i \(-0.215391\pi\)
0.779663 + 0.626200i \(0.215391\pi\)
\(444\) 0 0
\(445\) −12789.4 −1.36242
\(446\) −464.521 −0.0493177
\(447\) 0 0
\(448\) 2391.03 0.252155
\(449\) −7043.87 −0.740358 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(450\) 0 0
\(451\) −5107.33 −0.533248
\(452\) 9875.90 1.02771
\(453\) 0 0
\(454\) −1518.87 −0.157013
\(455\) 0 0
\(456\) 0 0
\(457\) −14098.9 −1.44314 −0.721572 0.692340i \(-0.756580\pi\)
−0.721572 + 0.692340i \(0.756580\pi\)
\(458\) −1019.23 −0.103986
\(459\) 0 0
\(460\) −19539.3 −1.98049
\(461\) 14449.7 1.45985 0.729924 0.683529i \(-0.239556\pi\)
0.729924 + 0.683529i \(0.239556\pi\)
\(462\) 0 0
\(463\) −15806.5 −1.58659 −0.793293 0.608840i \(-0.791635\pi\)
−0.793293 + 0.608840i \(0.791635\pi\)
\(464\) 6340.06 0.634332
\(465\) 0 0
\(466\) −1635.85 −0.162617
\(467\) 15071.3 1.49340 0.746699 0.665162i \(-0.231638\pi\)
0.746699 + 0.665162i \(0.231638\pi\)
\(468\) 0 0
\(469\) −2311.89 −0.227619
\(470\) 2487.69 0.244146
\(471\) 0 0
\(472\) −2017.68 −0.196761
\(473\) −617.299 −0.0600073
\(474\) 0 0
\(475\) 15524.5 1.49961
\(476\) −2886.78 −0.277973
\(477\) 0 0
\(478\) −2650.18 −0.253591
\(479\) −392.545 −0.0374443 −0.0187222 0.999825i \(-0.505960\pi\)
−0.0187222 + 0.999825i \(0.505960\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2268.51 −0.214373
\(483\) 0 0
\(484\) 6466.36 0.607284
\(485\) 13522.7 1.26605
\(486\) 0 0
\(487\) −9497.89 −0.883758 −0.441879 0.897075i \(-0.645688\pi\)
−0.441879 + 0.897075i \(0.645688\pi\)
\(488\) 4597.32 0.426457
\(489\) 0 0
\(490\) −2447.14 −0.225613
\(491\) 1893.82 0.174067 0.0870337 0.996205i \(-0.472261\pi\)
0.0870337 + 0.996205i \(0.472261\pi\)
\(492\) 0 0
\(493\) −7253.52 −0.662641
\(494\) 0 0
\(495\) 0 0
\(496\) 16410.1 1.48555
\(497\) 831.881 0.0750804
\(498\) 0 0
\(499\) 13370.1 1.19945 0.599727 0.800205i \(-0.295276\pi\)
0.599727 + 0.800205i \(0.295276\pi\)
\(500\) 9331.79 0.834661
\(501\) 0 0
\(502\) 2464.39 0.219106
\(503\) −5554.71 −0.492391 −0.246195 0.969220i \(-0.579180\pi\)
−0.246195 + 0.969220i \(0.579180\pi\)
\(504\) 0 0
\(505\) −6209.51 −0.547168
\(506\) −1381.62 −0.121385
\(507\) 0 0
\(508\) −20332.3 −1.77578
\(509\) 2197.55 0.191365 0.0956824 0.995412i \(-0.469497\pi\)
0.0956824 + 0.995412i \(0.469497\pi\)
\(510\) 0 0
\(511\) 637.756 0.0552107
\(512\) −8137.89 −0.702437
\(513\) 0 0
\(514\) −734.022 −0.0629890
\(515\) 10333.9 0.884206
\(516\) 0 0
\(517\) −7144.45 −0.607761
\(518\) 10.2334 0.000868010 0
\(519\) 0 0
\(520\) 0 0
\(521\) −17005.2 −1.42997 −0.714983 0.699142i \(-0.753565\pi\)
−0.714983 + 0.699142i \(0.753565\pi\)
\(522\) 0 0
\(523\) −14486.2 −1.21116 −0.605581 0.795783i \(-0.707059\pi\)
−0.605581 + 0.795783i \(0.707059\pi\)
\(524\) 16643.8 1.38758
\(525\) 0 0
\(526\) −2766.24 −0.229304
\(527\) −18774.4 −1.55185
\(528\) 0 0
\(529\) 7582.03 0.623163
\(530\) 528.244 0.0432933
\(531\) 0 0
\(532\) 3431.27 0.279632
\(533\) 0 0
\(534\) 0 0
\(535\) 10177.0 0.822413
\(536\) 2946.32 0.237429
\(537\) 0 0
\(538\) −1088.55 −0.0872315
\(539\) 7027.98 0.561626
\(540\) 0 0
\(541\) 15266.7 1.21325 0.606623 0.794990i \(-0.292524\pi\)
0.606623 + 0.794990i \(0.292524\pi\)
\(542\) 1243.31 0.0985330
\(543\) 0 0
\(544\) 5540.83 0.436693
\(545\) 3134.23 0.246341
\(546\) 0 0
\(547\) 15260.5 1.19286 0.596430 0.802665i \(-0.296586\pi\)
0.596430 + 0.802665i \(0.296586\pi\)
\(548\) −5371.62 −0.418731
\(549\) 0 0
\(550\) 1888.78 0.146432
\(551\) 8621.64 0.666595
\(552\) 0 0
\(553\) −1101.08 −0.0846703
\(554\) 1682.55 0.129033
\(555\) 0 0
\(556\) 5306.00 0.404721
\(557\) −10442.1 −0.794337 −0.397169 0.917746i \(-0.630007\pi\)
−0.397169 + 0.917746i \(0.630007\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 5754.94 0.434269
\(561\) 0 0
\(562\) 3999.57 0.300199
\(563\) −7145.26 −0.534879 −0.267440 0.963575i \(-0.586178\pi\)
−0.267440 + 0.963575i \(0.586178\pi\)
\(564\) 0 0
\(565\) 22524.7 1.67721
\(566\) −932.950 −0.0692841
\(567\) 0 0
\(568\) −1060.17 −0.0783162
\(569\) −4438.86 −0.327042 −0.163521 0.986540i \(-0.552285\pi\)
−0.163521 + 0.986540i \(0.552285\pi\)
\(570\) 0 0
\(571\) 10117.3 0.741497 0.370748 0.928733i \(-0.379101\pi\)
0.370748 + 0.928733i \(0.379101\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 543.109 0.0394929
\(575\) −26998.4 −1.95810
\(576\) 0 0
\(577\) −3105.60 −0.224069 −0.112035 0.993704i \(-0.535737\pi\)
−0.112035 + 0.993704i \(0.535737\pi\)
\(578\) 127.615 0.00918352
\(579\) 0 0
\(580\) 14834.5 1.06202
\(581\) −1828.16 −0.130542
\(582\) 0 0
\(583\) −1517.08 −0.107772
\(584\) −812.769 −0.0575901
\(585\) 0 0
\(586\) −3628.05 −0.255757
\(587\) 19662.3 1.38254 0.691270 0.722597i \(-0.257052\pi\)
0.691270 + 0.722597i \(0.257052\pi\)
\(588\) 0 0
\(589\) 22315.5 1.56111
\(590\) −2272.95 −0.158603
\(591\) 0 0
\(592\) 255.026 0.0177052
\(593\) 6395.51 0.442888 0.221444 0.975173i \(-0.428923\pi\)
0.221444 + 0.975173i \(0.428923\pi\)
\(594\) 0 0
\(595\) −6584.10 −0.453650
\(596\) 15423.9 1.06005
\(597\) 0 0
\(598\) 0 0
\(599\) −8878.48 −0.605618 −0.302809 0.953051i \(-0.597924\pi\)
−0.302809 + 0.953051i \(0.597924\pi\)
\(600\) 0 0
\(601\) 19100.6 1.29639 0.648194 0.761475i \(-0.275525\pi\)
0.648194 + 0.761475i \(0.275525\pi\)
\(602\) 65.6430 0.00444420
\(603\) 0 0
\(604\) 14079.3 0.948472
\(605\) 14748.3 0.991082
\(606\) 0 0
\(607\) 16595.8 1.10972 0.554861 0.831943i \(-0.312771\pi\)
0.554861 + 0.831943i \(0.312771\pi\)
\(608\) −6585.91 −0.439299
\(609\) 0 0
\(610\) 5178.97 0.343755
\(611\) 0 0
\(612\) 0 0
\(613\) −16469.2 −1.08513 −0.542564 0.840015i \(-0.682546\pi\)
−0.542564 + 0.840015i \(0.682546\pi\)
\(614\) −1583.01 −0.104047
\(615\) 0 0
\(616\) 845.205 0.0552829
\(617\) 10116.0 0.660055 0.330027 0.943971i \(-0.392942\pi\)
0.330027 + 0.943971i \(0.392942\pi\)
\(618\) 0 0
\(619\) −18854.8 −1.22430 −0.612148 0.790743i \(-0.709694\pi\)
−0.612148 + 0.790743i \(0.709694\pi\)
\(620\) 38396.3 2.48715
\(621\) 0 0
\(622\) 1460.49 0.0941487
\(623\) −3905.86 −0.251180
\(624\) 0 0
\(625\) −2730.82 −0.174773
\(626\) 157.019 0.0100252
\(627\) 0 0
\(628\) 3100.99 0.197043
\(629\) −291.769 −0.0184954
\(630\) 0 0
\(631\) −18946.2 −1.19531 −0.597653 0.801755i \(-0.703900\pi\)
−0.597653 + 0.801755i \(0.703900\pi\)
\(632\) 1403.24 0.0883194
\(633\) 0 0
\(634\) −1336.15 −0.0836991
\(635\) −46373.3 −2.89806
\(636\) 0 0
\(637\) 0 0
\(638\) 1048.95 0.0650912
\(639\) 0 0
\(640\) −15043.5 −0.929135
\(641\) 23586.9 1.45340 0.726698 0.686957i \(-0.241054\pi\)
0.726698 + 0.686957i \(0.241054\pi\)
\(642\) 0 0
\(643\) 27153.0 1.66534 0.832669 0.553772i \(-0.186812\pi\)
0.832669 + 0.553772i \(0.186812\pi\)
\(644\) −5967.25 −0.365128
\(645\) 0 0
\(646\) 2408.70 0.146701
\(647\) −6856.72 −0.416639 −0.208319 0.978061i \(-0.566799\pi\)
−0.208319 + 0.978061i \(0.566799\pi\)
\(648\) 0 0
\(649\) 6527.75 0.394817
\(650\) 0 0
\(651\) 0 0
\(652\) 7350.17 0.441495
\(653\) 8073.89 0.483853 0.241926 0.970295i \(-0.422221\pi\)
0.241926 + 0.970295i \(0.422221\pi\)
\(654\) 0 0
\(655\) 37960.9 2.26451
\(656\) 13534.8 0.805555
\(657\) 0 0
\(658\) 759.734 0.0450114
\(659\) −5305.73 −0.313629 −0.156815 0.987628i \(-0.550123\pi\)
−0.156815 + 0.987628i \(0.550123\pi\)
\(660\) 0 0
\(661\) −25848.3 −1.52100 −0.760502 0.649336i \(-0.775047\pi\)
−0.760502 + 0.649336i \(0.775047\pi\)
\(662\) 3373.75 0.198073
\(663\) 0 0
\(664\) 2329.85 0.136168
\(665\) 7825.96 0.456357
\(666\) 0 0
\(667\) −14993.7 −0.870404
\(668\) −28735.9 −1.66441
\(669\) 0 0
\(670\) 3319.09 0.191385
\(671\) −14873.6 −0.855722
\(672\) 0 0
\(673\) −14529.1 −0.832177 −0.416089 0.909324i \(-0.636599\pi\)
−0.416089 + 0.909324i \(0.636599\pi\)
\(674\) −2066.06 −0.118073
\(675\) 0 0
\(676\) 0 0
\(677\) −12058.1 −0.684535 −0.342267 0.939603i \(-0.611195\pi\)
−0.342267 + 0.939603i \(0.611195\pi\)
\(678\) 0 0
\(679\) 4129.78 0.233412
\(680\) 8390.91 0.473201
\(681\) 0 0
\(682\) 2715.00 0.152438
\(683\) 30028.8 1.68231 0.841156 0.540792i \(-0.181875\pi\)
0.841156 + 0.540792i \(0.181875\pi\)
\(684\) 0 0
\(685\) −12251.5 −0.683364
\(686\) −1565.22 −0.0871144
\(687\) 0 0
\(688\) 1635.89 0.0906505
\(689\) 0 0
\(690\) 0 0
\(691\) −449.696 −0.0247572 −0.0123786 0.999923i \(-0.503940\pi\)
−0.0123786 + 0.999923i \(0.503940\pi\)
\(692\) 11108.7 0.610244
\(693\) 0 0
\(694\) −2307.10 −0.126191
\(695\) 12101.8 0.660500
\(696\) 0 0
\(697\) −15484.8 −0.841506
\(698\) 22.0690 0.00119674
\(699\) 0 0
\(700\) 8157.67 0.440473
\(701\) 26986.0 1.45399 0.726994 0.686644i \(-0.240917\pi\)
0.726994 + 0.686644i \(0.240917\pi\)
\(702\) 0 0
\(703\) 346.801 0.0186057
\(704\) 9858.46 0.527776
\(705\) 0 0
\(706\) −3971.29 −0.211702
\(707\) −1896.37 −0.100877
\(708\) 0 0
\(709\) 9098.87 0.481968 0.240984 0.970529i \(-0.422530\pi\)
0.240984 + 0.970529i \(0.422530\pi\)
\(710\) −1194.30 −0.0631285
\(711\) 0 0
\(712\) 4977.71 0.262005
\(713\) −38808.4 −2.03841
\(714\) 0 0
\(715\) 0 0
\(716\) 9118.62 0.475948
\(717\) 0 0
\(718\) −3147.11 −0.163578
\(719\) 6293.55 0.326439 0.163220 0.986590i \(-0.447812\pi\)
0.163220 + 0.986590i \(0.447812\pi\)
\(720\) 0 0
\(721\) 3155.95 0.163015
\(722\) 144.295 0.00743783
\(723\) 0 0
\(724\) 8853.72 0.454483
\(725\) 20497.5 1.05001
\(726\) 0 0
\(727\) 18070.7 0.921878 0.460939 0.887432i \(-0.347513\pi\)
0.460939 + 0.887432i \(0.347513\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −915.601 −0.0464218
\(731\) −1871.58 −0.0946962
\(732\) 0 0
\(733\) −34771.5 −1.75214 −0.876068 0.482188i \(-0.839842\pi\)
−0.876068 + 0.482188i \(0.839842\pi\)
\(734\) −1755.61 −0.0882842
\(735\) 0 0
\(736\) 11453.4 0.573613
\(737\) −9532.17 −0.476421
\(738\) 0 0
\(739\) −23631.5 −1.17632 −0.588158 0.808746i \(-0.700147\pi\)
−0.588158 + 0.808746i \(0.700147\pi\)
\(740\) 596.710 0.0296425
\(741\) 0 0
\(742\) 161.324 0.00798167
\(743\) 32502.8 1.60486 0.802431 0.596745i \(-0.203540\pi\)
0.802431 + 0.596745i \(0.203540\pi\)
\(744\) 0 0
\(745\) 35178.6 1.72999
\(746\) 4390.69 0.215489
\(747\) 0 0
\(748\) −11902.5 −0.581816
\(749\) 3108.04 0.151622
\(750\) 0 0
\(751\) 2020.86 0.0981920 0.0490960 0.998794i \(-0.484366\pi\)
0.0490960 + 0.998794i \(0.484366\pi\)
\(752\) 18933.3 0.918120
\(753\) 0 0
\(754\) 0 0
\(755\) 32111.6 1.54790
\(756\) 0 0
\(757\) 12568.2 0.603434 0.301717 0.953398i \(-0.402440\pi\)
0.301717 + 0.953398i \(0.402440\pi\)
\(758\) −3581.73 −0.171628
\(759\) 0 0
\(760\) −9973.56 −0.476025
\(761\) −8704.81 −0.414651 −0.207325 0.978272i \(-0.566476\pi\)
−0.207325 + 0.978272i \(0.566476\pi\)
\(762\) 0 0
\(763\) 957.187 0.0454161
\(764\) 20941.4 0.991665
\(765\) 0 0
\(766\) 3205.16 0.151184
\(767\) 0 0
\(768\) 0 0
\(769\) 21915.9 1.02771 0.513853 0.857878i \(-0.328218\pi\)
0.513853 + 0.857878i \(0.328218\pi\)
\(770\) 952.140 0.0445620
\(771\) 0 0
\(772\) −15386.5 −0.717322
\(773\) −23077.5 −1.07379 −0.536896 0.843649i \(-0.680403\pi\)
−0.536896 + 0.843649i \(0.680403\pi\)
\(774\) 0 0
\(775\) 53054.0 2.45904
\(776\) −5263.08 −0.243471
\(777\) 0 0
\(778\) 3851.96 0.177506
\(779\) 18405.5 0.846528
\(780\) 0 0
\(781\) 3429.94 0.157148
\(782\) −4188.92 −0.191554
\(783\) 0 0
\(784\) −18624.6 −0.848426
\(785\) 7072.67 0.321572
\(786\) 0 0
\(787\) 16522.4 0.748362 0.374181 0.927356i \(-0.377924\pi\)
0.374181 + 0.927356i \(0.377924\pi\)
\(788\) 31356.4 1.41754
\(789\) 0 0
\(790\) 1580.78 0.0711918
\(791\) 6879.00 0.309215
\(792\) 0 0
\(793\) 0 0
\(794\) 4939.89 0.220794
\(795\) 0 0
\(796\) 32996.1 1.46924
\(797\) 11719.4 0.520855 0.260427 0.965493i \(-0.416137\pi\)
0.260427 + 0.965493i \(0.416137\pi\)
\(798\) 0 0
\(799\) −21661.2 −0.959095
\(800\) −15657.7 −0.691978
\(801\) 0 0
\(802\) −691.096 −0.0304282
\(803\) 2629.53 0.115559
\(804\) 0 0
\(805\) −13610.0 −0.595886
\(806\) 0 0
\(807\) 0 0
\(808\) 2416.77 0.105225
\(809\) 24096.0 1.04718 0.523592 0.851969i \(-0.324592\pi\)
0.523592 + 0.851969i \(0.324592\pi\)
\(810\) 0 0
\(811\) −16622.6 −0.719729 −0.359864 0.933005i \(-0.617177\pi\)
−0.359864 + 0.933005i \(0.617177\pi\)
\(812\) 4530.42 0.195796
\(813\) 0 0
\(814\) 42.1933 0.00181680
\(815\) 16764.1 0.720516
\(816\) 0 0
\(817\) 2224.59 0.0952613
\(818\) 2962.05 0.126609
\(819\) 0 0
\(820\) 31668.7 1.34868
\(821\) −38005.5 −1.61559 −0.807797 0.589461i \(-0.799340\pi\)
−0.807797 + 0.589461i \(0.799340\pi\)
\(822\) 0 0
\(823\) −15859.5 −0.671722 −0.335861 0.941912i \(-0.609027\pi\)
−0.335861 + 0.941912i \(0.609027\pi\)
\(824\) −4022.01 −0.170040
\(825\) 0 0
\(826\) −694.155 −0.0292406
\(827\) 12201.0 0.513023 0.256512 0.966541i \(-0.417427\pi\)
0.256512 + 0.966541i \(0.417427\pi\)
\(828\) 0 0
\(829\) −5431.41 −0.227552 −0.113776 0.993506i \(-0.536295\pi\)
−0.113776 + 0.993506i \(0.536295\pi\)
\(830\) 2624.62 0.109761
\(831\) 0 0
\(832\) 0 0
\(833\) 21308.0 0.886290
\(834\) 0 0
\(835\) −65540.3 −2.71630
\(836\) 14147.5 0.585288
\(837\) 0 0
\(838\) 4716.02 0.194406
\(839\) −7960.90 −0.327582 −0.163791 0.986495i \(-0.552372\pi\)
−0.163791 + 0.986495i \(0.552372\pi\)
\(840\) 0 0
\(841\) −13005.6 −0.533255
\(842\) 3457.61 0.141517
\(843\) 0 0
\(844\) −10655.0 −0.434550
\(845\) 0 0
\(846\) 0 0
\(847\) 4504.10 0.182719
\(848\) 4020.35 0.162806
\(849\) 0 0
\(850\) 5726.56 0.231082
\(851\) −603.115 −0.0242944
\(852\) 0 0
\(853\) −13576.7 −0.544969 −0.272485 0.962160i \(-0.587845\pi\)
−0.272485 + 0.962160i \(0.587845\pi\)
\(854\) 1581.65 0.0633756
\(855\) 0 0
\(856\) −3960.95 −0.158157
\(857\) 31223.9 1.24456 0.622281 0.782794i \(-0.286206\pi\)
0.622281 + 0.782794i \(0.286206\pi\)
\(858\) 0 0
\(859\) −11815.8 −0.469323 −0.234661 0.972077i \(-0.575398\pi\)
−0.234661 + 0.972077i \(0.575398\pi\)
\(860\) 3827.65 0.151770
\(861\) 0 0
\(862\) 6175.36 0.244006
\(863\) −1790.84 −0.0706384 −0.0353192 0.999376i \(-0.511245\pi\)
−0.0353192 + 0.999376i \(0.511245\pi\)
\(864\) 0 0
\(865\) 25336.4 0.995912
\(866\) −817.328 −0.0320715
\(867\) 0 0
\(868\) 11726.1 0.458538
\(869\) −4539.87 −0.177220
\(870\) 0 0
\(871\) 0 0
\(872\) −1219.86 −0.0473734
\(873\) 0 0
\(874\) 4979.01 0.192698
\(875\) 6500.00 0.251132
\(876\) 0 0
\(877\) −43542.5 −1.67654 −0.838270 0.545255i \(-0.816433\pi\)
−0.838270 + 0.545255i \(0.816433\pi\)
\(878\) −2698.42 −0.103721
\(879\) 0 0
\(880\) 23728.2 0.908953
\(881\) 1020.04 0.0390080 0.0195040 0.999810i \(-0.493791\pi\)
0.0195040 + 0.999810i \(0.493791\pi\)
\(882\) 0 0
\(883\) −34781.9 −1.32560 −0.662800 0.748797i \(-0.730632\pi\)
−0.662800 + 0.748797i \(0.730632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6374.70 −0.241718
\(887\) −49785.1 −1.88458 −0.942288 0.334802i \(-0.891330\pi\)
−0.942288 + 0.334802i \(0.891330\pi\)
\(888\) 0 0
\(889\) −14162.3 −0.534295
\(890\) 5607.49 0.211195
\(891\) 0 0
\(892\) −8272.08 −0.310504
\(893\) 25746.8 0.964818
\(894\) 0 0
\(895\) 20797.5 0.776743
\(896\) −4594.25 −0.171298
\(897\) 0 0
\(898\) 3088.36 0.114766
\(899\) 29463.9 1.09308
\(900\) 0 0
\(901\) −4599.60 −0.170072
\(902\) 2239.29 0.0826611
\(903\) 0 0
\(904\) −8766.74 −0.322541
\(905\) 20193.3 0.741712
\(906\) 0 0
\(907\) 17389.9 0.636627 0.318314 0.947985i \(-0.396884\pi\)
0.318314 + 0.947985i \(0.396884\pi\)
\(908\) −27047.6 −0.988553
\(909\) 0 0
\(910\) 0 0
\(911\) −20419.5 −0.742621 −0.371311 0.928509i \(-0.621091\pi\)
−0.371311 + 0.928509i \(0.621091\pi\)
\(912\) 0 0
\(913\) −7537.71 −0.273233
\(914\) 6181.60 0.223708
\(915\) 0 0
\(916\) −18150.2 −0.654695
\(917\) 11593.2 0.417492
\(918\) 0 0
\(919\) −33231.8 −1.19283 −0.596417 0.802674i \(-0.703410\pi\)
−0.596417 + 0.802674i \(0.703410\pi\)
\(920\) 17344.8 0.621567
\(921\) 0 0
\(922\) −6335.43 −0.226297
\(923\) 0 0
\(924\) 0 0
\(925\) 824.502 0.0293075
\(926\) 6930.31 0.245944
\(927\) 0 0
\(928\) −8695.59 −0.307594
\(929\) −25222.8 −0.890780 −0.445390 0.895337i \(-0.646935\pi\)
−0.445390 + 0.895337i \(0.646935\pi\)
\(930\) 0 0
\(931\) −25327.0 −0.891579
\(932\) −29130.9 −1.02383
\(933\) 0 0
\(934\) −6607.97 −0.231498
\(935\) −27146.9 −0.949519
\(936\) 0 0
\(937\) −26979.4 −0.940639 −0.470319 0.882496i \(-0.655861\pi\)
−0.470319 + 0.882496i \(0.655861\pi\)
\(938\) 1013.64 0.0352842
\(939\) 0 0
\(940\) 44300.2 1.53714
\(941\) 7641.67 0.264730 0.132365 0.991201i \(-0.457743\pi\)
0.132365 + 0.991201i \(0.457743\pi\)
\(942\) 0 0
\(943\) −32008.7 −1.10535
\(944\) −17299.0 −0.596434
\(945\) 0 0
\(946\) 270.653 0.00930200
\(947\) −2869.32 −0.0984587 −0.0492293 0.998788i \(-0.515677\pi\)
−0.0492293 + 0.998788i \(0.515677\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6806.67 −0.232461
\(951\) 0 0
\(952\) 2562.56 0.0872407
\(953\) −12313.6 −0.418548 −0.209274 0.977857i \(-0.567110\pi\)
−0.209274 + 0.977857i \(0.567110\pi\)
\(954\) 0 0
\(955\) 47762.6 1.61839
\(956\) −47193.8 −1.59661
\(957\) 0 0
\(958\) 172.110 0.00580442
\(959\) −3741.57 −0.125987
\(960\) 0 0
\(961\) 46470.7 1.55989
\(962\) 0 0
\(963\) 0 0
\(964\) −40397.1 −1.34969
\(965\) −35093.2 −1.17066
\(966\) 0 0
\(967\) 17838.0 0.593207 0.296603 0.955001i \(-0.404146\pi\)
0.296603 + 0.955001i \(0.404146\pi\)
\(968\) −5740.12 −0.190593
\(969\) 0 0
\(970\) −5928.97 −0.196255
\(971\) −41525.3 −1.37241 −0.686206 0.727408i \(-0.740725\pi\)
−0.686206 + 0.727408i \(0.740725\pi\)
\(972\) 0 0
\(973\) 3695.86 0.121772
\(974\) 4164.32 0.136995
\(975\) 0 0
\(976\) 39416.1 1.29270
\(977\) −31654.4 −1.03655 −0.518277 0.855213i \(-0.673426\pi\)
−0.518277 + 0.855213i \(0.673426\pi\)
\(978\) 0 0
\(979\) −16104.3 −0.525735
\(980\) −43578.0 −1.42046
\(981\) 0 0
\(982\) −830.342 −0.0269830
\(983\) −39913.2 −1.29505 −0.647525 0.762045i \(-0.724196\pi\)
−0.647525 + 0.762045i \(0.724196\pi\)
\(984\) 0 0
\(985\) 71516.8 2.31342
\(986\) 3180.28 0.102719
\(987\) 0 0
\(988\) 0 0
\(989\) −3868.74 −0.124387
\(990\) 0 0
\(991\) −2700.94 −0.0865773 −0.0432887 0.999063i \(-0.513784\pi\)
−0.0432887 + 0.999063i \(0.513784\pi\)
\(992\) −22506.9 −0.720358
\(993\) 0 0
\(994\) −364.736 −0.0116386
\(995\) 75256.6 2.39778
\(996\) 0 0
\(997\) −9729.08 −0.309050 −0.154525 0.987989i \(-0.549385\pi\)
−0.154525 + 0.987989i \(0.549385\pi\)
\(998\) −5862.08 −0.185933
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.l.1.2 2
3.2 odd 2 169.4.a.j.1.1 2
13.4 even 6 117.4.g.d.55.2 4
13.10 even 6 117.4.g.d.100.2 4
13.12 even 2 1521.4.a.t.1.1 2
39.2 even 12 169.4.e.g.147.3 8
39.5 even 4 169.4.b.e.168.2 4
39.8 even 4 169.4.b.e.168.3 4
39.11 even 12 169.4.e.g.147.2 8
39.17 odd 6 13.4.c.b.3.1 4
39.20 even 12 169.4.e.g.23.3 8
39.23 odd 6 13.4.c.b.9.1 yes 4
39.29 odd 6 169.4.c.f.22.2 4
39.32 even 12 169.4.e.g.23.2 8
39.35 odd 6 169.4.c.f.146.2 4
39.38 odd 2 169.4.a.f.1.2 2
156.23 even 6 208.4.i.e.113.1 4
156.95 even 6 208.4.i.e.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 39.17 odd 6
13.4.c.b.9.1 yes 4 39.23 odd 6
117.4.g.d.55.2 4 13.4 even 6
117.4.g.d.100.2 4 13.10 even 6
169.4.a.f.1.2 2 39.38 odd 2
169.4.a.j.1.1 2 3.2 odd 2
169.4.b.e.168.2 4 39.5 even 4
169.4.b.e.168.3 4 39.8 even 4
169.4.c.f.22.2 4 39.29 odd 6
169.4.c.f.146.2 4 39.35 odd 6
169.4.e.g.23.2 8 39.32 even 12
169.4.e.g.23.3 8 39.20 even 12
169.4.e.g.147.2 8 39.11 even 12
169.4.e.g.147.3 8 39.2 even 12
208.4.i.e.81.1 4 156.95 even 6
208.4.i.e.113.1 4 156.23 even 6
1521.4.a.l.1.2 2 1.1 even 1 trivial
1521.4.a.t.1.1 2 13.12 even 2