Properties

Label 1521.4.a.t.1.1
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 1.1
Root \(2.56155\) 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 \(-0.475304\pi\)
0.0775072 + 0.996992i \(0.475304\pi\)
\(3\) 0 0
\(4\) −7.80776 −0.975971
\(5\) 17.8078 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 5.43845 0.293649 0.146824 0.989163i \(-0.453095\pi\)
0.146824 + 0.989163i \(0.453095\pi\)
\(8\) −6.93087 −0.306304
\(9\) 0 0
\(10\) 7.80776 0.246903
\(11\) 22.4233 0.614625 0.307313 0.951609i \(-0.400570\pi\)
0.307313 + 0.951609i \(0.400570\pi\)
\(12\) 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 \(-0.662221\pi\)
−0.487857 + 0.872923i \(0.662221\pi\)
\(20\) −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 \(-0.795158\pi\)
−0.799983 + 0.600023i \(0.795158\pi\)
\(32\) 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 \(-0.503035\pi\)
−0.00953442 + 0.999955i \(0.503035\pi\)
\(38\) −35.4299 −0.151250
\(39\) 0 0
\(40\) −123.423 −0.487873
\(41\) −227.769 −0.867598 −0.433799 0.901010i \(-0.642827\pi\)
−0.433799 + 0.901010i \(0.642827\pi\)
\(42\) 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 \(-0.664618\pi\)
−0.494416 + 0.869225i \(0.664618\pi\)
\(48\) 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 \(-0.395919\pi\)
0.321186 + 0.947016i \(0.395919\pi\)
\(60\) 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 \(-0.626685\pi\)
−0.387570 + 0.921840i \(0.626685\pi\)
\(68\) 530.810 0.946619
\(69\) 0 0
\(70\) 42.4621 0.0725028
\(71\) 152.963 0.255681 0.127841 0.991795i \(-0.459195\pi\)
0.127841 + 0.991795i \(0.459195\pi\)
\(72\) 0 0
\(73\) 117.268 0.188016 0.0940081 0.995571i \(-0.470032\pi\)
0.0940081 + 0.995571i \(0.470032\pi\)
\(74\) −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 \(-0.571349\pi\)
−0.222276 + 0.974984i \(0.571349\pi\)
\(84\) 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 \(-0.640672\pi\)
−0.427688 + 0.903927i \(0.640672\pi\)
\(90\) 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 \(-0.369901\pi\)
0.397434 + 0.917631i \(0.369901\pi\)
\(98\) −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 \(-0.475360\pi\)
0.0773307 + 0.997005i \(0.475360\pi\)
\(110\) 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 \(-0.568819\pi\)
−0.214520 + 0.976720i \(0.568819\pi\)
\(138\) 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 \(-0.317260\pi\)
0.543074 + 0.839685i \(0.317260\pi\)
\(150\) 0 0
\(151\) 1803.24 0.971824 0.485912 0.874008i \(-0.338487\pi\)
0.485912 + 0.874008i \(0.338487\pi\)
\(152\) 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 \(-0.427375\pi\)
0.226183 + 0.974085i \(0.427375\pi\)
\(164\) 1778.37 0.846750
\(165\) 0 0
\(166\) −147.386 −0.0689120
\(167\) −3680.43 −1.70539 −0.852696 0.522408i \(-0.825034\pi\)
−0.852696 + 0.522408i \(0.825034\pi\)
\(168\) 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 \(-0.619783\pi\)
−0.367491 + 0.930027i \(0.619783\pi\)
\(194\) 332.943 0.123216
\(195\) 0 0
\(196\) 2447.14 0.891813
\(197\) 4016.05 1.45244 0.726222 0.687460i \(-0.241274\pi\)
0.726222 + 0.687460i \(0.241274\pi\)
\(198\) 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 \(-0.550851\pi\)
−0.159075 + 0.987267i \(0.550851\pi\)
\(224\) 443.239 0.132210
\(225\) 0 0
\(226\) −554.584 −0.163232
\(227\) −3464.19 −1.01289 −0.506446 0.862272i \(-0.669041\pi\)
−0.506446 + 0.862272i \(0.669041\pi\)
\(228\) 0 0
\(229\) −2324.64 −0.670815 −0.335407 0.942073i \(-0.608874\pi\)
−0.335407 + 0.942073i \(0.608874\pi\)
\(230\) −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 \(-0.804894\pi\)
−0.817958 + 0.575278i \(0.804894\pi\)
\(240\) 0 0
\(241\) −5173.96 −1.38292 −0.691461 0.722414i \(-0.743033\pi\)
−0.691461 + 0.722414i \(0.743033\pi\)
\(242\) −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 \(-0.397050\pi\)
0.317819 + 0.948151i \(0.397050\pi\)
\(272\) −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 \(-0.0803712\pi\)
0.968293 + 0.249819i \(0.0803712\pi\)
\(282\) 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 \(-0.808793\pi\)
−0.824944 + 0.565215i \(0.808793\pi\)
\(294\) 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 \(-0.608941\pi\)
−0.335605 + 0.942003i \(0.608941\pi\)
\(308\) −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 \(-0.587014\pi\)
−0.269972 + 0.962868i \(0.587014\pi\)
\(318\) 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 \(-0.279395\pi\)
0.638887 + 0.769301i \(0.279395\pi\)
\(332\) 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 \(-0.498771\pi\)
0.00386009 + 0.999993i \(0.498771\pi\)
\(350\) 458.096 0.0699608
\(351\) 0 0
\(352\) 1827.52 0.276725
\(353\) −9057.64 −1.36569 −0.682846 0.730562i \(-0.739258\pi\)
−0.682846 + 0.730562i \(0.739258\pi\)
\(354\) 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 \(-0.676917\pi\)
−0.527623 + 0.849479i \(0.676917\pi\)
\(360\) 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 \(-0.686742\pi\)
−0.553587 + 0.832791i \(0.686742\pi\)
\(380\) 11235.4 1.51675
\(381\) 0 0
\(382\) −1175.97 −0.157507
\(383\) 7310.25 0.975290 0.487645 0.873042i \(-0.337856\pi\)
0.487645 + 0.873042i \(0.337856\pi\)
\(384\) 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 \(-0.247712\pi\)
0.712171 + 0.702006i \(0.247712\pi\)
\(398\) −1852.90 −0.233361
\(399\) 0 0
\(400\) 11416.2 1.42702
\(401\) −1576.23 −0.196293 −0.0981464 0.995172i \(-0.531291\pi\)
−0.0981464 + 0.995172i \(0.531291\pi\)
\(402\) 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 \(-0.366095\pi\)
0.408377 + 0.912814i \(0.366095\pi\)
\(410\) −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 \(-0.349116\pi\)
0.456463 + 0.889743i \(0.349116\pi\)
\(422\) 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 \(-0.211610\pi\)
0.787044 + 0.616897i \(0.211610\pi\)
\(432\) 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 \(-0.379296\pi\)
0.370179 + 0.928960i \(0.379296\pi\)
\(450\) 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 \(-0.243420\pi\)
0.721572 + 0.692340i \(0.243420\pi\)
\(458\) −1019.23 −0.103986
\(459\) 0 0
\(460\) 19539.3 1.98049
\(461\) −14449.7 −1.45985 −0.729924 0.683529i \(-0.760444\pi\)
−0.729924 + 0.683529i \(0.760444\pi\)
\(462\) 0 0
\(463\) 15806.5 1.58659 0.793293 0.608840i \(-0.208365\pi\)
0.793293 + 0.608840i \(0.208365\pi\)
\(464\) 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 \(-0.494040\pi\)
0.0187222 + 0.999825i \(0.494040\pi\)
\(480\) 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 \(-0.354312\pi\)
0.441879 + 0.897075i \(0.354312\pi\)
\(488\) −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 \(-0.704724\pi\)
−0.599727 + 0.800205i \(0.704724\pi\)
\(500\) −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 \(-0.530503\pi\)
−0.0956824 + 0.995412i \(0.530503\pi\)
\(510\) 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 \(-0.707476\pi\)
−0.606623 + 0.794990i \(0.707476\pi\)
\(542\) 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 \(-0.369993\pi\)
0.397169 + 0.917746i \(0.369993\pi\)
\(558\) 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 \(-0.464263\pi\)
0.112035 + 0.993704i \(0.464263\pi\)
\(578\) −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 \(-0.742948\pi\)
−0.691270 + 0.722597i \(0.742948\pi\)
\(588\) 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 \(-0.571077\pi\)
−0.221444 + 0.975173i \(0.571077\pi\)
\(594\) 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 \(-0.317454\pi\)
0.542564 + 0.840015i \(0.317454\pi\)
\(614\) −1583.01 −0.104047
\(615\) 0 0
\(616\) −845.205 −0.0552829
\(617\) −10116.0 −0.660055 −0.330027 0.943971i \(-0.607058\pi\)
−0.330027 + 0.943971i \(0.607058\pi\)
\(618\) 0 0
\(619\) 18854.8 1.22430 0.612148 0.790743i \(-0.290306\pi\)
0.612148 + 0.790743i \(0.290306\pi\)
\(620\) 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 \(-0.296100\pi\)
0.597653 + 0.801755i \(0.296100\pi\)
\(632\) −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 \(-0.813188\pi\)
−0.832669 + 0.553772i \(0.813188\pi\)
\(644\) 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 \(-0.224953\pi\)
0.760502 + 0.649336i \(0.224953\pi\)
\(662\) 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 \(-0.818125\pi\)
−0.841156 + 0.540792i \(0.818125\pi\)
\(684\) 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 \(-0.496060\pi\)
0.0123786 + 0.999923i \(0.496060\pi\)
\(692\) 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 \(-0.577470\pi\)
−0.240984 + 0.970529i \(0.577470\pi\)
\(710\) 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 \(-0.160158\pi\)
0.876068 + 0.482188i \(0.160158\pi\)
\(734\) 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 \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(740\) 596.710 0.0296425
\(741\) 0 0
\(742\) 161.324 0.00798167
\(743\) −32502.8 −1.60486 −0.802431 0.596745i \(-0.796460\pi\)
−0.802431 + 0.596745i \(0.796460\pi\)
\(744\) 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 \(-0.433524\pi\)
0.207325 + 0.978272i \(0.433524\pi\)
\(762\) 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 \(-0.671782\pi\)
−0.513853 + 0.857878i \(0.671782\pi\)
\(770\) 952.140 0.0445620
\(771\) 0 0
\(772\) 15386.5 0.717322
\(773\) 23077.5 1.07379 0.536896 0.843649i \(-0.319597\pi\)
0.536896 + 0.843649i \(0.319597\pi\)
\(774\) 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 \(-0.622076\pi\)
−0.374181 + 0.927356i \(0.622076\pi\)
\(788\) −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 \(-0.382823\pi\)
0.359864 + 0.933005i \(0.382823\pi\)
\(812\) −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 \(-0.200660\pi\)
0.807797 + 0.589461i \(0.200660\pi\)
\(822\) 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 \(-0.582573\pi\)
−0.256512 + 0.966541i \(0.582573\pi\)
\(828\) 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 \(-0.447628\pi\)
0.163791 + 0.986495i \(0.447628\pi\)
\(840\) 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 \(-0.412155\pi\)
0.272485 + 0.962160i \(0.412155\pi\)
\(854\) 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 \(-0.488755\pi\)
0.0353192 + 0.999376i \(0.488755\pi\)
\(864\) 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 \(-0.183567\pi\)
0.838270 + 0.545255i \(0.183567\pi\)
\(878\) 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 \(-0.353065\pi\)
0.445390 + 0.895337i \(0.353065\pi\)
\(930\) 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 \(-0.542257\pi\)
−0.132365 + 0.991201i \(0.542257\pi\)
\(942\) 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 \(-0.484323\pi\)
0.0492293 + 0.998788i \(0.484323\pi\)
\(948\) 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 \(-0.595854\pi\)
−0.296603 + 0.955001i \(0.595854\pi\)
\(968\) 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 \(-0.326574\pi\)
0.518277 + 0.855213i \(0.326574\pi\)
\(978\) 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 \(-0.275804\pi\)
0.647525 + 0.762045i \(0.275804\pi\)
\(984\) 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.t.1.1 2
3.2 odd 2 169.4.a.f.1.2 2
13.3 even 3 117.4.g.d.100.2 4
13.9 even 3 117.4.g.d.55.2 4
13.12 even 2 1521.4.a.l.1.2 2
39.2 even 12 169.4.e.g.147.2 8
39.5 even 4 169.4.b.e.168.3 4
39.8 even 4 169.4.b.e.168.2 4
39.11 even 12 169.4.e.g.147.3 8
39.17 odd 6 169.4.c.f.146.2 4
39.20 even 12 169.4.e.g.23.2 8
39.23 odd 6 169.4.c.f.22.2 4
39.29 odd 6 13.4.c.b.9.1 yes 4
39.32 even 12 169.4.e.g.23.3 8
39.35 odd 6 13.4.c.b.3.1 4
39.38 odd 2 169.4.a.j.1.1 2
156.35 even 6 208.4.i.e.81.1 4
156.107 even 6 208.4.i.e.113.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 39.35 odd 6
13.4.c.b.9.1 yes 4 39.29 odd 6
117.4.g.d.55.2 4 13.9 even 3
117.4.g.d.100.2 4 13.3 even 3
169.4.a.f.1.2 2 3.2 odd 2
169.4.a.j.1.1 2 39.38 odd 2
169.4.b.e.168.2 4 39.8 even 4
169.4.b.e.168.3 4 39.5 even 4
169.4.c.f.22.2 4 39.23 odd 6
169.4.c.f.146.2 4 39.17 odd 6
169.4.e.g.23.2 8 39.20 even 12
169.4.e.g.23.3 8 39.32 even 12
169.4.e.g.147.2 8 39.2 even 12
169.4.e.g.147.3 8 39.11 even 12
208.4.i.e.81.1 4 156.35 even 6
208.4.i.e.113.1 4 156.107 even 6
1521.4.a.l.1.2 2 13.12 even 2
1521.4.a.t.1.1 2 1.1 even 1 trivial