Properties

Label 7623.2.a.db.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.10399\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} +O(q^{10})\) \(q-2.10399 q^{2} +2.42677 q^{4} +3.75046 q^{5} -1.00000 q^{7} -0.897910 q^{8} -7.89093 q^{10} -3.46575 q^{13} +2.10399 q^{14} -2.96434 q^{16} +1.52683 q^{17} +5.16008 q^{19} +9.10149 q^{20} +4.87487 q^{23} +9.06597 q^{25} +7.29190 q^{26} -2.42677 q^{28} +10.7274 q^{29} +8.12735 q^{31} +8.03275 q^{32} -3.21244 q^{34} -3.75046 q^{35} +9.25848 q^{37} -10.8568 q^{38} -3.36758 q^{40} -10.8973 q^{41} +0.137677 q^{43} -10.2567 q^{46} +2.73030 q^{47} +1.00000 q^{49} -19.0747 q^{50} -8.41057 q^{52} +2.79394 q^{53} +0.897910 q^{56} -22.5702 q^{58} +4.84485 q^{59} +0.297827 q^{61} -17.0998 q^{62} -10.9721 q^{64} -12.9982 q^{65} +0.816899 q^{67} +3.70527 q^{68} +7.89093 q^{70} -5.72995 q^{71} +2.47516 q^{73} -19.4797 q^{74} +12.5223 q^{76} -11.4243 q^{79} -11.1176 q^{80} +22.9278 q^{82} -10.6460 q^{83} +5.72633 q^{85} -0.289670 q^{86} -8.91589 q^{89} +3.46575 q^{91} +11.8302 q^{92} -5.74451 q^{94} +19.3527 q^{95} +8.08471 q^{97} -2.10399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} - 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} - 16q^{7} + 6q^{10} + 32q^{16} - 10q^{19} + 44q^{25} - 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} + 68q^{40} - 16q^{43} - 80q^{46} + 16q^{49} - 2q^{52} + 18q^{58} + 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} + 14q^{73} + 14q^{76} - 54q^{79} + 64q^{82} - 30q^{85} + 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.10399 −1.48774 −0.743872 0.668322i \(-0.767013\pi\)
−0.743872 + 0.668322i \(0.767013\pi\)
\(3\) 0 0
\(4\) 2.42677 1.21338
\(5\) 3.75046 1.67726 0.838629 0.544703i \(-0.183358\pi\)
0.838629 + 0.544703i \(0.183358\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.897910 −0.317459
\(9\) 0 0
\(10\) −7.89093 −2.49533
\(11\) 0 0
\(12\) 0 0
\(13\) −3.46575 −0.961227 −0.480613 0.876933i \(-0.659586\pi\)
−0.480613 + 0.876933i \(0.659586\pi\)
\(14\) 2.10399 0.562314
\(15\) 0 0
\(16\) −2.96434 −0.741085
\(17\) 1.52683 0.370311 0.185156 0.982709i \(-0.440721\pi\)
0.185156 + 0.982709i \(0.440721\pi\)
\(18\) 0 0
\(19\) 5.16008 1.18380 0.591902 0.806010i \(-0.298377\pi\)
0.591902 + 0.806010i \(0.298377\pi\)
\(20\) 9.10149 2.03516
\(21\) 0 0
\(22\) 0 0
\(23\) 4.87487 1.01648 0.508240 0.861215i \(-0.330296\pi\)
0.508240 + 0.861215i \(0.330296\pi\)
\(24\) 0 0
\(25\) 9.06597 1.81319
\(26\) 7.29190 1.43006
\(27\) 0 0
\(28\) −2.42677 −0.458616
\(29\) 10.7274 1.99202 0.996010 0.0892450i \(-0.0284454\pi\)
0.996010 + 0.0892450i \(0.0284454\pi\)
\(30\) 0 0
\(31\) 8.12735 1.45971 0.729857 0.683599i \(-0.239586\pi\)
0.729857 + 0.683599i \(0.239586\pi\)
\(32\) 8.03275 1.42000
\(33\) 0 0
\(34\) −3.21244 −0.550929
\(35\) −3.75046 −0.633944
\(36\) 0 0
\(37\) 9.25848 1.52208 0.761042 0.648703i \(-0.224688\pi\)
0.761042 + 0.648703i \(0.224688\pi\)
\(38\) −10.8568 −1.76120
\(39\) 0 0
\(40\) −3.36758 −0.532461
\(41\) −10.8973 −1.70187 −0.850937 0.525267i \(-0.823965\pi\)
−0.850937 + 0.525267i \(0.823965\pi\)
\(42\) 0 0
\(43\) 0.137677 0.0209955 0.0104978 0.999945i \(-0.496658\pi\)
0.0104978 + 0.999945i \(0.496658\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.2567 −1.51226
\(47\) 2.73030 0.398255 0.199127 0.979974i \(-0.436189\pi\)
0.199127 + 0.979974i \(0.436189\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −19.0747 −2.69757
\(51\) 0 0
\(52\) −8.41057 −1.16634
\(53\) 2.79394 0.383777 0.191889 0.981417i \(-0.438539\pi\)
0.191889 + 0.981417i \(0.438539\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.897910 0.119988
\(57\) 0 0
\(58\) −22.5702 −2.96362
\(59\) 4.84485 0.630746 0.315373 0.948968i \(-0.397870\pi\)
0.315373 + 0.948968i \(0.397870\pi\)
\(60\) 0 0
\(61\) 0.297827 0.0381328 0.0190664 0.999818i \(-0.493931\pi\)
0.0190664 + 0.999818i \(0.493931\pi\)
\(62\) −17.0998 −2.17168
\(63\) 0 0
\(64\) −10.9721 −1.37152
\(65\) −12.9982 −1.61223
\(66\) 0 0
\(67\) 0.816899 0.0998001 0.0499001 0.998754i \(-0.484110\pi\)
0.0499001 + 0.998754i \(0.484110\pi\)
\(68\) 3.70527 0.449329
\(69\) 0 0
\(70\) 7.89093 0.943146
\(71\) −5.72995 −0.680020 −0.340010 0.940422i \(-0.610430\pi\)
−0.340010 + 0.940422i \(0.610430\pi\)
\(72\) 0 0
\(73\) 2.47516 0.289695 0.144848 0.989454i \(-0.453731\pi\)
0.144848 + 0.989454i \(0.453731\pi\)
\(74\) −19.4797 −2.26447
\(75\) 0 0
\(76\) 12.5223 1.43641
\(77\) 0 0
\(78\) 0 0
\(79\) −11.4243 −1.28533 −0.642664 0.766148i \(-0.722171\pi\)
−0.642664 + 0.766148i \(0.722171\pi\)
\(80\) −11.1176 −1.24299
\(81\) 0 0
\(82\) 22.9278 2.53195
\(83\) −10.6460 −1.16855 −0.584273 0.811557i \(-0.698620\pi\)
−0.584273 + 0.811557i \(0.698620\pi\)
\(84\) 0 0
\(85\) 5.72633 0.621108
\(86\) −0.289670 −0.0312359
\(87\) 0 0
\(88\) 0 0
\(89\) −8.91589 −0.945082 −0.472541 0.881309i \(-0.656663\pi\)
−0.472541 + 0.881309i \(0.656663\pi\)
\(90\) 0 0
\(91\) 3.46575 0.363310
\(92\) 11.8302 1.23338
\(93\) 0 0
\(94\) −5.74451 −0.592501
\(95\) 19.3527 1.98555
\(96\) 0 0
\(97\) 8.08471 0.820878 0.410439 0.911888i \(-0.365375\pi\)
0.410439 + 0.911888i \(0.365375\pi\)
\(98\) −2.10399 −0.212535
\(99\) 0 0
\(100\) 22.0010 2.20010
\(101\) −1.72198 −0.171343 −0.0856715 0.996323i \(-0.527304\pi\)
−0.0856715 + 0.996323i \(0.527304\pi\)
\(102\) 0 0
\(103\) −2.69398 −0.265446 −0.132723 0.991153i \(-0.542372\pi\)
−0.132723 + 0.991153i \(0.542372\pi\)
\(104\) 3.11194 0.305150
\(105\) 0 0
\(106\) −5.87842 −0.570962
\(107\) −4.97994 −0.481429 −0.240715 0.970596i \(-0.577382\pi\)
−0.240715 + 0.970596i \(0.577382\pi\)
\(108\) 0 0
\(109\) −11.2061 −1.07335 −0.536676 0.843788i \(-0.680320\pi\)
−0.536676 + 0.843788i \(0.680320\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.96434 0.280104
\(113\) −16.4261 −1.54524 −0.772620 0.634869i \(-0.781054\pi\)
−0.772620 + 0.634869i \(0.781054\pi\)
\(114\) 0 0
\(115\) 18.2830 1.70490
\(116\) 26.0328 2.41708
\(117\) 0 0
\(118\) −10.1935 −0.938388
\(119\) −1.52683 −0.139965
\(120\) 0 0
\(121\) 0 0
\(122\) −0.626624 −0.0567318
\(123\) 0 0
\(124\) 19.7232 1.77119
\(125\) 15.2493 1.36394
\(126\) 0 0
\(127\) 19.6058 1.73973 0.869865 0.493291i \(-0.164206\pi\)
0.869865 + 0.493291i \(0.164206\pi\)
\(128\) 7.01975 0.620464
\(129\) 0 0
\(130\) 27.3480 2.39858
\(131\) −13.3659 −1.16778 −0.583891 0.811832i \(-0.698470\pi\)
−0.583891 + 0.811832i \(0.698470\pi\)
\(132\) 0 0
\(133\) −5.16008 −0.447436
\(134\) −1.71875 −0.148477
\(135\) 0 0
\(136\) −1.37096 −0.117559
\(137\) −5.27980 −0.451084 −0.225542 0.974233i \(-0.572415\pi\)
−0.225542 + 0.974233i \(0.572415\pi\)
\(138\) 0 0
\(139\) 21.3757 1.81306 0.906531 0.422139i \(-0.138720\pi\)
0.906531 + 0.422139i \(0.138720\pi\)
\(140\) −9.10149 −0.769217
\(141\) 0 0
\(142\) 12.0557 1.01170
\(143\) 0 0
\(144\) 0 0
\(145\) 40.2325 3.34113
\(146\) −5.20770 −0.430993
\(147\) 0 0
\(148\) 22.4682 1.84687
\(149\) 10.9966 0.900873 0.450437 0.892808i \(-0.351268\pi\)
0.450437 + 0.892808i \(0.351268\pi\)
\(150\) 0 0
\(151\) 3.54170 0.288219 0.144110 0.989562i \(-0.453968\pi\)
0.144110 + 0.989562i \(0.453968\pi\)
\(152\) −4.63329 −0.375810
\(153\) 0 0
\(154\) 0 0
\(155\) 30.4813 2.44832
\(156\) 0 0
\(157\) 12.7591 1.01828 0.509142 0.860682i \(-0.329963\pi\)
0.509142 + 0.860682i \(0.329963\pi\)
\(158\) 24.0365 1.91224
\(159\) 0 0
\(160\) 30.1265 2.38171
\(161\) −4.87487 −0.384194
\(162\) 0 0
\(163\) −16.2850 −1.27554 −0.637769 0.770228i \(-0.720142\pi\)
−0.637769 + 0.770228i \(0.720142\pi\)
\(164\) −26.4452 −2.06503
\(165\) 0 0
\(166\) 22.3990 1.73850
\(167\) −5.66281 −0.438201 −0.219101 0.975702i \(-0.570312\pi\)
−0.219101 + 0.975702i \(0.570312\pi\)
\(168\) 0 0
\(169\) −0.988556 −0.0760427
\(170\) −12.0481 −0.924049
\(171\) 0 0
\(172\) 0.334109 0.0254756
\(173\) 13.4214 1.02041 0.510205 0.860053i \(-0.329570\pi\)
0.510205 + 0.860053i \(0.329570\pi\)
\(174\) 0 0
\(175\) −9.06597 −0.685323
\(176\) 0 0
\(177\) 0 0
\(178\) 18.7589 1.40604
\(179\) −3.80842 −0.284654 −0.142327 0.989820i \(-0.545459\pi\)
−0.142327 + 0.989820i \(0.545459\pi\)
\(180\) 0 0
\(181\) −9.78859 −0.727580 −0.363790 0.931481i \(-0.618517\pi\)
−0.363790 + 0.931481i \(0.618517\pi\)
\(182\) −7.29190 −0.540512
\(183\) 0 0
\(184\) −4.37720 −0.322691
\(185\) 34.7236 2.55293
\(186\) 0 0
\(187\) 0 0
\(188\) 6.62579 0.483236
\(189\) 0 0
\(190\) −40.7179 −2.95398
\(191\) −8.98965 −0.650469 −0.325234 0.945633i \(-0.605443\pi\)
−0.325234 + 0.945633i \(0.605443\pi\)
\(192\) 0 0
\(193\) 5.84941 0.421049 0.210525 0.977589i \(-0.432483\pi\)
0.210525 + 0.977589i \(0.432483\pi\)
\(194\) −17.0101 −1.22126
\(195\) 0 0
\(196\) 2.42677 0.173340
\(197\) 6.19555 0.441414 0.220707 0.975340i \(-0.429163\pi\)
0.220707 + 0.975340i \(0.429163\pi\)
\(198\) 0 0
\(199\) −18.8634 −1.33719 −0.668596 0.743626i \(-0.733105\pi\)
−0.668596 + 0.743626i \(0.733105\pi\)
\(200\) −8.14043 −0.575615
\(201\) 0 0
\(202\) 3.62302 0.254914
\(203\) −10.7274 −0.752913
\(204\) 0 0
\(205\) −40.8700 −2.85448
\(206\) 5.66810 0.394915
\(207\) 0 0
\(208\) 10.2737 0.712351
\(209\) 0 0
\(210\) 0 0
\(211\) −0.640180 −0.0440718 −0.0220359 0.999757i \(-0.507015\pi\)
−0.0220359 + 0.999757i \(0.507015\pi\)
\(212\) 6.78024 0.465669
\(213\) 0 0
\(214\) 10.4777 0.716243
\(215\) 0.516352 0.0352149
\(216\) 0 0
\(217\) −8.12735 −0.551720
\(218\) 23.5775 1.59687
\(219\) 0 0
\(220\) 0 0
\(221\) −5.29163 −0.355953
\(222\) 0 0
\(223\) −16.7123 −1.11914 −0.559568 0.828785i \(-0.689033\pi\)
−0.559568 + 0.828785i \(0.689033\pi\)
\(224\) −8.03275 −0.536711
\(225\) 0 0
\(226\) 34.5604 2.29892
\(227\) 9.32634 0.619011 0.309506 0.950898i \(-0.399836\pi\)
0.309506 + 0.950898i \(0.399836\pi\)
\(228\) 0 0
\(229\) 21.3716 1.41228 0.706138 0.708074i \(-0.250436\pi\)
0.706138 + 0.708074i \(0.250436\pi\)
\(230\) −38.4672 −2.53646
\(231\) 0 0
\(232\) −9.63220 −0.632385
\(233\) −17.1164 −1.12133 −0.560666 0.828042i \(-0.689455\pi\)
−0.560666 + 0.828042i \(0.689455\pi\)
\(234\) 0 0
\(235\) 10.2399 0.667976
\(236\) 11.7573 0.765336
\(237\) 0 0
\(238\) 3.21244 0.208231
\(239\) 3.37044 0.218016 0.109008 0.994041i \(-0.465233\pi\)
0.109008 + 0.994041i \(0.465233\pi\)
\(240\) 0 0
\(241\) −5.77583 −0.372054 −0.186027 0.982545i \(-0.559561\pi\)
−0.186027 + 0.982545i \(0.559561\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.722755 0.0462697
\(245\) 3.75046 0.239608
\(246\) 0 0
\(247\) −17.8836 −1.13790
\(248\) −7.29763 −0.463400
\(249\) 0 0
\(250\) −32.0843 −2.02919
\(251\) −11.7616 −0.742383 −0.371191 0.928556i \(-0.621051\pi\)
−0.371191 + 0.928556i \(0.621051\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −41.2503 −2.58827
\(255\) 0 0
\(256\) 7.17482 0.448426
\(257\) 14.2660 0.889891 0.444945 0.895558i \(-0.353223\pi\)
0.444945 + 0.895558i \(0.353223\pi\)
\(258\) 0 0
\(259\) −9.25848 −0.575294
\(260\) −31.5435 −1.95625
\(261\) 0 0
\(262\) 28.1216 1.73736
\(263\) −4.85285 −0.299239 −0.149620 0.988744i \(-0.547805\pi\)
−0.149620 + 0.988744i \(0.547805\pi\)
\(264\) 0 0
\(265\) 10.4786 0.643693
\(266\) 10.8568 0.665670
\(267\) 0 0
\(268\) 1.98242 0.121096
\(269\) 6.35465 0.387450 0.193725 0.981056i \(-0.437943\pi\)
0.193725 + 0.981056i \(0.437943\pi\)
\(270\) 0 0
\(271\) 15.2155 0.924273 0.462137 0.886809i \(-0.347083\pi\)
0.462137 + 0.886809i \(0.347083\pi\)
\(272\) −4.52605 −0.274432
\(273\) 0 0
\(274\) 11.1086 0.671098
\(275\) 0 0
\(276\) 0 0
\(277\) 0.379296 0.0227897 0.0113948 0.999935i \(-0.496373\pi\)
0.0113948 + 0.999935i \(0.496373\pi\)
\(278\) −44.9742 −2.69737
\(279\) 0 0
\(280\) 3.36758 0.201251
\(281\) 5.04347 0.300868 0.150434 0.988620i \(-0.451933\pi\)
0.150434 + 0.988620i \(0.451933\pi\)
\(282\) 0 0
\(283\) 32.1351 1.91023 0.955116 0.296233i \(-0.0957305\pi\)
0.955116 + 0.296233i \(0.0957305\pi\)
\(284\) −13.9052 −0.825125
\(285\) 0 0
\(286\) 0 0
\(287\) 10.8973 0.643248
\(288\) 0 0
\(289\) −14.6688 −0.862870
\(290\) −84.6488 −4.97075
\(291\) 0 0
\(292\) 6.00663 0.351511
\(293\) 26.8246 1.56711 0.783554 0.621324i \(-0.213405\pi\)
0.783554 + 0.621324i \(0.213405\pi\)
\(294\) 0 0
\(295\) 18.1704 1.05792
\(296\) −8.31328 −0.483200
\(297\) 0 0
\(298\) −23.1366 −1.34027
\(299\) −16.8951 −0.977069
\(300\) 0 0
\(301\) −0.137677 −0.00793556
\(302\) −7.45169 −0.428797
\(303\) 0 0
\(304\) −15.2962 −0.877300
\(305\) 1.11699 0.0639585
\(306\) 0 0
\(307\) 5.75200 0.328284 0.164142 0.986437i \(-0.447514\pi\)
0.164142 + 0.986437i \(0.447514\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −64.1323 −3.64247
\(311\) −17.8807 −1.01392 −0.506960 0.861970i \(-0.669231\pi\)
−0.506960 + 0.861970i \(0.669231\pi\)
\(312\) 0 0
\(313\) 19.5005 1.10223 0.551117 0.834428i \(-0.314202\pi\)
0.551117 + 0.834428i \(0.314202\pi\)
\(314\) −26.8449 −1.51495
\(315\) 0 0
\(316\) −27.7240 −1.55960
\(317\) 17.5109 0.983508 0.491754 0.870734i \(-0.336356\pi\)
0.491754 + 0.870734i \(0.336356\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −41.1506 −2.30039
\(321\) 0 0
\(322\) 10.2567 0.571582
\(323\) 7.87859 0.438376
\(324\) 0 0
\(325\) −31.4204 −1.74289
\(326\) 34.2634 1.89767
\(327\) 0 0
\(328\) 9.78481 0.540276
\(329\) −2.73030 −0.150526
\(330\) 0 0
\(331\) −17.6231 −0.968652 −0.484326 0.874888i \(-0.660935\pi\)
−0.484326 + 0.874888i \(0.660935\pi\)
\(332\) −25.8352 −1.41789
\(333\) 0 0
\(334\) 11.9145 0.651931
\(335\) 3.06375 0.167391
\(336\) 0 0
\(337\) −1.60109 −0.0872167 −0.0436084 0.999049i \(-0.513885\pi\)
−0.0436084 + 0.999049i \(0.513885\pi\)
\(338\) 2.07991 0.113132
\(339\) 0 0
\(340\) 13.8965 0.753641
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −0.123621 −0.00666522
\(345\) 0 0
\(346\) −28.2384 −1.51811
\(347\) 3.44462 0.184917 0.0924585 0.995717i \(-0.470527\pi\)
0.0924585 + 0.995717i \(0.470527\pi\)
\(348\) 0 0
\(349\) −3.39745 −0.181862 −0.0909308 0.995857i \(-0.528984\pi\)
−0.0909308 + 0.995857i \(0.528984\pi\)
\(350\) 19.0747 1.01959
\(351\) 0 0
\(352\) 0 0
\(353\) −10.4083 −0.553977 −0.276989 0.960873i \(-0.589336\pi\)
−0.276989 + 0.960873i \(0.589336\pi\)
\(354\) 0 0
\(355\) −21.4900 −1.14057
\(356\) −21.6368 −1.14675
\(357\) 0 0
\(358\) 8.01286 0.423493
\(359\) 22.9948 1.21362 0.606811 0.794846i \(-0.292449\pi\)
0.606811 + 0.794846i \(0.292449\pi\)
\(360\) 0 0
\(361\) 7.62647 0.401393
\(362\) 20.5951 1.08245
\(363\) 0 0
\(364\) 8.41057 0.440834
\(365\) 9.28299 0.485894
\(366\) 0 0
\(367\) 19.6357 1.02498 0.512488 0.858695i \(-0.328724\pi\)
0.512488 + 0.858695i \(0.328724\pi\)
\(368\) −14.4508 −0.753298
\(369\) 0 0
\(370\) −73.0580 −3.79810
\(371\) −2.79394 −0.145054
\(372\) 0 0
\(373\) 31.9047 1.65196 0.825981 0.563698i \(-0.190622\pi\)
0.825981 + 0.563698i \(0.190622\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.45156 −0.126430
\(377\) −37.1784 −1.91478
\(378\) 0 0
\(379\) −1.71958 −0.0883287 −0.0441644 0.999024i \(-0.514063\pi\)
−0.0441644 + 0.999024i \(0.514063\pi\)
\(380\) 46.9645 2.40923
\(381\) 0 0
\(382\) 18.9141 0.967731
\(383\) −8.68039 −0.443547 −0.221774 0.975098i \(-0.571185\pi\)
−0.221774 + 0.975098i \(0.571185\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12.3071 −0.626414
\(387\) 0 0
\(388\) 19.6197 0.996039
\(389\) 26.8018 1.35890 0.679452 0.733720i \(-0.262218\pi\)
0.679452 + 0.733720i \(0.262218\pi\)
\(390\) 0 0
\(391\) 7.44311 0.376414
\(392\) −0.897910 −0.0453513
\(393\) 0 0
\(394\) −13.0354 −0.656712
\(395\) −42.8462 −2.15583
\(396\) 0 0
\(397\) −17.8714 −0.896938 −0.448469 0.893798i \(-0.648031\pi\)
−0.448469 + 0.893798i \(0.648031\pi\)
\(398\) 39.6884 1.98940
\(399\) 0 0
\(400\) −26.8746 −1.34373
\(401\) 28.9136 1.44388 0.721938 0.691958i \(-0.243252\pi\)
0.721938 + 0.691958i \(0.243252\pi\)
\(402\) 0 0
\(403\) −28.1674 −1.40312
\(404\) −4.17883 −0.207905
\(405\) 0 0
\(406\) 22.5702 1.12014
\(407\) 0 0
\(408\) 0 0
\(409\) −27.6583 −1.36761 −0.683807 0.729663i \(-0.739677\pi\)
−0.683807 + 0.729663i \(0.739677\pi\)
\(410\) 85.9899 4.24674
\(411\) 0 0
\(412\) −6.53766 −0.322087
\(413\) −4.84485 −0.238399
\(414\) 0 0
\(415\) −39.9273 −1.95995
\(416\) −27.8395 −1.36495
\(417\) 0 0
\(418\) 0 0
\(419\) 4.51003 0.220330 0.110165 0.993913i \(-0.464862\pi\)
0.110165 + 0.993913i \(0.464862\pi\)
\(420\) 0 0
\(421\) 35.9445 1.75183 0.875914 0.482467i \(-0.160259\pi\)
0.875914 + 0.482467i \(0.160259\pi\)
\(422\) 1.34693 0.0655676
\(423\) 0 0
\(424\) −2.50871 −0.121834
\(425\) 13.8422 0.671446
\(426\) 0 0
\(427\) −0.297827 −0.0144128
\(428\) −12.0851 −0.584158
\(429\) 0 0
\(430\) −1.08640 −0.0523907
\(431\) −22.2209 −1.07034 −0.535172 0.844743i \(-0.679753\pi\)
−0.535172 + 0.844743i \(0.679753\pi\)
\(432\) 0 0
\(433\) 25.4892 1.22493 0.612467 0.790496i \(-0.290177\pi\)
0.612467 + 0.790496i \(0.290177\pi\)
\(434\) 17.0998 0.820819
\(435\) 0 0
\(436\) −27.1946 −1.30239
\(437\) 25.1547 1.20331
\(438\) 0 0
\(439\) −11.1407 −0.531715 −0.265857 0.964012i \(-0.585655\pi\)
−0.265857 + 0.964012i \(0.585655\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.1335 0.529567
\(443\) −13.8512 −0.658092 −0.329046 0.944314i \(-0.606727\pi\)
−0.329046 + 0.944314i \(0.606727\pi\)
\(444\) 0 0
\(445\) −33.4387 −1.58515
\(446\) 35.1624 1.66499
\(447\) 0 0
\(448\) 10.9721 0.518385
\(449\) −10.6878 −0.504390 −0.252195 0.967676i \(-0.581152\pi\)
−0.252195 + 0.967676i \(0.581152\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −39.8624 −1.87497
\(453\) 0 0
\(454\) −19.6225 −0.920930
\(455\) 12.9982 0.609364
\(456\) 0 0
\(457\) −31.3778 −1.46779 −0.733895 0.679263i \(-0.762300\pi\)
−0.733895 + 0.679263i \(0.762300\pi\)
\(458\) −44.9656 −2.10111
\(459\) 0 0
\(460\) 44.3686 2.06870
\(461\) 41.1844 1.91815 0.959075 0.283151i \(-0.0913797\pi\)
0.959075 + 0.283151i \(0.0913797\pi\)
\(462\) 0 0
\(463\) 35.6274 1.65575 0.827874 0.560915i \(-0.189550\pi\)
0.827874 + 0.560915i \(0.189550\pi\)
\(464\) −31.7995 −1.47626
\(465\) 0 0
\(466\) 36.0127 1.66826
\(467\) −6.45945 −0.298908 −0.149454 0.988769i \(-0.547752\pi\)
−0.149454 + 0.988769i \(0.547752\pi\)
\(468\) 0 0
\(469\) −0.816899 −0.0377209
\(470\) −21.5446 −0.993777
\(471\) 0 0
\(472\) −4.35024 −0.200236
\(473\) 0 0
\(474\) 0 0
\(475\) 46.7812 2.14647
\(476\) −3.70527 −0.169831
\(477\) 0 0
\(478\) −7.09137 −0.324352
\(479\) −6.80306 −0.310840 −0.155420 0.987849i \(-0.549673\pi\)
−0.155420 + 0.987849i \(0.549673\pi\)
\(480\) 0 0
\(481\) −32.0876 −1.46307
\(482\) 12.1523 0.553521
\(483\) 0 0
\(484\) 0 0
\(485\) 30.3214 1.37682
\(486\) 0 0
\(487\) 26.7244 1.21100 0.605499 0.795846i \(-0.292974\pi\)
0.605499 + 0.795846i \(0.292974\pi\)
\(488\) −0.267422 −0.0121056
\(489\) 0 0
\(490\) −7.89093 −0.356476
\(491\) −6.09257 −0.274954 −0.137477 0.990505i \(-0.543899\pi\)
−0.137477 + 0.990505i \(0.543899\pi\)
\(492\) 0 0
\(493\) 16.3789 0.737667
\(494\) 37.6268 1.69291
\(495\) 0 0
\(496\) −24.0922 −1.08177
\(497\) 5.72995 0.257023
\(498\) 0 0
\(499\) 1.49552 0.0669488 0.0334744 0.999440i \(-0.489343\pi\)
0.0334744 + 0.999440i \(0.489343\pi\)
\(500\) 37.0064 1.65498
\(501\) 0 0
\(502\) 24.7462 1.10448
\(503\) 14.3833 0.641321 0.320660 0.947194i \(-0.396095\pi\)
0.320660 + 0.947194i \(0.396095\pi\)
\(504\) 0 0
\(505\) −6.45820 −0.287386
\(506\) 0 0
\(507\) 0 0
\(508\) 47.5786 2.11096
\(509\) 2.78397 0.123397 0.0616986 0.998095i \(-0.480348\pi\)
0.0616986 + 0.998095i \(0.480348\pi\)
\(510\) 0 0
\(511\) −2.47516 −0.109495
\(512\) −29.1352 −1.28761
\(513\) 0 0
\(514\) −30.0156 −1.32393
\(515\) −10.1037 −0.445221
\(516\) 0 0
\(517\) 0 0
\(518\) 19.4797 0.855890
\(519\) 0 0
\(520\) 11.6712 0.511816
\(521\) −40.5289 −1.77560 −0.887801 0.460227i \(-0.847768\pi\)
−0.887801 + 0.460227i \(0.847768\pi\)
\(522\) 0 0
\(523\) −0.895277 −0.0391477 −0.0195739 0.999808i \(-0.506231\pi\)
−0.0195739 + 0.999808i \(0.506231\pi\)
\(524\) −32.4358 −1.41697
\(525\) 0 0
\(526\) 10.2103 0.445192
\(527\) 12.4091 0.540549
\(528\) 0 0
\(529\) 0.764354 0.0332328
\(530\) −22.0468 −0.957651
\(531\) 0 0
\(532\) −12.5223 −0.542911
\(533\) 37.7674 1.63589
\(534\) 0 0
\(535\) −18.6771 −0.807481
\(536\) −0.733502 −0.0316825
\(537\) 0 0
\(538\) −13.3701 −0.576426
\(539\) 0 0
\(540\) 0 0
\(541\) 2.86106 0.123007 0.0615033 0.998107i \(-0.480411\pi\)
0.0615033 + 0.998107i \(0.480411\pi\)
\(542\) −32.0131 −1.37508
\(543\) 0 0
\(544\) 12.2647 0.525844
\(545\) −42.0281 −1.80029
\(546\) 0 0
\(547\) −26.7191 −1.14243 −0.571214 0.820801i \(-0.693527\pi\)
−0.571214 + 0.820801i \(0.693527\pi\)
\(548\) −12.8128 −0.547338
\(549\) 0 0
\(550\) 0 0
\(551\) 55.3540 2.35816
\(552\) 0 0
\(553\) 11.4243 0.485809
\(554\) −0.798034 −0.0339052
\(555\) 0 0
\(556\) 51.8738 2.19994
\(557\) −11.4407 −0.484756 −0.242378 0.970182i \(-0.577927\pi\)
−0.242378 + 0.970182i \(0.577927\pi\)
\(558\) 0 0
\(559\) −0.477154 −0.0201814
\(560\) 11.1176 0.469806
\(561\) 0 0
\(562\) −10.6114 −0.447615
\(563\) 12.2142 0.514769 0.257385 0.966309i \(-0.417139\pi\)
0.257385 + 0.966309i \(0.417139\pi\)
\(564\) 0 0
\(565\) −61.6056 −2.59177
\(566\) −67.6118 −2.84194
\(567\) 0 0
\(568\) 5.14498 0.215879
\(569\) 20.3769 0.854243 0.427121 0.904194i \(-0.359528\pi\)
0.427121 + 0.904194i \(0.359528\pi\)
\(570\) 0 0
\(571\) 20.3621 0.852129 0.426065 0.904693i \(-0.359900\pi\)
0.426065 + 0.904693i \(0.359900\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −22.9278 −0.956989
\(575\) 44.1954 1.84308
\(576\) 0 0
\(577\) 6.04479 0.251648 0.125824 0.992053i \(-0.459843\pi\)
0.125824 + 0.992053i \(0.459843\pi\)
\(578\) 30.8629 1.28373
\(579\) 0 0
\(580\) 97.6349 4.05407
\(581\) 10.6460 0.441669
\(582\) 0 0
\(583\) 0 0
\(584\) −2.22247 −0.0919665
\(585\) 0 0
\(586\) −56.4386 −2.33146
\(587\) −36.5418 −1.50824 −0.754120 0.656736i \(-0.771936\pi\)
−0.754120 + 0.656736i \(0.771936\pi\)
\(588\) 0 0
\(589\) 41.9378 1.72802
\(590\) −38.2304 −1.57392
\(591\) 0 0
\(592\) −27.4453 −1.12799
\(593\) 15.6666 0.643349 0.321675 0.946850i \(-0.395754\pi\)
0.321675 + 0.946850i \(0.395754\pi\)
\(594\) 0 0
\(595\) −5.72633 −0.234757
\(596\) 26.6861 1.09310
\(597\) 0 0
\(598\) 35.5471 1.45363
\(599\) −28.4274 −1.16151 −0.580756 0.814078i \(-0.697243\pi\)
−0.580756 + 0.814078i \(0.697243\pi\)
\(600\) 0 0
\(601\) 12.3125 0.502239 0.251120 0.967956i \(-0.419201\pi\)
0.251120 + 0.967956i \(0.419201\pi\)
\(602\) 0.289670 0.0118061
\(603\) 0 0
\(604\) 8.59487 0.349720
\(605\) 0 0
\(606\) 0 0
\(607\) 33.5574 1.36205 0.681026 0.732259i \(-0.261534\pi\)
0.681026 + 0.732259i \(0.261534\pi\)
\(608\) 41.4497 1.68101
\(609\) 0 0
\(610\) −2.35013 −0.0951539
\(611\) −9.46254 −0.382813
\(612\) 0 0
\(613\) 5.38046 0.217315 0.108657 0.994079i \(-0.465345\pi\)
0.108657 + 0.994079i \(0.465345\pi\)
\(614\) −12.1021 −0.488402
\(615\) 0 0
\(616\) 0 0
\(617\) 29.3058 1.17981 0.589904 0.807473i \(-0.299166\pi\)
0.589904 + 0.807473i \(0.299166\pi\)
\(618\) 0 0
\(619\) 3.35835 0.134984 0.0674918 0.997720i \(-0.478500\pi\)
0.0674918 + 0.997720i \(0.478500\pi\)
\(620\) 73.9710 2.97075
\(621\) 0 0
\(622\) 37.6207 1.50845
\(623\) 8.91589 0.357207
\(624\) 0 0
\(625\) 11.8620 0.474478
\(626\) −41.0289 −1.63984
\(627\) 0 0
\(628\) 30.9633 1.23557
\(629\) 14.1361 0.563645
\(630\) 0 0
\(631\) −31.1413 −1.23972 −0.619858 0.784714i \(-0.712810\pi\)
−0.619858 + 0.784714i \(0.712810\pi\)
\(632\) 10.2580 0.408040
\(633\) 0 0
\(634\) −36.8426 −1.46321
\(635\) 73.5306 2.91797
\(636\) 0 0
\(637\) −3.46575 −0.137318
\(638\) 0 0
\(639\) 0 0
\(640\) 26.3273 1.04068
\(641\) −21.2928 −0.841017 −0.420508 0.907289i \(-0.638148\pi\)
−0.420508 + 0.907289i \(0.638148\pi\)
\(642\) 0 0
\(643\) 22.7558 0.897399 0.448700 0.893683i \(-0.351887\pi\)
0.448700 + 0.893683i \(0.351887\pi\)
\(644\) −11.8302 −0.466174
\(645\) 0 0
\(646\) −16.5765 −0.652192
\(647\) −43.0823 −1.69374 −0.846870 0.531799i \(-0.821516\pi\)
−0.846870 + 0.531799i \(0.821516\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 66.1082 2.59298
\(651\) 0 0
\(652\) −39.5198 −1.54772
\(653\) −35.7629 −1.39951 −0.699756 0.714382i \(-0.746708\pi\)
−0.699756 + 0.714382i \(0.746708\pi\)
\(654\) 0 0
\(655\) −50.1282 −1.95867
\(656\) 32.3033 1.26123
\(657\) 0 0
\(658\) 5.74451 0.223944
\(659\) 16.6988 0.650495 0.325247 0.945629i \(-0.394552\pi\)
0.325247 + 0.945629i \(0.394552\pi\)
\(660\) 0 0
\(661\) 36.5422 1.42133 0.710664 0.703532i \(-0.248395\pi\)
0.710664 + 0.703532i \(0.248395\pi\)
\(662\) 37.0787 1.44111
\(663\) 0 0
\(664\) 9.55912 0.370966
\(665\) −19.3527 −0.750466
\(666\) 0 0
\(667\) 52.2944 2.02485
\(668\) −13.7423 −0.531706
\(669\) 0 0
\(670\) −6.44609 −0.249034
\(671\) 0 0
\(672\) 0 0
\(673\) −28.6689 −1.10510 −0.552552 0.833478i \(-0.686346\pi\)
−0.552552 + 0.833478i \(0.686346\pi\)
\(674\) 3.36867 0.129756
\(675\) 0 0
\(676\) −2.39899 −0.0922690
\(677\) 10.9806 0.422019 0.211010 0.977484i \(-0.432325\pi\)
0.211010 + 0.977484i \(0.432325\pi\)
\(678\) 0 0
\(679\) −8.08471 −0.310263
\(680\) −5.14173 −0.197176
\(681\) 0 0
\(682\) 0 0
\(683\) −25.3941 −0.971679 −0.485840 0.874048i \(-0.661486\pi\)
−0.485840 + 0.874048i \(0.661486\pi\)
\(684\) 0 0
\(685\) −19.8017 −0.756584
\(686\) 2.10399 0.0803306
\(687\) 0 0
\(688\) −0.408121 −0.0155595
\(689\) −9.68310 −0.368897
\(690\) 0 0
\(691\) −26.5965 −1.01178 −0.505889 0.862599i \(-0.668835\pi\)
−0.505889 + 0.862599i \(0.668835\pi\)
\(692\) 32.5706 1.23815
\(693\) 0 0
\(694\) −7.24744 −0.275109
\(695\) 80.1687 3.04097
\(696\) 0 0
\(697\) −16.6384 −0.630224
\(698\) 7.14820 0.270563
\(699\) 0 0
\(700\) −22.0010 −0.831559
\(701\) −45.1579 −1.70559 −0.852796 0.522245i \(-0.825095\pi\)
−0.852796 + 0.522245i \(0.825095\pi\)
\(702\) 0 0
\(703\) 47.7745 1.80185
\(704\) 0 0
\(705\) 0 0
\(706\) 21.8989 0.824177
\(707\) 1.72198 0.0647615
\(708\) 0 0
\(709\) 42.7235 1.60452 0.802258 0.596978i \(-0.203632\pi\)
0.802258 + 0.596978i \(0.203632\pi\)
\(710\) 45.2146 1.69687
\(711\) 0 0
\(712\) 8.00567 0.300025
\(713\) 39.6198 1.48377
\(714\) 0 0
\(715\) 0 0
\(716\) −9.24214 −0.345395
\(717\) 0 0
\(718\) −48.3809 −1.80556
\(719\) 15.4401 0.575819 0.287909 0.957658i \(-0.407040\pi\)
0.287909 + 0.957658i \(0.407040\pi\)
\(720\) 0 0
\(721\) 2.69398 0.100329
\(722\) −16.0460 −0.597171
\(723\) 0 0
\(724\) −23.7546 −0.882834
\(725\) 97.2539 3.61192
\(726\) 0 0
\(727\) 14.8932 0.552357 0.276178 0.961106i \(-0.410932\pi\)
0.276178 + 0.961106i \(0.410932\pi\)
\(728\) −3.11194 −0.115336
\(729\) 0 0
\(730\) −19.5313 −0.722886
\(731\) 0.210209 0.00777488
\(732\) 0 0
\(733\) 7.38664 0.272832 0.136416 0.990652i \(-0.456442\pi\)
0.136416 + 0.990652i \(0.456442\pi\)
\(734\) −41.3133 −1.52490
\(735\) 0 0
\(736\) 39.1586 1.44341
\(737\) 0 0
\(738\) 0 0
\(739\) −34.8552 −1.28217 −0.641084 0.767471i \(-0.721515\pi\)
−0.641084 + 0.767471i \(0.721515\pi\)
\(740\) 84.2660 3.09768
\(741\) 0 0
\(742\) 5.87842 0.215803
\(743\) −3.33330 −0.122287 −0.0611434 0.998129i \(-0.519475\pi\)
−0.0611434 + 0.998129i \(0.519475\pi\)
\(744\) 0 0
\(745\) 41.2422 1.51100
\(746\) −67.1271 −2.45770
\(747\) 0 0
\(748\) 0 0
\(749\) 4.97994 0.181963
\(750\) 0 0
\(751\) 51.2589 1.87046 0.935232 0.354036i \(-0.115191\pi\)
0.935232 + 0.354036i \(0.115191\pi\)
\(752\) −8.09353 −0.295141
\(753\) 0 0
\(754\) 78.2228 2.84871
\(755\) 13.2830 0.483418
\(756\) 0 0
\(757\) 4.07426 0.148081 0.0740407 0.997255i \(-0.476411\pi\)
0.0740407 + 0.997255i \(0.476411\pi\)
\(758\) 3.61797 0.131411
\(759\) 0 0
\(760\) −17.3770 −0.630330
\(761\) 6.01136 0.217912 0.108956 0.994047i \(-0.465249\pi\)
0.108956 + 0.994047i \(0.465249\pi\)
\(762\) 0 0
\(763\) 11.2061 0.405689
\(764\) −21.8158 −0.789267
\(765\) 0 0
\(766\) 18.2634 0.659885
\(767\) −16.7910 −0.606290
\(768\) 0 0
\(769\) −43.2310 −1.55895 −0.779474 0.626434i \(-0.784514\pi\)
−0.779474 + 0.626434i \(0.784514\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.1951 0.510894
\(773\) 9.64530 0.346917 0.173459 0.984841i \(-0.444506\pi\)
0.173459 + 0.984841i \(0.444506\pi\)
\(774\) 0 0
\(775\) 73.6823 2.64675
\(776\) −7.25935 −0.260595
\(777\) 0 0
\(778\) −56.3906 −2.02170
\(779\) −56.2311 −2.01469
\(780\) 0 0
\(781\) 0 0
\(782\) −15.6602 −0.560008
\(783\) 0 0
\(784\) −2.96434 −0.105869
\(785\) 47.8524 1.70793
\(786\) 0 0
\(787\) 24.1192 0.859757 0.429879 0.902887i \(-0.358556\pi\)
0.429879 + 0.902887i \(0.358556\pi\)
\(788\) 15.0351 0.535605
\(789\) 0 0
\(790\) 90.1480 3.20732
\(791\) 16.4261 0.584046
\(792\) 0 0
\(793\) −1.03219 −0.0366543
\(794\) 37.6011 1.33441
\(795\) 0 0
\(796\) −45.7771 −1.62253
\(797\) 42.4069 1.50213 0.751066 0.660228i \(-0.229540\pi\)
0.751066 + 0.660228i \(0.229540\pi\)
\(798\) 0 0
\(799\) 4.16871 0.147478
\(800\) 72.8247 2.57474
\(801\) 0 0
\(802\) −60.8338 −2.14812
\(803\) 0 0
\(804\) 0 0
\(805\) −18.2830 −0.644392
\(806\) 59.2638 2.08748
\(807\) 0 0
\(808\) 1.54618 0.0543944
\(809\) −3.46232 −0.121729 −0.0608643 0.998146i \(-0.519386\pi\)
−0.0608643 + 0.998146i \(0.519386\pi\)
\(810\) 0 0
\(811\) −15.7547 −0.553224 −0.276612 0.960982i \(-0.589212\pi\)
−0.276612 + 0.960982i \(0.589212\pi\)
\(812\) −26.0328 −0.913571
\(813\) 0 0
\(814\) 0 0
\(815\) −61.0762 −2.13941
\(816\) 0 0
\(817\) 0.710424 0.0248546
\(818\) 58.1927 2.03466
\(819\) 0 0
\(820\) −99.1819 −3.46358
\(821\) 7.93147 0.276810 0.138405 0.990376i \(-0.455802\pi\)
0.138405 + 0.990376i \(0.455802\pi\)
\(822\) 0 0
\(823\) 32.9402 1.14822 0.574111 0.818777i \(-0.305348\pi\)
0.574111 + 0.818777i \(0.305348\pi\)
\(824\) 2.41895 0.0842682
\(825\) 0 0
\(826\) 10.1935 0.354677
\(827\) 30.1775 1.04937 0.524687 0.851295i \(-0.324182\pi\)
0.524687 + 0.851295i \(0.324182\pi\)
\(828\) 0 0
\(829\) −21.7848 −0.756619 −0.378309 0.925679i \(-0.623494\pi\)
−0.378309 + 0.925679i \(0.623494\pi\)
\(830\) 84.0065 2.91591
\(831\) 0 0
\(832\) 38.0267 1.31834
\(833\) 1.52683 0.0529016
\(834\) 0 0
\(835\) −21.2381 −0.734976
\(836\) 0 0
\(837\) 0 0
\(838\) −9.48906 −0.327794
\(839\) 13.4974 0.465982 0.232991 0.972479i \(-0.425149\pi\)
0.232991 + 0.972479i \(0.425149\pi\)
\(840\) 0 0
\(841\) 86.0761 2.96814
\(842\) −75.6268 −2.60627
\(843\) 0 0
\(844\) −1.55357 −0.0534760
\(845\) −3.70754 −0.127543
\(846\) 0 0
\(847\) 0 0
\(848\) −8.28218 −0.284411
\(849\) 0 0
\(850\) −29.1239 −0.998940
\(851\) 45.1339 1.54717
\(852\) 0 0
\(853\) −22.5171 −0.770971 −0.385485 0.922714i \(-0.625966\pi\)
−0.385485 + 0.922714i \(0.625966\pi\)
\(854\) 0.626624 0.0214426
\(855\) 0 0
\(856\) 4.47154 0.152834
\(857\) 12.2772 0.419383 0.209691 0.977768i \(-0.432754\pi\)
0.209691 + 0.977768i \(0.432754\pi\)
\(858\) 0 0
\(859\) −46.4901 −1.58622 −0.793110 0.609078i \(-0.791540\pi\)
−0.793110 + 0.609078i \(0.791540\pi\)
\(860\) 1.25306 0.0427291
\(861\) 0 0
\(862\) 46.7526 1.59240
\(863\) −54.4619 −1.85391 −0.926953 0.375178i \(-0.877581\pi\)
−0.926953 + 0.375178i \(0.877581\pi\)
\(864\) 0 0
\(865\) 50.3364 1.71149
\(866\) −53.6290 −1.82239
\(867\) 0 0
\(868\) −19.7232 −0.669448
\(869\) 0 0
\(870\) 0 0
\(871\) −2.83117 −0.0959306
\(872\) 10.0621 0.340746
\(873\) 0 0
\(874\) −52.9253 −1.79022
\(875\) −15.2493 −0.515519
\(876\) 0 0
\(877\) 29.4571 0.994697 0.497348 0.867551i \(-0.334307\pi\)
0.497348 + 0.867551i \(0.334307\pi\)
\(878\) 23.4398 0.791056
\(879\) 0 0
\(880\) 0 0
\(881\) 29.9763 1.00993 0.504964 0.863141i \(-0.331506\pi\)
0.504964 + 0.863141i \(0.331506\pi\)
\(882\) 0 0
\(883\) 30.2544 1.01814 0.509071 0.860725i \(-0.329989\pi\)
0.509071 + 0.860725i \(0.329989\pi\)
\(884\) −12.8415 −0.431908
\(885\) 0 0
\(886\) 29.1428 0.979073
\(887\) 47.3156 1.58870 0.794352 0.607458i \(-0.207811\pi\)
0.794352 + 0.607458i \(0.207811\pi\)
\(888\) 0 0
\(889\) −19.6058 −0.657556
\(890\) 70.3546 2.35829
\(891\) 0 0
\(892\) −40.5567 −1.35794
\(893\) 14.0886 0.471456
\(894\) 0 0
\(895\) −14.2833 −0.477439
\(896\) −7.01975 −0.234513
\(897\) 0 0
\(898\) 22.4871 0.750404
\(899\) 87.1849 2.90778
\(900\) 0 0
\(901\) 4.26588 0.142117
\(902\) 0 0
\(903\) 0 0
\(904\) 14.7492 0.490551
\(905\) −36.7118 −1.22034
\(906\) 0 0
\(907\) −49.6479 −1.64853 −0.824265 0.566204i \(-0.808411\pi\)
−0.824265 + 0.566204i \(0.808411\pi\)
\(908\) 22.6328 0.751097
\(909\) 0 0
\(910\) −27.3480 −0.906578
\(911\) 21.5478 0.713909 0.356954 0.934122i \(-0.383815\pi\)
0.356954 + 0.934122i \(0.383815\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 66.0185 2.18370
\(915\) 0 0
\(916\) 51.8639 1.71363
\(917\) 13.3659 0.441380
\(918\) 0 0
\(919\) −44.2085 −1.45830 −0.729151 0.684353i \(-0.760085\pi\)
−0.729151 + 0.684353i \(0.760085\pi\)
\(920\) −16.4165 −0.541236
\(921\) 0 0
\(922\) −86.6516 −2.85372
\(923\) 19.8586 0.653654
\(924\) 0 0
\(925\) 83.9371 2.75983
\(926\) −74.9597 −2.46333
\(927\) 0 0
\(928\) 86.1702 2.82868
\(929\) −1.70847 −0.0560532 −0.0280266 0.999607i \(-0.508922\pi\)
−0.0280266 + 0.999607i \(0.508922\pi\)
\(930\) 0 0
\(931\) 5.16008 0.169115
\(932\) −41.5375 −1.36061
\(933\) 0 0
\(934\) 13.5906 0.444698
\(935\) 0 0
\(936\) 0 0
\(937\) 33.3716 1.09020 0.545101 0.838371i \(-0.316491\pi\)
0.545101 + 0.838371i \(0.316491\pi\)
\(938\) 1.71875 0.0561191
\(939\) 0 0
\(940\) 24.8498 0.810511
\(941\) −22.2132 −0.724128 −0.362064 0.932153i \(-0.617928\pi\)
−0.362064 + 0.932153i \(0.617928\pi\)
\(942\) 0 0
\(943\) −53.1230 −1.72992
\(944\) −14.3618 −0.467436
\(945\) 0 0
\(946\) 0 0
\(947\) −16.6969 −0.542578 −0.271289 0.962498i \(-0.587450\pi\)
−0.271289 + 0.962498i \(0.587450\pi\)
\(948\) 0 0
\(949\) −8.57829 −0.278463
\(950\) −98.4270 −3.19339
\(951\) 0 0
\(952\) 1.37096 0.0444330
\(953\) −19.0864 −0.618269 −0.309135 0.951018i \(-0.600039\pi\)
−0.309135 + 0.951018i \(0.600039\pi\)
\(954\) 0 0
\(955\) −33.7154 −1.09100
\(956\) 8.17927 0.264537
\(957\) 0 0
\(958\) 14.3136 0.462450
\(959\) 5.27980 0.170494
\(960\) 0 0
\(961\) 35.0538 1.13077
\(962\) 67.5119 2.17667
\(963\) 0 0
\(964\) −14.0166 −0.451444
\(965\) 21.9380 0.706208
\(966\) 0 0
\(967\) −36.9443 −1.18805 −0.594024 0.804447i \(-0.702462\pi\)
−0.594024 + 0.804447i \(0.702462\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −63.7959 −2.04836
\(971\) 31.3049 1.00462 0.502310 0.864687i \(-0.332484\pi\)
0.502310 + 0.864687i \(0.332484\pi\)
\(972\) 0 0
\(973\) −21.3757 −0.685273
\(974\) −56.2278 −1.80165
\(975\) 0 0
\(976\) −0.882859 −0.0282596
\(977\) 44.7409 1.43139 0.715695 0.698413i \(-0.246110\pi\)
0.715695 + 0.698413i \(0.246110\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 9.10149 0.290737
\(981\) 0 0
\(982\) 12.8187 0.409061
\(983\) −57.2232 −1.82514 −0.912568 0.408926i \(-0.865903\pi\)
−0.912568 + 0.408926i \(0.865903\pi\)
\(984\) 0 0
\(985\) 23.2362 0.740366
\(986\) −34.4610 −1.09746
\(987\) 0 0
\(988\) −43.3993 −1.38071
\(989\) 0.671156 0.0213415
\(990\) 0 0
\(991\) −26.9663 −0.856614 −0.428307 0.903633i \(-0.640890\pi\)
−0.428307 + 0.903633i \(0.640890\pi\)
\(992\) 65.2850 2.07280
\(993\) 0 0
\(994\) −12.0557 −0.382385
\(995\) −70.7466 −2.24282
\(996\) 0 0
\(997\) −25.3305 −0.802224 −0.401112 0.916029i \(-0.631376\pi\)
−0.401112 + 0.916029i \(0.631376\pi\)
\(998\) −3.14656 −0.0996027
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.db.1.3 16
3.2 odd 2 inner 7623.2.a.db.1.14 16
11.2 odd 10 693.2.m.k.631.2 yes 32
11.6 odd 10 693.2.m.k.190.2 32
11.10 odd 2 7623.2.a.dc.1.14 16
33.2 even 10 693.2.m.k.631.7 yes 32
33.17 even 10 693.2.m.k.190.7 yes 32
33.32 even 2 7623.2.a.dc.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.2 32 11.6 odd 10
693.2.m.k.190.7 yes 32 33.17 even 10
693.2.m.k.631.2 yes 32 11.2 odd 10
693.2.m.k.631.7 yes 32 33.2 even 10
7623.2.a.db.1.3 16 1.1 even 1 trivial
7623.2.a.db.1.14 16 3.2 odd 2 inner
7623.2.a.dc.1.3 16 33.32 even 2
7623.2.a.dc.1.14 16 11.10 odd 2