Properties

Label 3525.2.a.bd.1.6
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
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: \(1\)
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 1.6
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 \(-0.469888\pi\)
0.0944571 + 0.995529i \(0.469888\pi\)
\(3\) 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 \(-0.512469\pi\)
−0.0391631 + 0.999233i \(0.512469\pi\)
\(8\) −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 \(-0.767829\pi\)
−0.745584 + 0.666412i \(0.767829\pi\)
\(14\) −0.0553651 −0.0147969
\(15\) 0 0
\(16\) 3.57683 0.894208
\(17\) 0.969832 0.235219 0.117609 0.993060i \(-0.462477\pi\)
0.117609 + 0.993060i \(0.462477\pi\)
\(18\) 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 \(-0.290237\pi\)
0.612318 + 0.790611i \(0.290237\pi\)
\(24\) −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 \(-0.531208\pi\)
−0.0978846 + 0.995198i \(0.531208\pi\)
\(38\) 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 \(-0.656790\pi\)
−0.472892 + 0.881121i \(0.656790\pi\)
\(44\) 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 \(-0.702237\pi\)
−0.593455 + 0.804867i \(0.702237\pi\)
\(54\) 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 \(-0.414092\pi\)
0.266624 + 0.963801i \(0.414092\pi\)
\(68\) −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 \(-0.740843\pi\)
−0.686476 + 0.727153i \(0.740843\pi\)
\(74\) −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 \(-0.468618\pi\)
0.0984295 + 0.995144i \(0.468618\pi\)
\(84\) 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 \(-0.504688\pi\)
−0.0147262 + 0.999892i \(0.504688\pi\)
\(98\) −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 \(-0.688453\pi\)
−0.558058 + 0.829802i \(0.688453\pi\)
\(104\) 5.64311 0.553352
\(105\) 0 0
\(106\) −2.30853 −0.224224
\(107\) −8.10431 −0.783474 −0.391737 0.920077i \(-0.628126\pi\)
−0.391737 + 0.920077i \(0.628126\pi\)
\(108\) −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 \(-0.781661\pi\)
−0.773830 + 0.633394i \(0.781661\pi\)
\(114\) 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 \(-0.435732\pi\)
0.200536 + 0.979686i \(0.435732\pi\)
\(128\) −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 \(-0.717572\pi\)
−0.631528 + 0.775353i \(0.717572\pi\)
\(138\) 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 \(-0.728085\pi\)
−0.656787 + 0.754076i \(0.728085\pi\)
\(158\) −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 \(-0.597637\pi\)
−0.301948 + 0.953324i \(0.597637\pi\)
\(164\) −4.32128 −0.337435
\(165\) 0 0
\(166\) 0.479153 0.0371895
\(167\) 10.0987 0.781458 0.390729 0.920506i \(-0.372223\pi\)
0.390729 + 0.920506i \(0.372223\pi\)
\(168\) 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 \(-0.227660\pi\)
0.754952 + 0.655781i \(0.227660\pi\)
\(174\) −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 \(-0.534664\pi\)
−0.108686 + 0.994076i \(0.534664\pi\)
\(194\) −0.0774974 −0.00556399
\(195\) 0 0
\(196\) 13.4175 0.958395
\(197\) 7.63855 0.544224 0.272112 0.962266i \(-0.412278\pi\)
0.272112 + 0.962266i \(0.412278\pi\)
\(198\) −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 \(-0.289140\pi\)
0.615041 + 0.788495i \(0.289140\pi\)
\(224\) −0.633048 −0.0422973
\(225\) 0 0
\(226\) −4.39536 −0.292375
\(227\) −29.0700 −1.92944 −0.964721 0.263274i \(-0.915198\pi\)
−0.964721 + 0.263274i \(0.915198\pi\)
\(228\) −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 \(-0.847672\pi\)
−0.887663 + 0.460494i \(0.847672\pi\)
\(234\) −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 \(-0.467772\pi\)
0.101074 + 0.994879i \(0.467772\pi\)
\(258\) −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 \(-0.652396\pi\)
−0.460684 + 0.887564i \(0.652396\pi\)
\(264\) 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 \(-0.529189\pi\)
−0.0915720 + 0.995798i \(0.529189\pi\)
\(278\) 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 \(-0.563146\pi\)
−0.197079 + 0.980388i \(0.563146\pi\)
\(284\) −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 \(-0.320794\pi\)
0.533719 + 0.845662i \(0.320794\pi\)
\(294\) −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 \(-0.673442\pi\)
−0.518319 + 0.855187i \(0.673442\pi\)
\(308\) −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 \(-0.616713\pi\)
−0.358504 + 0.933528i \(0.616713\pi\)
\(314\) −4.39728 −0.248153
\(315\) 0 0
\(316\) 30.4378 1.71226
\(317\) −34.8739 −1.95871 −0.979355 0.202146i \(-0.935208\pi\)
−0.979355 + 0.202146i \(0.935208\pi\)
\(318\) −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 \(-0.205393\pi\)
0.798943 + 0.601407i \(0.205393\pi\)
\(338\) 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 \(-0.791548\pi\)
−0.793127 + 0.609057i \(0.791548\pi\)
\(348\) 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 \(-0.457578\pi\)
0.132878 + 0.991132i \(0.457578\pi\)
\(354\) −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 \(-0.550658\pi\)
−0.158475 + 0.987363i \(0.550658\pi\)
\(368\) 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 \(-0.666656\pi\)
−0.499972 + 0.866042i \(0.666656\pi\)
\(374\) −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 \(-0.452808\pi\)
0.147716 + 0.989030i \(0.452808\pi\)
\(384\) −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 \(-0.216333\pi\)
0.777805 + 0.628506i \(0.216333\pi\)
\(398\) 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 \(-0.318807\pi\)
0.538987 + 0.842314i \(0.318807\pi\)
\(434\) −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 \(-0.502891\pi\)
−0.00908196 + 0.999959i \(0.502891\pi\)
\(444\) 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 \(-0.582258\pi\)
−0.255555 + 0.966795i \(0.582258\pi\)
\(458\) −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 \(-0.297766\pi\)
0.593448 + 0.804873i \(0.297766\pi\)
\(464\) −35.5331 −1.64958
\(465\) 0 0
\(466\) −7.23996 −0.335384
\(467\) −27.9327 −1.29257 −0.646285 0.763096i \(-0.723678\pi\)
−0.646285 + 0.763096i \(0.723678\pi\)
\(468\) 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 \(-0.543403\pi\)
−0.135933 + 0.990718i \(0.543403\pi\)
\(488\) 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 \(-0.629868\pi\)
−0.396767 + 0.917919i \(0.629868\pi\)
\(504\) 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 \(-0.569878\pi\)
−0.217769 + 0.976000i \(0.569878\pi\)
\(524\) −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 \(-0.155109\pi\)
0.883605 + 0.468233i \(0.155109\pi\)
\(548\) 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 \(-0.865463\pi\)
−0.912001 + 0.410188i \(0.865463\pi\)
\(558\) 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 \(-0.738892\pi\)
−0.682006 + 0.731347i \(0.738892\pi\)
\(564\) 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 \(-0.517369\pi\)
−0.0545381 + 0.998512i \(0.517369\pi\)
\(578\) −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 \(-0.517326\pi\)
−0.0544028 + 0.998519i \(0.517326\pi\)
\(588\) 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 \(-0.550077\pi\)
−0.156673 + 0.987650i \(0.550077\pi\)
\(594\) −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 \(-0.387612\pi\)
0.345786 + 0.938313i \(0.387612\pi\)
\(608\) 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 \(-0.235126\pi\)
0.739366 + 0.673304i \(0.235126\pi\)
\(614\) −4.85262 −0.195836
\(615\) 0 0
\(616\) −0.276952 −0.0111587
\(617\) 23.1682 0.932716 0.466358 0.884596i \(-0.345566\pi\)
0.466358 + 0.884596i \(0.345566\pi\)
\(618\) −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 \(-0.224272\pi\)
0.761890 + 0.647707i \(0.224272\pi\)
\(644\) 2.34733 0.0924978
\(645\) 0 0
\(646\) 1.66558 0.0655314
\(647\) 8.32898 0.327446 0.163723 0.986506i \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(648\) −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 \(-0.379558\pi\)
0.369416 + 0.929264i \(0.379558\pi\)
\(654\) 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 \(-0.704264\pi\)
−0.598569 + 0.801071i \(0.704264\pi\)
\(674\) 7.83683 0.301863
\(675\) 0 0
\(676\) −30.6777 −1.17991
\(677\) −21.7287 −0.835101 −0.417550 0.908654i \(-0.637111\pi\)
−0.417550 + 0.908654i \(0.637111\pi\)
\(678\) −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 \(-0.488726\pi\)
0.0354101 + 0.999373i \(0.488726\pi\)
\(684\) −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 \(-0.513624\pi\)
−0.0427869 + 0.999084i \(0.513624\pi\)
\(728\) −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 \(-0.386205\pi\)
0.349932 + 0.936775i \(0.386205\pi\)
\(734\) −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 \(-0.476147\pi\)
0.0748676 + 0.997193i \(0.476147\pi\)
\(744\) −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 \(-0.649941\pi\)
−0.453825 + 0.891091i \(0.649941\pi\)
\(758\) −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 \(-0.237016\pi\)
0.735354 + 0.677683i \(0.237016\pi\)
\(774\) −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 \(-0.564995\pi\)
−0.202772 + 0.979226i \(0.564995\pi\)
\(788\) −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 \(-0.540111\pi\)
−0.125681 + 0.992071i \(0.540111\pi\)
\(798\) −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 \(-0.320985\pi\)
0.533213 + 0.845981i \(0.320985\pi\)
\(824\) 11.8891 0.414175
\(825\) 0 0
\(826\) 0.537482 0.0187014
\(827\) −33.5561 −1.16686 −0.583430 0.812163i \(-0.698290\pi\)
−0.583430 + 0.812163i \(0.698290\pi\)
\(828\) −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 \(-0.434643\pi\)
0.203886 + 0.978995i \(0.434643\pi\)
\(854\) 0.559264 0.0191376
\(855\) 0 0
\(856\) 8.50622 0.290736
\(857\) −1.29588 −0.0442665 −0.0221332 0.999755i \(-0.507046\pi\)
−0.0221332 + 0.999755i \(0.507046\pi\)
\(858\) 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 \(-0.523961\pi\)
−0.0752054 + 0.997168i \(0.523961\pi\)
\(864\) 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 \(-0.364130\pi\)
0.414004 + 0.910275i \(0.364130\pi\)
\(878\) 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 \(-0.441878\pi\)
0.181584 + 0.983375i \(0.441878\pi\)
\(884\) 10.0564 0.338233
\(885\) 0 0
\(886\) −0.102139 −0.00343142
\(887\) −30.3740 −1.01986 −0.509930 0.860216i \(-0.670329\pi\)
−0.509930 + 0.860216i \(0.670329\pi\)
\(888\) 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 \(-0.694871\pi\)
−0.574674 + 0.818382i \(0.694871\pi\)
\(908\) 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 \(-0.602260\pi\)
−0.315761 + 0.948839i \(0.602260\pi\)
\(938\) −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 \(-0.185747\pi\)
0.834517 + 0.550982i \(0.185747\pi\)
\(948\) 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 \(-0.403922\pi\)
0.297274 + 0.954792i \(0.403922\pi\)
\(954\) −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 \(-0.222304\pi\)
0.765879 + 0.642985i \(0.222304\pi\)
\(968\) 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 \(-0.874083\pi\)
−0.922774 + 0.385342i \(0.874083\pi\)
\(978\) −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 \(-0.642531\pi\)
−0.432960 + 0.901413i \(0.642531\pi\)
\(984\) −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 \(-0.204899\pi\)
0.799876 + 0.600166i \(0.204899\pi\)
\(998\) 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.bd.1.6 8
5.4 even 2 3525.2.a.be.1.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.6 8 1.1 even 1 trivial
3525.2.a.be.1.3 yes 8 5.4 even 2