Properties

Label 7168.2.a.bf.1.1
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.02908\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.42180 q^{3} +3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} +O(q^{10})\) \(q-3.42180 q^{3} +3.59402 q^{5} +1.00000 q^{7} +8.70871 q^{9} +1.08150 q^{11} +1.79014 q^{13} -12.2980 q^{15} -5.65390 q^{17} -0.0630276 q^{19} -3.42180 q^{21} +1.46467 q^{23} +7.91700 q^{25} -19.5341 q^{27} +5.04356 q^{29} -4.75455 q^{31} -3.70069 q^{33} +3.59402 q^{35} +7.19951 q^{37} -6.12551 q^{39} -7.50243 q^{41} +4.56166 q^{43} +31.2993 q^{45} -1.52393 q^{47} +1.00000 q^{49} +19.3465 q^{51} +6.59185 q^{53} +3.88695 q^{55} +0.215668 q^{57} +7.62071 q^{59} -9.62680 q^{61} +8.70871 q^{63} +6.43381 q^{65} +6.97006 q^{67} -5.01180 q^{69} -6.19187 q^{71} +8.59924 q^{73} -27.0904 q^{75} +1.08150 q^{77} +7.84435 q^{79} +40.7155 q^{81} +10.5197 q^{83} -20.3202 q^{85} -17.2580 q^{87} +9.32780 q^{89} +1.79014 q^{91} +16.2691 q^{93} -0.226523 q^{95} +0.485578 q^{97} +9.41852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q + 8q^{5} + 8q^{7} + 12q^{9} + 12q^{11} + 20q^{13} + 4q^{17} + 4q^{19} + 8q^{23} + 12q^{25} - 12q^{27} + 8q^{29} - 4q^{31} + 8q^{33} + 8q^{35} + 8q^{37} - 16q^{39} - 12q^{41} - 4q^{43} + 52q^{45} + 20q^{47} + 8q^{49} + 32q^{51} + 40q^{53} + 24q^{55} - 4q^{57} + 4q^{59} - 8q^{61} + 12q^{63} + 36q^{65} + 28q^{67} + 4q^{69} - 16q^{71} + 16q^{73} - 28q^{75} + 12q^{77} + 20q^{81} - 8q^{83} + 16q^{85} + 20q^{87} + 16q^{89} + 20q^{91} + 16q^{93} - 40q^{95} - 36q^{97} - 4q^{99} + O(q^{100}) \)

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 0
\(3\) −3.42180 −1.97558 −0.987788 0.155801i \(-0.950204\pi\)
−0.987788 + 0.155801i \(0.950204\pi\)
\(4\) 0 0
\(5\) 3.59402 1.60730 0.803648 0.595105i \(-0.202890\pi\)
0.803648 + 0.595105i \(0.202890\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 8.70871 2.90290
\(10\) 0 0
\(11\) 1.08150 0.326086 0.163043 0.986619i \(-0.447869\pi\)
0.163043 + 0.986619i \(0.447869\pi\)
\(12\) 0 0
\(13\) 1.79014 0.496496 0.248248 0.968696i \(-0.420145\pi\)
0.248248 + 0.968696i \(0.420145\pi\)
\(14\) 0 0
\(15\) −12.2980 −3.17534
\(16\) 0 0
\(17\) −5.65390 −1.37127 −0.685636 0.727945i \(-0.740476\pi\)
−0.685636 + 0.727945i \(0.740476\pi\)
\(18\) 0 0
\(19\) −0.0630276 −0.0144595 −0.00722977 0.999974i \(-0.502301\pi\)
−0.00722977 + 0.999974i \(0.502301\pi\)
\(20\) 0 0
\(21\) −3.42180 −0.746698
\(22\) 0 0
\(23\) 1.46467 0.305404 0.152702 0.988272i \(-0.451203\pi\)
0.152702 + 0.988272i \(0.451203\pi\)
\(24\) 0 0
\(25\) 7.91700 1.58340
\(26\) 0 0
\(27\) −19.5341 −3.75933
\(28\) 0 0
\(29\) 5.04356 0.936565 0.468282 0.883579i \(-0.344873\pi\)
0.468282 + 0.883579i \(0.344873\pi\)
\(30\) 0 0
\(31\) −4.75455 −0.853943 −0.426971 0.904265i \(-0.640419\pi\)
−0.426971 + 0.904265i \(0.640419\pi\)
\(32\) 0 0
\(33\) −3.70069 −0.644208
\(34\) 0 0
\(35\) 3.59402 0.607501
\(36\) 0 0
\(37\) 7.19951 1.18359 0.591796 0.806088i \(-0.298419\pi\)
0.591796 + 0.806088i \(0.298419\pi\)
\(38\) 0 0
\(39\) −6.12551 −0.980867
\(40\) 0 0
\(41\) −7.50243 −1.17168 −0.585841 0.810426i \(-0.699236\pi\)
−0.585841 + 0.810426i \(0.699236\pi\)
\(42\) 0 0
\(43\) 4.56166 0.695647 0.347824 0.937560i \(-0.386921\pi\)
0.347824 + 0.937560i \(0.386921\pi\)
\(44\) 0 0
\(45\) 31.2993 4.66583
\(46\) 0 0
\(47\) −1.52393 −0.222287 −0.111144 0.993804i \(-0.535451\pi\)
−0.111144 + 0.993804i \(0.535451\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 19.3465 2.70905
\(52\) 0 0
\(53\) 6.59185 0.905460 0.452730 0.891648i \(-0.350450\pi\)
0.452730 + 0.891648i \(0.350450\pi\)
\(54\) 0 0
\(55\) 3.88695 0.524117
\(56\) 0 0
\(57\) 0.215668 0.0285659
\(58\) 0 0
\(59\) 7.62071 0.992132 0.496066 0.868285i \(-0.334777\pi\)
0.496066 + 0.868285i \(0.334777\pi\)
\(60\) 0 0
\(61\) −9.62680 −1.23258 −0.616292 0.787517i \(-0.711366\pi\)
−0.616292 + 0.787517i \(0.711366\pi\)
\(62\) 0 0
\(63\) 8.70871 1.09719
\(64\) 0 0
\(65\) 6.43381 0.798016
\(66\) 0 0
\(67\) 6.97006 0.851529 0.425764 0.904834i \(-0.360005\pi\)
0.425764 + 0.904834i \(0.360005\pi\)
\(68\) 0 0
\(69\) −5.01180 −0.603349
\(70\) 0 0
\(71\) −6.19187 −0.734839 −0.367420 0.930055i \(-0.619759\pi\)
−0.367420 + 0.930055i \(0.619759\pi\)
\(72\) 0 0
\(73\) 8.59924 1.00647 0.503233 0.864151i \(-0.332144\pi\)
0.503233 + 0.864151i \(0.332144\pi\)
\(74\) 0 0
\(75\) −27.0904 −3.12813
\(76\) 0 0
\(77\) 1.08150 0.123249
\(78\) 0 0
\(79\) 7.84435 0.882558 0.441279 0.897370i \(-0.354525\pi\)
0.441279 + 0.897370i \(0.354525\pi\)
\(80\) 0 0
\(81\) 40.7155 4.52395
\(82\) 0 0
\(83\) 10.5197 1.15469 0.577345 0.816500i \(-0.304089\pi\)
0.577345 + 0.816500i \(0.304089\pi\)
\(84\) 0 0
\(85\) −20.3202 −2.20404
\(86\) 0 0
\(87\) −17.2580 −1.85026
\(88\) 0 0
\(89\) 9.32780 0.988745 0.494373 0.869250i \(-0.335398\pi\)
0.494373 + 0.869250i \(0.335398\pi\)
\(90\) 0 0
\(91\) 1.79014 0.187658
\(92\) 0 0
\(93\) 16.2691 1.68703
\(94\) 0 0
\(95\) −0.226523 −0.0232407
\(96\) 0 0
\(97\) 0.485578 0.0493030 0.0246515 0.999696i \(-0.492152\pi\)
0.0246515 + 0.999696i \(0.492152\pi\)
\(98\) 0 0
\(99\) 9.41852 0.946597
\(100\) 0 0
\(101\) 6.44571 0.641372 0.320686 0.947186i \(-0.396087\pi\)
0.320686 + 0.947186i \(0.396087\pi\)
\(102\) 0 0
\(103\) 4.26862 0.420599 0.210300 0.977637i \(-0.432556\pi\)
0.210300 + 0.977637i \(0.432556\pi\)
\(104\) 0 0
\(105\) −12.2980 −1.20016
\(106\) 0 0
\(107\) 2.55775 0.247267 0.123633 0.992328i \(-0.460545\pi\)
0.123633 + 0.992328i \(0.460545\pi\)
\(108\) 0 0
\(109\) −15.5780 −1.49210 −0.746050 0.665890i \(-0.768052\pi\)
−0.746050 + 0.665890i \(0.768052\pi\)
\(110\) 0 0
\(111\) −24.6353 −2.33828
\(112\) 0 0
\(113\) 11.6345 1.09449 0.547243 0.836974i \(-0.315677\pi\)
0.547243 + 0.836974i \(0.315677\pi\)
\(114\) 0 0
\(115\) 5.26404 0.490875
\(116\) 0 0
\(117\) 15.5898 1.44128
\(118\) 0 0
\(119\) −5.65390 −0.518292
\(120\) 0 0
\(121\) −9.83035 −0.893668
\(122\) 0 0
\(123\) 25.6718 2.31475
\(124\) 0 0
\(125\) 10.4838 0.937695
\(126\) 0 0
\(127\) −11.7630 −1.04380 −0.521899 0.853007i \(-0.674776\pi\)
−0.521899 + 0.853007i \(0.674776\pi\)
\(128\) 0 0
\(129\) −15.6091 −1.37430
\(130\) 0 0
\(131\) −15.6871 −1.37059 −0.685294 0.728267i \(-0.740326\pi\)
−0.685294 + 0.728267i \(0.740326\pi\)
\(132\) 0 0
\(133\) −0.0630276 −0.00546519
\(134\) 0 0
\(135\) −70.2059 −6.04236
\(136\) 0 0
\(137\) −7.47159 −0.638341 −0.319170 0.947697i \(-0.603404\pi\)
−0.319170 + 0.947697i \(0.603404\pi\)
\(138\) 0 0
\(139\) 0.352269 0.0298790 0.0149395 0.999888i \(-0.495244\pi\)
0.0149395 + 0.999888i \(0.495244\pi\)
\(140\) 0 0
\(141\) 5.21457 0.439146
\(142\) 0 0
\(143\) 1.93605 0.161901
\(144\) 0 0
\(145\) 18.1267 1.50534
\(146\) 0 0
\(147\) −3.42180 −0.282225
\(148\) 0 0
\(149\) 6.26539 0.513281 0.256640 0.966507i \(-0.417384\pi\)
0.256640 + 0.966507i \(0.417384\pi\)
\(150\) 0 0
\(151\) −12.4475 −1.01297 −0.506483 0.862250i \(-0.669055\pi\)
−0.506483 + 0.862250i \(0.669055\pi\)
\(152\) 0 0
\(153\) −49.2382 −3.98067
\(154\) 0 0
\(155\) −17.0880 −1.37254
\(156\) 0 0
\(157\) 5.24808 0.418842 0.209421 0.977826i \(-0.432842\pi\)
0.209421 + 0.977826i \(0.432842\pi\)
\(158\) 0 0
\(159\) −22.5560 −1.78881
\(160\) 0 0
\(161\) 1.46467 0.115432
\(162\) 0 0
\(163\) 7.72263 0.604883 0.302442 0.953168i \(-0.402198\pi\)
0.302442 + 0.953168i \(0.402198\pi\)
\(164\) 0 0
\(165\) −13.3004 −1.03543
\(166\) 0 0
\(167\) 8.39368 0.649523 0.324761 0.945796i \(-0.394716\pi\)
0.324761 + 0.945796i \(0.394716\pi\)
\(168\) 0 0
\(169\) −9.79539 −0.753491
\(170\) 0 0
\(171\) −0.548890 −0.0419746
\(172\) 0 0
\(173\) 11.5546 0.878482 0.439241 0.898369i \(-0.355247\pi\)
0.439241 + 0.898369i \(0.355247\pi\)
\(174\) 0 0
\(175\) 7.91700 0.598469
\(176\) 0 0
\(177\) −26.0765 −1.96003
\(178\) 0 0
\(179\) 19.4675 1.45507 0.727536 0.686070i \(-0.240666\pi\)
0.727536 + 0.686070i \(0.240666\pi\)
\(180\) 0 0
\(181\) −11.4791 −0.853233 −0.426617 0.904433i \(-0.640295\pi\)
−0.426617 + 0.904433i \(0.640295\pi\)
\(182\) 0 0
\(183\) 32.9410 2.43507
\(184\) 0 0
\(185\) 25.8752 1.90238
\(186\) 0 0
\(187\) −6.11472 −0.447153
\(188\) 0 0
\(189\) −19.5341 −1.42089
\(190\) 0 0
\(191\) −16.5900 −1.20041 −0.600206 0.799846i \(-0.704915\pi\)
−0.600206 + 0.799846i \(0.704915\pi\)
\(192\) 0 0
\(193\) −1.32261 −0.0952033 −0.0476016 0.998866i \(-0.515158\pi\)
−0.0476016 + 0.998866i \(0.515158\pi\)
\(194\) 0 0
\(195\) −22.0152 −1.57654
\(196\) 0 0
\(197\) −16.3452 −1.16454 −0.582272 0.812994i \(-0.697836\pi\)
−0.582272 + 0.812994i \(0.697836\pi\)
\(198\) 0 0
\(199\) 26.1422 1.85317 0.926587 0.376081i \(-0.122728\pi\)
0.926587 + 0.376081i \(0.122728\pi\)
\(200\) 0 0
\(201\) −23.8502 −1.68226
\(202\) 0 0
\(203\) 5.04356 0.353988
\(204\) 0 0
\(205\) −26.9639 −1.88324
\(206\) 0 0
\(207\) 12.7554 0.886559
\(208\) 0 0
\(209\) −0.0681647 −0.00471505
\(210\) 0 0
\(211\) 24.7253 1.70216 0.851081 0.525034i \(-0.175947\pi\)
0.851081 + 0.525034i \(0.175947\pi\)
\(212\) 0 0
\(213\) 21.1873 1.45173
\(214\) 0 0
\(215\) 16.3947 1.11811
\(216\) 0 0
\(217\) −4.75455 −0.322760
\(218\) 0 0
\(219\) −29.4249 −1.98835
\(220\) 0 0
\(221\) −10.1213 −0.680831
\(222\) 0 0
\(223\) −9.70637 −0.649987 −0.324993 0.945716i \(-0.605362\pi\)
−0.324993 + 0.945716i \(0.605362\pi\)
\(224\) 0 0
\(225\) 68.9469 4.59646
\(226\) 0 0
\(227\) 23.7536 1.57658 0.788292 0.615301i \(-0.210966\pi\)
0.788292 + 0.615301i \(0.210966\pi\)
\(228\) 0 0
\(229\) 18.6093 1.22974 0.614868 0.788630i \(-0.289209\pi\)
0.614868 + 0.788630i \(0.289209\pi\)
\(230\) 0 0
\(231\) −3.70069 −0.243488
\(232\) 0 0
\(233\) −10.7832 −0.706431 −0.353216 0.935542i \(-0.614912\pi\)
−0.353216 + 0.935542i \(0.614912\pi\)
\(234\) 0 0
\(235\) −5.47702 −0.357282
\(236\) 0 0
\(237\) −26.8418 −1.74356
\(238\) 0 0
\(239\) −2.08310 −0.134745 −0.0673724 0.997728i \(-0.521462\pi\)
−0.0673724 + 0.997728i \(0.521462\pi\)
\(240\) 0 0
\(241\) 15.8817 1.02303 0.511516 0.859274i \(-0.329084\pi\)
0.511516 + 0.859274i \(0.329084\pi\)
\(242\) 0 0
\(243\) −80.7182 −5.17808
\(244\) 0 0
\(245\) 3.59402 0.229614
\(246\) 0 0
\(247\) −0.112828 −0.00717911
\(248\) 0 0
\(249\) −35.9964 −2.28118
\(250\) 0 0
\(251\) −24.2151 −1.52844 −0.764220 0.644955i \(-0.776876\pi\)
−0.764220 + 0.644955i \(0.776876\pi\)
\(252\) 0 0
\(253\) 1.58404 0.0995880
\(254\) 0 0
\(255\) 69.5318 4.35425
\(256\) 0 0
\(257\) −16.3998 −1.02299 −0.511497 0.859285i \(-0.670909\pi\)
−0.511497 + 0.859285i \(0.670909\pi\)
\(258\) 0 0
\(259\) 7.19951 0.447356
\(260\) 0 0
\(261\) 43.9229 2.71876
\(262\) 0 0
\(263\) 24.4978 1.51060 0.755300 0.655379i \(-0.227491\pi\)
0.755300 + 0.655379i \(0.227491\pi\)
\(264\) 0 0
\(265\) 23.6912 1.45534
\(266\) 0 0
\(267\) −31.9179 −1.95334
\(268\) 0 0
\(269\) 1.48885 0.0907768 0.0453884 0.998969i \(-0.485547\pi\)
0.0453884 + 0.998969i \(0.485547\pi\)
\(270\) 0 0
\(271\) −28.0458 −1.70366 −0.851829 0.523820i \(-0.824507\pi\)
−0.851829 + 0.523820i \(0.824507\pi\)
\(272\) 0 0
\(273\) −6.12551 −0.370733
\(274\) 0 0
\(275\) 8.56227 0.516324
\(276\) 0 0
\(277\) 5.08702 0.305650 0.152825 0.988253i \(-0.451163\pi\)
0.152825 + 0.988253i \(0.451163\pi\)
\(278\) 0 0
\(279\) −41.4060 −2.47891
\(280\) 0 0
\(281\) 2.33236 0.139137 0.0695683 0.997577i \(-0.477838\pi\)
0.0695683 + 0.997577i \(0.477838\pi\)
\(282\) 0 0
\(283\) −6.35544 −0.377791 −0.188896 0.981997i \(-0.560491\pi\)
−0.188896 + 0.981997i \(0.560491\pi\)
\(284\) 0 0
\(285\) 0.775116 0.0459139
\(286\) 0 0
\(287\) −7.50243 −0.442854
\(288\) 0 0
\(289\) 14.9666 0.880386
\(290\) 0 0
\(291\) −1.66155 −0.0974019
\(292\) 0 0
\(293\) −1.45568 −0.0850415 −0.0425207 0.999096i \(-0.513539\pi\)
−0.0425207 + 0.999096i \(0.513539\pi\)
\(294\) 0 0
\(295\) 27.3890 1.59465
\(296\) 0 0
\(297\) −21.1262 −1.22587
\(298\) 0 0
\(299\) 2.62196 0.151632
\(300\) 0 0
\(301\) 4.56166 0.262930
\(302\) 0 0
\(303\) −22.0559 −1.26708
\(304\) 0 0
\(305\) −34.5989 −1.98113
\(306\) 0 0
\(307\) 7.44089 0.424674 0.212337 0.977196i \(-0.431893\pi\)
0.212337 + 0.977196i \(0.431893\pi\)
\(308\) 0 0
\(309\) −14.6064 −0.830926
\(310\) 0 0
\(311\) −8.74660 −0.495974 −0.247987 0.968763i \(-0.579769\pi\)
−0.247987 + 0.968763i \(0.579769\pi\)
\(312\) 0 0
\(313\) −8.77036 −0.495730 −0.247865 0.968795i \(-0.579729\pi\)
−0.247865 + 0.968795i \(0.579729\pi\)
\(314\) 0 0
\(315\) 31.2993 1.76352
\(316\) 0 0
\(317\) 21.5025 1.20770 0.603851 0.797097i \(-0.293632\pi\)
0.603851 + 0.797097i \(0.293632\pi\)
\(318\) 0 0
\(319\) 5.45463 0.305401
\(320\) 0 0
\(321\) −8.75210 −0.488495
\(322\) 0 0
\(323\) 0.356352 0.0198279
\(324\) 0 0
\(325\) 14.1726 0.786152
\(326\) 0 0
\(327\) 53.3047 2.94776
\(328\) 0 0
\(329\) −1.52393 −0.0840167
\(330\) 0 0
\(331\) 6.96891 0.383046 0.191523 0.981488i \(-0.438657\pi\)
0.191523 + 0.981488i \(0.438657\pi\)
\(332\) 0 0
\(333\) 62.6985 3.43586
\(334\) 0 0
\(335\) 25.0506 1.36866
\(336\) 0 0
\(337\) 12.5793 0.685237 0.342619 0.939475i \(-0.388686\pi\)
0.342619 + 0.939475i \(0.388686\pi\)
\(338\) 0 0
\(339\) −39.8111 −2.16224
\(340\) 0 0
\(341\) −5.14207 −0.278459
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −18.0125 −0.969761
\(346\) 0 0
\(347\) 4.78717 0.256989 0.128494 0.991710i \(-0.458986\pi\)
0.128494 + 0.991710i \(0.458986\pi\)
\(348\) 0 0
\(349\) −35.1475 −1.88140 −0.940702 0.339233i \(-0.889832\pi\)
−0.940702 + 0.339233i \(0.889832\pi\)
\(350\) 0 0
\(351\) −34.9688 −1.86650
\(352\) 0 0
\(353\) 8.70479 0.463309 0.231655 0.972798i \(-0.425586\pi\)
0.231655 + 0.972798i \(0.425586\pi\)
\(354\) 0 0
\(355\) −22.2537 −1.18110
\(356\) 0 0
\(357\) 19.3465 1.02393
\(358\) 0 0
\(359\) −4.04735 −0.213611 −0.106805 0.994280i \(-0.534062\pi\)
−0.106805 + 0.994280i \(0.534062\pi\)
\(360\) 0 0
\(361\) −18.9960 −0.999791
\(362\) 0 0
\(363\) 33.6375 1.76551
\(364\) 0 0
\(365\) 30.9059 1.61769
\(366\) 0 0
\(367\) 31.6904 1.65423 0.827113 0.562036i \(-0.189982\pi\)
0.827113 + 0.562036i \(0.189982\pi\)
\(368\) 0 0
\(369\) −65.3365 −3.40128
\(370\) 0 0
\(371\) 6.59185 0.342232
\(372\) 0 0
\(373\) 21.4087 1.10850 0.554251 0.832349i \(-0.313005\pi\)
0.554251 + 0.832349i \(0.313005\pi\)
\(374\) 0 0
\(375\) −35.8733 −1.85249
\(376\) 0 0
\(377\) 9.02869 0.465001
\(378\) 0 0
\(379\) 19.5360 1.00350 0.501749 0.865014i \(-0.332690\pi\)
0.501749 + 0.865014i \(0.332690\pi\)
\(380\) 0 0
\(381\) 40.2507 2.06210
\(382\) 0 0
\(383\) 17.3513 0.886610 0.443305 0.896371i \(-0.353806\pi\)
0.443305 + 0.896371i \(0.353806\pi\)
\(384\) 0 0
\(385\) 3.88695 0.198097
\(386\) 0 0
\(387\) 39.7262 2.01940
\(388\) 0 0
\(389\) −25.2975 −1.28263 −0.641316 0.767277i \(-0.721611\pi\)
−0.641316 + 0.767277i \(0.721611\pi\)
\(390\) 0 0
\(391\) −8.28108 −0.418792
\(392\) 0 0
\(393\) 53.6781 2.70770
\(394\) 0 0
\(395\) 28.1928 1.41853
\(396\) 0 0
\(397\) −6.15919 −0.309121 −0.154560 0.987983i \(-0.549396\pi\)
−0.154560 + 0.987983i \(0.549396\pi\)
\(398\) 0 0
\(399\) 0.215668 0.0107969
\(400\) 0 0
\(401\) −17.6318 −0.880492 −0.440246 0.897877i \(-0.645109\pi\)
−0.440246 + 0.897877i \(0.645109\pi\)
\(402\) 0 0
\(403\) −8.51133 −0.423979
\(404\) 0 0
\(405\) 146.333 7.27132
\(406\) 0 0
\(407\) 7.78631 0.385953
\(408\) 0 0
\(409\) 24.7264 1.22264 0.611320 0.791383i \(-0.290639\pi\)
0.611320 + 0.791383i \(0.290639\pi\)
\(410\) 0 0
\(411\) 25.5663 1.26109
\(412\) 0 0
\(413\) 7.62071 0.374991
\(414\) 0 0
\(415\) 37.8081 1.85593
\(416\) 0 0
\(417\) −1.20539 −0.0590283
\(418\) 0 0
\(419\) 2.36688 0.115630 0.0578149 0.998327i \(-0.481587\pi\)
0.0578149 + 0.998327i \(0.481587\pi\)
\(420\) 0 0
\(421\) 31.4952 1.53498 0.767492 0.641059i \(-0.221505\pi\)
0.767492 + 0.641059i \(0.221505\pi\)
\(422\) 0 0
\(423\) −13.2714 −0.645279
\(424\) 0 0
\(425\) −44.7619 −2.17127
\(426\) 0 0
\(427\) −9.62680 −0.465873
\(428\) 0 0
\(429\) −6.62477 −0.319847
\(430\) 0 0
\(431\) 23.0028 1.10800 0.554002 0.832515i \(-0.313100\pi\)
0.554002 + 0.832515i \(0.313100\pi\)
\(432\) 0 0
\(433\) 1.31078 0.0629921 0.0314960 0.999504i \(-0.489973\pi\)
0.0314960 + 0.999504i \(0.489973\pi\)
\(434\) 0 0
\(435\) −62.0258 −2.97391
\(436\) 0 0
\(437\) −0.0923145 −0.00441600
\(438\) 0 0
\(439\) 36.2799 1.73155 0.865773 0.500437i \(-0.166827\pi\)
0.865773 + 0.500437i \(0.166827\pi\)
\(440\) 0 0
\(441\) 8.70871 0.414701
\(442\) 0 0
\(443\) 13.5410 0.643352 0.321676 0.946850i \(-0.395754\pi\)
0.321676 + 0.946850i \(0.395754\pi\)
\(444\) 0 0
\(445\) 33.5243 1.58921
\(446\) 0 0
\(447\) −21.4389 −1.01403
\(448\) 0 0
\(449\) −3.60363 −0.170066 −0.0850329 0.996378i \(-0.527100\pi\)
−0.0850329 + 0.996378i \(0.527100\pi\)
\(450\) 0 0
\(451\) −8.11391 −0.382069
\(452\) 0 0
\(453\) 42.5929 2.00119
\(454\) 0 0
\(455\) 6.43381 0.301622
\(456\) 0 0
\(457\) −6.35277 −0.297170 −0.148585 0.988900i \(-0.547472\pi\)
−0.148585 + 0.988900i \(0.547472\pi\)
\(458\) 0 0
\(459\) 110.444 5.15507
\(460\) 0 0
\(461\) −5.88792 −0.274228 −0.137114 0.990555i \(-0.543783\pi\)
−0.137114 + 0.990555i \(0.543783\pi\)
\(462\) 0 0
\(463\) 12.7205 0.591171 0.295585 0.955316i \(-0.404485\pi\)
0.295585 + 0.955316i \(0.404485\pi\)
\(464\) 0 0
\(465\) 58.4716 2.71156
\(466\) 0 0
\(467\) 21.7910 1.00836 0.504182 0.863597i \(-0.331794\pi\)
0.504182 + 0.863597i \(0.331794\pi\)
\(468\) 0 0
\(469\) 6.97006 0.321848
\(470\) 0 0
\(471\) −17.9579 −0.827455
\(472\) 0 0
\(473\) 4.93346 0.226841
\(474\) 0 0
\(475\) −0.498990 −0.0228952
\(476\) 0 0
\(477\) 57.4065 2.62846
\(478\) 0 0
\(479\) −26.3235 −1.20275 −0.601375 0.798967i \(-0.705380\pi\)
−0.601375 + 0.798967i \(0.705380\pi\)
\(480\) 0 0
\(481\) 12.8882 0.587649
\(482\) 0 0
\(483\) −5.01180 −0.228045
\(484\) 0 0
\(485\) 1.74518 0.0792445
\(486\) 0 0
\(487\) 4.62575 0.209613 0.104806 0.994493i \(-0.466578\pi\)
0.104806 + 0.994493i \(0.466578\pi\)
\(488\) 0 0
\(489\) −26.4253 −1.19499
\(490\) 0 0
\(491\) −3.87808 −0.175015 −0.0875076 0.996164i \(-0.527890\pi\)
−0.0875076 + 0.996164i \(0.527890\pi\)
\(492\) 0 0
\(493\) −28.5157 −1.28428
\(494\) 0 0
\(495\) 33.8504 1.52146
\(496\) 0 0
\(497\) −6.19187 −0.277743
\(498\) 0 0
\(499\) 0.994539 0.0445217 0.0222608 0.999752i \(-0.492914\pi\)
0.0222608 + 0.999752i \(0.492914\pi\)
\(500\) 0 0
\(501\) −28.7215 −1.28318
\(502\) 0 0
\(503\) 23.3204 1.03980 0.519902 0.854226i \(-0.325968\pi\)
0.519902 + 0.854226i \(0.325968\pi\)
\(504\) 0 0
\(505\) 23.1660 1.03087
\(506\) 0 0
\(507\) 33.5179 1.48858
\(508\) 0 0
\(509\) −6.79254 −0.301074 −0.150537 0.988604i \(-0.548100\pi\)
−0.150537 + 0.988604i \(0.548100\pi\)
\(510\) 0 0
\(511\) 8.59924 0.380408
\(512\) 0 0
\(513\) 1.23119 0.0543582
\(514\) 0 0
\(515\) 15.3415 0.676027
\(516\) 0 0
\(517\) −1.64813 −0.0724848
\(518\) 0 0
\(519\) −39.5376 −1.73551
\(520\) 0 0
\(521\) 33.2330 1.45596 0.727982 0.685596i \(-0.240458\pi\)
0.727982 + 0.685596i \(0.240458\pi\)
\(522\) 0 0
\(523\) −28.8901 −1.26328 −0.631638 0.775264i \(-0.717617\pi\)
−0.631638 + 0.775264i \(0.717617\pi\)
\(524\) 0 0
\(525\) −27.0904 −1.18232
\(526\) 0 0
\(527\) 26.8818 1.17099
\(528\) 0 0
\(529\) −20.8548 −0.906728
\(530\) 0 0
\(531\) 66.3666 2.88006
\(532\) 0 0
\(533\) −13.4304 −0.581736
\(534\) 0 0
\(535\) 9.19260 0.397431
\(536\) 0 0
\(537\) −66.6140 −2.87461
\(538\) 0 0
\(539\) 1.08150 0.0465837
\(540\) 0 0
\(541\) −4.95435 −0.213004 −0.106502 0.994312i \(-0.533965\pi\)
−0.106502 + 0.994312i \(0.533965\pi\)
\(542\) 0 0
\(543\) 39.2791 1.68563
\(544\) 0 0
\(545\) −55.9876 −2.39824
\(546\) 0 0
\(547\) −31.0796 −1.32887 −0.664434 0.747347i \(-0.731328\pi\)
−0.664434 + 0.747347i \(0.731328\pi\)
\(548\) 0 0
\(549\) −83.8370 −3.57808
\(550\) 0 0
\(551\) −0.317883 −0.0135423
\(552\) 0 0
\(553\) 7.84435 0.333576
\(554\) 0 0
\(555\) −88.5398 −3.75830
\(556\) 0 0
\(557\) 6.01930 0.255046 0.127523 0.991836i \(-0.459297\pi\)
0.127523 + 0.991836i \(0.459297\pi\)
\(558\) 0 0
\(559\) 8.16603 0.345386
\(560\) 0 0
\(561\) 20.9233 0.883384
\(562\) 0 0
\(563\) −12.6945 −0.535008 −0.267504 0.963557i \(-0.586199\pi\)
−0.267504 + 0.963557i \(0.586199\pi\)
\(564\) 0 0
\(565\) 41.8148 1.75916
\(566\) 0 0
\(567\) 40.7155 1.70989
\(568\) 0 0
\(569\) 12.3968 0.519701 0.259851 0.965649i \(-0.416327\pi\)
0.259851 + 0.965649i \(0.416327\pi\)
\(570\) 0 0
\(571\) 7.41186 0.310177 0.155088 0.987901i \(-0.450434\pi\)
0.155088 + 0.987901i \(0.450434\pi\)
\(572\) 0 0
\(573\) 56.7677 2.37151
\(574\) 0 0
\(575\) 11.5958 0.483577
\(576\) 0 0
\(577\) 43.5232 1.81189 0.905947 0.423390i \(-0.139160\pi\)
0.905947 + 0.423390i \(0.139160\pi\)
\(578\) 0 0
\(579\) 4.52569 0.188081
\(580\) 0 0
\(581\) 10.5197 0.436432
\(582\) 0 0
\(583\) 7.12912 0.295258
\(584\) 0 0
\(585\) 56.0302 2.31657
\(586\) 0 0
\(587\) 3.74658 0.154638 0.0773190 0.997006i \(-0.475364\pi\)
0.0773190 + 0.997006i \(0.475364\pi\)
\(588\) 0 0
\(589\) 0.299668 0.0123476
\(590\) 0 0
\(591\) 55.9299 2.30065
\(592\) 0 0
\(593\) −5.12318 −0.210384 −0.105192 0.994452i \(-0.533546\pi\)
−0.105192 + 0.994452i \(0.533546\pi\)
\(594\) 0 0
\(595\) −20.3202 −0.833048
\(596\) 0 0
\(597\) −89.4535 −3.66109
\(598\) 0 0
\(599\) 39.6945 1.62187 0.810936 0.585135i \(-0.198959\pi\)
0.810936 + 0.585135i \(0.198959\pi\)
\(600\) 0 0
\(601\) −19.6274 −0.800620 −0.400310 0.916380i \(-0.631097\pi\)
−0.400310 + 0.916380i \(0.631097\pi\)
\(602\) 0 0
\(603\) 60.7003 2.47191
\(604\) 0 0
\(605\) −35.3305 −1.43639
\(606\) 0 0
\(607\) 25.7837 1.04653 0.523264 0.852170i \(-0.324714\pi\)
0.523264 + 0.852170i \(0.324714\pi\)
\(608\) 0 0
\(609\) −17.2580 −0.699331
\(610\) 0 0
\(611\) −2.72804 −0.110365
\(612\) 0 0
\(613\) −5.78639 −0.233710 −0.116855 0.993149i \(-0.537281\pi\)
−0.116855 + 0.993149i \(0.537281\pi\)
\(614\) 0 0
\(615\) 92.2651 3.72049
\(616\) 0 0
\(617\) −10.4658 −0.421337 −0.210668 0.977558i \(-0.567564\pi\)
−0.210668 + 0.977558i \(0.567564\pi\)
\(618\) 0 0
\(619\) −30.4411 −1.22353 −0.611765 0.791040i \(-0.709540\pi\)
−0.611765 + 0.791040i \(0.709540\pi\)
\(620\) 0 0
\(621\) −28.6109 −1.14812
\(622\) 0 0
\(623\) 9.32780 0.373711
\(624\) 0 0
\(625\) −1.90615 −0.0762459
\(626\) 0 0
\(627\) 0.233246 0.00931495
\(628\) 0 0
\(629\) −40.7053 −1.62303
\(630\) 0 0
\(631\) −0.948167 −0.0377459 −0.0188730 0.999822i \(-0.506008\pi\)
−0.0188730 + 0.999822i \(0.506008\pi\)
\(632\) 0 0
\(633\) −84.6052 −3.36275
\(634\) 0 0
\(635\) −42.2765 −1.67769
\(636\) 0 0
\(637\) 1.79014 0.0709280
\(638\) 0 0
\(639\) −53.9232 −2.13317
\(640\) 0 0
\(641\) 12.1984 0.481808 0.240904 0.970549i \(-0.422556\pi\)
0.240904 + 0.970549i \(0.422556\pi\)
\(642\) 0 0
\(643\) 50.4676 1.99025 0.995124 0.0986311i \(-0.0314464\pi\)
0.995124 + 0.0986311i \(0.0314464\pi\)
\(644\) 0 0
\(645\) −56.0995 −2.20891
\(646\) 0 0
\(647\) −29.4906 −1.15939 −0.579697 0.814832i \(-0.696829\pi\)
−0.579697 + 0.814832i \(0.696829\pi\)
\(648\) 0 0
\(649\) 8.24183 0.323520
\(650\) 0 0
\(651\) 16.2691 0.637637
\(652\) 0 0
\(653\) 43.6924 1.70982 0.854908 0.518779i \(-0.173613\pi\)
0.854908 + 0.518779i \(0.173613\pi\)
\(654\) 0 0
\(655\) −56.3798 −2.20294
\(656\) 0 0
\(657\) 74.8883 2.92167
\(658\) 0 0
\(659\) −17.8415 −0.695006 −0.347503 0.937679i \(-0.612970\pi\)
−0.347503 + 0.937679i \(0.612970\pi\)
\(660\) 0 0
\(661\) −29.0216 −1.12881 −0.564405 0.825498i \(-0.690894\pi\)
−0.564405 + 0.825498i \(0.690894\pi\)
\(662\) 0 0
\(663\) 34.6330 1.34503
\(664\) 0 0
\(665\) −0.226523 −0.00878418
\(666\) 0 0
\(667\) 7.38713 0.286031
\(668\) 0 0
\(669\) 33.2133 1.28410
\(670\) 0 0
\(671\) −10.4114 −0.401929
\(672\) 0 0
\(673\) 19.4620 0.750205 0.375103 0.926983i \(-0.377607\pi\)
0.375103 + 0.926983i \(0.377607\pi\)
\(674\) 0 0
\(675\) −154.651 −5.95253
\(676\) 0 0
\(677\) 5.21145 0.200292 0.100146 0.994973i \(-0.468069\pi\)
0.100146 + 0.994973i \(0.468069\pi\)
\(678\) 0 0
\(679\) 0.485578 0.0186348
\(680\) 0 0
\(681\) −81.2802 −3.11466
\(682\) 0 0
\(683\) −1.15851 −0.0443292 −0.0221646 0.999754i \(-0.507056\pi\)
−0.0221646 + 0.999754i \(0.507056\pi\)
\(684\) 0 0
\(685\) −26.8531 −1.02600
\(686\) 0 0
\(687\) −63.6772 −2.42944
\(688\) 0 0
\(689\) 11.8003 0.449558
\(690\) 0 0
\(691\) −12.8345 −0.488247 −0.244123 0.969744i \(-0.578500\pi\)
−0.244123 + 0.969744i \(0.578500\pi\)
\(692\) 0 0
\(693\) 9.41852 0.357780
\(694\) 0 0
\(695\) 1.26606 0.0480244
\(696\) 0 0
\(697\) 42.4180 1.60670
\(698\) 0 0
\(699\) 36.8980 1.39561
\(700\) 0 0
\(701\) 31.9498 1.20673 0.603364 0.797466i \(-0.293827\pi\)
0.603364 + 0.797466i \(0.293827\pi\)
\(702\) 0 0
\(703\) −0.453768 −0.0171142
\(704\) 0 0
\(705\) 18.7413 0.705837
\(706\) 0 0
\(707\) 6.44571 0.242416
\(708\) 0 0
\(709\) −19.2624 −0.723415 −0.361708 0.932292i \(-0.617806\pi\)
−0.361708 + 0.932292i \(0.617806\pi\)
\(710\) 0 0
\(711\) 68.3142 2.56198
\(712\) 0 0
\(713\) −6.96383 −0.260798
\(714\) 0 0
\(715\) 6.95820 0.260222
\(716\) 0 0
\(717\) 7.12796 0.266199
\(718\) 0 0
\(719\) 25.9785 0.968836 0.484418 0.874837i \(-0.339031\pi\)
0.484418 + 0.874837i \(0.339031\pi\)
\(720\) 0 0
\(721\) 4.26862 0.158972
\(722\) 0 0
\(723\) −54.3441 −2.02108
\(724\) 0 0
\(725\) 39.9298 1.48296
\(726\) 0 0
\(727\) −27.5763 −1.02275 −0.511375 0.859358i \(-0.670864\pi\)
−0.511375 + 0.859358i \(0.670864\pi\)
\(728\) 0 0
\(729\) 154.055 5.70574
\(730\) 0 0
\(731\) −25.7912 −0.953922
\(732\) 0 0
\(733\) −42.7081 −1.57746 −0.788730 0.614740i \(-0.789261\pi\)
−0.788730 + 0.614740i \(0.789261\pi\)
\(734\) 0 0
\(735\) −12.2980 −0.453619
\(736\) 0 0
\(737\) 7.53816 0.277672
\(738\) 0 0
\(739\) −51.6817 −1.90114 −0.950571 0.310508i \(-0.899501\pi\)
−0.950571 + 0.310508i \(0.899501\pi\)
\(740\) 0 0
\(741\) 0.386077 0.0141829
\(742\) 0 0
\(743\) −43.1375 −1.58256 −0.791281 0.611452i \(-0.790586\pi\)
−0.791281 + 0.611452i \(0.790586\pi\)
\(744\) 0 0
\(745\) 22.5180 0.824994
\(746\) 0 0
\(747\) 91.6133 3.35196
\(748\) 0 0
\(749\) 2.55775 0.0934581
\(750\) 0 0
\(751\) −9.04305 −0.329986 −0.164993 0.986295i \(-0.552760\pi\)
−0.164993 + 0.986295i \(0.552760\pi\)
\(752\) 0 0
\(753\) 82.8591 3.01955
\(754\) 0 0
\(755\) −44.7367 −1.62813
\(756\) 0 0
\(757\) −1.51946 −0.0552258 −0.0276129 0.999619i \(-0.508791\pi\)
−0.0276129 + 0.999619i \(0.508791\pi\)
\(758\) 0 0
\(759\) −5.42028 −0.196744
\(760\) 0 0
\(761\) 34.1138 1.23662 0.618312 0.785932i \(-0.287817\pi\)
0.618312 + 0.785932i \(0.287817\pi\)
\(762\) 0 0
\(763\) −15.5780 −0.563961
\(764\) 0 0
\(765\) −176.963 −6.39811
\(766\) 0 0
\(767\) 13.6422 0.492590
\(768\) 0 0
\(769\) −54.2612 −1.95671 −0.978354 0.206939i \(-0.933650\pi\)
−0.978354 + 0.206939i \(0.933650\pi\)
\(770\) 0 0
\(771\) 56.1169 2.02100
\(772\) 0 0
\(773\) −36.3141 −1.30613 −0.653063 0.757303i \(-0.726516\pi\)
−0.653063 + 0.757303i \(0.726516\pi\)
\(774\) 0 0
\(775\) −37.6418 −1.35213
\(776\) 0 0
\(777\) −24.6353 −0.883786
\(778\) 0 0
\(779\) 0.472860 0.0169420
\(780\) 0 0
\(781\) −6.69653 −0.239621
\(782\) 0 0
\(783\) −98.5212 −3.52086
\(784\) 0 0
\(785\) 18.8617 0.673203
\(786\) 0 0
\(787\) 23.1195 0.824120 0.412060 0.911157i \(-0.364809\pi\)
0.412060 + 0.911157i \(0.364809\pi\)
\(788\) 0 0
\(789\) −83.8266 −2.98431
\(790\) 0 0
\(791\) 11.6345 0.413677
\(792\) 0 0
\(793\) −17.2333 −0.611974
\(794\) 0 0
\(795\) −81.0667 −2.87514
\(796\) 0 0
\(797\) −42.1734 −1.49386 −0.746929 0.664904i \(-0.768472\pi\)
−0.746929 + 0.664904i \(0.768472\pi\)
\(798\) 0 0
\(799\) 8.61612 0.304816
\(800\) 0 0
\(801\) 81.2332 2.87023
\(802\) 0 0
\(803\) 9.30012 0.328194
\(804\) 0 0
\(805\) 5.26404 0.185533
\(806\) 0 0
\(807\) −5.09455 −0.179337
\(808\) 0 0
\(809\) 33.6378 1.18264 0.591320 0.806437i \(-0.298607\pi\)
0.591320 + 0.806437i \(0.298607\pi\)
\(810\) 0 0
\(811\) −28.5082 −1.00106 −0.500528 0.865720i \(-0.666861\pi\)
−0.500528 + 0.865720i \(0.666861\pi\)
\(812\) 0 0
\(813\) 95.9670 3.36571
\(814\) 0 0
\(815\) 27.7553 0.972226
\(816\) 0 0
\(817\) −0.287511 −0.0100587
\(818\) 0 0
\(819\) 15.5898 0.544753
\(820\) 0 0
\(821\) 55.8856 1.95042 0.975210 0.221282i \(-0.0710242\pi\)
0.975210 + 0.221282i \(0.0710242\pi\)
\(822\) 0 0
\(823\) −9.35542 −0.326109 −0.163055 0.986617i \(-0.552135\pi\)
−0.163055 + 0.986617i \(0.552135\pi\)
\(824\) 0 0
\(825\) −29.2984 −1.02004
\(826\) 0 0
\(827\) −45.6961 −1.58901 −0.794504 0.607259i \(-0.792269\pi\)
−0.794504 + 0.607259i \(0.792269\pi\)
\(828\) 0 0
\(829\) −7.18848 −0.249666 −0.124833 0.992178i \(-0.539840\pi\)
−0.124833 + 0.992178i \(0.539840\pi\)
\(830\) 0 0
\(831\) −17.4068 −0.603834
\(832\) 0 0
\(833\) −5.65390 −0.195896
\(834\) 0 0
\(835\) 30.1671 1.04397
\(836\) 0 0
\(837\) 92.8758 3.21026
\(838\) 0 0
\(839\) −1.28514 −0.0443681 −0.0221841 0.999754i \(-0.507062\pi\)
−0.0221841 + 0.999754i \(0.507062\pi\)
\(840\) 0 0
\(841\) −3.56255 −0.122846
\(842\) 0 0
\(843\) −7.98085 −0.274875
\(844\) 0 0
\(845\) −35.2048 −1.21108
\(846\) 0 0
\(847\) −9.83035 −0.337775
\(848\) 0 0
\(849\) 21.7470 0.746356
\(850\) 0 0
\(851\) 10.5449 0.361474
\(852\) 0 0
\(853\) 13.5876 0.465232 0.232616 0.972569i \(-0.425271\pi\)
0.232616 + 0.972569i \(0.425271\pi\)
\(854\) 0 0
\(855\) −1.97272 −0.0674657
\(856\) 0 0
\(857\) −35.7439 −1.22099 −0.610494 0.792021i \(-0.709029\pi\)
−0.610494 + 0.792021i \(0.709029\pi\)
\(858\) 0 0
\(859\) 35.4359 1.20906 0.604529 0.796583i \(-0.293361\pi\)
0.604529 + 0.796583i \(0.293361\pi\)
\(860\) 0 0
\(861\) 25.6718 0.874893
\(862\) 0 0
\(863\) −18.5689 −0.632092 −0.316046 0.948744i \(-0.602355\pi\)
−0.316046 + 0.948744i \(0.602355\pi\)
\(864\) 0 0
\(865\) 41.5276 1.41198
\(866\) 0 0
\(867\) −51.2126 −1.73927
\(868\) 0 0
\(869\) 8.48370 0.287790
\(870\) 0 0
\(871\) 12.4774 0.422781
\(872\) 0 0
\(873\) 4.22876 0.143122
\(874\) 0 0
\(875\) 10.4838 0.354415
\(876\) 0 0
\(877\) 56.4896 1.90752 0.953759 0.300572i \(-0.0971775\pi\)
0.953759 + 0.300572i \(0.0971775\pi\)
\(878\) 0 0
\(879\) 4.98103 0.168006
\(880\) 0 0
\(881\) −41.1798 −1.38738 −0.693691 0.720273i \(-0.744017\pi\)
−0.693691 + 0.720273i \(0.744017\pi\)
\(882\) 0 0
\(883\) 38.1836 1.28498 0.642490 0.766294i \(-0.277901\pi\)
0.642490 + 0.766294i \(0.277901\pi\)
\(884\) 0 0
\(885\) −93.7197 −3.15035
\(886\) 0 0
\(887\) −47.5720 −1.59731 −0.798656 0.601788i \(-0.794455\pi\)
−0.798656 + 0.601788i \(0.794455\pi\)
\(888\) 0 0
\(889\) −11.7630 −0.394519
\(890\) 0 0
\(891\) 44.0341 1.47520
\(892\) 0 0
\(893\) 0.0960494 0.00321417
\(894\) 0 0
\(895\) 69.9667 2.33873
\(896\) 0 0
\(897\) −8.97183 −0.299561
\(898\) 0 0
\(899\) −23.9798 −0.799773
\(900\) 0 0
\(901\) −37.2696 −1.24163
\(902\) 0 0
\(903\) −15.6091 −0.519438
\(904\) 0 0
\(905\) −41.2561 −1.37140
\(906\) 0 0
\(907\) −33.0681 −1.09801 −0.549004 0.835820i \(-0.684993\pi\)
−0.549004 + 0.835820i \(0.684993\pi\)
\(908\) 0 0
\(909\) 56.1338 1.86184
\(910\) 0 0
\(911\) 36.5816 1.21200 0.606002 0.795463i \(-0.292772\pi\)
0.606002 + 0.795463i \(0.292772\pi\)
\(912\) 0 0
\(913\) 11.3771 0.376528
\(914\) 0 0
\(915\) 118.391 3.91387
\(916\) 0 0
\(917\) −15.6871 −0.518033
\(918\) 0 0
\(919\) −16.2676 −0.536618 −0.268309 0.963333i \(-0.586465\pi\)
−0.268309 + 0.963333i \(0.586465\pi\)
\(920\) 0 0
\(921\) −25.4612 −0.838976
\(922\) 0 0
\(923\) −11.0843 −0.364845
\(924\) 0 0
\(925\) 56.9985 1.87410
\(926\) 0 0
\(927\) 37.1742 1.22096
\(928\) 0 0
\(929\) 11.1820 0.366871 0.183435 0.983032i \(-0.441278\pi\)
0.183435 + 0.983032i \(0.441278\pi\)
\(930\) 0 0
\(931\) −0.0630276 −0.00206565
\(932\) 0 0
\(933\) 29.9291 0.979835
\(934\) 0 0
\(935\) −21.9764 −0.718706
\(936\) 0 0
\(937\) 37.5933 1.22812 0.614059 0.789260i \(-0.289536\pi\)
0.614059 + 0.789260i \(0.289536\pi\)
\(938\) 0 0
\(939\) 30.0104 0.979352
\(940\) 0 0
\(941\) 19.9354 0.649876 0.324938 0.945735i \(-0.394656\pi\)
0.324938 + 0.945735i \(0.394656\pi\)
\(942\) 0 0
\(943\) −10.9886 −0.357837
\(944\) 0 0
\(945\) −70.2059 −2.28380
\(946\) 0 0
\(947\) 11.2482 0.365518 0.182759 0.983158i \(-0.441497\pi\)
0.182759 + 0.983158i \(0.441497\pi\)
\(948\) 0 0
\(949\) 15.3939 0.499706
\(950\) 0 0
\(951\) −73.5774 −2.38591
\(952\) 0 0
\(953\) −29.7695 −0.964329 −0.482164 0.876081i \(-0.660149\pi\)
−0.482164 + 0.876081i \(0.660149\pi\)
\(954\) 0 0
\(955\) −59.6249 −1.92942
\(956\) 0 0
\(957\) −18.6647 −0.603343
\(958\) 0 0
\(959\) −7.47159 −0.241270
\(960\) 0 0
\(961\) −8.39424 −0.270782
\(962\) 0 0
\(963\) 22.2747 0.717792
\(964\) 0 0
\(965\) −4.75348 −0.153020
\(966\) 0 0
\(967\) 55.3211 1.77901 0.889504 0.456927i \(-0.151050\pi\)
0.889504 + 0.456927i \(0.151050\pi\)
\(968\) 0 0
\(969\) −1.21936 −0.0391716
\(970\) 0 0
\(971\) 11.9746 0.384284 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(972\) 0 0
\(973\) 0.352269 0.0112932
\(974\) 0 0
\(975\) −48.4956 −1.55310
\(976\) 0 0
\(977\) −39.7381 −1.27133 −0.635667 0.771963i \(-0.719275\pi\)
−0.635667 + 0.771963i \(0.719275\pi\)
\(978\) 0 0
\(979\) 10.0881 0.322416
\(980\) 0 0
\(981\) −135.664 −4.33142
\(982\) 0 0
\(983\) 5.03380 0.160553 0.0802766 0.996773i \(-0.474420\pi\)
0.0802766 + 0.996773i \(0.474420\pi\)
\(984\) 0 0
\(985\) −58.7449 −1.87177
\(986\) 0 0
\(987\) 5.21457 0.165982
\(988\) 0 0
\(989\) 6.68132 0.212454
\(990\) 0 0
\(991\) 28.5560 0.907110 0.453555 0.891228i \(-0.350156\pi\)
0.453555 + 0.891228i \(0.350156\pi\)
\(992\) 0 0
\(993\) −23.8462 −0.756737
\(994\) 0 0
\(995\) 93.9558 2.97860
\(996\) 0 0
\(997\) 26.4015 0.836144 0.418072 0.908414i \(-0.362706\pi\)
0.418072 + 0.908414i \(0.362706\pi\)
\(998\) 0 0
\(999\) −140.636 −4.44952
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 7168.2.a.bf.1.1 8
4.3 odd 2 7168.2.a.be.1.8 8
8.3 odd 2 7168.2.a.ba.1.1 8
8.5 even 2 7168.2.a.bb.1.8 8
32.3 odd 8 1792.2.m.g.1345.8 yes 16
32.5 even 8 1792.2.m.h.449.8 yes 16
32.11 odd 8 1792.2.m.g.449.8 yes 16
32.13 even 8 1792.2.m.h.1345.8 yes 16
32.19 odd 8 1792.2.m.f.1345.1 yes 16
32.21 even 8 1792.2.m.e.449.1 16
32.27 odd 8 1792.2.m.f.449.1 yes 16
32.29 even 8 1792.2.m.e.1345.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.1 16 32.21 even 8
1792.2.m.e.1345.1 yes 16 32.29 even 8
1792.2.m.f.449.1 yes 16 32.27 odd 8
1792.2.m.f.1345.1 yes 16 32.19 odd 8
1792.2.m.g.449.8 yes 16 32.11 odd 8
1792.2.m.g.1345.8 yes 16 32.3 odd 8
1792.2.m.h.449.8 yes 16 32.5 even 8
1792.2.m.h.1345.8 yes 16 32.13 even 8
7168.2.a.ba.1.1 8 8.3 odd 2
7168.2.a.bb.1.8 8 8.5 even 2
7168.2.a.be.1.8 8 4.3 odd 2
7168.2.a.bf.1.1 8 1.1 even 1 trivial