Properties

Label 3850.2.a.bt.1.3
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.24914 q^{3} +1.00000 q^{4} -2.24914 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.05863 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.24914 q^{3} +1.00000 q^{4} -2.24914 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.05863 q^{9} +1.00000 q^{11} +2.24914 q^{12} -0.941367 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.49828 q^{17} -2.05863 q^{18} -4.36641 q^{19} +2.24914 q^{21} -1.00000 q^{22} -6.24914 q^{23} -2.24914 q^{24} +0.941367 q^{26} -2.11727 q^{27} +1.00000 q^{28} +8.74742 q^{29} -9.55691 q^{31} -1.00000 q^{32} +2.24914 q^{33} +6.49828 q^{34} +2.05863 q^{36} -4.24914 q^{37} +4.36641 q^{38} -2.11727 q^{39} -2.13187 q^{41} -2.24914 q^{42} -7.67418 q^{43} +1.00000 q^{44} +6.24914 q^{46} -11.1138 q^{47} +2.24914 q^{48} +1.00000 q^{49} -14.6155 q^{51} -0.941367 q^{52} +4.74742 q^{53} +2.11727 q^{54} -1.00000 q^{56} -9.82066 q^{57} -8.74742 q^{58} -1.88273 q^{59} +9.11383 q^{61} +9.55691 q^{62} +2.05863 q^{63} +1.00000 q^{64} -2.24914 q^{66} -12.9966 q^{67} -6.49828 q^{68} -14.0552 q^{69} -14.6155 q^{71} -2.05863 q^{72} +10.4983 q^{73} +4.24914 q^{74} -4.36641 q^{76} +1.00000 q^{77} +2.11727 q^{78} +8.36641 q^{79} -10.9379 q^{81} +2.13187 q^{82} +8.49828 q^{83} +2.24914 q^{84} +7.67418 q^{86} +19.6742 q^{87} -1.00000 q^{88} +12.3810 q^{89} -0.941367 q^{91} -6.24914 q^{92} -21.4948 q^{93} +11.1138 q^{94} -2.24914 q^{96} +15.3630 q^{97} -1.00000 q^{98} +2.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + 3 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{16} - 2 q^{17} - 7 q^{18} - 6 q^{19} - 2 q^{21} - 3 q^{22} - 10 q^{23} + 2 q^{24} + 2 q^{26} - 8 q^{27} + 3 q^{28} - 12 q^{31} - 3 q^{32} - 2 q^{33} + 2 q^{34} + 7 q^{36} - 4 q^{37} + 6 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} - 8 q^{43} + 3 q^{44} + 10 q^{46} - 2 q^{48} + 3 q^{49} - 28 q^{51} - 2 q^{52} - 12 q^{53} + 8 q^{54} - 3 q^{56} + 8 q^{57} - 4 q^{59} - 6 q^{61} + 12 q^{62} + 7 q^{63} + 3 q^{64} + 2 q^{66} - 4 q^{67} - 2 q^{68} - 8 q^{69} - 28 q^{71} - 7 q^{72} + 14 q^{73} + 4 q^{74} - 6 q^{76} + 3 q^{77} + 8 q^{78} + 18 q^{79} + 3 q^{81} - 4 q^{82} + 8 q^{83} - 2 q^{84} + 8 q^{86} + 44 q^{87} - 3 q^{88} + 18 q^{89} - 2 q^{91} - 10 q^{92} - 12 q^{93} + 2 q^{96} + 4 q^{97} - 3 q^{98} + 7 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.24914 1.29854 0.649271 0.760557i \(-0.275074\pi\)
0.649271 + 0.760557i \(0.275074\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.24914 −0.918208
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 2.05863 0.686211
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.24914 0.649271
\(13\) −0.941367 −0.261088 −0.130544 0.991443i \(-0.541672\pi\)
−0.130544 + 0.991443i \(0.541672\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.49828 −1.57606 −0.788032 0.615634i \(-0.788900\pi\)
−0.788032 + 0.615634i \(0.788900\pi\)
\(18\) −2.05863 −0.485224
\(19\) −4.36641 −1.00172 −0.500861 0.865528i \(-0.666983\pi\)
−0.500861 + 0.865528i \(0.666983\pi\)
\(20\) 0 0
\(21\) 2.24914 0.490803
\(22\) −1.00000 −0.213201
\(23\) −6.24914 −1.30304 −0.651518 0.758633i \(-0.725867\pi\)
−0.651518 + 0.758633i \(0.725867\pi\)
\(24\) −2.24914 −0.459104
\(25\) 0 0
\(26\) 0.941367 0.184617
\(27\) −2.11727 −0.407468
\(28\) 1.00000 0.188982
\(29\) 8.74742 1.62436 0.812178 0.583410i \(-0.198282\pi\)
0.812178 + 0.583410i \(0.198282\pi\)
\(30\) 0 0
\(31\) −9.55691 −1.71647 −0.858236 0.513255i \(-0.828440\pi\)
−0.858236 + 0.513255i \(0.828440\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.24914 0.391525
\(34\) 6.49828 1.11445
\(35\) 0 0
\(36\) 2.05863 0.343106
\(37\) −4.24914 −0.698554 −0.349277 0.937019i \(-0.613573\pi\)
−0.349277 + 0.937019i \(0.613573\pi\)
\(38\) 4.36641 0.708325
\(39\) −2.11727 −0.339034
\(40\) 0 0
\(41\) −2.13187 −0.332943 −0.166471 0.986046i \(-0.553237\pi\)
−0.166471 + 0.986046i \(0.553237\pi\)
\(42\) −2.24914 −0.347050
\(43\) −7.67418 −1.17030 −0.585151 0.810925i \(-0.698965\pi\)
−0.585151 + 0.810925i \(0.698965\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.24914 0.921386
\(47\) −11.1138 −1.62112 −0.810559 0.585657i \(-0.800837\pi\)
−0.810559 + 0.585657i \(0.800837\pi\)
\(48\) 2.24914 0.324635
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −14.6155 −2.04659
\(52\) −0.941367 −0.130544
\(53\) 4.74742 0.652109 0.326054 0.945351i \(-0.394281\pi\)
0.326054 + 0.945351i \(0.394281\pi\)
\(54\) 2.11727 0.288123
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −9.82066 −1.30078
\(58\) −8.74742 −1.14859
\(59\) −1.88273 −0.245111 −0.122556 0.992462i \(-0.539109\pi\)
−0.122556 + 0.992462i \(0.539109\pi\)
\(60\) 0 0
\(61\) 9.11383 1.16691 0.583453 0.812147i \(-0.301701\pi\)
0.583453 + 0.812147i \(0.301701\pi\)
\(62\) 9.55691 1.21373
\(63\) 2.05863 0.259363
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.24914 −0.276850
\(67\) −12.9966 −1.58778 −0.793891 0.608060i \(-0.791948\pi\)
−0.793891 + 0.608060i \(0.791948\pi\)
\(68\) −6.49828 −0.788032
\(69\) −14.0552 −1.69205
\(70\) 0 0
\(71\) −14.6155 −1.73455 −0.867273 0.497833i \(-0.834129\pi\)
−0.867273 + 0.497833i \(0.834129\pi\)
\(72\) −2.05863 −0.242612
\(73\) 10.4983 1.22873 0.614365 0.789022i \(-0.289412\pi\)
0.614365 + 0.789022i \(0.289412\pi\)
\(74\) 4.24914 0.493953
\(75\) 0 0
\(76\) −4.36641 −0.500861
\(77\) 1.00000 0.113961
\(78\) 2.11727 0.239733
\(79\) 8.36641 0.941294 0.470647 0.882322i \(-0.344020\pi\)
0.470647 + 0.882322i \(0.344020\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) 2.13187 0.235426
\(83\) 8.49828 0.932808 0.466404 0.884572i \(-0.345549\pi\)
0.466404 + 0.884572i \(0.345549\pi\)
\(84\) 2.24914 0.245401
\(85\) 0 0
\(86\) 7.67418 0.827528
\(87\) 19.6742 2.10929
\(88\) −1.00000 −0.106600
\(89\) 12.3810 1.31238 0.656192 0.754594i \(-0.272166\pi\)
0.656192 + 0.754594i \(0.272166\pi\)
\(90\) 0 0
\(91\) −0.941367 −0.0986821
\(92\) −6.24914 −0.651518
\(93\) −21.4948 −2.22891
\(94\) 11.1138 1.14630
\(95\) 0 0
\(96\) −2.24914 −0.229552
\(97\) 15.3630 1.55987 0.779937 0.625859i \(-0.215251\pi\)
0.779937 + 0.625859i \(0.215251\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.05863 0.206900
\(100\) 0 0
\(101\) −16.8793 −1.67955 −0.839776 0.542932i \(-0.817314\pi\)
−0.839776 + 0.542932i \(0.817314\pi\)
\(102\) 14.6155 1.44715
\(103\) 6.61555 0.651849 0.325925 0.945396i \(-0.394324\pi\)
0.325925 + 0.945396i \(0.394324\pi\)
\(104\) 0.941367 0.0923086
\(105\) 0 0
\(106\) −4.74742 −0.461110
\(107\) 14.5535 1.40694 0.703469 0.710726i \(-0.251633\pi\)
0.703469 + 0.710726i \(0.251633\pi\)
\(108\) −2.11727 −0.203734
\(109\) −0.249141 −0.0238633 −0.0119317 0.999929i \(-0.503798\pi\)
−0.0119317 + 0.999929i \(0.503798\pi\)
\(110\) 0 0
\(111\) −9.55691 −0.907102
\(112\) 1.00000 0.0944911
\(113\) −10.9966 −1.03447 −0.517235 0.855844i \(-0.673039\pi\)
−0.517235 + 0.855844i \(0.673039\pi\)
\(114\) 9.82066 0.919789
\(115\) 0 0
\(116\) 8.74742 0.812178
\(117\) −1.93793 −0.179162
\(118\) 1.88273 0.173320
\(119\) −6.49828 −0.595696
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −9.11383 −0.825127
\(123\) −4.79488 −0.432340
\(124\) −9.55691 −0.858236
\(125\) 0 0
\(126\) −2.05863 −0.183398
\(127\) 5.88273 0.522008 0.261004 0.965338i \(-0.415946\pi\)
0.261004 + 0.965338i \(0.415946\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −17.2603 −1.51969
\(130\) 0 0
\(131\) −1.25258 −0.109438 −0.0547191 0.998502i \(-0.517426\pi\)
−0.0547191 + 0.998502i \(0.517426\pi\)
\(132\) 2.24914 0.195763
\(133\) −4.36641 −0.378615
\(134\) 12.9966 1.12273
\(135\) 0 0
\(136\) 6.49828 0.557223
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 14.0552 1.19646
\(139\) 10.9820 0.931477 0.465739 0.884922i \(-0.345789\pi\)
0.465739 + 0.884922i \(0.345789\pi\)
\(140\) 0 0
\(141\) −24.9966 −2.10509
\(142\) 14.6155 1.22651
\(143\) −0.941367 −0.0787210
\(144\) 2.05863 0.171553
\(145\) 0 0
\(146\) −10.4983 −0.868844
\(147\) 2.24914 0.185506
\(148\) −4.24914 −0.349277
\(149\) 0.0146079 0.00119673 0.000598363 1.00000i \(-0.499810\pi\)
0.000598363 1.00000i \(0.499810\pi\)
\(150\) 0 0
\(151\) −15.2457 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(152\) 4.36641 0.354162
\(153\) −13.3776 −1.08151
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −2.11727 −0.169517
\(157\) 5.50172 0.439085 0.219542 0.975603i \(-0.429544\pi\)
0.219542 + 0.975603i \(0.429544\pi\)
\(158\) −8.36641 −0.665596
\(159\) 10.6776 0.846790
\(160\) 0 0
\(161\) −6.24914 −0.492501
\(162\) 10.9379 0.859365
\(163\) 18.2277 1.42770 0.713850 0.700298i \(-0.246950\pi\)
0.713850 + 0.700298i \(0.246950\pi\)
\(164\) −2.13187 −0.166471
\(165\) 0 0
\(166\) −8.49828 −0.659595
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −2.24914 −0.173525
\(169\) −12.1138 −0.931833
\(170\) 0 0
\(171\) −8.98883 −0.687393
\(172\) −7.67418 −0.585151
\(173\) −0.117266 −0.00891559 −0.00445780 0.999990i \(-0.501419\pi\)
−0.00445780 + 0.999990i \(0.501419\pi\)
\(174\) −19.6742 −1.49150
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −4.23453 −0.318287
\(178\) −12.3810 −0.927996
\(179\) −22.5535 −1.68573 −0.842863 0.538128i \(-0.819132\pi\)
−0.842863 + 0.538128i \(0.819132\pi\)
\(180\) 0 0
\(181\) 20.8793 1.55195 0.775973 0.630766i \(-0.217259\pi\)
0.775973 + 0.630766i \(0.217259\pi\)
\(182\) 0.941367 0.0697788
\(183\) 20.4983 1.51528
\(184\) 6.24914 0.460693
\(185\) 0 0
\(186\) 21.4948 1.57608
\(187\) −6.49828 −0.475201
\(188\) −11.1138 −0.810559
\(189\) −2.11727 −0.154008
\(190\) 0 0
\(191\) 3.11383 0.225309 0.112654 0.993634i \(-0.464065\pi\)
0.112654 + 0.993634i \(0.464065\pi\)
\(192\) 2.24914 0.162318
\(193\) 6.17246 0.444304 0.222152 0.975012i \(-0.428692\pi\)
0.222152 + 0.975012i \(0.428692\pi\)
\(194\) −15.3630 −1.10300
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.73281 −0.479693 −0.239847 0.970811i \(-0.577097\pi\)
−0.239847 + 0.970811i \(0.577097\pi\)
\(198\) −2.05863 −0.146301
\(199\) −13.2311 −0.937927 −0.468964 0.883217i \(-0.655373\pi\)
−0.468964 + 0.883217i \(0.655373\pi\)
\(200\) 0 0
\(201\) −29.2311 −2.06180
\(202\) 16.8793 1.18762
\(203\) 8.74742 0.613949
\(204\) −14.6155 −1.02329
\(205\) 0 0
\(206\) −6.61555 −0.460927
\(207\) −12.8647 −0.894158
\(208\) −0.941367 −0.0652720
\(209\) −4.36641 −0.302031
\(210\) 0 0
\(211\) 23.1138 1.59122 0.795611 0.605808i \(-0.207150\pi\)
0.795611 + 0.605808i \(0.207150\pi\)
\(212\) 4.74742 0.326054
\(213\) −32.8724 −2.25238
\(214\) −14.5535 −0.994855
\(215\) 0 0
\(216\) 2.11727 0.144062
\(217\) −9.55691 −0.648766
\(218\) 0.249141 0.0168739
\(219\) 23.6121 1.59556
\(220\) 0 0
\(221\) 6.11727 0.411492
\(222\) 9.55691 0.641418
\(223\) −10.3810 −0.695164 −0.347582 0.937650i \(-0.612997\pi\)
−0.347582 + 0.937650i \(0.612997\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.9966 0.731480
\(227\) −16.1104 −1.06928 −0.534642 0.845079i \(-0.679554\pi\)
−0.534642 + 0.845079i \(0.679554\pi\)
\(228\) −9.82066 −0.650389
\(229\) 24.2897 1.60511 0.802555 0.596578i \(-0.203473\pi\)
0.802555 + 0.596578i \(0.203473\pi\)
\(230\) 0 0
\(231\) 2.24914 0.147983
\(232\) −8.74742 −0.574296
\(233\) −6.88617 −0.451128 −0.225564 0.974228i \(-0.572422\pi\)
−0.225564 + 0.974228i \(0.572422\pi\)
\(234\) 1.93793 0.126686
\(235\) 0 0
\(236\) −1.88273 −0.122556
\(237\) 18.8172 1.22231
\(238\) 6.49828 0.421221
\(239\) −11.1353 −0.720283 −0.360142 0.932898i \(-0.617272\pi\)
−0.360142 + 0.932898i \(0.617272\pi\)
\(240\) 0 0
\(241\) 2.36641 0.152434 0.0762168 0.997091i \(-0.475716\pi\)
0.0762168 + 0.997091i \(0.475716\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −18.2491 −1.17068
\(244\) 9.11383 0.583453
\(245\) 0 0
\(246\) 4.79488 0.305711
\(247\) 4.11039 0.261538
\(248\) 9.55691 0.606865
\(249\) 19.1138 1.21129
\(250\) 0 0
\(251\) 25.1070 1.58474 0.792368 0.610043i \(-0.208848\pi\)
0.792368 + 0.610043i \(0.208848\pi\)
\(252\) 2.05863 0.129682
\(253\) −6.24914 −0.392880
\(254\) −5.88273 −0.369116
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.86469 0.178694 0.0893472 0.996001i \(-0.471522\pi\)
0.0893472 + 0.996001i \(0.471522\pi\)
\(258\) 17.2603 1.07458
\(259\) −4.24914 −0.264029
\(260\) 0 0
\(261\) 18.0077 1.11465
\(262\) 1.25258 0.0773846
\(263\) −3.76547 −0.232189 −0.116094 0.993238i \(-0.537037\pi\)
−0.116094 + 0.993238i \(0.537037\pi\)
\(264\) −2.24914 −0.138425
\(265\) 0 0
\(266\) 4.36641 0.267722
\(267\) 27.8466 1.70419
\(268\) −12.9966 −0.793891
\(269\) 8.94137 0.545165 0.272582 0.962132i \(-0.412122\pi\)
0.272582 + 0.962132i \(0.412122\pi\)
\(270\) 0 0
\(271\) 21.4948 1.30572 0.652859 0.757479i \(-0.273569\pi\)
0.652859 + 0.757479i \(0.273569\pi\)
\(272\) −6.49828 −0.394016
\(273\) −2.11727 −0.128143
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) −14.0552 −0.846023
\(277\) −7.64820 −0.459536 −0.229768 0.973245i \(-0.573797\pi\)
−0.229768 + 0.973245i \(0.573797\pi\)
\(278\) −10.9820 −0.658654
\(279\) −19.6742 −1.17786
\(280\) 0 0
\(281\) −28.6155 −1.70706 −0.853530 0.521043i \(-0.825543\pi\)
−0.853530 + 0.521043i \(0.825543\pi\)
\(282\) 24.9966 1.48852
\(283\) 2.87930 0.171156 0.0855782 0.996331i \(-0.472726\pi\)
0.0855782 + 0.996331i \(0.472726\pi\)
\(284\) −14.6155 −0.867273
\(285\) 0 0
\(286\) 0.941367 0.0556642
\(287\) −2.13187 −0.125841
\(288\) −2.05863 −0.121306
\(289\) 25.2277 1.48398
\(290\) 0 0
\(291\) 34.5535 2.02556
\(292\) 10.4983 0.614365
\(293\) 12.9414 0.756043 0.378021 0.925797i \(-0.376605\pi\)
0.378021 + 0.925797i \(0.376605\pi\)
\(294\) −2.24914 −0.131173
\(295\) 0 0
\(296\) 4.24914 0.246976
\(297\) −2.11727 −0.122856
\(298\) −0.0146079 −0.000846213 0
\(299\) 5.88273 0.340207
\(300\) 0 0
\(301\) −7.67418 −0.442332
\(302\) 15.2457 0.877292
\(303\) −37.9639 −2.18097
\(304\) −4.36641 −0.250431
\(305\) 0 0
\(306\) 13.3776 0.764745
\(307\) −0.498281 −0.0284384 −0.0142192 0.999899i \(-0.504526\pi\)
−0.0142192 + 0.999899i \(0.504526\pi\)
\(308\) 1.00000 0.0569803
\(309\) 14.8793 0.846454
\(310\) 0 0
\(311\) −23.4396 −1.32914 −0.664570 0.747226i \(-0.731385\pi\)
−0.664570 + 0.747226i \(0.731385\pi\)
\(312\) 2.11727 0.119867
\(313\) −9.36984 −0.529615 −0.264807 0.964301i \(-0.585308\pi\)
−0.264807 + 0.964301i \(0.585308\pi\)
\(314\) −5.50172 −0.310480
\(315\) 0 0
\(316\) 8.36641 0.470647
\(317\) 9.97852 0.560449 0.280225 0.959934i \(-0.409591\pi\)
0.280225 + 0.959934i \(0.409591\pi\)
\(318\) −10.6776 −0.598771
\(319\) 8.74742 0.489762
\(320\) 0 0
\(321\) 32.7328 1.82697
\(322\) 6.24914 0.348251
\(323\) 28.3741 1.57878
\(324\) −10.9379 −0.607663
\(325\) 0 0
\(326\) −18.2277 −1.00954
\(327\) −0.560352 −0.0309875
\(328\) 2.13187 0.117713
\(329\) −11.1138 −0.612725
\(330\) 0 0
\(331\) −20.4362 −1.12328 −0.561638 0.827383i \(-0.689829\pi\)
−0.561638 + 0.827383i \(0.689829\pi\)
\(332\) 8.49828 0.466404
\(333\) −8.74742 −0.479356
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 2.24914 0.122701
\(337\) −7.88273 −0.429400 −0.214700 0.976680i \(-0.568877\pi\)
−0.214700 + 0.976680i \(0.568877\pi\)
\(338\) 12.1138 0.658905
\(339\) −24.7328 −1.34330
\(340\) 0 0
\(341\) −9.55691 −0.517536
\(342\) 8.98883 0.486060
\(343\) 1.00000 0.0539949
\(344\) 7.67418 0.413764
\(345\) 0 0
\(346\) 0.117266 0.00630428
\(347\) −13.5569 −0.727773 −0.363887 0.931443i \(-0.618550\pi\)
−0.363887 + 0.931443i \(0.618550\pi\)
\(348\) 19.6742 1.05465
\(349\) −10.7328 −0.574514 −0.287257 0.957853i \(-0.592743\pi\)
−0.287257 + 0.957853i \(0.592743\pi\)
\(350\) 0 0
\(351\) 1.99312 0.106385
\(352\) −1.00000 −0.0533002
\(353\) 20.8578 1.11015 0.555075 0.831801i \(-0.312690\pi\)
0.555075 + 0.831801i \(0.312690\pi\)
\(354\) 4.23453 0.225063
\(355\) 0 0
\(356\) 12.3810 0.656192
\(357\) −14.6155 −0.773537
\(358\) 22.5535 1.19199
\(359\) 0.366407 0.0193382 0.00966911 0.999953i \(-0.496922\pi\)
0.00966911 + 0.999953i \(0.496922\pi\)
\(360\) 0 0
\(361\) 0.0655089 0.00344783
\(362\) −20.8793 −1.09739
\(363\) 2.24914 0.118049
\(364\) −0.941367 −0.0493410
\(365\) 0 0
\(366\) −20.4983 −1.07146
\(367\) −20.8432 −1.08801 −0.544003 0.839083i \(-0.683092\pi\)
−0.544003 + 0.839083i \(0.683092\pi\)
\(368\) −6.24914 −0.325759
\(369\) −4.38875 −0.228469
\(370\) 0 0
\(371\) 4.74742 0.246474
\(372\) −21.4948 −1.11446
\(373\) 5.37758 0.278440 0.139220 0.990261i \(-0.455540\pi\)
0.139220 + 0.990261i \(0.455540\pi\)
\(374\) 6.49828 0.336018
\(375\) 0 0
\(376\) 11.1138 0.573152
\(377\) −8.23453 −0.424100
\(378\) 2.11727 0.108900
\(379\) 3.53093 0.181372 0.0906860 0.995880i \(-0.471094\pi\)
0.0906860 + 0.995880i \(0.471094\pi\)
\(380\) 0 0
\(381\) 13.2311 0.677850
\(382\) −3.11383 −0.159317
\(383\) −1.38445 −0.0707422 −0.0353711 0.999374i \(-0.511261\pi\)
−0.0353711 + 0.999374i \(0.511261\pi\)
\(384\) −2.24914 −0.114776
\(385\) 0 0
\(386\) −6.17246 −0.314170
\(387\) −15.7983 −0.803074
\(388\) 15.3630 0.779937
\(389\) 15.7294 0.797511 0.398756 0.917057i \(-0.369442\pi\)
0.398756 + 0.917057i \(0.369442\pi\)
\(390\) 0 0
\(391\) 40.6087 2.05367
\(392\) −1.00000 −0.0505076
\(393\) −2.81722 −0.142110
\(394\) 6.73281 0.339194
\(395\) 0 0
\(396\) 2.05863 0.103450
\(397\) 17.6121 0.883926 0.441963 0.897033i \(-0.354282\pi\)
0.441963 + 0.897033i \(0.354282\pi\)
\(398\) 13.2311 0.663215
\(399\) −9.82066 −0.491648
\(400\) 0 0
\(401\) −32.8172 −1.63881 −0.819407 0.573212i \(-0.805697\pi\)
−0.819407 + 0.573212i \(0.805697\pi\)
\(402\) 29.2311 1.45791
\(403\) 8.99656 0.448151
\(404\) −16.8793 −0.839776
\(405\) 0 0
\(406\) −8.74742 −0.434127
\(407\) −4.24914 −0.210622
\(408\) 14.6155 0.723577
\(409\) −33.3561 −1.64935 −0.824676 0.565605i \(-0.808643\pi\)
−0.824676 + 0.565605i \(0.808643\pi\)
\(410\) 0 0
\(411\) −22.4914 −1.10942
\(412\) 6.61555 0.325925
\(413\) −1.88273 −0.0926433
\(414\) 12.8647 0.632265
\(415\) 0 0
\(416\) 0.941367 0.0461543
\(417\) 24.7000 1.20956
\(418\) 4.36641 0.213568
\(419\) 12.3449 0.603089 0.301544 0.953452i \(-0.402498\pi\)
0.301544 + 0.953452i \(0.402498\pi\)
\(420\) 0 0
\(421\) −8.87930 −0.432750 −0.216375 0.976310i \(-0.569423\pi\)
−0.216375 + 0.976310i \(0.569423\pi\)
\(422\) −23.1138 −1.12516
\(423\) −22.8793 −1.11243
\(424\) −4.74742 −0.230555
\(425\) 0 0
\(426\) 32.8724 1.59267
\(427\) 9.11383 0.441049
\(428\) 14.5535 0.703469
\(429\) −2.11727 −0.102223
\(430\) 0 0
\(431\) 18.2784 0.880437 0.440219 0.897891i \(-0.354901\pi\)
0.440219 + 0.897891i \(0.354901\pi\)
\(432\) −2.11727 −0.101867
\(433\) −6.86469 −0.329896 −0.164948 0.986302i \(-0.552746\pi\)
−0.164948 + 0.986302i \(0.552746\pi\)
\(434\) 9.55691 0.458747
\(435\) 0 0
\(436\) −0.249141 −0.0119317
\(437\) 27.2863 1.30528
\(438\) −23.6121 −1.12823
\(439\) −11.8466 −0.565409 −0.282705 0.959207i \(-0.591232\pi\)
−0.282705 + 0.959207i \(0.591232\pi\)
\(440\) 0 0
\(441\) 2.05863 0.0980302
\(442\) −6.11727 −0.290969
\(443\) −17.8827 −0.849634 −0.424817 0.905279i \(-0.639662\pi\)
−0.424817 + 0.905279i \(0.639662\pi\)
\(444\) −9.55691 −0.453551
\(445\) 0 0
\(446\) 10.3810 0.491555
\(447\) 0.0328552 0.00155400
\(448\) 1.00000 0.0472456
\(449\) −16.8172 −0.793654 −0.396827 0.917893i \(-0.629889\pi\)
−0.396827 + 0.917893i \(0.629889\pi\)
\(450\) 0 0
\(451\) −2.13187 −0.100386
\(452\) −10.9966 −0.517235
\(453\) −34.2897 −1.61107
\(454\) 16.1104 0.756098
\(455\) 0 0
\(456\) 9.82066 0.459895
\(457\) 12.2277 0.571986 0.285993 0.958232i \(-0.407677\pi\)
0.285993 + 0.958232i \(0.407677\pi\)
\(458\) −24.2897 −1.13498
\(459\) 13.7586 0.642196
\(460\) 0 0
\(461\) −16.6155 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(462\) −2.24914 −0.104639
\(463\) −18.4837 −0.859009 −0.429505 0.903065i \(-0.641312\pi\)
−0.429505 + 0.903065i \(0.641312\pi\)
\(464\) 8.74742 0.406089
\(465\) 0 0
\(466\) 6.88617 0.318996
\(467\) −24.3956 −1.12889 −0.564447 0.825469i \(-0.690911\pi\)
−0.564447 + 0.825469i \(0.690911\pi\)
\(468\) −1.93793 −0.0895808
\(469\) −12.9966 −0.600125
\(470\) 0 0
\(471\) 12.3741 0.570170
\(472\) 1.88273 0.0866598
\(473\) −7.67418 −0.352859
\(474\) −18.8172 −0.864304
\(475\) 0 0
\(476\) −6.49828 −0.297848
\(477\) 9.77320 0.447484
\(478\) 11.1353 0.509317
\(479\) −27.7655 −1.26864 −0.634318 0.773072i \(-0.718719\pi\)
−0.634318 + 0.773072i \(0.718719\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −2.36641 −0.107787
\(483\) −14.0552 −0.639534
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 18.2491 0.827798
\(487\) −10.0958 −0.457484 −0.228742 0.973487i \(-0.573461\pi\)
−0.228742 + 0.973487i \(0.573461\pi\)
\(488\) −9.11383 −0.412564
\(489\) 40.9966 1.85393
\(490\) 0 0
\(491\) −32.1104 −1.44912 −0.724561 0.689211i \(-0.757957\pi\)
−0.724561 + 0.689211i \(0.757957\pi\)
\(492\) −4.79488 −0.216170
\(493\) −56.8432 −2.56009
\(494\) −4.11039 −0.184935
\(495\) 0 0
\(496\) −9.55691 −0.429118
\(497\) −14.6155 −0.655597
\(498\) −19.1138 −0.856511
\(499\) −8.79488 −0.393713 −0.196857 0.980432i \(-0.563073\pi\)
−0.196857 + 0.980432i \(0.563073\pi\)
\(500\) 0 0
\(501\) 17.9931 0.803874
\(502\) −25.1070 −1.12058
\(503\) 15.0034 0.668970 0.334485 0.942401i \(-0.391438\pi\)
0.334485 + 0.942401i \(0.391438\pi\)
\(504\) −2.05863 −0.0916988
\(505\) 0 0
\(506\) 6.24914 0.277808
\(507\) −27.2457 −1.21002
\(508\) 5.88273 0.261004
\(509\) 21.8759 0.969630 0.484815 0.874617i \(-0.338887\pi\)
0.484815 + 0.874617i \(0.338887\pi\)
\(510\) 0 0
\(511\) 10.4983 0.464417
\(512\) −1.00000 −0.0441942
\(513\) 9.24485 0.408170
\(514\) −2.86469 −0.126356
\(515\) 0 0
\(516\) −17.2603 −0.759843
\(517\) −11.1138 −0.488786
\(518\) 4.24914 0.186697
\(519\) −0.263748 −0.0115773
\(520\) 0 0
\(521\) 14.0292 0.614631 0.307316 0.951608i \(-0.400569\pi\)
0.307316 + 0.951608i \(0.400569\pi\)
\(522\) −18.0077 −0.788177
\(523\) −1.14992 −0.0502825 −0.0251412 0.999684i \(-0.508004\pi\)
−0.0251412 + 0.999684i \(0.508004\pi\)
\(524\) −1.25258 −0.0547191
\(525\) 0 0
\(526\) 3.76547 0.164182
\(527\) 62.1035 2.70527
\(528\) 2.24914 0.0978813
\(529\) 16.0518 0.697902
\(530\) 0 0
\(531\) −3.87586 −0.168198
\(532\) −4.36641 −0.189308
\(533\) 2.00688 0.0869274
\(534\) −27.8466 −1.20504
\(535\) 0 0
\(536\) 12.9966 0.561366
\(537\) −50.7259 −2.18899
\(538\) −8.94137 −0.385490
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −6.47680 −0.278459 −0.139230 0.990260i \(-0.544463\pi\)
−0.139230 + 0.990260i \(0.544463\pi\)
\(542\) −21.4948 −0.923283
\(543\) 46.9605 2.01527
\(544\) 6.49828 0.278612
\(545\) 0 0
\(546\) 2.11727 0.0906106
\(547\) −12.9966 −0.555693 −0.277846 0.960626i \(-0.589621\pi\)
−0.277846 + 0.960626i \(0.589621\pi\)
\(548\) −10.0000 −0.427179
\(549\) 18.7620 0.800744
\(550\) 0 0
\(551\) −38.1948 −1.62715
\(552\) 14.0552 0.598229
\(553\) 8.36641 0.355776
\(554\) 7.64820 0.324941
\(555\) 0 0
\(556\) 10.9820 0.465739
\(557\) 16.4914 0.698763 0.349382 0.936981i \(-0.386392\pi\)
0.349382 + 0.936981i \(0.386392\pi\)
\(558\) 19.6742 0.832874
\(559\) 7.22422 0.305552
\(560\) 0 0
\(561\) −14.6155 −0.617069
\(562\) 28.6155 1.20707
\(563\) 16.7620 0.706435 0.353218 0.935541i \(-0.385088\pi\)
0.353218 + 0.935541i \(0.385088\pi\)
\(564\) −24.9966 −1.05255
\(565\) 0 0
\(566\) −2.87930 −0.121026
\(567\) −10.9379 −0.459350
\(568\) 14.6155 0.613255
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 12.7328 0.532852 0.266426 0.963855i \(-0.414157\pi\)
0.266426 + 0.963855i \(0.414157\pi\)
\(572\) −0.941367 −0.0393605
\(573\) 7.00344 0.292573
\(574\) 2.13187 0.0889827
\(575\) 0 0
\(576\) 2.05863 0.0857764
\(577\) 11.8613 0.493790 0.246895 0.969042i \(-0.420590\pi\)
0.246895 + 0.969042i \(0.420590\pi\)
\(578\) −25.2277 −1.04933
\(579\) 13.8827 0.576947
\(580\) 0 0
\(581\) 8.49828 0.352568
\(582\) −34.5535 −1.43229
\(583\) 4.74742 0.196618
\(584\) −10.4983 −0.434422
\(585\) 0 0
\(586\) −12.9414 −0.534603
\(587\) 8.60094 0.354999 0.177499 0.984121i \(-0.443199\pi\)
0.177499 + 0.984121i \(0.443199\pi\)
\(588\) 2.24914 0.0927530
\(589\) 41.7294 1.71943
\(590\) 0 0
\(591\) −15.1430 −0.622902
\(592\) −4.24914 −0.174639
\(593\) −10.7328 −0.440744 −0.220372 0.975416i \(-0.570727\pi\)
−0.220372 + 0.975416i \(0.570727\pi\)
\(594\) 2.11727 0.0868725
\(595\) 0 0
\(596\) 0.0146079 0.000598363 0
\(597\) −29.7586 −1.21794
\(598\) −5.88273 −0.240563
\(599\) 16.0812 0.657059 0.328529 0.944494i \(-0.393447\pi\)
0.328529 + 0.944494i \(0.393447\pi\)
\(600\) 0 0
\(601\) −34.0889 −1.39052 −0.695258 0.718760i \(-0.744710\pi\)
−0.695258 + 0.718760i \(0.744710\pi\)
\(602\) 7.67418 0.312776
\(603\) −26.7552 −1.08955
\(604\) −15.2457 −0.620339
\(605\) 0 0
\(606\) 37.9639 1.54218
\(607\) −11.6742 −0.473840 −0.236920 0.971529i \(-0.576138\pi\)
−0.236920 + 0.971529i \(0.576138\pi\)
\(608\) 4.36641 0.177081
\(609\) 19.6742 0.797238
\(610\) 0 0
\(611\) 10.4622 0.423255
\(612\) −13.3776 −0.540756
\(613\) 34.2637 1.38390 0.691950 0.721946i \(-0.256752\pi\)
0.691950 + 0.721946i \(0.256752\pi\)
\(614\) 0.498281 0.0201090
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −39.3707 −1.58500 −0.792502 0.609869i \(-0.791222\pi\)
−0.792502 + 0.609869i \(0.791222\pi\)
\(618\) −14.8793 −0.598533
\(619\) 16.1104 0.647531 0.323766 0.946137i \(-0.395051\pi\)
0.323766 + 0.946137i \(0.395051\pi\)
\(620\) 0 0
\(621\) 13.2311 0.530946
\(622\) 23.4396 0.939844
\(623\) 12.3810 0.496035
\(624\) −2.11727 −0.0847585
\(625\) 0 0
\(626\) 9.36984 0.374494
\(627\) −9.82066 −0.392199
\(628\) 5.50172 0.219542
\(629\) 27.6121 1.10097
\(630\) 0 0
\(631\) 13.8827 0.552663 0.276331 0.961062i \(-0.410881\pi\)
0.276331 + 0.961062i \(0.410881\pi\)
\(632\) −8.36641 −0.332798
\(633\) 51.9862 2.06627
\(634\) −9.97852 −0.396298
\(635\) 0 0
\(636\) 10.6776 0.423395
\(637\) −0.941367 −0.0372983
\(638\) −8.74742 −0.346314
\(639\) −30.0881 −1.19026
\(640\) 0 0
\(641\) −39.3415 −1.55390 −0.776948 0.629565i \(-0.783233\pi\)
−0.776948 + 0.629565i \(0.783233\pi\)
\(642\) −32.7328 −1.29186
\(643\) −38.3595 −1.51275 −0.756376 0.654137i \(-0.773032\pi\)
−0.756376 + 0.654137i \(0.773032\pi\)
\(644\) −6.24914 −0.246251
\(645\) 0 0
\(646\) −28.3741 −1.11637
\(647\) −25.7294 −1.01153 −0.505763 0.862672i \(-0.668789\pi\)
−0.505763 + 0.862672i \(0.668789\pi\)
\(648\) 10.9379 0.429682
\(649\) −1.88273 −0.0739038
\(650\) 0 0
\(651\) −21.4948 −0.842449
\(652\) 18.2277 0.713850
\(653\) −18.4768 −0.723053 −0.361526 0.932362i \(-0.617744\pi\)
−0.361526 + 0.932362i \(0.617744\pi\)
\(654\) 0.560352 0.0219115
\(655\) 0 0
\(656\) −2.13187 −0.0832357
\(657\) 21.6121 0.843169
\(658\) 11.1138 0.433262
\(659\) −39.3776 −1.53393 −0.766966 0.641687i \(-0.778235\pi\)
−0.766966 + 0.641687i \(0.778235\pi\)
\(660\) 0 0
\(661\) −34.2208 −1.33103 −0.665517 0.746383i \(-0.731789\pi\)
−0.665517 + 0.746383i \(0.731789\pi\)
\(662\) 20.4362 0.794276
\(663\) 13.7586 0.534339
\(664\) −8.49828 −0.329797
\(665\) 0 0
\(666\) 8.74742 0.338956
\(667\) −54.6639 −2.11659
\(668\) 8.00000 0.309529
\(669\) −23.3484 −0.902700
\(670\) 0 0
\(671\) 9.11383 0.351835
\(672\) −2.24914 −0.0867625
\(673\) 24.4622 0.942948 0.471474 0.881880i \(-0.343722\pi\)
0.471474 + 0.881880i \(0.343722\pi\)
\(674\) 7.88273 0.303632
\(675\) 0 0
\(676\) −12.1138 −0.465916
\(677\) 39.1070 1.50300 0.751501 0.659732i \(-0.229330\pi\)
0.751501 + 0.659732i \(0.229330\pi\)
\(678\) 24.7328 0.949858
\(679\) 15.3630 0.589577
\(680\) 0 0
\(681\) −36.2345 −1.38851
\(682\) 9.55691 0.365953
\(683\) −23.3776 −0.894518 −0.447259 0.894404i \(-0.647600\pi\)
−0.447259 + 0.894404i \(0.647600\pi\)
\(684\) −8.98883 −0.343697
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 54.6310 2.08430
\(688\) −7.67418 −0.292575
\(689\) −4.46907 −0.170258
\(690\) 0 0
\(691\) 8.49828 0.323290 0.161645 0.986849i \(-0.448320\pi\)
0.161645 + 0.986849i \(0.448320\pi\)
\(692\) −0.117266 −0.00445780
\(693\) 2.05863 0.0782010
\(694\) 13.5569 0.514613
\(695\) 0 0
\(696\) −19.6742 −0.745748
\(697\) 13.8535 0.524739
\(698\) 10.7328 0.406243
\(699\) −15.4880 −0.585809
\(700\) 0 0
\(701\) 14.9751 0.565601 0.282800 0.959179i \(-0.408737\pi\)
0.282800 + 0.959179i \(0.408737\pi\)
\(702\) −1.99312 −0.0752256
\(703\) 18.5535 0.699758
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −20.8578 −0.784994
\(707\) −16.8793 −0.634811
\(708\) −4.23453 −0.159143
\(709\) 40.7259 1.52949 0.764747 0.644330i \(-0.222864\pi\)
0.764747 + 0.644330i \(0.222864\pi\)
\(710\) 0 0
\(711\) 17.2234 0.645927
\(712\) −12.3810 −0.463998
\(713\) 59.7225 2.23663
\(714\) 14.6155 0.546973
\(715\) 0 0
\(716\) −22.5535 −0.842863
\(717\) −25.0449 −0.935318
\(718\) −0.366407 −0.0136742
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 6.61555 0.246376
\(722\) −0.0655089 −0.00243799
\(723\) 5.32238 0.197942
\(724\) 20.8793 0.775973
\(725\) 0 0
\(726\) −2.24914 −0.0834734
\(727\) 51.6413 1.91527 0.957635 0.287984i \(-0.0929848\pi\)
0.957635 + 0.287984i \(0.0929848\pi\)
\(728\) 0.941367 0.0348894
\(729\) −8.23109 −0.304855
\(730\) 0 0
\(731\) 49.8690 1.84447
\(732\) 20.4983 0.757638
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 20.8432 0.769337
\(735\) 0 0
\(736\) 6.24914 0.230346
\(737\) −12.9966 −0.478735
\(738\) 4.38875 0.161552
\(739\) −49.1070 −1.80643 −0.903214 0.429190i \(-0.858799\pi\)
−0.903214 + 0.429190i \(0.858799\pi\)
\(740\) 0 0
\(741\) 9.24485 0.339618
\(742\) −4.74742 −0.174283
\(743\) 53.2311 1.95286 0.976430 0.215836i \(-0.0692475\pi\)
0.976430 + 0.215836i \(0.0692475\pi\)
\(744\) 21.4948 0.788039
\(745\) 0 0
\(746\) −5.37758 −0.196887
\(747\) 17.4948 0.640103
\(748\) −6.49828 −0.237601
\(749\) 14.5535 0.531772
\(750\) 0 0
\(751\) −31.6121 −1.15354 −0.576771 0.816906i \(-0.695688\pi\)
−0.576771 + 0.816906i \(0.695688\pi\)
\(752\) −11.1138 −0.405280
\(753\) 56.4691 2.05785
\(754\) 8.23453 0.299884
\(755\) 0 0
\(756\) −2.11727 −0.0770042
\(757\) 1.24570 0.0452758 0.0226379 0.999744i \(-0.492794\pi\)
0.0226379 + 0.999744i \(0.492794\pi\)
\(758\) −3.53093 −0.128249
\(759\) −14.0552 −0.510171
\(760\) 0 0
\(761\) −10.6009 −0.384284 −0.192142 0.981367i \(-0.561543\pi\)
−0.192142 + 0.981367i \(0.561543\pi\)
\(762\) −13.2311 −0.479312
\(763\) −0.249141 −0.00901949
\(764\) 3.11383 0.112654
\(765\) 0 0
\(766\) 1.38445 0.0500223
\(767\) 1.77234 0.0639956
\(768\) 2.24914 0.0811589
\(769\) −19.1284 −0.689789 −0.344895 0.938641i \(-0.612085\pi\)
−0.344895 + 0.938641i \(0.612085\pi\)
\(770\) 0 0
\(771\) 6.44309 0.232042
\(772\) 6.17246 0.222152
\(773\) −39.9931 −1.43845 −0.719226 0.694776i \(-0.755504\pi\)
−0.719226 + 0.694776i \(0.755504\pi\)
\(774\) 15.7983 0.567859
\(775\) 0 0
\(776\) −15.3630 −0.551498
\(777\) −9.55691 −0.342852
\(778\) −15.7294 −0.563925
\(779\) 9.30863 0.333516
\(780\) 0 0
\(781\) −14.6155 −0.522985
\(782\) −40.6087 −1.45216
\(783\) −18.5206 −0.661873
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 2.81722 0.100487
\(787\) −37.9931 −1.35431 −0.677154 0.735841i \(-0.736787\pi\)
−0.677154 + 0.735841i \(0.736787\pi\)
\(788\) −6.73281 −0.239847
\(789\) −8.46907 −0.301507
\(790\) 0 0
\(791\) −10.9966 −0.390993
\(792\) −2.05863 −0.0731503
\(793\) −8.57946 −0.304665
\(794\) −17.6121 −0.625030
\(795\) 0 0
\(796\) −13.2311 −0.468964
\(797\) 20.3810 0.721933 0.360966 0.932579i \(-0.382447\pi\)
0.360966 + 0.932579i \(0.382447\pi\)
\(798\) 9.82066 0.347648
\(799\) 72.2208 2.55499
\(800\) 0 0
\(801\) 25.4880 0.900573
\(802\) 32.8172 1.15882
\(803\) 10.4983 0.370476
\(804\) −29.2311 −1.03090
\(805\) 0 0
\(806\) −8.99656 −0.316890
\(807\) 20.1104 0.707919
\(808\) 16.8793 0.593812
\(809\) −31.7294 −1.11555 −0.557773 0.829994i \(-0.688344\pi\)
−0.557773 + 0.829994i \(0.688344\pi\)
\(810\) 0 0
\(811\) 43.5095 1.52782 0.763912 0.645321i \(-0.223276\pi\)
0.763912 + 0.645321i \(0.223276\pi\)
\(812\) 8.74742 0.306974
\(813\) 48.3449 1.69553
\(814\) 4.24914 0.148932
\(815\) 0 0
\(816\) −14.6155 −0.511646
\(817\) 33.5086 1.17232
\(818\) 33.3561 1.16627
\(819\) −1.93793 −0.0677167
\(820\) 0 0
\(821\) −24.7766 −0.864711 −0.432355 0.901703i \(-0.642317\pi\)
−0.432355 + 0.901703i \(0.642317\pi\)
\(822\) 22.4914 0.784478
\(823\) −28.8647 −1.00616 −0.503080 0.864240i \(-0.667800\pi\)
−0.503080 + 0.864240i \(0.667800\pi\)
\(824\) −6.61555 −0.230464
\(825\) 0 0
\(826\) 1.88273 0.0655087
\(827\) −34.2277 −1.19021 −0.595106 0.803647i \(-0.702890\pi\)
−0.595106 + 0.803647i \(0.702890\pi\)
\(828\) −12.8647 −0.447079
\(829\) 23.6381 0.820985 0.410492 0.911864i \(-0.365357\pi\)
0.410492 + 0.911864i \(0.365357\pi\)
\(830\) 0 0
\(831\) −17.2019 −0.596727
\(832\) −0.941367 −0.0326360
\(833\) −6.49828 −0.225152
\(834\) −24.7000 −0.855290
\(835\) 0 0
\(836\) −4.36641 −0.151015
\(837\) 20.2345 0.699408
\(838\) −12.3449 −0.426448
\(839\) −24.9053 −0.859826 −0.429913 0.902870i \(-0.641456\pi\)
−0.429913 + 0.902870i \(0.641456\pi\)
\(840\) 0 0
\(841\) 47.5174 1.63853
\(842\) 8.87930 0.306001
\(843\) −64.3604 −2.21669
\(844\) 23.1138 0.795611
\(845\) 0 0
\(846\) 22.8793 0.786606
\(847\) 1.00000 0.0343604
\(848\) 4.74742 0.163027
\(849\) 6.47594 0.222254
\(850\) 0 0
\(851\) 26.5535 0.910241
\(852\) −32.8724 −1.12619
\(853\) −22.9966 −0.787387 −0.393694 0.919242i \(-0.628803\pi\)
−0.393694 + 0.919242i \(0.628803\pi\)
\(854\) −9.11383 −0.311869
\(855\) 0 0
\(856\) −14.5535 −0.497428
\(857\) 31.2603 1.06783 0.533916 0.845538i \(-0.320720\pi\)
0.533916 + 0.845538i \(0.320720\pi\)
\(858\) 2.11727 0.0722823
\(859\) 20.3449 0.694160 0.347080 0.937836i \(-0.387173\pi\)
0.347080 + 0.937836i \(0.387173\pi\)
\(860\) 0 0
\(861\) −4.79488 −0.163409
\(862\) −18.2784 −0.622563
\(863\) 58.3595 1.98658 0.993291 0.115644i \(-0.0368930\pi\)
0.993291 + 0.115644i \(0.0368930\pi\)
\(864\) 2.11727 0.0720309
\(865\) 0 0
\(866\) 6.86469 0.233272
\(867\) 56.7405 1.92701
\(868\) −9.55691 −0.324383
\(869\) 8.36641 0.283811
\(870\) 0 0
\(871\) 12.2345 0.414551
\(872\) 0.249141 0.00843696
\(873\) 31.6267 1.07040
\(874\) −27.2863 −0.922973
\(875\) 0 0
\(876\) 23.6121 0.797779
\(877\) −11.9639 −0.403992 −0.201996 0.979386i \(-0.564743\pi\)
−0.201996 + 0.979386i \(0.564743\pi\)
\(878\) 11.8466 0.399805
\(879\) 29.1070 0.981753
\(880\) 0 0
\(881\) 5.88961 0.198426 0.0992130 0.995066i \(-0.468367\pi\)
0.0992130 + 0.995066i \(0.468367\pi\)
\(882\) −2.05863 −0.0693178
\(883\) 19.4880 0.655822 0.327911 0.944709i \(-0.393655\pi\)
0.327911 + 0.944709i \(0.393655\pi\)
\(884\) 6.11727 0.205746
\(885\) 0 0
\(886\) 17.8827 0.600782
\(887\) −50.4622 −1.69435 −0.847177 0.531310i \(-0.821700\pi\)
−0.847177 + 0.531310i \(0.821700\pi\)
\(888\) 9.55691 0.320709
\(889\) 5.88273 0.197301
\(890\) 0 0
\(891\) −10.9379 −0.366434
\(892\) −10.3810 −0.347582
\(893\) 48.5275 1.62391
\(894\) −0.0328552 −0.00109884
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 13.2311 0.441773
\(898\) 16.8172 0.561198
\(899\) −83.5984 −2.78816
\(900\) 0 0
\(901\) −30.8501 −1.02777
\(902\) 2.13187 0.0709836
\(903\) −17.2603 −0.574387
\(904\) 10.9966 0.365740
\(905\) 0 0
\(906\) 34.2897 1.13920
\(907\) 16.8432 0.559269 0.279635 0.960106i \(-0.409787\pi\)
0.279635 + 0.960106i \(0.409787\pi\)
\(908\) −16.1104 −0.534642
\(909\) −34.7483 −1.15253
\(910\) 0 0
\(911\) −38.6155 −1.27939 −0.639695 0.768629i \(-0.720939\pi\)
−0.639695 + 0.768629i \(0.720939\pi\)
\(912\) −9.82066 −0.325195
\(913\) 8.49828 0.281252
\(914\) −12.2277 −0.404455
\(915\) 0 0
\(916\) 24.2897 0.802555
\(917\) −1.25258 −0.0413638
\(918\) −13.7586 −0.454101
\(919\) 17.2818 0.570074 0.285037 0.958517i \(-0.407994\pi\)
0.285037 + 0.958517i \(0.407994\pi\)
\(920\) 0 0
\(921\) −1.12070 −0.0369285
\(922\) 16.6155 0.547204
\(923\) 13.7586 0.452870
\(924\) 2.24914 0.0739913
\(925\) 0 0
\(926\) 18.4837 0.607411
\(927\) 13.6190 0.447306
\(928\) −8.74742 −0.287148
\(929\) 2.91539 0.0956508 0.0478254 0.998856i \(-0.484771\pi\)
0.0478254 + 0.998856i \(0.484771\pi\)
\(930\) 0 0
\(931\) −4.36641 −0.143103
\(932\) −6.88617 −0.225564
\(933\) −52.7191 −1.72594
\(934\) 24.3956 0.798249
\(935\) 0 0
\(936\) 1.93793 0.0633432
\(937\) −22.5795 −0.737639 −0.368819 0.929501i \(-0.620238\pi\)
−0.368819 + 0.929501i \(0.620238\pi\)
\(938\) 12.9966 0.424353
\(939\) −21.0741 −0.687727
\(940\) 0 0
\(941\) −21.7655 −0.709534 −0.354767 0.934955i \(-0.615440\pi\)
−0.354767 + 0.934955i \(0.615440\pi\)
\(942\) −12.3741 −0.403171
\(943\) 13.3224 0.433836
\(944\) −1.88273 −0.0612778
\(945\) 0 0
\(946\) 7.67418 0.249509
\(947\) 1.49484 0.0485759 0.0242879 0.999705i \(-0.492268\pi\)
0.0242879 + 0.999705i \(0.492268\pi\)
\(948\) 18.8172 0.611155
\(949\) −9.88273 −0.320807
\(950\) 0 0
\(951\) 22.4431 0.727767
\(952\) 6.49828 0.210610
\(953\) 4.83098 0.156491 0.0782453 0.996934i \(-0.475068\pi\)
0.0782453 + 0.996934i \(0.475068\pi\)
\(954\) −9.77320 −0.316419
\(955\) 0 0
\(956\) −11.1353 −0.360142
\(957\) 19.6742 0.635976
\(958\) 27.7655 0.897062
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 60.3346 1.94628
\(962\) −4.00000 −0.128965
\(963\) 29.9603 0.965456
\(964\) 2.36641 0.0762168
\(965\) 0 0
\(966\) 14.0552 0.452218
\(967\) 51.9278 1.66989 0.834943 0.550336i \(-0.185501\pi\)
0.834943 + 0.550336i \(0.185501\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 63.8174 2.05011
\(970\) 0 0
\(971\) 59.6933 1.91565 0.957824 0.287354i \(-0.0927757\pi\)
0.957824 + 0.287354i \(0.0927757\pi\)
\(972\) −18.2491 −0.585341
\(973\) 10.9820 0.352065
\(974\) 10.0958 0.323490
\(975\) 0 0
\(976\) 9.11383 0.291727
\(977\) −31.2311 −0.999171 −0.499586 0.866265i \(-0.666514\pi\)
−0.499586 + 0.866265i \(0.666514\pi\)
\(978\) −40.9966 −1.31093
\(979\) 12.3810 0.395699
\(980\) 0 0
\(981\) −0.512889 −0.0163753
\(982\) 32.1104 1.02468
\(983\) 22.6155 0.721324 0.360662 0.932697i \(-0.382551\pi\)
0.360662 + 0.932697i \(0.382551\pi\)
\(984\) 4.79488 0.152855
\(985\) 0 0
\(986\) 56.8432 1.81026
\(987\) −24.9966 −0.795649
\(988\) 4.11039 0.130769
\(989\) 47.9570 1.52494
\(990\) 0 0
\(991\) 22.4102 0.711884 0.355942 0.934508i \(-0.384160\pi\)
0.355942 + 0.934508i \(0.384160\pi\)
\(992\) 9.55691 0.303432
\(993\) −45.9639 −1.45862
\(994\) 14.6155 0.463577
\(995\) 0 0
\(996\) 19.1138 0.605645
\(997\) 27.3415 0.865914 0.432957 0.901415i \(-0.357470\pi\)
0.432957 + 0.901415i \(0.357470\pi\)
\(998\) 8.79488 0.278397
\(999\) 8.99656 0.284639
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 3850.2.a.bt.1.3 3
5.2 odd 4 3850.2.c.ba.1849.1 6
5.3 odd 4 3850.2.c.ba.1849.6 6
5.4 even 2 770.2.a.m.1.1 3
15.14 odd 2 6930.2.a.ce.1.2 3
20.19 odd 2 6160.2.a.bf.1.3 3
35.34 odd 2 5390.2.a.ca.1.3 3
55.54 odd 2 8470.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 5.4 even 2
3850.2.a.bt.1.3 3 1.1 even 1 trivial
3850.2.c.ba.1849.1 6 5.2 odd 4
3850.2.c.ba.1849.6 6 5.3 odd 4
5390.2.a.ca.1.3 3 35.34 odd 2
6160.2.a.bf.1.3 3 20.19 odd 2
6930.2.a.ce.1.2 3 15.14 odd 2
8470.2.a.ci.1.1 3 55.54 odd 2