Properties

Label 7168.2.a.bb.1.5
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
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: \(1\)
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 1.5
Root \(-2.02908\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.242103 q^{3} +0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} +O(q^{10})\) \(q+0.242103 q^{3} +0.379610 q^{5} +1.00000 q^{7} -2.94139 q^{9} -2.61527 q^{11} -2.31328 q^{13} +0.0919045 q^{15} +7.37134 q^{17} -5.44437 q^{19} +0.242103 q^{21} +6.44892 q^{23} -4.85590 q^{25} -1.43843 q^{27} +5.07278 q^{29} +6.10161 q^{31} -0.633164 q^{33} +0.379610 q^{35} -10.4914 q^{37} -0.560052 q^{39} -0.836588 q^{41} +5.50056 q^{43} -1.11658 q^{45} -6.02070 q^{47} +1.00000 q^{49} +1.78462 q^{51} +0.813824 q^{53} -0.992782 q^{55} -1.31810 q^{57} +7.53795 q^{59} +1.31502 q^{61} -2.94139 q^{63} -0.878144 q^{65} +8.79384 q^{67} +1.56130 q^{69} -11.4285 q^{71} -3.68616 q^{73} -1.17563 q^{75} -2.61527 q^{77} -4.21672 q^{79} +8.47591 q^{81} -16.9913 q^{83} +2.79823 q^{85} +1.22813 q^{87} +9.32780 q^{89} -2.31328 q^{91} +1.47722 q^{93} -2.06673 q^{95} -13.9032 q^{97} +7.69252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 0.242103 0.139778 0.0698890 0.997555i \(-0.477735\pi\)
0.0698890 + 0.997555i \(0.477735\pi\)
\(4\) 0 0
\(5\) 0.379610 0.169767 0.0848833 0.996391i \(-0.472948\pi\)
0.0848833 + 0.996391i \(0.472948\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.94139 −0.980462
\(10\) 0 0
\(11\) −2.61527 −0.788533 −0.394267 0.918996i \(-0.629001\pi\)
−0.394267 + 0.918996i \(0.629001\pi\)
\(12\) 0 0
\(13\) −2.31328 −0.641589 −0.320794 0.947149i \(-0.603950\pi\)
−0.320794 + 0.947149i \(0.603950\pi\)
\(14\) 0 0
\(15\) 0.0919045 0.0237296
\(16\) 0 0
\(17\) 7.37134 1.78781 0.893906 0.448255i \(-0.147954\pi\)
0.893906 + 0.448255i \(0.147954\pi\)
\(18\) 0 0
\(19\) −5.44437 −1.24902 −0.624512 0.781015i \(-0.714702\pi\)
−0.624512 + 0.781015i \(0.714702\pi\)
\(20\) 0 0
\(21\) 0.242103 0.0528312
\(22\) 0 0
\(23\) 6.44892 1.34469 0.672347 0.740236i \(-0.265286\pi\)
0.672347 + 0.740236i \(0.265286\pi\)
\(24\) 0 0
\(25\) −4.85590 −0.971179
\(26\) 0 0
\(27\) −1.43843 −0.276825
\(28\) 0 0
\(29\) 5.07278 0.941991 0.470996 0.882135i \(-0.343895\pi\)
0.470996 + 0.882135i \(0.343895\pi\)
\(30\) 0 0
\(31\) 6.10161 1.09588 0.547941 0.836517i \(-0.315412\pi\)
0.547941 + 0.836517i \(0.315412\pi\)
\(32\) 0 0
\(33\) −0.633164 −0.110220
\(34\) 0 0
\(35\) 0.379610 0.0641657
\(36\) 0 0
\(37\) −10.4914 −1.72477 −0.862386 0.506252i \(-0.831031\pi\)
−0.862386 + 0.506252i \(0.831031\pi\)
\(38\) 0 0
\(39\) −0.560052 −0.0896801
\(40\) 0 0
\(41\) −0.836588 −0.130653 −0.0653265 0.997864i \(-0.520809\pi\)
−0.0653265 + 0.997864i \(0.520809\pi\)
\(42\) 0 0
\(43\) 5.50056 0.838828 0.419414 0.907795i \(-0.362236\pi\)
0.419414 + 0.907795i \(0.362236\pi\)
\(44\) 0 0
\(45\) −1.11658 −0.166450
\(46\) 0 0
\(47\) −6.02070 −0.878209 −0.439104 0.898436i \(-0.644704\pi\)
−0.439104 + 0.898436i \(0.644704\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.78462 0.249897
\(52\) 0 0
\(53\) 0.813824 0.111787 0.0558936 0.998437i \(-0.482199\pi\)
0.0558936 + 0.998437i \(0.482199\pi\)
\(54\) 0 0
\(55\) −0.992782 −0.133867
\(56\) 0 0
\(57\) −1.31810 −0.174586
\(58\) 0 0
\(59\) 7.53795 0.981357 0.490679 0.871341i \(-0.336749\pi\)
0.490679 + 0.871341i \(0.336749\pi\)
\(60\) 0 0
\(61\) 1.31502 0.168371 0.0841857 0.996450i \(-0.473171\pi\)
0.0841857 + 0.996450i \(0.473171\pi\)
\(62\) 0 0
\(63\) −2.94139 −0.370580
\(64\) 0 0
\(65\) −0.878144 −0.108920
\(66\) 0 0
\(67\) 8.79384 1.07434 0.537170 0.843474i \(-0.319494\pi\)
0.537170 + 0.843474i \(0.319494\pi\)
\(68\) 0 0
\(69\) 1.56130 0.187959
\(70\) 0 0
\(71\) −11.4285 −1.35631 −0.678155 0.734919i \(-0.737220\pi\)
−0.678155 + 0.734919i \(0.737220\pi\)
\(72\) 0 0
\(73\) −3.68616 −0.431433 −0.215716 0.976456i \(-0.569209\pi\)
−0.215716 + 0.976456i \(0.569209\pi\)
\(74\) 0 0
\(75\) −1.17563 −0.135750
\(76\) 0 0
\(77\) −2.61527 −0.298038
\(78\) 0 0
\(79\) −4.21672 −0.474418 −0.237209 0.971459i \(-0.576233\pi\)
−0.237209 + 0.971459i \(0.576233\pi\)
\(80\) 0 0
\(81\) 8.47591 0.941768
\(82\) 0 0
\(83\) −16.9913 −1.86504 −0.932518 0.361123i \(-0.882393\pi\)
−0.932518 + 0.361123i \(0.882393\pi\)
\(84\) 0 0
\(85\) 2.79823 0.303511
\(86\) 0 0
\(87\) 1.22813 0.131670
\(88\) 0 0
\(89\) 9.32780 0.988745 0.494373 0.869250i \(-0.335398\pi\)
0.494373 + 0.869250i \(0.335398\pi\)
\(90\) 0 0
\(91\) −2.31328 −0.242498
\(92\) 0 0
\(93\) 1.47722 0.153180
\(94\) 0 0
\(95\) −2.06673 −0.212043
\(96\) 0 0
\(97\) −13.9032 −1.41166 −0.705828 0.708384i \(-0.749425\pi\)
−0.705828 + 0.708384i \(0.749425\pi\)
\(98\) 0 0
\(99\) 7.69252 0.773127
\(100\) 0 0
\(101\) −1.15104 −0.114533 −0.0572666 0.998359i \(-0.518238\pi\)
−0.0572666 + 0.998359i \(0.518238\pi\)
\(102\) 0 0
\(103\) −8.39975 −0.827652 −0.413826 0.910356i \(-0.635808\pi\)
−0.413826 + 0.910356i \(0.635808\pi\)
\(104\) 0 0
\(105\) 0.0919045 0.00896896
\(106\) 0 0
\(107\) 0.997270 0.0964097 0.0482048 0.998837i \(-0.484650\pi\)
0.0482048 + 0.998837i \(0.484650\pi\)
\(108\) 0 0
\(109\) −17.1491 −1.64259 −0.821295 0.570503i \(-0.806748\pi\)
−0.821295 + 0.570503i \(0.806748\pi\)
\(110\) 0 0
\(111\) −2.53999 −0.241085
\(112\) 0 0
\(113\) 3.51705 0.330856 0.165428 0.986222i \(-0.447099\pi\)
0.165428 + 0.986222i \(0.447099\pi\)
\(114\) 0 0
\(115\) 2.44807 0.228284
\(116\) 0 0
\(117\) 6.80425 0.629054
\(118\) 0 0
\(119\) 7.37134 0.675729
\(120\) 0 0
\(121\) −4.16036 −0.378215
\(122\) 0 0
\(123\) −0.202540 −0.0182624
\(124\) 0 0
\(125\) −3.74139 −0.334640
\(126\) 0 0
\(127\) 5.86352 0.520303 0.260152 0.965568i \(-0.416227\pi\)
0.260152 + 0.965568i \(0.416227\pi\)
\(128\) 0 0
\(129\) 1.33170 0.117250
\(130\) 0 0
\(131\) −11.0171 −0.962572 −0.481286 0.876564i \(-0.659830\pi\)
−0.481286 + 0.876564i \(0.659830\pi\)
\(132\) 0 0
\(133\) −5.44437 −0.472087
\(134\) 0 0
\(135\) −0.546040 −0.0469957
\(136\) 0 0
\(137\) −6.34879 −0.542414 −0.271207 0.962521i \(-0.587423\pi\)
−0.271207 + 0.962521i \(0.587423\pi\)
\(138\) 0 0
\(139\) −19.9573 −1.69275 −0.846377 0.532584i \(-0.821221\pi\)
−0.846377 + 0.532584i \(0.821221\pi\)
\(140\) 0 0
\(141\) −1.45763 −0.122754
\(142\) 0 0
\(143\) 6.04986 0.505914
\(144\) 0 0
\(145\) 1.92568 0.159919
\(146\) 0 0
\(147\) 0.242103 0.0199683
\(148\) 0 0
\(149\) −10.0223 −0.821061 −0.410531 0.911847i \(-0.634657\pi\)
−0.410531 + 0.911847i \(0.634657\pi\)
\(150\) 0 0
\(151\) −8.28521 −0.674241 −0.337120 0.941462i \(-0.609453\pi\)
−0.337120 + 0.941462i \(0.609453\pi\)
\(152\) 0 0
\(153\) −21.6819 −1.75288
\(154\) 0 0
\(155\) 2.31623 0.186044
\(156\) 0 0
\(157\) −10.8753 −0.867942 −0.433971 0.900927i \(-0.642888\pi\)
−0.433971 + 0.900927i \(0.642888\pi\)
\(158\) 0 0
\(159\) 0.197029 0.0156254
\(160\) 0 0
\(161\) 6.44892 0.508246
\(162\) 0 0
\(163\) 6.32087 0.495089 0.247545 0.968877i \(-0.420376\pi\)
0.247545 + 0.968877i \(0.420376\pi\)
\(164\) 0 0
\(165\) −0.240355 −0.0187116
\(166\) 0 0
\(167\) 12.8905 0.997493 0.498747 0.866748i \(-0.333794\pi\)
0.498747 + 0.866748i \(0.333794\pi\)
\(168\) 0 0
\(169\) −7.64873 −0.588364
\(170\) 0 0
\(171\) 16.0140 1.22462
\(172\) 0 0
\(173\) −8.78590 −0.667980 −0.333990 0.942577i \(-0.608395\pi\)
−0.333990 + 0.942577i \(0.608395\pi\)
\(174\) 0 0
\(175\) −4.85590 −0.367071
\(176\) 0 0
\(177\) 1.82496 0.137172
\(178\) 0 0
\(179\) 2.88010 0.215269 0.107635 0.994191i \(-0.465672\pi\)
0.107635 + 0.994191i \(0.465672\pi\)
\(180\) 0 0
\(181\) −23.6338 −1.75669 −0.878343 0.478030i \(-0.841351\pi\)
−0.878343 + 0.478030i \(0.841351\pi\)
\(182\) 0 0
\(183\) 0.318371 0.0235346
\(184\) 0 0
\(185\) −3.98263 −0.292809
\(186\) 0 0
\(187\) −19.2780 −1.40975
\(188\) 0 0
\(189\) −1.43843 −0.104630
\(190\) 0 0
\(191\) −5.11015 −0.369758 −0.184879 0.982761i \(-0.559189\pi\)
−0.184879 + 0.982761i \(0.559189\pi\)
\(192\) 0 0
\(193\) 0.676235 0.0486765 0.0243382 0.999704i \(-0.492252\pi\)
0.0243382 + 0.999704i \(0.492252\pi\)
\(194\) 0 0
\(195\) −0.212601 −0.0152247
\(196\) 0 0
\(197\) −20.2867 −1.44537 −0.722684 0.691178i \(-0.757092\pi\)
−0.722684 + 0.691178i \(0.757092\pi\)
\(198\) 0 0
\(199\) −4.94660 −0.350655 −0.175328 0.984510i \(-0.556098\pi\)
−0.175328 + 0.984510i \(0.556098\pi\)
\(200\) 0 0
\(201\) 2.12901 0.150169
\(202\) 0 0
\(203\) 5.07278 0.356039
\(204\) 0 0
\(205\) −0.317577 −0.0221805
\(206\) 0 0
\(207\) −18.9688 −1.31842
\(208\) 0 0
\(209\) 14.2385 0.984897
\(210\) 0 0
\(211\) 6.61583 0.455453 0.227726 0.973725i \(-0.426871\pi\)
0.227726 + 0.973725i \(0.426871\pi\)
\(212\) 0 0
\(213\) −2.76686 −0.189582
\(214\) 0 0
\(215\) 2.08807 0.142405
\(216\) 0 0
\(217\) 6.10161 0.414204
\(218\) 0 0
\(219\) −0.892431 −0.0603049
\(220\) 0 0
\(221\) −17.0520 −1.14704
\(222\) 0 0
\(223\) 4.16691 0.279037 0.139518 0.990219i \(-0.455445\pi\)
0.139518 + 0.990219i \(0.455445\pi\)
\(224\) 0 0
\(225\) 14.2831 0.952204
\(226\) 0 0
\(227\) 17.1150 1.13596 0.567982 0.823041i \(-0.307724\pi\)
0.567982 + 0.823041i \(0.307724\pi\)
\(228\) 0 0
\(229\) −19.1324 −1.26431 −0.632153 0.774844i \(-0.717829\pi\)
−0.632153 + 0.774844i \(0.717829\pi\)
\(230\) 0 0
\(231\) −0.633164 −0.0416591
\(232\) 0 0
\(233\) −13.3857 −0.876924 −0.438462 0.898750i \(-0.644477\pi\)
−0.438462 + 0.898750i \(0.644477\pi\)
\(234\) 0 0
\(235\) −2.28551 −0.149091
\(236\) 0 0
\(237\) −1.02088 −0.0663133
\(238\) 0 0
\(239\) 20.6475 1.33558 0.667788 0.744352i \(-0.267241\pi\)
0.667788 + 0.744352i \(0.267241\pi\)
\(240\) 0 0
\(241\) 0.401861 0.0258861 0.0129431 0.999916i \(-0.495880\pi\)
0.0129431 + 0.999916i \(0.495880\pi\)
\(242\) 0 0
\(243\) 6.36732 0.408464
\(244\) 0 0
\(245\) 0.379610 0.0242524
\(246\) 0 0
\(247\) 12.5944 0.801360
\(248\) 0 0
\(249\) −4.11364 −0.260691
\(250\) 0 0
\(251\) 13.0881 0.826113 0.413056 0.910705i \(-0.364461\pi\)
0.413056 + 0.910705i \(0.364461\pi\)
\(252\) 0 0
\(253\) −16.8657 −1.06034
\(254\) 0 0
\(255\) 0.677459 0.0424241
\(256\) 0 0
\(257\) 16.3273 1.01847 0.509234 0.860628i \(-0.329929\pi\)
0.509234 + 0.860628i \(0.329929\pi\)
\(258\) 0 0
\(259\) −10.4914 −0.651902
\(260\) 0 0
\(261\) −14.9210 −0.923587
\(262\) 0 0
\(263\) 13.3352 0.822284 0.411142 0.911571i \(-0.365130\pi\)
0.411142 + 0.911571i \(0.365130\pi\)
\(264\) 0 0
\(265\) 0.308935 0.0189777
\(266\) 0 0
\(267\) 2.25829 0.138205
\(268\) 0 0
\(269\) −27.3334 −1.66655 −0.833275 0.552859i \(-0.813537\pi\)
−0.833275 + 0.552859i \(0.813537\pi\)
\(270\) 0 0
\(271\) 24.5968 1.49415 0.747074 0.664741i \(-0.231458\pi\)
0.747074 + 0.664741i \(0.231458\pi\)
\(272\) 0 0
\(273\) −0.560052 −0.0338959
\(274\) 0 0
\(275\) 12.6995 0.765807
\(276\) 0 0
\(277\) −25.4522 −1.52928 −0.764638 0.644460i \(-0.777082\pi\)
−0.764638 + 0.644460i \(0.777082\pi\)
\(278\) 0 0
\(279\) −17.9472 −1.07447
\(280\) 0 0
\(281\) 13.7357 0.819404 0.409702 0.912219i \(-0.365633\pi\)
0.409702 + 0.912219i \(0.365633\pi\)
\(282\) 0 0
\(283\) 14.1905 0.843540 0.421770 0.906703i \(-0.361409\pi\)
0.421770 + 0.906703i \(0.361409\pi\)
\(284\) 0 0
\(285\) −0.500362 −0.0296389
\(286\) 0 0
\(287\) −0.836588 −0.0493822
\(288\) 0 0
\(289\) 37.3366 2.19627
\(290\) 0 0
\(291\) −3.36600 −0.197318
\(292\) 0 0
\(293\) 27.9132 1.63070 0.815352 0.578965i \(-0.196543\pi\)
0.815352 + 0.578965i \(0.196543\pi\)
\(294\) 0 0
\(295\) 2.86148 0.166602
\(296\) 0 0
\(297\) 3.76187 0.218286
\(298\) 0 0
\(299\) −14.9182 −0.862741
\(300\) 0 0
\(301\) 5.50056 0.317047
\(302\) 0 0
\(303\) −0.278671 −0.0160092
\(304\) 0 0
\(305\) 0.499195 0.0285838
\(306\) 0 0
\(307\) 4.45200 0.254089 0.127045 0.991897i \(-0.459451\pi\)
0.127045 + 0.991897i \(0.459451\pi\)
\(308\) 0 0
\(309\) −2.03360 −0.115688
\(310\) 0 0
\(311\) −32.2711 −1.82993 −0.914964 0.403535i \(-0.867781\pi\)
−0.914964 + 0.403535i \(0.867781\pi\)
\(312\) 0 0
\(313\) −22.3372 −1.26257 −0.631286 0.775550i \(-0.717473\pi\)
−0.631286 + 0.775550i \(0.717473\pi\)
\(314\) 0 0
\(315\) −1.11658 −0.0629121
\(316\) 0 0
\(317\) 0.209793 0.0117832 0.00589158 0.999983i \(-0.498125\pi\)
0.00589158 + 0.999983i \(0.498125\pi\)
\(318\) 0 0
\(319\) −13.2667 −0.742792
\(320\) 0 0
\(321\) 0.241442 0.0134760
\(322\) 0 0
\(323\) −40.1323 −2.23302
\(324\) 0 0
\(325\) 11.2331 0.623098
\(326\) 0 0
\(327\) −4.15186 −0.229598
\(328\) 0 0
\(329\) −6.02070 −0.331932
\(330\) 0 0
\(331\) 28.4861 1.56574 0.782868 0.622188i \(-0.213756\pi\)
0.782868 + 0.622188i \(0.213756\pi\)
\(332\) 0 0
\(333\) 30.8592 1.69107
\(334\) 0 0
\(335\) 3.33823 0.182387
\(336\) 0 0
\(337\) −6.83335 −0.372236 −0.186118 0.982527i \(-0.559591\pi\)
−0.186118 + 0.982527i \(0.559591\pi\)
\(338\) 0 0
\(339\) 0.851487 0.0462464
\(340\) 0 0
\(341\) −15.9574 −0.864140
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.592685 0.0319091
\(346\) 0 0
\(347\) 28.0706 1.50691 0.753454 0.657501i \(-0.228386\pi\)
0.753454 + 0.657501i \(0.228386\pi\)
\(348\) 0 0
\(349\) −23.4899 −1.25738 −0.628692 0.777654i \(-0.716409\pi\)
−0.628692 + 0.777654i \(0.716409\pi\)
\(350\) 0 0
\(351\) 3.32748 0.177608
\(352\) 0 0
\(353\) 5.41293 0.288101 0.144051 0.989570i \(-0.453987\pi\)
0.144051 + 0.989570i \(0.453987\pi\)
\(354\) 0 0
\(355\) −4.33836 −0.230256
\(356\) 0 0
\(357\) 1.78462 0.0944522
\(358\) 0 0
\(359\) −29.1561 −1.53880 −0.769399 0.638768i \(-0.779444\pi\)
−0.769399 + 0.638768i \(0.779444\pi\)
\(360\) 0 0
\(361\) 10.6412 0.560061
\(362\) 0 0
\(363\) −1.00724 −0.0528662
\(364\) 0 0
\(365\) −1.39930 −0.0732429
\(366\) 0 0
\(367\) 6.14461 0.320746 0.160373 0.987056i \(-0.448730\pi\)
0.160373 + 0.987056i \(0.448730\pi\)
\(368\) 0 0
\(369\) 2.46073 0.128100
\(370\) 0 0
\(371\) 0.813824 0.0422516
\(372\) 0 0
\(373\) 29.6063 1.53296 0.766478 0.642270i \(-0.222007\pi\)
0.766478 + 0.642270i \(0.222007\pi\)
\(374\) 0 0
\(375\) −0.905802 −0.0467754
\(376\) 0 0
\(377\) −11.7348 −0.604371
\(378\) 0 0
\(379\) −33.4317 −1.71727 −0.858635 0.512587i \(-0.828687\pi\)
−0.858635 + 0.512587i \(0.828687\pi\)
\(380\) 0 0
\(381\) 1.41957 0.0727270
\(382\) 0 0
\(383\) 6.99982 0.357674 0.178837 0.983879i \(-0.442767\pi\)
0.178837 + 0.983879i \(0.442767\pi\)
\(384\) 0 0
\(385\) −0.992782 −0.0505968
\(386\) 0 0
\(387\) −16.1793 −0.822439
\(388\) 0 0
\(389\) 21.8974 1.11024 0.555121 0.831770i \(-0.312672\pi\)
0.555121 + 0.831770i \(0.312672\pi\)
\(390\) 0 0
\(391\) 47.5372 2.40406
\(392\) 0 0
\(393\) −2.66728 −0.134546
\(394\) 0 0
\(395\) −1.60071 −0.0805403
\(396\) 0 0
\(397\) 8.00336 0.401677 0.200839 0.979624i \(-0.435633\pi\)
0.200839 + 0.979624i \(0.435633\pi\)
\(398\) 0 0
\(399\) −1.31810 −0.0659874
\(400\) 0 0
\(401\) −26.7191 −1.33429 −0.667144 0.744929i \(-0.732483\pi\)
−0.667144 + 0.744929i \(0.732483\pi\)
\(402\) 0 0
\(403\) −14.1147 −0.703106
\(404\) 0 0
\(405\) 3.21754 0.159881
\(406\) 0 0
\(407\) 27.4378 1.36004
\(408\) 0 0
\(409\) −20.7456 −1.02581 −0.512903 0.858447i \(-0.671430\pi\)
−0.512903 + 0.858447i \(0.671430\pi\)
\(410\) 0 0
\(411\) −1.53706 −0.0758176
\(412\) 0 0
\(413\) 7.53795 0.370918
\(414\) 0 0
\(415\) −6.45006 −0.316621
\(416\) 0 0
\(417\) −4.83171 −0.236610
\(418\) 0 0
\(419\) 8.65187 0.422671 0.211336 0.977414i \(-0.432219\pi\)
0.211336 + 0.977414i \(0.432219\pi\)
\(420\) 0 0
\(421\) −12.7739 −0.622562 −0.311281 0.950318i \(-0.600758\pi\)
−0.311281 + 0.950318i \(0.600758\pi\)
\(422\) 0 0
\(423\) 17.7092 0.861050
\(424\) 0 0
\(425\) −35.7945 −1.73629
\(426\) 0 0
\(427\) 1.31502 0.0636384
\(428\) 0 0
\(429\) 1.46469 0.0707157
\(430\) 0 0
\(431\) 2.81338 0.135516 0.0677579 0.997702i \(-0.478415\pi\)
0.0677579 + 0.997702i \(0.478415\pi\)
\(432\) 0 0
\(433\) 20.6954 0.994558 0.497279 0.867591i \(-0.334333\pi\)
0.497279 + 0.867591i \(0.334333\pi\)
\(434\) 0 0
\(435\) 0.466211 0.0223531
\(436\) 0 0
\(437\) −35.1103 −1.67955
\(438\) 0 0
\(439\) −15.6336 −0.746151 −0.373075 0.927801i \(-0.621697\pi\)
−0.373075 + 0.927801i \(0.621697\pi\)
\(440\) 0 0
\(441\) −2.94139 −0.140066
\(442\) 0 0
\(443\) −26.3904 −1.25385 −0.626923 0.779081i \(-0.715686\pi\)
−0.626923 + 0.779081i \(0.715686\pi\)
\(444\) 0 0
\(445\) 3.54092 0.167856
\(446\) 0 0
\(447\) −2.42643 −0.114766
\(448\) 0 0
\(449\) 26.8536 1.26730 0.633650 0.773620i \(-0.281556\pi\)
0.633650 + 0.773620i \(0.281556\pi\)
\(450\) 0 0
\(451\) 2.18790 0.103024
\(452\) 0 0
\(453\) −2.00587 −0.0942441
\(454\) 0 0
\(455\) −0.878144 −0.0411680
\(456\) 0 0
\(457\) 0.385896 0.0180514 0.00902572 0.999959i \(-0.497127\pi\)
0.00902572 + 0.999959i \(0.497127\pi\)
\(458\) 0 0
\(459\) −10.6031 −0.494911
\(460\) 0 0
\(461\) −3.02146 −0.140723 −0.0703617 0.997522i \(-0.522415\pi\)
−0.0703617 + 0.997522i \(0.522415\pi\)
\(462\) 0 0
\(463\) −32.5560 −1.51301 −0.756503 0.653991i \(-0.773094\pi\)
−0.756503 + 0.653991i \(0.773094\pi\)
\(464\) 0 0
\(465\) 0.560766 0.0260049
\(466\) 0 0
\(467\) 11.3179 0.523729 0.261864 0.965105i \(-0.415663\pi\)
0.261864 + 0.965105i \(0.415663\pi\)
\(468\) 0 0
\(469\) 8.79384 0.406062
\(470\) 0 0
\(471\) −2.63294 −0.121319
\(472\) 0 0
\(473\) −14.3855 −0.661444
\(474\) 0 0
\(475\) 26.4373 1.21303
\(476\) 0 0
\(477\) −2.39377 −0.109603
\(478\) 0 0
\(479\) −20.2388 −0.924735 −0.462368 0.886688i \(-0.653000\pi\)
−0.462368 + 0.886688i \(0.653000\pi\)
\(480\) 0 0
\(481\) 24.2695 1.10659
\(482\) 0 0
\(483\) 1.56130 0.0710417
\(484\) 0 0
\(485\) −5.27779 −0.239652
\(486\) 0 0
\(487\) 36.6988 1.66298 0.831492 0.555537i \(-0.187487\pi\)
0.831492 + 0.555537i \(0.187487\pi\)
\(488\) 0 0
\(489\) 1.53030 0.0692026
\(490\) 0 0
\(491\) −19.2355 −0.868085 −0.434043 0.900892i \(-0.642913\pi\)
−0.434043 + 0.900892i \(0.642913\pi\)
\(492\) 0 0
\(493\) 37.3932 1.68410
\(494\) 0 0
\(495\) 2.92015 0.131251
\(496\) 0 0
\(497\) −11.4285 −0.512637
\(498\) 0 0
\(499\) 34.2197 1.53188 0.765942 0.642910i \(-0.222273\pi\)
0.765942 + 0.642910i \(0.222273\pi\)
\(500\) 0 0
\(501\) 3.12081 0.139428
\(502\) 0 0
\(503\) −13.4123 −0.598026 −0.299013 0.954249i \(-0.596657\pi\)
−0.299013 + 0.954249i \(0.596657\pi\)
\(504\) 0 0
\(505\) −0.436947 −0.0194439
\(506\) 0 0
\(507\) −1.85178 −0.0822404
\(508\) 0 0
\(509\) −26.8407 −1.18969 −0.594846 0.803840i \(-0.702787\pi\)
−0.594846 + 0.803840i \(0.702787\pi\)
\(510\) 0 0
\(511\) −3.68616 −0.163066
\(512\) 0 0
\(513\) 7.83132 0.345761
\(514\) 0 0
\(515\) −3.18863 −0.140508
\(516\) 0 0
\(517\) 15.7457 0.692497
\(518\) 0 0
\(519\) −2.12709 −0.0933689
\(520\) 0 0
\(521\) −0.00960703 −0.000420892 0 −0.000210446 1.00000i \(-0.500067\pi\)
−0.000210446 1.00000i \(0.500067\pi\)
\(522\) 0 0
\(523\) 33.3512 1.45835 0.729173 0.684329i \(-0.239905\pi\)
0.729173 + 0.684329i \(0.239905\pi\)
\(524\) 0 0
\(525\) −1.17563 −0.0513085
\(526\) 0 0
\(527\) 44.9770 1.95923
\(528\) 0 0
\(529\) 18.5886 0.808201
\(530\) 0 0
\(531\) −22.1720 −0.962183
\(532\) 0 0
\(533\) 1.93526 0.0838256
\(534\) 0 0
\(535\) 0.378573 0.0163671
\(536\) 0 0
\(537\) 0.697281 0.0300899
\(538\) 0 0
\(539\) −2.61527 −0.112648
\(540\) 0 0
\(541\) −16.4724 −0.708205 −0.354102 0.935207i \(-0.615214\pi\)
−0.354102 + 0.935207i \(0.615214\pi\)
\(542\) 0 0
\(543\) −5.72181 −0.245546
\(544\) 0 0
\(545\) −6.50998 −0.278857
\(546\) 0 0
\(547\) −19.8048 −0.846792 −0.423396 0.905945i \(-0.639162\pi\)
−0.423396 + 0.905945i \(0.639162\pi\)
\(548\) 0 0
\(549\) −3.86799 −0.165082
\(550\) 0 0
\(551\) −27.6181 −1.17657
\(552\) 0 0
\(553\) −4.21672 −0.179313
\(554\) 0 0
\(555\) −0.964205 −0.0409282
\(556\) 0 0
\(557\) −34.4796 −1.46095 −0.730473 0.682941i \(-0.760701\pi\)
−0.730473 + 0.682941i \(0.760701\pi\)
\(558\) 0 0
\(559\) −12.7244 −0.538183
\(560\) 0 0
\(561\) −4.66727 −0.197052
\(562\) 0 0
\(563\) 37.1639 1.56627 0.783136 0.621850i \(-0.213619\pi\)
0.783136 + 0.621850i \(0.213619\pi\)
\(564\) 0 0
\(565\) 1.33510 0.0561683
\(566\) 0 0
\(567\) 8.47591 0.355955
\(568\) 0 0
\(569\) 36.8053 1.54296 0.771479 0.636255i \(-0.219517\pi\)
0.771479 + 0.636255i \(0.219517\pi\)
\(570\) 0 0
\(571\) −32.6027 −1.36438 −0.682190 0.731175i \(-0.738972\pi\)
−0.682190 + 0.731175i \(0.738972\pi\)
\(572\) 0 0
\(573\) −1.23718 −0.0516840
\(574\) 0 0
\(575\) −31.3153 −1.30594
\(576\) 0 0
\(577\) −3.51057 −0.146147 −0.0730735 0.997327i \(-0.523281\pi\)
−0.0730735 + 0.997327i \(0.523281\pi\)
\(578\) 0 0
\(579\) 0.163718 0.00680390
\(580\) 0 0
\(581\) −16.9913 −0.704918
\(582\) 0 0
\(583\) −2.12837 −0.0881480
\(584\) 0 0
\(585\) 2.58296 0.106792
\(586\) 0 0
\(587\) −39.6426 −1.63622 −0.818112 0.575058i \(-0.804979\pi\)
−0.818112 + 0.575058i \(0.804979\pi\)
\(588\) 0 0
\(589\) −33.2194 −1.36878
\(590\) 0 0
\(591\) −4.91147 −0.202031
\(592\) 0 0
\(593\) −6.27767 −0.257793 −0.128896 0.991658i \(-0.541143\pi\)
−0.128896 + 0.991658i \(0.541143\pi\)
\(594\) 0 0
\(595\) 2.79823 0.114716
\(596\) 0 0
\(597\) −1.19759 −0.0490139
\(598\) 0 0
\(599\) 10.3600 0.423300 0.211650 0.977346i \(-0.432116\pi\)
0.211650 + 0.977346i \(0.432116\pi\)
\(600\) 0 0
\(601\) −23.9656 −0.977576 −0.488788 0.872403i \(-0.662561\pi\)
−0.488788 + 0.872403i \(0.662561\pi\)
\(602\) 0 0
\(603\) −25.8661 −1.05335
\(604\) 0 0
\(605\) −1.57931 −0.0642083
\(606\) 0 0
\(607\) 14.4285 0.585633 0.292817 0.956169i \(-0.405407\pi\)
0.292817 + 0.956169i \(0.405407\pi\)
\(608\) 0 0
\(609\) 1.22813 0.0497665
\(610\) 0 0
\(611\) 13.9276 0.563449
\(612\) 0 0
\(613\) −4.43815 −0.179255 −0.0896277 0.995975i \(-0.528568\pi\)
−0.0896277 + 0.995975i \(0.528568\pi\)
\(614\) 0 0
\(615\) −0.0768862 −0.00310035
\(616\) 0 0
\(617\) −15.7644 −0.634651 −0.317325 0.948317i \(-0.602785\pi\)
−0.317325 + 0.948317i \(0.602785\pi\)
\(618\) 0 0
\(619\) −8.72071 −0.350515 −0.175257 0.984523i \(-0.556076\pi\)
−0.175257 + 0.984523i \(0.556076\pi\)
\(620\) 0 0
\(621\) −9.27630 −0.372245
\(622\) 0 0
\(623\) 9.32780 0.373711
\(624\) 0 0
\(625\) 22.8592 0.914369
\(626\) 0 0
\(627\) 3.44718 0.137667
\(628\) 0 0
\(629\) −77.3355 −3.08357
\(630\) 0 0
\(631\) 30.5796 1.21736 0.608678 0.793417i \(-0.291700\pi\)
0.608678 + 0.793417i \(0.291700\pi\)
\(632\) 0 0
\(633\) 1.60171 0.0636623
\(634\) 0 0
\(635\) 2.22585 0.0883301
\(636\) 0 0
\(637\) −2.31328 −0.0916556
\(638\) 0 0
\(639\) 33.6155 1.32981
\(640\) 0 0
\(641\) −18.6228 −0.735556 −0.367778 0.929914i \(-0.619881\pi\)
−0.367778 + 0.929914i \(0.619881\pi\)
\(642\) 0 0
\(643\) 39.3632 1.55233 0.776166 0.630529i \(-0.217162\pi\)
0.776166 + 0.630529i \(0.217162\pi\)
\(644\) 0 0
\(645\) 0.505527 0.0199051
\(646\) 0 0
\(647\) −29.2607 −1.15036 −0.575178 0.818029i \(-0.695067\pi\)
−0.575178 + 0.818029i \(0.695067\pi\)
\(648\) 0 0
\(649\) −19.7138 −0.773833
\(650\) 0 0
\(651\) 1.47722 0.0578967
\(652\) 0 0
\(653\) −24.5826 −0.961990 −0.480995 0.876723i \(-0.659724\pi\)
−0.480995 + 0.876723i \(0.659724\pi\)
\(654\) 0 0
\(655\) −4.18221 −0.163413
\(656\) 0 0
\(657\) 10.8424 0.423004
\(658\) 0 0
\(659\) −39.1780 −1.52616 −0.763079 0.646305i \(-0.776313\pi\)
−0.763079 + 0.646305i \(0.776313\pi\)
\(660\) 0 0
\(661\) 26.5220 1.03159 0.515794 0.856713i \(-0.327497\pi\)
0.515794 + 0.856713i \(0.327497\pi\)
\(662\) 0 0
\(663\) −4.12833 −0.160331
\(664\) 0 0
\(665\) −2.06673 −0.0801445
\(666\) 0 0
\(667\) 32.7140 1.26669
\(668\) 0 0
\(669\) 1.00882 0.0390032
\(670\) 0 0
\(671\) −3.43914 −0.132767
\(672\) 0 0
\(673\) 8.89179 0.342753 0.171377 0.985206i \(-0.445179\pi\)
0.171377 + 0.985206i \(0.445179\pi\)
\(674\) 0 0
\(675\) 6.98485 0.268847
\(676\) 0 0
\(677\) −32.5301 −1.25023 −0.625117 0.780531i \(-0.714949\pi\)
−0.625117 + 0.780531i \(0.714949\pi\)
\(678\) 0 0
\(679\) −13.9032 −0.533556
\(680\) 0 0
\(681\) 4.14360 0.158783
\(682\) 0 0
\(683\) −11.4983 −0.439969 −0.219984 0.975503i \(-0.570601\pi\)
−0.219984 + 0.975503i \(0.570601\pi\)
\(684\) 0 0
\(685\) −2.41006 −0.0920838
\(686\) 0 0
\(687\) −4.63201 −0.176722
\(688\) 0 0
\(689\) −1.88260 −0.0717215
\(690\) 0 0
\(691\) −1.54743 −0.0588671 −0.0294335 0.999567i \(-0.509370\pi\)
−0.0294335 + 0.999567i \(0.509370\pi\)
\(692\) 0 0
\(693\) 7.69252 0.292215
\(694\) 0 0
\(695\) −7.57597 −0.287373
\(696\) 0 0
\(697\) −6.16677 −0.233583
\(698\) 0 0
\(699\) −3.24071 −0.122575
\(700\) 0 0
\(701\) −16.3562 −0.617764 −0.308882 0.951100i \(-0.599955\pi\)
−0.308882 + 0.951100i \(0.599955\pi\)
\(702\) 0 0
\(703\) 57.1189 2.15428
\(704\) 0 0
\(705\) −0.553329 −0.0208396
\(706\) 0 0
\(707\) −1.15104 −0.0432895
\(708\) 0 0
\(709\) −3.81344 −0.143217 −0.0716085 0.997433i \(-0.522813\pi\)
−0.0716085 + 0.997433i \(0.522813\pi\)
\(710\) 0 0
\(711\) 12.4030 0.465149
\(712\) 0 0
\(713\) 39.3488 1.47363
\(714\) 0 0
\(715\) 2.29658 0.0858873
\(716\) 0 0
\(717\) 4.99882 0.186684
\(718\) 0 0
\(719\) −6.08527 −0.226942 −0.113471 0.993541i \(-0.536197\pi\)
−0.113471 + 0.993541i \(0.536197\pi\)
\(720\) 0 0
\(721\) −8.39975 −0.312823
\(722\) 0 0
\(723\) 0.0972917 0.00361832
\(724\) 0 0
\(725\) −24.6329 −0.914843
\(726\) 0 0
\(727\) −41.6162 −1.54346 −0.771729 0.635951i \(-0.780608\pi\)
−0.771729 + 0.635951i \(0.780608\pi\)
\(728\) 0 0
\(729\) −23.8862 −0.884674
\(730\) 0 0
\(731\) 40.5465 1.49967
\(732\) 0 0
\(733\) 33.3653 1.23238 0.616188 0.787599i \(-0.288676\pi\)
0.616188 + 0.787599i \(0.288676\pi\)
\(734\) 0 0
\(735\) 0.0919045 0.00338995
\(736\) 0 0
\(737\) −22.9983 −0.847152
\(738\) 0 0
\(739\) −10.6509 −0.391801 −0.195900 0.980624i \(-0.562763\pi\)
−0.195900 + 0.980624i \(0.562763\pi\)
\(740\) 0 0
\(741\) 3.04913 0.112013
\(742\) 0 0
\(743\) 14.4179 0.528943 0.264472 0.964393i \(-0.414802\pi\)
0.264472 + 0.964393i \(0.414802\pi\)
\(744\) 0 0
\(745\) −3.80457 −0.139389
\(746\) 0 0
\(747\) 49.9779 1.82860
\(748\) 0 0
\(749\) 0.997270 0.0364394
\(750\) 0 0
\(751\) −17.6760 −0.645008 −0.322504 0.946568i \(-0.604525\pi\)
−0.322504 + 0.946568i \(0.604525\pi\)
\(752\) 0 0
\(753\) 3.16866 0.115472
\(754\) 0 0
\(755\) −3.14514 −0.114464
\(756\) 0 0
\(757\) 30.1724 1.09663 0.548317 0.836271i \(-0.315269\pi\)
0.548317 + 0.836271i \(0.315269\pi\)
\(758\) 0 0
\(759\) −4.08323 −0.148212
\(760\) 0 0
\(761\) 22.2510 0.806597 0.403299 0.915068i \(-0.367864\pi\)
0.403299 + 0.915068i \(0.367864\pi\)
\(762\) 0 0
\(763\) −17.1491 −0.620841
\(764\) 0 0
\(765\) −8.23068 −0.297581
\(766\) 0 0
\(767\) −17.4374 −0.629628
\(768\) 0 0
\(769\) −30.5537 −1.10179 −0.550897 0.834573i \(-0.685714\pi\)
−0.550897 + 0.834573i \(0.685714\pi\)
\(770\) 0 0
\(771\) 3.95288 0.142360
\(772\) 0 0
\(773\) −27.0949 −0.974534 −0.487267 0.873253i \(-0.662006\pi\)
−0.487267 + 0.873253i \(0.662006\pi\)
\(774\) 0 0
\(775\) −29.6288 −1.06430
\(776\) 0 0
\(777\) −2.53999 −0.0911217
\(778\) 0 0
\(779\) 4.55469 0.163189
\(780\) 0 0
\(781\) 29.8885 1.06950
\(782\) 0 0
\(783\) −7.29682 −0.260767
\(784\) 0 0
\(785\) −4.12836 −0.147348
\(786\) 0 0
\(787\) −19.5404 −0.696540 −0.348270 0.937394i \(-0.613231\pi\)
−0.348270 + 0.937394i \(0.613231\pi\)
\(788\) 0 0
\(789\) 3.22849 0.114937
\(790\) 0 0
\(791\) 3.51705 0.125052
\(792\) 0 0
\(793\) −3.04202 −0.108025
\(794\) 0 0
\(795\) 0.0747941 0.00265267
\(796\) 0 0
\(797\) 12.4240 0.440082 0.220041 0.975491i \(-0.429381\pi\)
0.220041 + 0.975491i \(0.429381\pi\)
\(798\) 0 0
\(799\) −44.3806 −1.57007
\(800\) 0 0
\(801\) −27.4367 −0.969427
\(802\) 0 0
\(803\) 9.64032 0.340199
\(804\) 0 0
\(805\) 2.44807 0.0862833
\(806\) 0 0
\(807\) −6.61750 −0.232947
\(808\) 0 0
\(809\) −2.11808 −0.0744676 −0.0372338 0.999307i \(-0.511855\pi\)
−0.0372338 + 0.999307i \(0.511855\pi\)
\(810\) 0 0
\(811\) 0.287353 0.0100903 0.00504517 0.999987i \(-0.498394\pi\)
0.00504517 + 0.999987i \(0.498394\pi\)
\(812\) 0 0
\(813\) 5.95495 0.208849
\(814\) 0 0
\(815\) 2.39946 0.0840496
\(816\) 0 0
\(817\) −29.9471 −1.04772
\(818\) 0 0
\(819\) 6.80425 0.237760
\(820\) 0 0
\(821\) 29.4076 1.02633 0.513166 0.858290i \(-0.328473\pi\)
0.513166 + 0.858290i \(0.328473\pi\)
\(822\) 0 0
\(823\) 10.1430 0.353563 0.176782 0.984250i \(-0.443431\pi\)
0.176782 + 0.984250i \(0.443431\pi\)
\(824\) 0 0
\(825\) 3.07458 0.107043
\(826\) 0 0
\(827\) −27.7379 −0.964542 −0.482271 0.876022i \(-0.660188\pi\)
−0.482271 + 0.876022i \(0.660188\pi\)
\(828\) 0 0
\(829\) 13.0904 0.454650 0.227325 0.973819i \(-0.427002\pi\)
0.227325 + 0.973819i \(0.427002\pi\)
\(830\) 0 0
\(831\) −6.16205 −0.213759
\(832\) 0 0
\(833\) 7.37134 0.255402
\(834\) 0 0
\(835\) 4.89334 0.169341
\(836\) 0 0
\(837\) −8.77672 −0.303368
\(838\) 0 0
\(839\) 9.57102 0.330428 0.165214 0.986258i \(-0.447168\pi\)
0.165214 + 0.986258i \(0.447168\pi\)
\(840\) 0 0
\(841\) −3.26691 −0.112652
\(842\) 0 0
\(843\) 3.32545 0.114535
\(844\) 0 0
\(845\) −2.90353 −0.0998845
\(846\) 0 0
\(847\) −4.16036 −0.142952
\(848\) 0 0
\(849\) 3.43557 0.117908
\(850\) 0 0
\(851\) −67.6581 −2.31929
\(852\) 0 0
\(853\) 20.2042 0.691779 0.345890 0.938275i \(-0.387577\pi\)
0.345890 + 0.938275i \(0.387577\pi\)
\(854\) 0 0
\(855\) 6.07907 0.207900
\(856\) 0 0
\(857\) 37.1981 1.27066 0.635332 0.772239i \(-0.280863\pi\)
0.635332 + 0.772239i \(0.280863\pi\)
\(858\) 0 0
\(859\) 51.1291 1.74450 0.872251 0.489058i \(-0.162659\pi\)
0.872251 + 0.489058i \(0.162659\pi\)
\(860\) 0 0
\(861\) −0.202540 −0.00690255
\(862\) 0 0
\(863\) 27.6826 0.942325 0.471163 0.882046i \(-0.343835\pi\)
0.471163 + 0.882046i \(0.343835\pi\)
\(864\) 0 0
\(865\) −3.33521 −0.113401
\(866\) 0 0
\(867\) 9.03930 0.306991
\(868\) 0 0
\(869\) 11.0279 0.374095
\(870\) 0 0
\(871\) −20.3426 −0.689284
\(872\) 0 0
\(873\) 40.8947 1.38407
\(874\) 0 0
\(875\) −3.74139 −0.126482
\(876\) 0 0
\(877\) −3.08114 −0.104043 −0.0520214 0.998646i \(-0.516566\pi\)
−0.0520214 + 0.998646i \(0.516566\pi\)
\(878\) 0 0
\(879\) 6.75785 0.227937
\(880\) 0 0
\(881\) 39.9678 1.34655 0.673276 0.739392i \(-0.264887\pi\)
0.673276 + 0.739392i \(0.264887\pi\)
\(882\) 0 0
\(883\) 7.82110 0.263201 0.131601 0.991303i \(-0.457988\pi\)
0.131601 + 0.991303i \(0.457988\pi\)
\(884\) 0 0
\(885\) 0.692772 0.0232873
\(886\) 0 0
\(887\) 47.8800 1.60765 0.803826 0.594865i \(-0.202794\pi\)
0.803826 + 0.594865i \(0.202794\pi\)
\(888\) 0 0
\(889\) 5.86352 0.196656
\(890\) 0 0
\(891\) −22.1668 −0.742616
\(892\) 0 0
\(893\) 32.7789 1.09690
\(894\) 0 0
\(895\) 1.09332 0.0365455
\(896\) 0 0
\(897\) −3.61173 −0.120592
\(898\) 0 0
\(899\) 30.9521 1.03231
\(900\) 0 0
\(901\) 5.99897 0.199855
\(902\) 0 0
\(903\) 1.33170 0.0443163
\(904\) 0 0
\(905\) −8.97162 −0.298227
\(906\) 0 0
\(907\) −40.1949 −1.33465 −0.667325 0.744767i \(-0.732561\pi\)
−0.667325 + 0.744767i \(0.732561\pi\)
\(908\) 0 0
\(909\) 3.38566 0.112295
\(910\) 0 0
\(911\) −50.4192 −1.67046 −0.835231 0.549899i \(-0.814666\pi\)
−0.835231 + 0.549899i \(0.814666\pi\)
\(912\) 0 0
\(913\) 44.4368 1.47064
\(914\) 0 0
\(915\) 0.120857 0.00399539
\(916\) 0 0
\(917\) −11.0171 −0.363818
\(918\) 0 0
\(919\) 41.1782 1.35834 0.679171 0.733980i \(-0.262339\pi\)
0.679171 + 0.733980i \(0.262339\pi\)
\(920\) 0 0
\(921\) 1.07784 0.0355161
\(922\) 0 0
\(923\) 26.4373 0.870193
\(924\) 0 0
\(925\) 50.9450 1.67506
\(926\) 0 0
\(927\) 24.7069 0.811482
\(928\) 0 0
\(929\) −57.2128 −1.87709 −0.938546 0.345155i \(-0.887826\pi\)
−0.938546 + 0.345155i \(0.887826\pi\)
\(930\) 0 0
\(931\) −5.44437 −0.178432
\(932\) 0 0
\(933\) −7.81293 −0.255784
\(934\) 0 0
\(935\) −7.31813 −0.239328
\(936\) 0 0
\(937\) 22.3565 0.730353 0.365177 0.930938i \(-0.381009\pi\)
0.365177 + 0.930938i \(0.381009\pi\)
\(938\) 0 0
\(939\) −5.40789 −0.176480
\(940\) 0 0
\(941\) −21.2245 −0.691899 −0.345949 0.938253i \(-0.612443\pi\)
−0.345949 + 0.938253i \(0.612443\pi\)
\(942\) 0 0
\(943\) −5.39509 −0.175688
\(944\) 0 0
\(945\) −0.546040 −0.0177627
\(946\) 0 0
\(947\) 10.2010 0.331487 0.165743 0.986169i \(-0.446998\pi\)
0.165743 + 0.986169i \(0.446998\pi\)
\(948\) 0 0
\(949\) 8.52714 0.276803
\(950\) 0 0
\(951\) 0.0507915 0.00164703
\(952\) 0 0
\(953\) 49.0686 1.58949 0.794744 0.606944i \(-0.207605\pi\)
0.794744 + 0.606944i \(0.207605\pi\)
\(954\) 0 0
\(955\) −1.93986 −0.0627725
\(956\) 0 0
\(957\) −3.21190 −0.103826
\(958\) 0 0
\(959\) −6.34879 −0.205013
\(960\) 0 0
\(961\) 6.22968 0.200957
\(962\) 0 0
\(963\) −2.93335 −0.0945260
\(964\) 0 0
\(965\) 0.256705 0.00826364
\(966\) 0 0
\(967\) 3.07877 0.0990064 0.0495032 0.998774i \(-0.484236\pi\)
0.0495032 + 0.998774i \(0.484236\pi\)
\(968\) 0 0
\(969\) −9.71614 −0.312127
\(970\) 0 0
\(971\) −22.6421 −0.726619 −0.363310 0.931668i \(-0.618353\pi\)
−0.363310 + 0.931668i \(0.618353\pi\)
\(972\) 0 0
\(973\) −19.9573 −0.639801
\(974\) 0 0
\(975\) 2.71955 0.0870954
\(976\) 0 0
\(977\) −10.9770 −0.351185 −0.175592 0.984463i \(-0.556184\pi\)
−0.175592 + 0.984463i \(0.556184\pi\)
\(978\) 0 0
\(979\) −24.3947 −0.779659
\(980\) 0 0
\(981\) 50.4423 1.61050
\(982\) 0 0
\(983\) −15.7349 −0.501864 −0.250932 0.968005i \(-0.580737\pi\)
−0.250932 + 0.968005i \(0.580737\pi\)
\(984\) 0 0
\(985\) −7.70103 −0.245375
\(986\) 0 0
\(987\) −1.45763 −0.0463968
\(988\) 0 0
\(989\) 35.4727 1.12797
\(990\) 0 0
\(991\) 46.7969 1.48655 0.743276 0.668984i \(-0.233271\pi\)
0.743276 + 0.668984i \(0.233271\pi\)
\(992\) 0 0
\(993\) 6.89656 0.218856
\(994\) 0 0
\(995\) −1.87778 −0.0595295
\(996\) 0 0
\(997\) 41.2295 1.30575 0.652875 0.757466i \(-0.273563\pi\)
0.652875 + 0.757466i \(0.273563\pi\)
\(998\) 0 0
\(999\) 15.0911 0.477460
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.bb.1.5 8
4.3 odd 2 7168.2.a.ba.1.4 8
8.3 odd 2 7168.2.a.be.1.5 8
8.5 even 2 7168.2.a.bf.1.4 8
32.3 odd 8 1792.2.m.f.1345.4 yes 16
32.5 even 8 1792.2.m.e.449.4 16
32.11 odd 8 1792.2.m.f.449.4 yes 16
32.13 even 8 1792.2.m.e.1345.4 yes 16
32.19 odd 8 1792.2.m.g.1345.5 yes 16
32.21 even 8 1792.2.m.h.449.5 yes 16
32.27 odd 8 1792.2.m.g.449.5 yes 16
32.29 even 8 1792.2.m.h.1345.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.4 16 32.5 even 8
1792.2.m.e.1345.4 yes 16 32.13 even 8
1792.2.m.f.449.4 yes 16 32.11 odd 8
1792.2.m.f.1345.4 yes 16 32.3 odd 8
1792.2.m.g.449.5 yes 16 32.27 odd 8
1792.2.m.g.1345.5 yes 16 32.19 odd 8
1792.2.m.h.449.5 yes 16 32.21 even 8
1792.2.m.h.1345.5 yes 16 32.29 even 8
7168.2.a.ba.1.4 8 4.3 odd 2
7168.2.a.bb.1.5 8 1.1 even 1 trivial
7168.2.a.be.1.5 8 8.3 odd 2
7168.2.a.bf.1.4 8 8.5 even 2