Properties

Label 3525.2.a.be.1.3
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.267165\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.267165 q^{2} -1.00000 q^{3} -1.92862 q^{4} +0.267165 q^{6} +0.207232 q^{7} +1.04959 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.267165 q^{2} -1.00000 q^{3} -1.92862 q^{4} +0.267165 q^{6} +0.207232 q^{7} +1.04959 q^{8} +1.00000 q^{9} -1.27329 q^{11} +1.92862 q^{12} +5.37648 q^{13} -0.0553651 q^{14} +3.57683 q^{16} -0.969832 q^{17} -0.267165 q^{18} +6.42820 q^{19} -0.207232 q^{21} +0.340179 q^{22} -5.87315 q^{23} -1.04959 q^{24} -1.43641 q^{26} -1.00000 q^{27} -0.399672 q^{28} -9.93423 q^{29} +5.97938 q^{31} -3.05479 q^{32} +1.27329 q^{33} +0.259105 q^{34} -1.92862 q^{36} +1.19082 q^{37} -1.71739 q^{38} -5.37648 q^{39} +2.24060 q^{41} +0.0553651 q^{42} +6.20192 q^{43} +2.45570 q^{44} +1.56910 q^{46} +1.00000 q^{47} -3.57683 q^{48} -6.95706 q^{49} +0.969832 q^{51} -10.3692 q^{52} +8.64084 q^{53} +0.267165 q^{54} +0.217508 q^{56} -6.42820 q^{57} +2.65408 q^{58} -9.70796 q^{59} -10.1014 q^{61} -1.59748 q^{62} +0.207232 q^{63} -6.33753 q^{64} -0.340179 q^{66} -4.36482 q^{67} +1.87044 q^{68} +5.87315 q^{69} +9.54225 q^{71} +1.04959 q^{72} +11.7305 q^{73} -0.318145 q^{74} -12.3976 q^{76} -0.263866 q^{77} +1.43641 q^{78} -15.7821 q^{79} +1.00000 q^{81} -0.598611 q^{82} -1.79347 q^{83} +0.399672 q^{84} -1.65694 q^{86} +9.93423 q^{87} -1.33644 q^{88} -7.28256 q^{89} +1.11418 q^{91} +11.3271 q^{92} -5.97938 q^{93} -0.267165 q^{94} +3.05479 q^{96} +0.290073 q^{97} +1.85868 q^{98} -1.27329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 3q^{2} - 8q^{3} + 7q^{4} - 3q^{6} + 8q^{7} + 6q^{8} + 8q^{9} - 8q^{11} - 7q^{12} + 10q^{13} + q^{14} + 5q^{16} + 6q^{17} + 3q^{18} - 2q^{19} - 8q^{21} + 10q^{23} - 6q^{24} - 14q^{26} - 8q^{27} + 44q^{28} - 13q^{29} + 10q^{32} + 8q^{33} + 28q^{34} + 7q^{36} + 3q^{37} + 36q^{38} - 10q^{39} - 16q^{41} - q^{42} + 25q^{43} - 17q^{44} - 5q^{46} + 8q^{47} - 5q^{48} + 16q^{49} - 6q^{51} - 17q^{52} + 4q^{53} - 3q^{54} + 37q^{56} + 2q^{57} + 15q^{58} - 8q^{59} + 15q^{61} + 6q^{62} + 8q^{63} - 14q^{64} + 27q^{67} + 14q^{68} - 10q^{69} + 14q^{71} + 6q^{72} + 28q^{73} - 21q^{74} + 6q^{76} + 4q^{77} + 14q^{78} + 7q^{79} + 8q^{81} - 53q^{82} + 60q^{83} - 44q^{84} - 3q^{86} + 13q^{87} + 54q^{88} - 34q^{89} + 23q^{91} - 43q^{92} + 3q^{94} - 10q^{96} + 7q^{97} + 40q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.267165 −0.188914 −0.0944571 0.995529i \(-0.530112\pi\)
−0.0944571 + 0.995529i \(0.530112\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.92862 −0.964311
\(5\) 0 0
\(6\) 0.267165 0.109070
\(7\) 0.207232 0.0783262 0.0391631 0.999233i \(-0.487531\pi\)
0.0391631 + 0.999233i \(0.487531\pi\)
\(8\) 1.04959 0.371086
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.27329 −0.383912 −0.191956 0.981404i \(-0.561483\pi\)
−0.191956 + 0.981404i \(0.561483\pi\)
\(12\) 1.92862 0.556745
\(13\) 5.37648 1.49117 0.745584 0.666412i \(-0.232171\pi\)
0.745584 + 0.666412i \(0.232171\pi\)
\(14\) −0.0553651 −0.0147969
\(15\) 0 0
\(16\) 3.57683 0.894208
\(17\) −0.969832 −0.235219 −0.117609 0.993060i \(-0.537523\pi\)
−0.117609 + 0.993060i \(0.537523\pi\)
\(18\) −0.267165 −0.0629714
\(19\) 6.42820 1.47473 0.737365 0.675494i \(-0.236070\pi\)
0.737365 + 0.675494i \(0.236070\pi\)
\(20\) 0 0
\(21\) −0.207232 −0.0452216
\(22\) 0.340179 0.0725265
\(23\) −5.87315 −1.22464 −0.612318 0.790611i \(-0.709763\pi\)
−0.612318 + 0.790611i \(0.709763\pi\)
\(24\) −1.04959 −0.214247
\(25\) 0 0
\(26\) −1.43641 −0.281703
\(27\) −1.00000 −0.192450
\(28\) −0.399672 −0.0755308
\(29\) −9.93423 −1.84474 −0.922370 0.386307i \(-0.873751\pi\)
−0.922370 + 0.386307i \(0.873751\pi\)
\(30\) 0 0
\(31\) 5.97938 1.07393 0.536964 0.843605i \(-0.319571\pi\)
0.536964 + 0.843605i \(0.319571\pi\)
\(32\) −3.05479 −0.540015
\(33\) 1.27329 0.221652
\(34\) 0.259105 0.0444362
\(35\) 0 0
\(36\) −1.92862 −0.321437
\(37\) 1.19082 0.195769 0.0978846 0.995198i \(-0.468792\pi\)
0.0978846 + 0.995198i \(0.468792\pi\)
\(38\) −1.71739 −0.278598
\(39\) −5.37648 −0.860926
\(40\) 0 0
\(41\) 2.24060 0.349923 0.174962 0.984575i \(-0.444020\pi\)
0.174962 + 0.984575i \(0.444020\pi\)
\(42\) 0.0553651 0.00854301
\(43\) 6.20192 0.945783 0.472892 0.881121i \(-0.343210\pi\)
0.472892 + 0.881121i \(0.343210\pi\)
\(44\) 2.45570 0.370211
\(45\) 0 0
\(46\) 1.56910 0.231351
\(47\) 1.00000 0.145865
\(48\) −3.57683 −0.516271
\(49\) −6.95706 −0.993865
\(50\) 0 0
\(51\) 0.969832 0.135804
\(52\) −10.3692 −1.43795
\(53\) 8.64084 1.18691 0.593455 0.804867i \(-0.297763\pi\)
0.593455 + 0.804867i \(0.297763\pi\)
\(54\) 0.267165 0.0363566
\(55\) 0 0
\(56\) 0.217508 0.0290658
\(57\) −6.42820 −0.851436
\(58\) 2.65408 0.348498
\(59\) −9.70796 −1.26387 −0.631934 0.775022i \(-0.717739\pi\)
−0.631934 + 0.775022i \(0.717739\pi\)
\(60\) 0 0
\(61\) −10.1014 −1.29335 −0.646676 0.762765i \(-0.723841\pi\)
−0.646676 + 0.762765i \(0.723841\pi\)
\(62\) −1.59748 −0.202880
\(63\) 0.207232 0.0261087
\(64\) −6.33753 −0.792191
\(65\) 0 0
\(66\) −0.340179 −0.0418732
\(67\) −4.36482 −0.533248 −0.266624 0.963801i \(-0.585908\pi\)
−0.266624 + 0.963801i \(0.585908\pi\)
\(68\) 1.87044 0.226824
\(69\) 5.87315 0.707044
\(70\) 0 0
\(71\) 9.54225 1.13246 0.566229 0.824248i \(-0.308402\pi\)
0.566229 + 0.824248i \(0.308402\pi\)
\(72\) 1.04959 0.123695
\(73\) 11.7305 1.37295 0.686476 0.727153i \(-0.259157\pi\)
0.686476 + 0.727153i \(0.259157\pi\)
\(74\) −0.318145 −0.0369836
\(75\) 0 0
\(76\) −12.3976 −1.42210
\(77\) −0.263866 −0.0300704
\(78\) 1.43641 0.162641
\(79\) −15.7821 −1.77563 −0.887814 0.460202i \(-0.847777\pi\)
−0.887814 + 0.460202i \(0.847777\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.598611 −0.0661055
\(83\) −1.79347 −0.196859 −0.0984295 0.995144i \(-0.531382\pi\)
−0.0984295 + 0.995144i \(0.531382\pi\)
\(84\) 0.399672 0.0436077
\(85\) 0 0
\(86\) −1.65694 −0.178672
\(87\) 9.93423 1.06506
\(88\) −1.33644 −0.142465
\(89\) −7.28256 −0.771949 −0.385975 0.922509i \(-0.626135\pi\)
−0.385975 + 0.922509i \(0.626135\pi\)
\(90\) 0 0
\(91\) 1.11418 0.116797
\(92\) 11.3271 1.18093
\(93\) −5.97938 −0.620033
\(94\) −0.267165 −0.0275560
\(95\) 0 0
\(96\) 3.05479 0.311778
\(97\) 0.290073 0.0294524 0.0147262 0.999892i \(-0.495312\pi\)
0.0147262 + 0.999892i \(0.495312\pi\)
\(98\) 1.85868 0.187755
\(99\) −1.27329 −0.127971
\(100\) 0 0
\(101\) −5.63344 −0.560548 −0.280274 0.959920i \(-0.590425\pi\)
−0.280274 + 0.959920i \(0.590425\pi\)
\(102\) −0.259105 −0.0256552
\(103\) 11.3273 1.11612 0.558058 0.829802i \(-0.311547\pi\)
0.558058 + 0.829802i \(0.311547\pi\)
\(104\) 5.64311 0.553352
\(105\) 0 0
\(106\) −2.30853 −0.224224
\(107\) 8.10431 0.783474 0.391737 0.920077i \(-0.371874\pi\)
0.391737 + 0.920077i \(0.371874\pi\)
\(108\) 1.92862 0.185582
\(109\) 13.2362 1.26780 0.633899 0.773416i \(-0.281454\pi\)
0.633899 + 0.773416i \(0.281454\pi\)
\(110\) 0 0
\(111\) −1.19082 −0.113027
\(112\) 0.741232 0.0700399
\(113\) 16.4518 1.54766 0.773830 0.633394i \(-0.218339\pi\)
0.773830 + 0.633394i \(0.218339\pi\)
\(114\) 1.71739 0.160848
\(115\) 0 0
\(116\) 19.1594 1.77890
\(117\) 5.37648 0.497056
\(118\) 2.59363 0.238763
\(119\) −0.200980 −0.0184238
\(120\) 0 0
\(121\) −9.37873 −0.852611
\(122\) 2.69874 0.244333
\(123\) −2.24060 −0.202028
\(124\) −11.5320 −1.03560
\(125\) 0 0
\(126\) −0.0553651 −0.00493231
\(127\) −4.51985 −0.401072 −0.200536 0.979686i \(-0.564268\pi\)
−0.200536 + 0.979686i \(0.564268\pi\)
\(128\) 7.80274 0.689671
\(129\) −6.20192 −0.546048
\(130\) 0 0
\(131\) 20.5471 1.79520 0.897602 0.440806i \(-0.145307\pi\)
0.897602 + 0.440806i \(0.145307\pi\)
\(132\) −2.45570 −0.213741
\(133\) 1.33213 0.115510
\(134\) 1.16613 0.100738
\(135\) 0 0
\(136\) −1.01793 −0.0872865
\(137\) 14.7837 1.26306 0.631528 0.775353i \(-0.282428\pi\)
0.631528 + 0.775353i \(0.282428\pi\)
\(138\) −1.56910 −0.133571
\(139\) 18.5218 1.57100 0.785500 0.618862i \(-0.212406\pi\)
0.785500 + 0.618862i \(0.212406\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −2.54936 −0.213937
\(143\) −6.84583 −0.572477
\(144\) 3.57683 0.298069
\(145\) 0 0
\(146\) −3.13398 −0.259370
\(147\) 6.95706 0.573808
\(148\) −2.29664 −0.188783
\(149\) −1.52423 −0.124870 −0.0624349 0.998049i \(-0.519887\pi\)
−0.0624349 + 0.998049i \(0.519887\pi\)
\(150\) 0 0
\(151\) 22.2961 1.81443 0.907215 0.420666i \(-0.138204\pi\)
0.907215 + 0.420666i \(0.138204\pi\)
\(152\) 6.74698 0.547252
\(153\) −0.969832 −0.0784063
\(154\) 0.0704959 0.00568072
\(155\) 0 0
\(156\) 10.3692 0.830201
\(157\) 16.4590 1.31357 0.656787 0.754076i \(-0.271915\pi\)
0.656787 + 0.754076i \(0.271915\pi\)
\(158\) 4.21644 0.335442
\(159\) −8.64084 −0.685263
\(160\) 0 0
\(161\) −1.21710 −0.0959211
\(162\) −0.267165 −0.0209905
\(163\) 7.71004 0.603897 0.301948 0.953324i \(-0.402363\pi\)
0.301948 + 0.953324i \(0.402363\pi\)
\(164\) −4.32128 −0.337435
\(165\) 0 0
\(166\) 0.479153 0.0371895
\(167\) −10.0987 −0.781458 −0.390729 0.920506i \(-0.627777\pi\)
−0.390729 + 0.920506i \(0.627777\pi\)
\(168\) −0.217508 −0.0167811
\(169\) 15.9066 1.22358
\(170\) 0 0
\(171\) 6.42820 0.491577
\(172\) −11.9612 −0.912030
\(173\) −19.8597 −1.50990 −0.754952 0.655781i \(-0.772340\pi\)
−0.754952 + 0.655781i \(0.772340\pi\)
\(174\) −2.65408 −0.201205
\(175\) 0 0
\(176\) −4.55435 −0.343297
\(177\) 9.70796 0.729695
\(178\) 1.94565 0.145832
\(179\) −12.2276 −0.913931 −0.456966 0.889484i \(-0.651064\pi\)
−0.456966 + 0.889484i \(0.651064\pi\)
\(180\) 0 0
\(181\) −10.1323 −0.753126 −0.376563 0.926391i \(-0.622894\pi\)
−0.376563 + 0.926391i \(0.622894\pi\)
\(182\) −0.297669 −0.0220647
\(183\) 10.1014 0.746717
\(184\) −6.16441 −0.454446
\(185\) 0 0
\(186\) 1.59748 0.117133
\(187\) 1.23488 0.0903034
\(188\) −1.92862 −0.140659
\(189\) −0.207232 −0.0150739
\(190\) 0 0
\(191\) −4.90791 −0.355124 −0.177562 0.984110i \(-0.556821\pi\)
−0.177562 + 0.984110i \(0.556821\pi\)
\(192\) 6.33753 0.457372
\(193\) 3.01983 0.217372 0.108686 0.994076i \(-0.465336\pi\)
0.108686 + 0.994076i \(0.465336\pi\)
\(194\) −0.0774974 −0.00556399
\(195\) 0 0
\(196\) 13.4175 0.958395
\(197\) −7.63855 −0.544224 −0.272112 0.962266i \(-0.587722\pi\)
−0.272112 + 0.962266i \(0.587722\pi\)
\(198\) 0.340179 0.0241755
\(199\) 14.8893 1.05547 0.527737 0.849408i \(-0.323041\pi\)
0.527737 + 0.849408i \(0.323041\pi\)
\(200\) 0 0
\(201\) 4.36482 0.307871
\(202\) 1.50506 0.105896
\(203\) −2.05869 −0.144491
\(204\) −1.87044 −0.130957
\(205\) 0 0
\(206\) −3.02627 −0.210850
\(207\) −5.87315 −0.408212
\(208\) 19.2308 1.33341
\(209\) −8.18498 −0.566167
\(210\) 0 0
\(211\) −4.94144 −0.340183 −0.170091 0.985428i \(-0.554406\pi\)
−0.170091 + 0.985428i \(0.554406\pi\)
\(212\) −16.6649 −1.14455
\(213\) −9.54225 −0.653824
\(214\) −2.16519 −0.148009
\(215\) 0 0
\(216\) −1.04959 −0.0714156
\(217\) 1.23912 0.0841167
\(218\) −3.53625 −0.239505
\(219\) −11.7305 −0.792674
\(220\) 0 0
\(221\) −5.21428 −0.350751
\(222\) 0.318145 0.0213525
\(223\) −18.3690 −1.23008 −0.615041 0.788495i \(-0.710860\pi\)
−0.615041 + 0.788495i \(0.710860\pi\)
\(224\) −0.633048 −0.0422973
\(225\) 0 0
\(226\) −4.39536 −0.292375
\(227\) 29.0700 1.92944 0.964721 0.263274i \(-0.0848024\pi\)
0.964721 + 0.263274i \(0.0848024\pi\)
\(228\) 12.3976 0.821049
\(229\) −0.499997 −0.0330407 −0.0165204 0.999864i \(-0.505259\pi\)
−0.0165204 + 0.999864i \(0.505259\pi\)
\(230\) 0 0
\(231\) 0.263866 0.0173611
\(232\) −10.4269 −0.684558
\(233\) 27.0992 1.77533 0.887663 0.460494i \(-0.152328\pi\)
0.887663 + 0.460494i \(0.152328\pi\)
\(234\) −1.43641 −0.0939010
\(235\) 0 0
\(236\) 18.7230 1.21876
\(237\) 15.7821 1.02516
\(238\) 0.0536948 0.00348052
\(239\) 25.1215 1.62497 0.812487 0.582979i \(-0.198113\pi\)
0.812487 + 0.582979i \(0.198113\pi\)
\(240\) 0 0
\(241\) −7.98036 −0.514060 −0.257030 0.966403i \(-0.582744\pi\)
−0.257030 + 0.966403i \(0.582744\pi\)
\(242\) 2.50567 0.161070
\(243\) −1.00000 −0.0641500
\(244\) 19.4818 1.24719
\(245\) 0 0
\(246\) 0.598611 0.0381660
\(247\) 34.5611 2.19907
\(248\) 6.27590 0.398520
\(249\) 1.79347 0.113657
\(250\) 0 0
\(251\) −29.7735 −1.87929 −0.939645 0.342151i \(-0.888844\pi\)
−0.939645 + 0.342151i \(0.888844\pi\)
\(252\) −0.399672 −0.0251769
\(253\) 7.47824 0.470153
\(254\) 1.20755 0.0757682
\(255\) 0 0
\(256\) 10.5904 0.661902
\(257\) −3.24067 −0.202148 −0.101074 0.994879i \(-0.532228\pi\)
−0.101074 + 0.994879i \(0.532228\pi\)
\(258\) 1.65694 0.103156
\(259\) 0.246775 0.0153339
\(260\) 0 0
\(261\) −9.93423 −0.614914
\(262\) −5.48946 −0.339140
\(263\) 14.9421 0.921367 0.460684 0.887564i \(-0.347604\pi\)
0.460684 + 0.887564i \(0.347604\pi\)
\(264\) 1.33644 0.0822520
\(265\) 0 0
\(266\) −0.355898 −0.0218215
\(267\) 7.28256 0.445685
\(268\) 8.41809 0.514217
\(269\) 12.7592 0.777941 0.388970 0.921250i \(-0.372831\pi\)
0.388970 + 0.921250i \(0.372831\pi\)
\(270\) 0 0
\(271\) 9.13747 0.555062 0.277531 0.960717i \(-0.410484\pi\)
0.277531 + 0.960717i \(0.410484\pi\)
\(272\) −3.46893 −0.210334
\(273\) −1.11418 −0.0674331
\(274\) −3.94968 −0.238609
\(275\) 0 0
\(276\) −11.3271 −0.681811
\(277\) 3.04812 0.183144 0.0915720 0.995798i \(-0.470811\pi\)
0.0915720 + 0.995798i \(0.470811\pi\)
\(278\) −4.94838 −0.296784
\(279\) 5.97938 0.357976
\(280\) 0 0
\(281\) −8.19048 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(282\) 0.267165 0.0159095
\(283\) 6.63076 0.394158 0.197079 0.980388i \(-0.436854\pi\)
0.197079 + 0.980388i \(0.436854\pi\)
\(284\) −18.4034 −1.09204
\(285\) 0 0
\(286\) 1.82897 0.108149
\(287\) 0.464324 0.0274082
\(288\) −3.05479 −0.180005
\(289\) −16.0594 −0.944672
\(290\) 0 0
\(291\) −0.290073 −0.0170044
\(292\) −22.6237 −1.32395
\(293\) −18.2716 −1.06744 −0.533719 0.845662i \(-0.679206\pi\)
−0.533719 + 0.845662i \(0.679206\pi\)
\(294\) −1.85868 −0.108401
\(295\) 0 0
\(296\) 1.24987 0.0726473
\(297\) 1.27329 0.0738839
\(298\) 0.407221 0.0235897
\(299\) −31.5769 −1.82614
\(300\) 0 0
\(301\) 1.28523 0.0740796
\(302\) −5.95674 −0.342772
\(303\) 5.63344 0.323633
\(304\) 22.9926 1.31872
\(305\) 0 0
\(306\) 0.259105 0.0148121
\(307\) 18.1634 1.03664 0.518319 0.855187i \(-0.326558\pi\)
0.518319 + 0.855187i \(0.326558\pi\)
\(308\) 0.508899 0.0289972
\(309\) −11.3273 −0.644389
\(310\) 0 0
\(311\) 28.0180 1.58876 0.794379 0.607423i \(-0.207797\pi\)
0.794379 + 0.607423i \(0.207797\pi\)
\(312\) −5.64311 −0.319478
\(313\) 12.6852 0.717009 0.358504 0.933528i \(-0.383287\pi\)
0.358504 + 0.933528i \(0.383287\pi\)
\(314\) −4.39728 −0.248153
\(315\) 0 0
\(316\) 30.4378 1.71226
\(317\) 34.8739 1.95871 0.979355 0.202146i \(-0.0647915\pi\)
0.979355 + 0.202146i \(0.0647915\pi\)
\(318\) 2.30853 0.129456
\(319\) 12.6492 0.708218
\(320\) 0 0
\(321\) −8.10431 −0.452339
\(322\) 0.325167 0.0181209
\(323\) −6.23427 −0.346884
\(324\) −1.92862 −0.107146
\(325\) 0 0
\(326\) −2.05985 −0.114085
\(327\) −13.2362 −0.731963
\(328\) 2.35172 0.129852
\(329\) 0.207232 0.0114250
\(330\) 0 0
\(331\) 20.3577 1.11896 0.559480 0.828844i \(-0.311001\pi\)
0.559480 + 0.828844i \(0.311001\pi\)
\(332\) 3.45893 0.189833
\(333\) 1.19082 0.0652564
\(334\) 2.69801 0.147629
\(335\) 0 0
\(336\) −0.741232 −0.0404375
\(337\) −29.3333 −1.59789 −0.798943 0.601407i \(-0.794607\pi\)
−0.798943 + 0.601407i \(0.794607\pi\)
\(338\) −4.24968 −0.231152
\(339\) −16.4518 −0.893542
\(340\) 0 0
\(341\) −7.61350 −0.412294
\(342\) −1.71739 −0.0928659
\(343\) −2.89234 −0.156172
\(344\) 6.50948 0.350967
\(345\) 0 0
\(346\) 5.30581 0.285242
\(347\) 29.5486 1.58625 0.793127 0.609057i \(-0.208452\pi\)
0.793127 + 0.609057i \(0.208452\pi\)
\(348\) −19.1594 −1.02705
\(349\) 0.661771 0.0354238 0.0177119 0.999843i \(-0.494362\pi\)
0.0177119 + 0.999843i \(0.494362\pi\)
\(350\) 0 0
\(351\) −5.37648 −0.286975
\(352\) 3.88964 0.207318
\(353\) −4.99309 −0.265756 −0.132878 0.991132i \(-0.542422\pi\)
−0.132878 + 0.991132i \(0.542422\pi\)
\(354\) −2.59363 −0.137850
\(355\) 0 0
\(356\) 14.0453 0.744400
\(357\) 0.200980 0.0106370
\(358\) 3.26678 0.172655
\(359\) −17.0698 −0.900910 −0.450455 0.892799i \(-0.648738\pi\)
−0.450455 + 0.892799i \(0.648738\pi\)
\(360\) 0 0
\(361\) 22.3218 1.17483
\(362\) 2.70699 0.142276
\(363\) 9.37873 0.492255
\(364\) −2.14883 −0.112629
\(365\) 0 0
\(366\) −2.69874 −0.141065
\(367\) 6.07188 0.316949 0.158475 0.987363i \(-0.449342\pi\)
0.158475 + 0.987363i \(0.449342\pi\)
\(368\) −21.0073 −1.09508
\(369\) 2.24060 0.116641
\(370\) 0 0
\(371\) 1.79065 0.0929661
\(372\) 11.5320 0.597905
\(373\) 19.3121 0.999944 0.499972 0.866042i \(-0.333344\pi\)
0.499972 + 0.866042i \(0.333344\pi\)
\(374\) −0.329917 −0.0170596
\(375\) 0 0
\(376\) 1.04959 0.0541285
\(377\) −53.4112 −2.75082
\(378\) 0.0553651 0.00284767
\(379\) −4.68181 −0.240488 −0.120244 0.992744i \(-0.538368\pi\)
−0.120244 + 0.992744i \(0.538368\pi\)
\(380\) 0 0
\(381\) 4.51985 0.231559
\(382\) 1.31122 0.0670880
\(383\) −5.78171 −0.295432 −0.147716 0.989030i \(-0.547192\pi\)
−0.147716 + 0.989030i \(0.547192\pi\)
\(384\) −7.80274 −0.398182
\(385\) 0 0
\(386\) −0.806794 −0.0410647
\(387\) 6.20192 0.315261
\(388\) −0.559441 −0.0284013
\(389\) 1.93198 0.0979553 0.0489777 0.998800i \(-0.484404\pi\)
0.0489777 + 0.998800i \(0.484404\pi\)
\(390\) 0 0
\(391\) 5.69597 0.288058
\(392\) −7.30206 −0.368810
\(393\) −20.5471 −1.03646
\(394\) 2.04076 0.102812
\(395\) 0 0
\(396\) 2.45570 0.123404
\(397\) −30.9953 −1.55561 −0.777805 0.628506i \(-0.783667\pi\)
−0.777805 + 0.628506i \(0.783667\pi\)
\(398\) −3.97790 −0.199394
\(399\) −1.33213 −0.0666897
\(400\) 0 0
\(401\) 36.2628 1.81088 0.905438 0.424478i \(-0.139542\pi\)
0.905438 + 0.424478i \(0.139542\pi\)
\(402\) −1.16613 −0.0581612
\(403\) 32.1480 1.60141
\(404\) 10.8648 0.540543
\(405\) 0 0
\(406\) 0.550009 0.0272965
\(407\) −1.51626 −0.0751582
\(408\) 1.01793 0.0503949
\(409\) −0.0369376 −0.00182645 −0.000913223 1.00000i \(-0.500291\pi\)
−0.000913223 1.00000i \(0.500291\pi\)
\(410\) 0 0
\(411\) −14.7837 −0.729225
\(412\) −21.8462 −1.07628
\(413\) −2.01180 −0.0989940
\(414\) 1.56910 0.0771171
\(415\) 0 0
\(416\) −16.4240 −0.805253
\(417\) −18.5218 −0.907017
\(418\) 2.18674 0.106957
\(419\) 0.362309 0.0176999 0.00884997 0.999961i \(-0.497183\pi\)
0.00884997 + 0.999961i \(0.497183\pi\)
\(420\) 0 0
\(421\) 12.6406 0.616063 0.308032 0.951376i \(-0.400330\pi\)
0.308032 + 0.951376i \(0.400330\pi\)
\(422\) 1.32018 0.0642654
\(423\) 1.00000 0.0486217
\(424\) 9.06934 0.440446
\(425\) 0 0
\(426\) 2.54936 0.123517
\(427\) −2.09333 −0.101303
\(428\) −15.6302 −0.755513
\(429\) 6.84583 0.330520
\(430\) 0 0
\(431\) 6.98498 0.336455 0.168227 0.985748i \(-0.446196\pi\)
0.168227 + 0.985748i \(0.446196\pi\)
\(432\) −3.57683 −0.172090
\(433\) −22.4312 −1.07797 −0.538987 0.842314i \(-0.681193\pi\)
−0.538987 + 0.842314i \(0.681193\pi\)
\(434\) −0.331049 −0.0158908
\(435\) 0 0
\(436\) −25.5276 −1.22255
\(437\) −37.7538 −1.80601
\(438\) 3.13398 0.149747
\(439\) 19.9828 0.953726 0.476863 0.878978i \(-0.341774\pi\)
0.476863 + 0.878978i \(0.341774\pi\)
\(440\) 0 0
\(441\) −6.95706 −0.331288
\(442\) 1.39307 0.0662618
\(443\) 0.382306 0.0181639 0.00908196 0.999959i \(-0.497109\pi\)
0.00908196 + 0.999959i \(0.497109\pi\)
\(444\) 2.29664 0.108994
\(445\) 0 0
\(446\) 4.90756 0.232380
\(447\) 1.52423 0.0720936
\(448\) −1.31334 −0.0620493
\(449\) 33.6928 1.59006 0.795030 0.606570i \(-0.207455\pi\)
0.795030 + 0.606570i \(0.207455\pi\)
\(450\) 0 0
\(451\) −2.85294 −0.134340
\(452\) −31.7294 −1.49243
\(453\) −22.2961 −1.04756
\(454\) −7.76648 −0.364499
\(455\) 0 0
\(456\) −6.74698 −0.315956
\(457\) 10.9263 0.511110 0.255555 0.966795i \(-0.417742\pi\)
0.255555 + 0.966795i \(0.417742\pi\)
\(458\) 0.133582 0.00624186
\(459\) 0.969832 0.0452679
\(460\) 0 0
\(461\) −35.9233 −1.67311 −0.836557 0.547880i \(-0.815435\pi\)
−0.836557 + 0.547880i \(0.815435\pi\)
\(462\) −0.0704959 −0.00327977
\(463\) −25.5389 −1.18690 −0.593448 0.804873i \(-0.702234\pi\)
−0.593448 + 0.804873i \(0.702234\pi\)
\(464\) −35.5331 −1.64958
\(465\) 0 0
\(466\) −7.23996 −0.335384
\(467\) 27.9327 1.29257 0.646285 0.763096i \(-0.276322\pi\)
0.646285 + 0.763096i \(0.276322\pi\)
\(468\) −10.3692 −0.479317
\(469\) −0.904529 −0.0417673
\(470\) 0 0
\(471\) −16.4590 −0.758393
\(472\) −10.1894 −0.469005
\(473\) −7.89685 −0.363098
\(474\) −4.21644 −0.193667
\(475\) 0 0
\(476\) 0.387614 0.0177663
\(477\) 8.64084 0.395637
\(478\) −6.71159 −0.306981
\(479\) 17.2921 0.790096 0.395048 0.918660i \(-0.370728\pi\)
0.395048 + 0.918660i \(0.370728\pi\)
\(480\) 0 0
\(481\) 6.40241 0.291925
\(482\) 2.13207 0.0971133
\(483\) 1.21710 0.0553801
\(484\) 18.0880 0.822183
\(485\) 0 0
\(486\) 0.267165 0.0121189
\(487\) 5.99955 0.271866 0.135933 0.990718i \(-0.456597\pi\)
0.135933 + 0.990718i \(0.456597\pi\)
\(488\) −10.6023 −0.479945
\(489\) −7.71004 −0.348660
\(490\) 0 0
\(491\) −21.8555 −0.986323 −0.493161 0.869938i \(-0.664159\pi\)
−0.493161 + 0.869938i \(0.664159\pi\)
\(492\) 4.32128 0.194818
\(493\) 9.63454 0.433918
\(494\) −9.23352 −0.415436
\(495\) 0 0
\(496\) 21.3872 0.960315
\(497\) 1.97746 0.0887010
\(498\) −0.479153 −0.0214713
\(499\) 34.1218 1.52750 0.763752 0.645510i \(-0.223355\pi\)
0.763752 + 0.645510i \(0.223355\pi\)
\(500\) 0 0
\(501\) 10.0987 0.451175
\(502\) 7.95445 0.355025
\(503\) 17.7971 0.793533 0.396767 0.917919i \(-0.370132\pi\)
0.396767 + 0.917919i \(0.370132\pi\)
\(504\) 0.217508 0.00968860
\(505\) 0 0
\(506\) −1.99793 −0.0888186
\(507\) −15.9066 −0.706435
\(508\) 8.71709 0.386758
\(509\) −21.7120 −0.962369 −0.481184 0.876619i \(-0.659793\pi\)
−0.481184 + 0.876619i \(0.659793\pi\)
\(510\) 0 0
\(511\) 2.43093 0.107538
\(512\) −18.4349 −0.814714
\(513\) −6.42820 −0.283812
\(514\) 0.865795 0.0381886
\(515\) 0 0
\(516\) 11.9612 0.526561
\(517\) −1.27329 −0.0559993
\(518\) −0.0659297 −0.00289678
\(519\) 19.8597 0.871743
\(520\) 0 0
\(521\) −22.2821 −0.976197 −0.488099 0.872788i \(-0.662309\pi\)
−0.488099 + 0.872788i \(0.662309\pi\)
\(522\) 2.65408 0.116166
\(523\) 9.96039 0.435538 0.217769 0.976000i \(-0.430122\pi\)
0.217769 + 0.976000i \(0.430122\pi\)
\(524\) −39.6275 −1.73114
\(525\) 0 0
\(526\) −3.99200 −0.174059
\(527\) −5.79899 −0.252608
\(528\) 4.55435 0.198203
\(529\) 11.4939 0.499736
\(530\) 0 0
\(531\) −9.70796 −0.421290
\(532\) −2.56917 −0.111388
\(533\) 12.0466 0.521794
\(534\) −1.94565 −0.0841963
\(535\) 0 0
\(536\) −4.58128 −0.197881
\(537\) 12.2276 0.527658
\(538\) −3.40881 −0.146964
\(539\) 8.85837 0.381557
\(540\) 0 0
\(541\) −4.70353 −0.202221 −0.101110 0.994875i \(-0.532240\pi\)
−0.101110 + 0.994875i \(0.532240\pi\)
\(542\) −2.44121 −0.104859
\(543\) 10.1323 0.434818
\(544\) 2.96263 0.127022
\(545\) 0 0
\(546\) 0.297669 0.0127391
\(547\) −41.3316 −1.76721 −0.883605 0.468233i \(-0.844891\pi\)
−0.883605 + 0.468233i \(0.844891\pi\)
\(548\) −28.5121 −1.21798
\(549\) −10.1014 −0.431117
\(550\) 0 0
\(551\) −63.8592 −2.72049
\(552\) 6.16441 0.262375
\(553\) −3.27056 −0.139078
\(554\) −0.814352 −0.0345985
\(555\) 0 0
\(556\) −35.7216 −1.51493
\(557\) 43.0480 1.82400 0.912001 0.410188i \(-0.134537\pi\)
0.912001 + 0.410188i \(0.134537\pi\)
\(558\) −1.59748 −0.0676268
\(559\) 33.3445 1.41032
\(560\) 0 0
\(561\) −1.23488 −0.0521367
\(562\) 2.18821 0.0923041
\(563\) 32.3647 1.36401 0.682006 0.731347i \(-0.261108\pi\)
0.682006 + 0.731347i \(0.261108\pi\)
\(564\) 1.92862 0.0812097
\(565\) 0 0
\(566\) −1.77151 −0.0744620
\(567\) 0.207232 0.00870291
\(568\) 10.0155 0.420240
\(569\) −34.4813 −1.44553 −0.722766 0.691093i \(-0.757129\pi\)
−0.722766 + 0.691093i \(0.757129\pi\)
\(570\) 0 0
\(571\) 36.6501 1.53376 0.766879 0.641792i \(-0.221809\pi\)
0.766879 + 0.641792i \(0.221809\pi\)
\(572\) 13.2030 0.552047
\(573\) 4.90791 0.205031
\(574\) −0.124051 −0.00517779
\(575\) 0 0
\(576\) −6.33753 −0.264064
\(577\) 2.62010 0.109076 0.0545381 0.998512i \(-0.482631\pi\)
0.0545381 + 0.998512i \(0.482631\pi\)
\(578\) 4.29052 0.178462
\(579\) −3.01983 −0.125500
\(580\) 0 0
\(581\) −0.371664 −0.0154192
\(582\) 0.0774974 0.00321237
\(583\) −11.0023 −0.455669
\(584\) 12.3122 0.509484
\(585\) 0 0
\(586\) 4.88154 0.201654
\(587\) 2.63615 0.108806 0.0544028 0.998519i \(-0.482674\pi\)
0.0544028 + 0.998519i \(0.482674\pi\)
\(588\) −13.4175 −0.553330
\(589\) 38.4366 1.58375
\(590\) 0 0
\(591\) 7.63855 0.314208
\(592\) 4.25935 0.175058
\(593\) 7.63049 0.313347 0.156673 0.987650i \(-0.449923\pi\)
0.156673 + 0.987650i \(0.449923\pi\)
\(594\) −0.340179 −0.0139577
\(595\) 0 0
\(596\) 2.93967 0.120413
\(597\) −14.8893 −0.609378
\(598\) 8.43625 0.344984
\(599\) 20.3053 0.829650 0.414825 0.909901i \(-0.363843\pi\)
0.414825 + 0.909901i \(0.363843\pi\)
\(600\) 0 0
\(601\) −1.07475 −0.0438398 −0.0219199 0.999760i \(-0.506978\pi\)
−0.0219199 + 0.999760i \(0.506978\pi\)
\(602\) −0.343369 −0.0139947
\(603\) −4.36482 −0.177749
\(604\) −43.0008 −1.74968
\(605\) 0 0
\(606\) −1.50506 −0.0611389
\(607\) −17.0385 −0.691573 −0.345786 0.938313i \(-0.612388\pi\)
−0.345786 + 0.938313i \(0.612388\pi\)
\(608\) −19.6368 −0.796377
\(609\) 2.05869 0.0834222
\(610\) 0 0
\(611\) 5.37648 0.217509
\(612\) 1.87044 0.0756081
\(613\) −36.6117 −1.47873 −0.739366 0.673304i \(-0.764874\pi\)
−0.739366 + 0.673304i \(0.764874\pi\)
\(614\) −4.85262 −0.195836
\(615\) 0 0
\(616\) −0.276952 −0.0111587
\(617\) −23.1682 −0.932716 −0.466358 0.884596i \(-0.654434\pi\)
−0.466358 + 0.884596i \(0.654434\pi\)
\(618\) 3.02627 0.121734
\(619\) −26.4232 −1.06204 −0.531018 0.847360i \(-0.678190\pi\)
−0.531018 + 0.847360i \(0.678190\pi\)
\(620\) 0 0
\(621\) 5.87315 0.235681
\(622\) −7.48545 −0.300139
\(623\) −1.50918 −0.0604639
\(624\) −19.2308 −0.769847
\(625\) 0 0
\(626\) −3.38904 −0.135453
\(627\) 8.18498 0.326877
\(628\) −31.7433 −1.26669
\(629\) −1.15489 −0.0460486
\(630\) 0 0
\(631\) −42.2696 −1.68273 −0.841363 0.540471i \(-0.818246\pi\)
−0.841363 + 0.540471i \(0.818246\pi\)
\(632\) −16.5648 −0.658912
\(633\) 4.94144 0.196405
\(634\) −9.31708 −0.370028
\(635\) 0 0
\(636\) 16.6649 0.660807
\(637\) −37.4045 −1.48202
\(638\) −3.37942 −0.133793
\(639\) 9.54225 0.377486
\(640\) 0 0
\(641\) −2.47629 −0.0978077 −0.0489039 0.998803i \(-0.515573\pi\)
−0.0489039 + 0.998803i \(0.515573\pi\)
\(642\) 2.16519 0.0854532
\(643\) −38.6391 −1.52378 −0.761890 0.647707i \(-0.775728\pi\)
−0.761890 + 0.647707i \(0.775728\pi\)
\(644\) 2.34733 0.0924978
\(645\) 0 0
\(646\) 1.66558 0.0655314
\(647\) −8.32898 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(648\) 1.04959 0.0412318
\(649\) 12.3611 0.485215
\(650\) 0 0
\(651\) −1.23912 −0.0485648
\(652\) −14.8698 −0.582345
\(653\) −18.8800 −0.738832 −0.369416 0.929264i \(-0.620442\pi\)
−0.369416 + 0.929264i \(0.620442\pi\)
\(654\) 3.53625 0.138278
\(655\) 0 0
\(656\) 8.01426 0.312904
\(657\) 11.7305 0.457651
\(658\) −0.0553651 −0.00215835
\(659\) −13.9987 −0.545311 −0.272656 0.962112i \(-0.587902\pi\)
−0.272656 + 0.962112i \(0.587902\pi\)
\(660\) 0 0
\(661\) 5.55730 0.216154 0.108077 0.994143i \(-0.465531\pi\)
0.108077 + 0.994143i \(0.465531\pi\)
\(662\) −5.43887 −0.211388
\(663\) 5.21428 0.202506
\(664\) −1.88241 −0.0730517
\(665\) 0 0
\(666\) −0.318145 −0.0123279
\(667\) 58.3453 2.25914
\(668\) 19.4765 0.753569
\(669\) 18.3690 0.710188
\(670\) 0 0
\(671\) 12.8620 0.496533
\(672\) 0.633048 0.0244204
\(673\) 31.0565 1.19714 0.598569 0.801071i \(-0.295736\pi\)
0.598569 + 0.801071i \(0.295736\pi\)
\(674\) 7.83683 0.301863
\(675\) 0 0
\(676\) −30.6777 −1.17991
\(677\) 21.7287 0.835101 0.417550 0.908654i \(-0.362889\pi\)
0.417550 + 0.908654i \(0.362889\pi\)
\(678\) 4.39536 0.168803
\(679\) 0.0601123 0.00230690
\(680\) 0 0
\(681\) −29.0700 −1.11396
\(682\) 2.03406 0.0778882
\(683\) −1.85083 −0.0708202 −0.0354101 0.999373i \(-0.511274\pi\)
−0.0354101 + 0.999373i \(0.511274\pi\)
\(684\) −12.3976 −0.474033
\(685\) 0 0
\(686\) 0.772733 0.0295031
\(687\) 0.499997 0.0190761
\(688\) 22.1832 0.845727
\(689\) 46.4573 1.76988
\(690\) 0 0
\(691\) 6.56714 0.249826 0.124913 0.992168i \(-0.460135\pi\)
0.124913 + 0.992168i \(0.460135\pi\)
\(692\) 38.3018 1.45602
\(693\) −0.263866 −0.0100235
\(694\) −7.89436 −0.299666
\(695\) 0 0
\(696\) 10.4269 0.395230
\(697\) −2.17301 −0.0823085
\(698\) −0.176802 −0.00669206
\(699\) −27.0992 −1.02498
\(700\) 0 0
\(701\) −28.3117 −1.06932 −0.534659 0.845068i \(-0.679560\pi\)
−0.534659 + 0.845068i \(0.679560\pi\)
\(702\) 1.43641 0.0542137
\(703\) 7.65482 0.288707
\(704\) 8.06953 0.304132
\(705\) 0 0
\(706\) 1.33398 0.0502050
\(707\) −1.16743 −0.0439056
\(708\) −18.7230 −0.703653
\(709\) 18.8498 0.707920 0.353960 0.935260i \(-0.384835\pi\)
0.353960 + 0.935260i \(0.384835\pi\)
\(710\) 0 0
\(711\) −15.7821 −0.591876
\(712\) −7.64371 −0.286460
\(713\) −35.1178 −1.31517
\(714\) −0.0536948 −0.00200948
\(715\) 0 0
\(716\) 23.5824 0.881314
\(717\) −25.1215 −0.938179
\(718\) 4.56046 0.170195
\(719\) −21.6400 −0.807036 −0.403518 0.914972i \(-0.632213\pi\)
−0.403518 + 0.914972i \(0.632213\pi\)
\(720\) 0 0
\(721\) 2.34738 0.0874210
\(722\) −5.96360 −0.221942
\(723\) 7.98036 0.296793
\(724\) 19.5413 0.726248
\(725\) 0 0
\(726\) −2.50567 −0.0929941
\(727\) 2.30732 0.0855737 0.0427869 0.999084i \(-0.486376\pi\)
0.0427869 + 0.999084i \(0.486376\pi\)
\(728\) 1.16943 0.0433420
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.01482 −0.222466
\(732\) −19.4818 −0.720068
\(733\) −18.9481 −0.699863 −0.349932 0.936775i \(-0.613795\pi\)
−0.349932 + 0.936775i \(0.613795\pi\)
\(734\) −1.62219 −0.0598763
\(735\) 0 0
\(736\) 17.9412 0.661322
\(737\) 5.55769 0.204720
\(738\) −0.598611 −0.0220352
\(739\) 18.6083 0.684519 0.342259 0.939605i \(-0.388808\pi\)
0.342259 + 0.939605i \(0.388808\pi\)
\(740\) 0 0
\(741\) −34.5611 −1.26963
\(742\) −0.478400 −0.0175626
\(743\) −4.08148 −0.149735 −0.0748676 0.997193i \(-0.523853\pi\)
−0.0748676 + 0.997193i \(0.523853\pi\)
\(744\) −6.27590 −0.230086
\(745\) 0 0
\(746\) −5.15953 −0.188904
\(747\) −1.79347 −0.0656196
\(748\) −2.38162 −0.0870806
\(749\) 1.67947 0.0613665
\(750\) 0 0
\(751\) −0.378534 −0.0138129 −0.00690645 0.999976i \(-0.502198\pi\)
−0.00690645 + 0.999976i \(0.502198\pi\)
\(752\) 3.57683 0.130434
\(753\) 29.7735 1.08501
\(754\) 14.2696 0.519669
\(755\) 0 0
\(756\) 0.399672 0.0145359
\(757\) 24.9727 0.907649 0.453825 0.891091i \(-0.350059\pi\)
0.453825 + 0.891091i \(0.350059\pi\)
\(758\) 1.25082 0.0454317
\(759\) −7.47824 −0.271443
\(760\) 0 0
\(761\) −13.0020 −0.471322 −0.235661 0.971835i \(-0.575725\pi\)
−0.235661 + 0.971835i \(0.575725\pi\)
\(762\) −1.20755 −0.0437448
\(763\) 2.74296 0.0993018
\(764\) 9.46551 0.342450
\(765\) 0 0
\(766\) 1.54467 0.0558113
\(767\) −52.1947 −1.88464
\(768\) −10.5904 −0.382150
\(769\) 30.0353 1.08310 0.541550 0.840669i \(-0.317838\pi\)
0.541550 + 0.840669i \(0.317838\pi\)
\(770\) 0 0
\(771\) 3.24067 0.116710
\(772\) −5.82412 −0.209615
\(773\) −40.8899 −1.47071 −0.735354 0.677683i \(-0.762984\pi\)
−0.735354 + 0.677683i \(0.762984\pi\)
\(774\) −1.65694 −0.0595573
\(775\) 0 0
\(776\) 0.304458 0.0109294
\(777\) −0.246775 −0.00885301
\(778\) −0.516158 −0.0185052
\(779\) 14.4030 0.516042
\(780\) 0 0
\(781\) −12.1501 −0.434764
\(782\) −1.52176 −0.0544182
\(783\) 9.93423 0.355021
\(784\) −24.8842 −0.888722
\(785\) 0 0
\(786\) 5.48946 0.195802
\(787\) 11.3769 0.405543 0.202772 0.979226i \(-0.435005\pi\)
0.202772 + 0.979226i \(0.435005\pi\)
\(788\) 14.7319 0.524802
\(789\) −14.9421 −0.531952
\(790\) 0 0
\(791\) 3.40934 0.121222
\(792\) −1.33644 −0.0474882
\(793\) −54.3100 −1.92860
\(794\) 8.28086 0.293877
\(795\) 0 0
\(796\) −28.7158 −1.01780
\(797\) 7.09623 0.251361 0.125681 0.992071i \(-0.459889\pi\)
0.125681 + 0.992071i \(0.459889\pi\)
\(798\) 0.355898 0.0125986
\(799\) −0.969832 −0.0343102
\(800\) 0 0
\(801\) −7.28256 −0.257316
\(802\) −9.68815 −0.342100
\(803\) −14.9364 −0.527093
\(804\) −8.41809 −0.296883
\(805\) 0 0
\(806\) −8.58883 −0.302529
\(807\) −12.7592 −0.449144
\(808\) −5.91281 −0.208012
\(809\) 11.6552 0.409776 0.204888 0.978785i \(-0.434317\pi\)
0.204888 + 0.978785i \(0.434317\pi\)
\(810\) 0 0
\(811\) 0.0224294 0.000787602 0 0.000393801 1.00000i \(-0.499875\pi\)
0.000393801 1.00000i \(0.499875\pi\)
\(812\) 3.97043 0.139335
\(813\) −9.13747 −0.320465
\(814\) 0.405092 0.0141985
\(815\) 0 0
\(816\) 3.46893 0.121437
\(817\) 39.8672 1.39478
\(818\) 0.00986844 0.000345042 0
\(819\) 1.11418 0.0389325
\(820\) 0 0
\(821\) −41.9487 −1.46402 −0.732009 0.681294i \(-0.761417\pi\)
−0.732009 + 0.681294i \(0.761417\pi\)
\(822\) 3.94968 0.137761
\(823\) −30.5936 −1.06643 −0.533213 0.845981i \(-0.679015\pi\)
−0.533213 + 0.845981i \(0.679015\pi\)
\(824\) 11.8891 0.414175
\(825\) 0 0
\(826\) 0.537482 0.0187014
\(827\) 33.5561 1.16686 0.583430 0.812163i \(-0.301710\pi\)
0.583430 + 0.812163i \(0.301710\pi\)
\(828\) 11.3271 0.393644
\(829\) 33.2808 1.15589 0.577945 0.816076i \(-0.303855\pi\)
0.577945 + 0.816076i \(0.303855\pi\)
\(830\) 0 0
\(831\) −3.04812 −0.105738
\(832\) −34.0736 −1.18129
\(833\) 6.74717 0.233776
\(834\) 4.94838 0.171348
\(835\) 0 0
\(836\) 15.7857 0.545961
\(837\) −5.97938 −0.206678
\(838\) −0.0967963 −0.00334377
\(839\) 30.8908 1.06647 0.533235 0.845967i \(-0.320976\pi\)
0.533235 + 0.845967i \(0.320976\pi\)
\(840\) 0 0
\(841\) 69.6890 2.40307
\(842\) −3.37712 −0.116383
\(843\) 8.19048 0.282095
\(844\) 9.53018 0.328042
\(845\) 0 0
\(846\) −0.267165 −0.00918533
\(847\) −1.94357 −0.0667818
\(848\) 30.9068 1.06134
\(849\) −6.63076 −0.227567
\(850\) 0 0
\(851\) −6.99385 −0.239746
\(852\) 18.4034 0.630490
\(853\) −11.9094 −0.407772 −0.203886 0.978995i \(-0.565357\pi\)
−0.203886 + 0.978995i \(0.565357\pi\)
\(854\) 0.559264 0.0191376
\(855\) 0 0
\(856\) 8.50622 0.290736
\(857\) 1.29588 0.0442665 0.0221332 0.999755i \(-0.492954\pi\)
0.0221332 + 0.999755i \(0.492954\pi\)
\(858\) −1.82897 −0.0624400
\(859\) 15.9383 0.543808 0.271904 0.962324i \(-0.412347\pi\)
0.271904 + 0.962324i \(0.412347\pi\)
\(860\) 0 0
\(861\) −0.464324 −0.0158241
\(862\) −1.86614 −0.0635611
\(863\) 4.41860 0.150411 0.0752054 0.997168i \(-0.476039\pi\)
0.0752054 + 0.997168i \(0.476039\pi\)
\(864\) 3.05479 0.103926
\(865\) 0 0
\(866\) 5.99283 0.203645
\(867\) 16.0594 0.545407
\(868\) −2.38979 −0.0811147
\(869\) 20.0953 0.681685
\(870\) 0 0
\(871\) −23.4674 −0.795162
\(872\) 13.8926 0.470463
\(873\) 0.290073 0.00981748
\(874\) 10.0865 0.341181
\(875\) 0 0
\(876\) 22.6237 0.764385
\(877\) −24.5208 −0.828007 −0.414004 0.910275i \(-0.635870\pi\)
−0.414004 + 0.910275i \(0.635870\pi\)
\(878\) −5.33870 −0.180172
\(879\) 18.2716 0.616286
\(880\) 0 0
\(881\) 28.7702 0.969293 0.484646 0.874710i \(-0.338948\pi\)
0.484646 + 0.874710i \(0.338948\pi\)
\(882\) 1.85868 0.0625851
\(883\) −10.7916 −0.363167 −0.181584 0.983375i \(-0.558122\pi\)
−0.181584 + 0.983375i \(0.558122\pi\)
\(884\) 10.0564 0.338233
\(885\) 0 0
\(886\) −0.102139 −0.00343142
\(887\) 30.3740 1.01986 0.509930 0.860216i \(-0.329671\pi\)
0.509930 + 0.860216i \(0.329671\pi\)
\(888\) −1.24987 −0.0419430
\(889\) −0.936656 −0.0314144
\(890\) 0 0
\(891\) −1.27329 −0.0426569
\(892\) 35.4269 1.18618
\(893\) 6.42820 0.215112
\(894\) −0.407221 −0.0136195
\(895\) 0 0
\(896\) 1.61697 0.0540193
\(897\) 31.5769 1.05432
\(898\) −9.00153 −0.300385
\(899\) −59.4005 −1.98112
\(900\) 0 0
\(901\) −8.38016 −0.279184
\(902\) 0.762207 0.0253787
\(903\) −1.28523 −0.0427699
\(904\) 17.2677 0.574316
\(905\) 0 0
\(906\) 5.95674 0.197899
\(907\) 34.6143 1.14935 0.574674 0.818382i \(-0.305129\pi\)
0.574674 + 0.818382i \(0.305129\pi\)
\(908\) −56.0650 −1.86058
\(909\) −5.63344 −0.186849
\(910\) 0 0
\(911\) 27.9342 0.925502 0.462751 0.886488i \(-0.346862\pi\)
0.462751 + 0.886488i \(0.346862\pi\)
\(912\) −22.9926 −0.761361
\(913\) 2.28361 0.0755765
\(914\) −2.91912 −0.0965560
\(915\) 0 0
\(916\) 0.964305 0.0318615
\(917\) 4.25800 0.140612
\(918\) −0.259105 −0.00855175
\(919\) 22.8657 0.754269 0.377135 0.926158i \(-0.376909\pi\)
0.377135 + 0.926158i \(0.376909\pi\)
\(920\) 0 0
\(921\) −18.1634 −0.598504
\(922\) 9.59745 0.316075
\(923\) 51.3038 1.68868
\(924\) −0.508899 −0.0167415
\(925\) 0 0
\(926\) 6.82311 0.224221
\(927\) 11.3273 0.372038
\(928\) 30.3470 0.996188
\(929\) −19.6394 −0.644348 −0.322174 0.946680i \(-0.604414\pi\)
−0.322174 + 0.946680i \(0.604414\pi\)
\(930\) 0 0
\(931\) −44.7213 −1.46568
\(932\) −52.2641 −1.71197
\(933\) −28.0180 −0.917270
\(934\) −7.46263 −0.244185
\(935\) 0 0
\(936\) 5.64311 0.184451
\(937\) 19.3312 0.631521 0.315761 0.948839i \(-0.397740\pi\)
0.315761 + 0.948839i \(0.397740\pi\)
\(938\) 0.241659 0.00789043
\(939\) −12.6852 −0.413965
\(940\) 0 0
\(941\) −16.7500 −0.546034 −0.273017 0.962009i \(-0.588021\pi\)
−0.273017 + 0.962009i \(0.588021\pi\)
\(942\) 4.39728 0.143271
\(943\) −13.1594 −0.428529
\(944\) −34.7237 −1.13016
\(945\) 0 0
\(946\) 2.10976 0.0685943
\(947\) −51.3618 −1.66903 −0.834517 0.550982i \(-0.814253\pi\)
−0.834517 + 0.550982i \(0.814253\pi\)
\(948\) −30.4378 −0.988573
\(949\) 63.0688 2.04730
\(950\) 0 0
\(951\) −34.8739 −1.13086
\(952\) −0.210947 −0.00683682
\(953\) −18.3541 −0.594548 −0.297274 0.954792i \(-0.596078\pi\)
−0.297274 + 0.954792i \(0.596078\pi\)
\(954\) −2.30853 −0.0747414
\(955\) 0 0
\(956\) −48.4499 −1.56698
\(957\) −12.6492 −0.408890
\(958\) −4.61985 −0.149260
\(959\) 3.06365 0.0989303
\(960\) 0 0
\(961\) 4.75296 0.153321
\(962\) −1.71050 −0.0551488
\(963\) 8.10431 0.261158
\(964\) 15.3911 0.495714
\(965\) 0 0
\(966\) −0.325167 −0.0104621
\(967\) −47.6325 −1.53176 −0.765879 0.642985i \(-0.777696\pi\)
−0.765879 + 0.642985i \(0.777696\pi\)
\(968\) −9.84383 −0.316393
\(969\) 6.23427 0.200274
\(970\) 0 0
\(971\) −15.0570 −0.483203 −0.241601 0.970376i \(-0.577673\pi\)
−0.241601 + 0.970376i \(0.577673\pi\)
\(972\) 1.92862 0.0618606
\(973\) 3.83830 0.123050
\(974\) −1.60287 −0.0513593
\(975\) 0 0
\(976\) −36.1310 −1.15653
\(977\) 57.6863 1.84555 0.922774 0.385342i \(-0.125917\pi\)
0.922774 + 0.385342i \(0.125917\pi\)
\(978\) 2.05985 0.0658669
\(979\) 9.27283 0.296361
\(980\) 0 0
\(981\) 13.2362 0.422599
\(982\) 5.83902 0.186330
\(983\) 27.1491 0.865920 0.432960 0.901413i \(-0.357469\pi\)
0.432960 + 0.901413i \(0.357469\pi\)
\(984\) −2.35172 −0.0749700
\(985\) 0 0
\(986\) −2.57401 −0.0819732
\(987\) −0.207232 −0.00659625
\(988\) −66.6553 −2.12059
\(989\) −36.4248 −1.15824
\(990\) 0 0
\(991\) 15.3572 0.487837 0.243918 0.969796i \(-0.421567\pi\)
0.243918 + 0.969796i \(0.421567\pi\)
\(992\) −18.2657 −0.579937
\(993\) −20.3577 −0.646032
\(994\) −0.528307 −0.0167569
\(995\) 0 0
\(996\) −3.45893 −0.109600
\(997\) −50.5126 −1.59975 −0.799876 0.600166i \(-0.795101\pi\)
−0.799876 + 0.600166i \(0.795101\pi\)
\(998\) −9.11617 −0.288567
\(999\) −1.19082 −0.0376758
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 3525.2.a.be.1.3 yes 8
5.4 even 2 3525.2.a.bd.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.6 8 5.4 even 2
3525.2.a.be.1.3 yes 8 1.1 even 1 trivial