Properties

Label 7168.2.a.bf.1.4
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 1.4
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 \(-0.522265\pi\)
−0.0698890 + 0.997555i \(0.522265\pi\)
\(4\) 0 0
\(5\) −0.379610 −0.169767 −0.0848833 0.996391i \(-0.527052\pi\)
−0.0848833 + 0.996391i \(0.527052\pi\)
\(6\) 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 \(-0.370999\pi\)
0.394267 + 0.918996i \(0.370999\pi\)
\(12\) 0 0
\(13\) 2.31328 0.641589 0.320794 0.947149i \(-0.396050\pi\)
0.320794 + 0.947149i \(0.396050\pi\)
\(14\) 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 \(-0.285298\pi\)
0.624512 + 0.781015i \(0.285298\pi\)
\(20\) 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 \(-0.656105\pi\)
−0.470996 + 0.882135i \(0.656105\pi\)
\(30\) 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 \(-0.168969\pi\)
0.862386 + 0.506252i \(0.168969\pi\)
\(38\) 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 \(-0.637764\pi\)
−0.419414 + 0.907795i \(0.637764\pi\)
\(44\) 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 \(-0.517801\pi\)
−0.0558936 + 0.998437i \(0.517801\pi\)
\(54\) 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 \(-0.663251\pi\)
−0.490679 + 0.871341i \(0.663251\pi\)
\(60\) 0 0
\(61\) −1.31502 −0.168371 −0.0841857 0.996450i \(-0.526829\pi\)
−0.0841857 + 0.996450i \(0.526829\pi\)
\(62\) 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 \(-0.680506\pi\)
−0.537170 + 0.843474i \(0.680506\pi\)
\(68\) 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 \(-0.117607\pi\)
0.932518 + 0.361123i \(0.117607\pi\)
\(84\) 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 \(-0.481762\pi\)
0.0572666 + 0.998359i \(0.481762\pi\)
\(102\) 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 \(-0.515350\pi\)
−0.0482048 + 0.998837i \(0.515350\pi\)
\(108\) 0 0
\(109\) 17.1491 1.64259 0.821295 0.570503i \(-0.193252\pi\)
0.821295 + 0.570503i \(0.193252\pi\)
\(110\) 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 \(-0.340170\pi\)
0.481286 + 0.876564i \(0.340170\pi\)
\(132\) 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 \(-0.178779\pi\)
0.846377 + 0.532584i \(0.178779\pi\)
\(140\) 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 \(-0.365343\pi\)
0.410531 + 0.911847i \(0.365343\pi\)
\(150\) 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 \(-0.357112\pi\)
0.433971 + 0.900927i \(0.357112\pi\)
\(158\) 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 \(-0.579624\pi\)
−0.247545 + 0.968877i \(0.579624\pi\)
\(164\) 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 \(-0.391605\pi\)
0.333990 + 0.942577i \(0.391605\pi\)
\(174\) 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 \(-0.534328\pi\)
−0.107635 + 0.994191i \(0.534328\pi\)
\(180\) 0 0
\(181\) 23.6338 1.75669 0.878343 0.478030i \(-0.158649\pi\)
0.878343 + 0.478030i \(0.158649\pi\)
\(182\) 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 \(-0.242908\pi\)
0.722684 + 0.691178i \(0.242908\pi\)
\(198\) 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 \(-0.573129\pi\)
−0.227726 + 0.973725i \(0.573129\pi\)
\(212\) 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 \(-0.692276\pi\)
−0.567982 + 0.823041i \(0.692276\pi\)
\(228\) 0 0
\(229\) 19.1324 1.26431 0.632153 0.774844i \(-0.282171\pi\)
0.632153 + 0.774844i \(0.282171\pi\)
\(230\) 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 \(-0.635539\pi\)
−0.413056 + 0.910705i \(0.635539\pi\)
\(252\) 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 \(-0.186463\pi\)
0.833275 + 0.552859i \(0.186463\pi\)
\(270\) 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 \(-0.222918\pi\)
0.764638 + 0.644460i \(0.222918\pi\)
\(278\) 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 \(-0.638591\pi\)
−0.421770 + 0.906703i \(0.638591\pi\)
\(284\) 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 \(-0.803457\pi\)
−0.815352 + 0.578965i \(0.803457\pi\)
\(294\) 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 \(-0.540549\pi\)
−0.127045 + 0.991897i \(0.540549\pi\)
\(308\) 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 \(-0.501875\pi\)
−0.00589158 + 0.999983i \(0.501875\pi\)
\(318\) 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 \(-0.786244\pi\)
−0.782868 + 0.622188i \(0.786244\pi\)
\(332\) 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 \(-0.771614\pi\)
−0.753454 + 0.657501i \(0.771614\pi\)
\(348\) 0 0
\(349\) 23.4899 1.25738 0.628692 0.777654i \(-0.283591\pi\)
0.628692 + 0.777654i \(0.283591\pi\)
\(350\) 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 \(-0.777993\pi\)
−0.766478 + 0.642270i \(0.777993\pi\)
\(374\) 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 \(-0.171313\pi\)
0.858635 + 0.512587i \(0.171313\pi\)
\(380\) 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 \(-0.687328\pi\)
−0.555121 + 0.831770i \(0.687328\pi\)
\(390\) 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 \(-0.564367\pi\)
−0.200839 + 0.979624i \(0.564367\pi\)
\(398\) 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 \(-0.567781\pi\)
−0.211336 + 0.977414i \(0.567781\pi\)
\(420\) 0 0
\(421\) 12.7739 0.622562 0.311281 0.950318i \(-0.399242\pi\)
0.311281 + 0.950318i \(0.399242\pi\)
\(422\) 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 \(-0.284314\pi\)
0.626923 + 0.779081i \(0.284314\pi\)
\(444\) 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 \(-0.477585\pi\)
0.0703617 + 0.997522i \(0.477585\pi\)
\(462\) 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 \(-0.584337\pi\)
−0.261864 + 0.965105i \(0.584337\pi\)
\(468\) 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 \(-0.357087\pi\)
0.434043 + 0.900892i \(0.357087\pi\)
\(492\) 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 \(-0.777727\pi\)
−0.765942 + 0.642910i \(0.777727\pi\)
\(500\) 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 \(-0.297213\pi\)
0.594846 + 0.803840i \(0.297213\pi\)
\(510\) 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 \(-0.760095\pi\)
−0.729173 + 0.684329i \(0.760095\pi\)
\(524\) 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 \(-0.384786\pi\)
0.354102 + 0.935207i \(0.384786\pi\)
\(542\) 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 \(-0.360838\pi\)
0.423396 + 0.905945i \(0.360838\pi\)
\(548\) 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 \(-0.239299\pi\)
0.730473 + 0.682941i \(0.239299\pi\)
\(558\) 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 \(-0.786381\pi\)
−0.783136 + 0.621850i \(0.786381\pi\)
\(564\) 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 \(-0.261028\pi\)
0.682190 + 0.731175i \(0.261028\pi\)
\(572\) 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 \(-0.195021\pi\)
0.818112 + 0.575058i \(0.195021\pi\)
\(588\) 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 \(-0.471432\pi\)
0.0896277 + 0.995975i \(0.471432\pi\)
\(614\) 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 \(-0.443924\pi\)
0.175257 + 0.984523i \(0.443924\pi\)
\(620\) 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 \(-0.782838\pi\)
−0.776166 + 0.630529i \(0.782838\pi\)
\(644\) 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 \(-0.340276\pi\)
0.480995 + 0.876723i \(0.340276\pi\)
\(654\) 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 \(-0.223687\pi\)
0.763079 + 0.646305i \(0.223687\pi\)
\(660\) 0 0
\(661\) −26.5220 −1.03159 −0.515794 0.856713i \(-0.672503\pi\)
−0.515794 + 0.856713i \(0.672503\pi\)
\(662\) 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 \(-0.285051\pi\)
0.625117 + 0.780531i \(0.285051\pi\)
\(678\) 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 \(-0.429399\pi\)
0.219984 + 0.975503i \(0.429399\pi\)
\(684\) 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 \(-0.490630\pi\)
0.0294335 + 0.999567i \(0.490630\pi\)
\(692\) 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 \(-0.400045\pi\)
0.308882 + 0.951100i \(0.400045\pi\)
\(702\) 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 \(-0.477187\pi\)
0.0716085 + 0.997433i \(0.477187\pi\)
\(710\) 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 \(-0.711324\pi\)
−0.616188 + 0.787599i \(0.711324\pi\)
\(734\) 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 \(-0.437237\pi\)
0.195900 + 0.980624i \(0.437237\pi\)
\(740\) 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 \(-0.684731\pi\)
−0.548317 + 0.836271i \(0.684731\pi\)
\(758\) 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 \(-0.337994\pi\)
0.487267 + 0.873253i \(0.337994\pi\)
\(774\) 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 \(-0.386769\pi\)
0.348270 + 0.937394i \(0.386769\pi\)
\(788\) 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 \(-0.570619\pi\)
−0.220041 + 0.975491i \(0.570619\pi\)
\(798\) 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 \(-0.501606\pi\)
−0.00504517 + 0.999987i \(0.501606\pi\)
\(812\) 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 \(-0.671527\pi\)
−0.513166 + 0.858290i \(0.671527\pi\)
\(822\) 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 \(-0.339812\pi\)
0.482271 + 0.876022i \(0.339812\pi\)
\(828\) 0 0
\(829\) −13.0904 −0.454650 −0.227325 0.973819i \(-0.572998\pi\)
−0.227325 + 0.973819i \(0.572998\pi\)
\(830\) 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 \(-0.612423\pi\)
−0.345890 + 0.938275i \(0.612423\pi\)
\(854\) 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 \(-0.837341\pi\)
−0.872251 + 0.489058i \(0.837341\pi\)
\(860\) 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 \(-0.483434\pi\)
0.0520214 + 0.998646i \(0.483434\pi\)
\(878\) 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 \(-0.542012\pi\)
−0.131601 + 0.991303i \(0.542012\pi\)
\(884\) 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 \(-0.267439\pi\)
0.667325 + 0.744767i \(0.267439\pi\)
\(908\) 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 \(-0.387557\pi\)
0.345949 + 0.938253i \(0.387557\pi\)
\(942\) 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 \(-0.553002\pi\)
−0.165743 + 0.986169i \(0.553002\pi\)
\(948\) 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 \(-0.381647\pi\)
0.363310 + 0.931668i \(0.381647\pi\)
\(972\) 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 \(-0.726437\pi\)
−0.652875 + 0.757466i \(0.726437\pi\)
\(998\) 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.bf.1.4 8
4.3 odd 2 7168.2.a.be.1.5 8
8.3 odd 2 7168.2.a.ba.1.4 8
8.5 even 2 7168.2.a.bb.1.5 8
32.3 odd 8 1792.2.m.g.1345.5 yes 16
32.5 even 8 1792.2.m.h.449.5 yes 16
32.11 odd 8 1792.2.m.g.449.5 yes 16
32.13 even 8 1792.2.m.h.1345.5 yes 16
32.19 odd 8 1792.2.m.f.1345.4 yes 16
32.21 even 8 1792.2.m.e.449.4 16
32.27 odd 8 1792.2.m.f.449.4 yes 16
32.29 even 8 1792.2.m.e.1345.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.4 16 32.21 even 8
1792.2.m.e.1345.4 yes 16 32.29 even 8
1792.2.m.f.449.4 yes 16 32.27 odd 8
1792.2.m.f.1345.4 yes 16 32.19 odd 8
1792.2.m.g.449.5 yes 16 32.11 odd 8
1792.2.m.g.1345.5 yes 16 32.3 odd 8
1792.2.m.h.449.5 yes 16 32.5 even 8
1792.2.m.h.1345.5 yes 16 32.13 even 8
7168.2.a.ba.1.4 8 8.3 odd 2
7168.2.a.bb.1.5 8 8.5 even 2
7168.2.a.be.1.5 8 4.3 odd 2
7168.2.a.bf.1.4 8 1.1 even 1 trivial