Properties

Label 1148.2.d.a.1065.15
Level $1148$
Weight $2$
Character 1148.1065
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1065.15
Root \(4.38287i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} +O(q^{10})\) \(q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} -1.15173i q^{11} +1.66501i q^{13} -1.90790i q^{15} +2.28853i q^{17} +5.03870i q^{19} +1.40464 q^{21} +4.00693 q^{23} -3.15507 q^{25} +5.65647i q^{27} +2.40457i q^{29} +5.53514 q^{31} +1.61777 q^{33} +1.35828i q^{35} -0.838223 q^{37} -2.33875 q^{39} +(6.21611 - 1.53621i) q^{41} -10.5361 q^{43} -1.39492 q^{45} +13.4336i q^{47} -1.00000 q^{49} -3.21457 q^{51} +6.80004i q^{53} +1.56437i q^{55} -7.07758 q^{57} -7.72782 q^{59} +3.76583 q^{61} -1.02697i q^{63} -2.26155i q^{65} -6.05601i q^{67} +5.62831i q^{69} +13.1629i q^{71} +8.45968 q^{73} -4.43176i q^{75} -1.15173 q^{77} +1.01755i q^{79} -4.86441 q^{81} -17.7169 q^{83} -3.10846i q^{85} -3.37756 q^{87} +7.88443i q^{89} +1.66501 q^{91} +7.77491i q^{93} -6.84396i q^{95} -9.62923i q^{97} -1.18280i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.40464i 0.810972i 0.914101 + 0.405486i \(0.132898\pi\)
−0.914101 + 0.405486i \(0.867102\pi\)
\(4\) 0 0
\(5\) −1.35828 −0.607442 −0.303721 0.952761i \(-0.598229\pi\)
−0.303721 + 0.952761i \(0.598229\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.02697 0.342324
\(10\) 0 0
\(11\) 1.15173i 0.347260i −0.984811 0.173630i \(-0.944450\pi\)
0.984811 0.173630i \(-0.0555496\pi\)
\(12\) 0 0
\(13\) 1.66501i 0.461791i 0.972979 + 0.230896i \(0.0741656\pi\)
−0.972979 + 0.230896i \(0.925834\pi\)
\(14\) 0 0
\(15\) 1.90790i 0.492618i
\(16\) 0 0
\(17\) 2.28853i 0.555049i 0.960719 + 0.277525i \(0.0895140\pi\)
−0.960719 + 0.277525i \(0.910486\pi\)
\(18\) 0 0
\(19\) 5.03870i 1.15596i 0.816052 + 0.577978i \(0.196158\pi\)
−0.816052 + 0.577978i \(0.803842\pi\)
\(20\) 0 0
\(21\) 1.40464 0.306519
\(22\) 0 0
\(23\) 4.00693 0.835502 0.417751 0.908562i \(-0.362818\pi\)
0.417751 + 0.908562i \(0.362818\pi\)
\(24\) 0 0
\(25\) −3.15507 −0.631015
\(26\) 0 0
\(27\) 5.65647i 1.08859i
\(28\) 0 0
\(29\) 2.40457i 0.446517i 0.974759 + 0.223259i \(0.0716694\pi\)
−0.974759 + 0.223259i \(0.928331\pi\)
\(30\) 0 0
\(31\) 5.53514 0.994141 0.497070 0.867710i \(-0.334409\pi\)
0.497070 + 0.867710i \(0.334409\pi\)
\(32\) 0 0
\(33\) 1.61777 0.281618
\(34\) 0 0
\(35\) 1.35828i 0.229591i
\(36\) 0 0
\(37\) −0.838223 −0.137803 −0.0689015 0.997623i \(-0.521949\pi\)
−0.0689015 + 0.997623i \(0.521949\pi\)
\(38\) 0 0
\(39\) −2.33875 −0.374500
\(40\) 0 0
\(41\) 6.21611 1.53621i 0.970794 0.239916i
\(42\) 0 0
\(43\) −10.5361 −1.60674 −0.803371 0.595479i \(-0.796962\pi\)
−0.803371 + 0.595479i \(0.796962\pi\)
\(44\) 0 0
\(45\) −1.39492 −0.207942
\(46\) 0 0
\(47\) 13.4336i 1.95950i 0.200234 + 0.979748i \(0.435830\pi\)
−0.200234 + 0.979748i \(0.564170\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.21457 −0.450129
\(52\) 0 0
\(53\) 6.80004i 0.934057i 0.884242 + 0.467028i \(0.154675\pi\)
−0.884242 + 0.467028i \(0.845325\pi\)
\(54\) 0 0
\(55\) 1.56437i 0.210940i
\(56\) 0 0
\(57\) −7.07758 −0.937448
\(58\) 0 0
\(59\) −7.72782 −1.00608 −0.503039 0.864264i \(-0.667785\pi\)
−0.503039 + 0.864264i \(0.667785\pi\)
\(60\) 0 0
\(61\) 3.76583 0.482165 0.241083 0.970505i \(-0.422497\pi\)
0.241083 + 0.970505i \(0.422497\pi\)
\(62\) 0 0
\(63\) 1.02697i 0.129386i
\(64\) 0 0
\(65\) 2.26155i 0.280511i
\(66\) 0 0
\(67\) 6.05601i 0.739859i −0.929060 0.369930i \(-0.879382\pi\)
0.929060 0.369930i \(-0.120618\pi\)
\(68\) 0 0
\(69\) 5.62831i 0.677569i
\(70\) 0 0
\(71\) 13.1629i 1.56215i 0.624436 + 0.781076i \(0.285329\pi\)
−0.624436 + 0.781076i \(0.714671\pi\)
\(72\) 0 0
\(73\) 8.45968 0.990130 0.495065 0.868856i \(-0.335144\pi\)
0.495065 + 0.868856i \(0.335144\pi\)
\(74\) 0 0
\(75\) 4.43176i 0.511735i
\(76\) 0 0
\(77\) −1.15173 −0.131252
\(78\) 0 0
\(79\) 1.01755i 0.114484i 0.998360 + 0.0572418i \(0.0182306\pi\)
−0.998360 + 0.0572418i \(0.981769\pi\)
\(80\) 0 0
\(81\) −4.86441 −0.540490
\(82\) 0 0
\(83\) −17.7169 −1.94468 −0.972340 0.233571i \(-0.924959\pi\)
−0.972340 + 0.233571i \(0.924959\pi\)
\(84\) 0 0
\(85\) 3.10846i 0.337160i
\(86\) 0 0
\(87\) −3.37756 −0.362113
\(88\) 0 0
\(89\) 7.88443i 0.835748i 0.908505 + 0.417874i \(0.137225\pi\)
−0.908505 + 0.417874i \(0.862775\pi\)
\(90\) 0 0
\(91\) 1.66501 0.174541
\(92\) 0 0
\(93\) 7.77491i 0.806220i
\(94\) 0 0
\(95\) 6.84396i 0.702176i
\(96\) 0 0
\(97\) 9.62923i 0.977700i −0.872368 0.488850i \(-0.837417\pi\)
0.872368 0.488850i \(-0.162583\pi\)
\(98\) 0 0
\(99\) 1.18280i 0.118875i
\(100\) 0 0
\(101\) 3.15207i 0.313643i −0.987627 0.156821i \(-0.949875\pi\)
0.987627 0.156821i \(-0.0501247\pi\)
\(102\) 0 0
\(103\) 6.42106 0.632686 0.316343 0.948645i \(-0.397545\pi\)
0.316343 + 0.948645i \(0.397545\pi\)
\(104\) 0 0
\(105\) −1.90790 −0.186192
\(106\) 0 0
\(107\) 2.68916 0.259971 0.129985 0.991516i \(-0.458507\pi\)
0.129985 + 0.991516i \(0.458507\pi\)
\(108\) 0 0
\(109\) 0.843693i 0.0808111i −0.999183 0.0404056i \(-0.987135\pi\)
0.999183 0.0404056i \(-0.0128650\pi\)
\(110\) 0 0
\(111\) 1.17741i 0.111754i
\(112\) 0 0
\(113\) 2.46309 0.231708 0.115854 0.993266i \(-0.463039\pi\)
0.115854 + 0.993266i \(0.463039\pi\)
\(114\) 0 0
\(115\) −5.44253 −0.507519
\(116\) 0 0
\(117\) 1.70992i 0.158082i
\(118\) 0 0
\(119\) 2.28853 0.209789
\(120\) 0 0
\(121\) 9.67352 0.879411
\(122\) 0 0
\(123\) 2.15783 + 8.73143i 0.194565 + 0.787286i
\(124\) 0 0
\(125\) 11.0769 0.990746
\(126\) 0 0
\(127\) −18.6916 −1.65861 −0.829305 0.558796i \(-0.811264\pi\)
−0.829305 + 0.558796i \(0.811264\pi\)
\(128\) 0 0
\(129\) 14.7995i 1.30302i
\(130\) 0 0
\(131\) −3.67323 −0.320931 −0.160466 0.987041i \(-0.551300\pi\)
−0.160466 + 0.987041i \(0.551300\pi\)
\(132\) 0 0
\(133\) 5.03870 0.436910
\(134\) 0 0
\(135\) 7.68307i 0.661253i
\(136\) 0 0
\(137\) 13.6977i 1.17027i −0.810935 0.585136i \(-0.801041\pi\)
0.810935 0.585136i \(-0.198959\pi\)
\(138\) 0 0
\(139\) 17.9939 1.52623 0.763113 0.646265i \(-0.223670\pi\)
0.763113 + 0.646265i \(0.223670\pi\)
\(140\) 0 0
\(141\) −18.8695 −1.58910
\(142\) 0 0
\(143\) 1.91764 0.160361
\(144\) 0 0
\(145\) 3.26608i 0.271233i
\(146\) 0 0
\(147\) 1.40464i 0.115853i
\(148\) 0 0
\(149\) 7.97880i 0.653649i −0.945085 0.326824i \(-0.894021\pi\)
0.945085 0.326824i \(-0.105979\pi\)
\(150\) 0 0
\(151\) 4.53026i 0.368667i 0.982864 + 0.184333i \(0.0590126\pi\)
−0.982864 + 0.184333i \(0.940987\pi\)
\(152\) 0 0
\(153\) 2.35025i 0.190007i
\(154\) 0 0
\(155\) −7.51828 −0.603882
\(156\) 0 0
\(157\) 2.54513i 0.203124i 0.994829 + 0.101562i \(0.0323840\pi\)
−0.994829 + 0.101562i \(0.967616\pi\)
\(158\) 0 0
\(159\) −9.55164 −0.757494
\(160\) 0 0
\(161\) 4.00693i 0.315790i
\(162\) 0 0
\(163\) 22.3553 1.75100 0.875502 0.483215i \(-0.160531\pi\)
0.875502 + 0.483215i \(0.160531\pi\)
\(164\) 0 0
\(165\) −2.19739 −0.171066
\(166\) 0 0
\(167\) 22.9254i 1.77402i 0.461748 + 0.887011i \(0.347223\pi\)
−0.461748 + 0.887011i \(0.652777\pi\)
\(168\) 0 0
\(169\) 10.2277 0.786749
\(170\) 0 0
\(171\) 5.17461i 0.395712i
\(172\) 0 0
\(173\) 18.7451 1.42516 0.712581 0.701590i \(-0.247526\pi\)
0.712581 + 0.701590i \(0.247526\pi\)
\(174\) 0 0
\(175\) 3.15507i 0.238501i
\(176\) 0 0
\(177\) 10.8548i 0.815900i
\(178\) 0 0
\(179\) 13.3583i 0.998444i 0.866474 + 0.499222i \(0.166381\pi\)
−0.866474 + 0.499222i \(0.833619\pi\)
\(180\) 0 0
\(181\) 13.1556i 0.977846i −0.872327 0.488923i \(-0.837390\pi\)
0.872327 0.488923i \(-0.162610\pi\)
\(182\) 0 0
\(183\) 5.28966i 0.391023i
\(184\) 0 0
\(185\) 1.13854 0.0837073
\(186\) 0 0
\(187\) 2.63576 0.192746
\(188\) 0 0
\(189\) 5.65647 0.411447
\(190\) 0 0
\(191\) 3.95650i 0.286283i 0.989702 + 0.143141i \(0.0457203\pi\)
−0.989702 + 0.143141i \(0.954280\pi\)
\(192\) 0 0
\(193\) 25.2978i 1.82098i −0.413534 0.910488i \(-0.635706\pi\)
0.413534 0.910488i \(-0.364294\pi\)
\(194\) 0 0
\(195\) 3.17668 0.227487
\(196\) 0 0
\(197\) −5.25097 −0.374116 −0.187058 0.982349i \(-0.559895\pi\)
−0.187058 + 0.982349i \(0.559895\pi\)
\(198\) 0 0
\(199\) 10.7283i 0.760511i −0.924882 0.380255i \(-0.875836\pi\)
0.924882 0.380255i \(-0.124164\pi\)
\(200\) 0 0
\(201\) 8.50654 0.600005
\(202\) 0 0
\(203\) 2.40457 0.168768
\(204\) 0 0
\(205\) −8.44323 + 2.08661i −0.589700 + 0.145735i
\(206\) 0 0
\(207\) 4.11501 0.286013
\(208\) 0 0
\(209\) 5.80322 0.401417
\(210\) 0 0
\(211\) 14.2255i 0.979326i −0.871912 0.489663i \(-0.837120\pi\)
0.871912 0.489663i \(-0.162880\pi\)
\(212\) 0 0
\(213\) −18.4892 −1.26686
\(214\) 0 0
\(215\) 14.3110 0.976002
\(216\) 0 0
\(217\) 5.53514i 0.375750i
\(218\) 0 0
\(219\) 11.8828i 0.802968i
\(220\) 0 0
\(221\) −3.81042 −0.256317
\(222\) 0 0
\(223\) 18.8314 1.26105 0.630523 0.776171i \(-0.282841\pi\)
0.630523 + 0.776171i \(0.282841\pi\)
\(224\) 0 0
\(225\) −3.24018 −0.216012
\(226\) 0 0
\(227\) 1.40671i 0.0933667i −0.998910 0.0466833i \(-0.985135\pi\)
0.998910 0.0466833i \(-0.0148652\pi\)
\(228\) 0 0
\(229\) 15.8047i 1.04440i 0.852822 + 0.522202i \(0.174889\pi\)
−0.852822 + 0.522202i \(0.825111\pi\)
\(230\) 0 0
\(231\) 1.61777i 0.106442i
\(232\) 0 0
\(233\) 1.10843i 0.0726159i −0.999341 0.0363080i \(-0.988440\pi\)
0.999341 0.0363080i \(-0.0115597\pi\)
\(234\) 0 0
\(235\) 18.2466i 1.19028i
\(236\) 0 0
\(237\) −1.42930 −0.0928430
\(238\) 0 0
\(239\) 17.6798i 1.14361i −0.820389 0.571806i \(-0.806243\pi\)
0.820389 0.571806i \(-0.193757\pi\)
\(240\) 0 0
\(241\) −8.94266 −0.576047 −0.288024 0.957623i \(-0.592998\pi\)
−0.288024 + 0.957623i \(0.592998\pi\)
\(242\) 0 0
\(243\) 10.1366i 0.650266i
\(244\) 0 0
\(245\) 1.35828 0.0867774
\(246\) 0 0
\(247\) −8.38948 −0.533810
\(248\) 0 0
\(249\) 24.8859i 1.57708i
\(250\) 0 0
\(251\) −7.20705 −0.454905 −0.227453 0.973789i \(-0.573040\pi\)
−0.227453 + 0.973789i \(0.573040\pi\)
\(252\) 0 0
\(253\) 4.61490i 0.290136i
\(254\) 0 0
\(255\) 4.36628 0.273427
\(256\) 0 0
\(257\) 5.59872i 0.349239i 0.984636 + 0.174619i \(0.0558695\pi\)
−0.984636 + 0.174619i \(0.944131\pi\)
\(258\) 0 0
\(259\) 0.838223i 0.0520847i
\(260\) 0 0
\(261\) 2.46943i 0.152854i
\(262\) 0 0
\(263\) 14.4777i 0.892730i 0.894851 + 0.446365i \(0.147282\pi\)
−0.894851 + 0.446365i \(0.852718\pi\)
\(264\) 0 0
\(265\) 9.23636i 0.567385i
\(266\) 0 0
\(267\) −11.0748 −0.677768
\(268\) 0 0
\(269\) 0.913652 0.0557063 0.0278532 0.999612i \(-0.491133\pi\)
0.0278532 + 0.999612i \(0.491133\pi\)
\(270\) 0 0
\(271\) 0.314962 0.0191326 0.00956630 0.999954i \(-0.496955\pi\)
0.00956630 + 0.999954i \(0.496955\pi\)
\(272\) 0 0
\(273\) 2.33875i 0.141548i
\(274\) 0 0
\(275\) 3.63379i 0.219126i
\(276\) 0 0
\(277\) 5.19368 0.312058 0.156029 0.987752i \(-0.450131\pi\)
0.156029 + 0.987752i \(0.450131\pi\)
\(278\) 0 0
\(279\) 5.68444 0.340319
\(280\) 0 0
\(281\) 21.4131i 1.27740i −0.769456 0.638699i \(-0.779473\pi\)
0.769456 0.638699i \(-0.220527\pi\)
\(282\) 0 0
\(283\) −21.3292 −1.26789 −0.633945 0.773379i \(-0.718565\pi\)
−0.633945 + 0.773379i \(0.718565\pi\)
\(284\) 0 0
\(285\) 9.61334 0.569445
\(286\) 0 0
\(287\) −1.53621 6.21611i −0.0906798 0.366925i
\(288\) 0 0
\(289\) 11.7627 0.691921
\(290\) 0 0
\(291\) 13.5256 0.792887
\(292\) 0 0
\(293\) 21.2087i 1.23903i −0.784986 0.619513i \(-0.787330\pi\)
0.784986 0.619513i \(-0.212670\pi\)
\(294\) 0 0
\(295\) 10.4966 0.611133
\(296\) 0 0
\(297\) 6.51472 0.378022
\(298\) 0 0
\(299\) 6.67158i 0.385827i
\(300\) 0 0
\(301\) 10.5361i 0.607291i
\(302\) 0 0
\(303\) 4.42754 0.254355
\(304\) 0 0
\(305\) −5.11506 −0.292887
\(306\) 0 0
\(307\) −6.94680 −0.396475 −0.198237 0.980154i \(-0.563522\pi\)
−0.198237 + 0.980154i \(0.563522\pi\)
\(308\) 0 0
\(309\) 9.01931i 0.513090i
\(310\) 0 0
\(311\) 18.5322i 1.05087i −0.850835 0.525433i \(-0.823903\pi\)
0.850835 0.525433i \(-0.176097\pi\)
\(312\) 0 0
\(313\) 26.7851i 1.51398i −0.653426 0.756991i \(-0.726669\pi\)
0.653426 0.756991i \(-0.273331\pi\)
\(314\) 0 0
\(315\) 1.39492i 0.0785947i
\(316\) 0 0
\(317\) 4.59346i 0.257995i −0.991645 0.128997i \(-0.958824\pi\)
0.991645 0.128997i \(-0.0411758\pi\)
\(318\) 0 0
\(319\) 2.76941 0.155057
\(320\) 0 0
\(321\) 3.77731i 0.210829i
\(322\) 0 0
\(323\) −11.5312 −0.641612
\(324\) 0 0
\(325\) 5.25323i 0.291397i
\(326\) 0 0
\(327\) 1.18509 0.0655355
\(328\) 0 0
\(329\) 13.4336 0.740620
\(330\) 0 0
\(331\) 29.1959i 1.60475i −0.596820 0.802375i \(-0.703569\pi\)
0.596820 0.802375i \(-0.296431\pi\)
\(332\) 0 0
\(333\) −0.860833 −0.0471734
\(334\) 0 0
\(335\) 8.22576i 0.449421i
\(336\) 0 0
\(337\) −25.3198 −1.37925 −0.689627 0.724164i \(-0.742226\pi\)
−0.689627 + 0.724164i \(0.742226\pi\)
\(338\) 0 0
\(339\) 3.45977i 0.187909i
\(340\) 0 0
\(341\) 6.37499i 0.345225i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 7.64482i 0.411583i
\(346\) 0 0
\(347\) 14.5714i 0.782233i 0.920341 + 0.391116i \(0.127911\pi\)
−0.920341 + 0.391116i \(0.872089\pi\)
\(348\) 0 0
\(349\) 24.3554 1.30371 0.651857 0.758342i \(-0.273990\pi\)
0.651857 + 0.758342i \(0.273990\pi\)
\(350\) 0 0
\(351\) −9.41808 −0.502700
\(352\) 0 0
\(353\) −10.7267 −0.570924 −0.285462 0.958390i \(-0.592147\pi\)
−0.285462 + 0.958390i \(0.592147\pi\)
\(354\) 0 0
\(355\) 17.8790i 0.948917i
\(356\) 0 0
\(357\) 3.21457i 0.170133i
\(358\) 0 0
\(359\) 30.6428 1.61727 0.808634 0.588313i \(-0.200208\pi\)
0.808634 + 0.588313i \(0.200208\pi\)
\(360\) 0 0
\(361\) −6.38845 −0.336234
\(362\) 0 0
\(363\) 13.5879i 0.713178i
\(364\) 0 0
\(365\) −11.4906 −0.601446
\(366\) 0 0
\(367\) −35.9390 −1.87600 −0.938002 0.346631i \(-0.887326\pi\)
−0.938002 + 0.346631i \(0.887326\pi\)
\(368\) 0 0
\(369\) 6.38378 1.57765i 0.332326 0.0821292i
\(370\) 0 0
\(371\) 6.80004 0.353040
\(372\) 0 0
\(373\) −17.7915 −0.921207 −0.460603 0.887606i \(-0.652367\pi\)
−0.460603 + 0.887606i \(0.652367\pi\)
\(374\) 0 0
\(375\) 15.5591i 0.803467i
\(376\) 0 0
\(377\) −4.00363 −0.206198
\(378\) 0 0
\(379\) −9.86593 −0.506779 −0.253389 0.967364i \(-0.581545\pi\)
−0.253389 + 0.967364i \(0.581545\pi\)
\(380\) 0 0
\(381\) 26.2551i 1.34509i
\(382\) 0 0
\(383\) 3.98164i 0.203452i −0.994812 0.101726i \(-0.967563\pi\)
0.994812 0.101726i \(-0.0324365\pi\)
\(384\) 0 0
\(385\) 1.56437 0.0797278
\(386\) 0 0
\(387\) −10.8203 −0.550027
\(388\) 0 0
\(389\) −34.3705 −1.74265 −0.871326 0.490705i \(-0.836739\pi\)
−0.871326 + 0.490705i \(0.836739\pi\)
\(390\) 0 0
\(391\) 9.16995i 0.463744i
\(392\) 0 0
\(393\) 5.15958i 0.260266i
\(394\) 0 0
\(395\) 1.38212i 0.0695421i
\(396\) 0 0
\(397\) 18.4252i 0.924733i 0.886689 + 0.462367i \(0.153000\pi\)
−0.886689 + 0.462367i \(0.847000\pi\)
\(398\) 0 0
\(399\) 7.07758i 0.354322i
\(400\) 0 0
\(401\) −20.2007 −1.00877 −0.504387 0.863478i \(-0.668281\pi\)
−0.504387 + 0.863478i \(0.668281\pi\)
\(402\) 0 0
\(403\) 9.21607i 0.459085i
\(404\) 0 0
\(405\) 6.60723 0.328316
\(406\) 0 0
\(407\) 0.965407i 0.0478534i
\(408\) 0 0
\(409\) −1.11502 −0.0551344 −0.0275672 0.999620i \(-0.508776\pi\)
−0.0275672 + 0.999620i \(0.508776\pi\)
\(410\) 0 0
\(411\) 19.2404 0.949057
\(412\) 0 0
\(413\) 7.72782i 0.380261i
\(414\) 0 0
\(415\) 24.0645 1.18128
\(416\) 0 0
\(417\) 25.2751i 1.23773i
\(418\) 0 0
\(419\) −22.4070 −1.09465 −0.547327 0.836919i \(-0.684354\pi\)
−0.547327 + 0.836919i \(0.684354\pi\)
\(420\) 0 0
\(421\) 7.94066i 0.387004i 0.981100 + 0.193502i \(0.0619846\pi\)
−0.981100 + 0.193502i \(0.938015\pi\)
\(422\) 0 0
\(423\) 13.7960i 0.670783i
\(424\) 0 0
\(425\) 7.22047i 0.350244i
\(426\) 0 0
\(427\) 3.76583i 0.182241i
\(428\) 0 0
\(429\) 2.69361i 0.130049i
\(430\) 0 0
\(431\) 37.7371 1.81773 0.908866 0.417089i \(-0.136950\pi\)
0.908866 + 0.417089i \(0.136950\pi\)
\(432\) 0 0
\(433\) 26.4996 1.27349 0.636744 0.771075i \(-0.280281\pi\)
0.636744 + 0.771075i \(0.280281\pi\)
\(434\) 0 0
\(435\) 4.58768 0.219962
\(436\) 0 0
\(437\) 20.1897i 0.965803i
\(438\) 0 0
\(439\) 27.4781i 1.31146i −0.754996 0.655729i \(-0.772361\pi\)
0.754996 0.655729i \(-0.227639\pi\)
\(440\) 0 0
\(441\) −1.02697 −0.0489035
\(442\) 0 0
\(443\) 6.04591 0.287250 0.143625 0.989632i \(-0.454124\pi\)
0.143625 + 0.989632i \(0.454124\pi\)
\(444\) 0 0
\(445\) 10.7093i 0.507668i
\(446\) 0 0
\(447\) 11.2074 0.530091
\(448\) 0 0
\(449\) −5.55217 −0.262023 −0.131012 0.991381i \(-0.541822\pi\)
−0.131012 + 0.991381i \(0.541822\pi\)
\(450\) 0 0
\(451\) −1.76930 7.15928i −0.0833132 0.337117i
\(452\) 0 0
\(453\) −6.36340 −0.298979
\(454\) 0 0
\(455\) −2.26155 −0.106023
\(456\) 0 0
\(457\) 23.5007i 1.09932i 0.835389 + 0.549659i \(0.185242\pi\)
−0.835389 + 0.549659i \(0.814758\pi\)
\(458\) 0 0
\(459\) −12.9450 −0.604219
\(460\) 0 0
\(461\) 25.8753 1.20513 0.602567 0.798069i \(-0.294145\pi\)
0.602567 + 0.798069i \(0.294145\pi\)
\(462\) 0 0
\(463\) 38.5903i 1.79344i −0.442594 0.896722i \(-0.645942\pi\)
0.442594 0.896722i \(-0.354058\pi\)
\(464\) 0 0
\(465\) 10.5605i 0.489732i
\(466\) 0 0
\(467\) 11.2032 0.518424 0.259212 0.965820i \(-0.416537\pi\)
0.259212 + 0.965820i \(0.416537\pi\)
\(468\) 0 0
\(469\) −6.05601 −0.279640
\(470\) 0 0
\(471\) −3.57500 −0.164728
\(472\) 0 0
\(473\) 12.1348i 0.557956i
\(474\) 0 0
\(475\) 15.8975i 0.729425i
\(476\) 0 0
\(477\) 6.98346i 0.319750i
\(478\) 0 0
\(479\) 31.4877i 1.43871i 0.694643 + 0.719355i \(0.255562\pi\)
−0.694643 + 0.719355i \(0.744438\pi\)
\(480\) 0 0
\(481\) 1.39565i 0.0636362i
\(482\) 0 0
\(483\) 5.62831 0.256097
\(484\) 0 0
\(485\) 13.0792i 0.593896i
\(486\) 0 0
\(487\) −19.9431 −0.903708 −0.451854 0.892092i \(-0.649237\pi\)
−0.451854 + 0.892092i \(0.649237\pi\)
\(488\) 0 0
\(489\) 31.4013i 1.42001i
\(490\) 0 0
\(491\) −21.7545 −0.981768 −0.490884 0.871225i \(-0.663326\pi\)
−0.490884 + 0.871225i \(0.663326\pi\)
\(492\) 0 0
\(493\) −5.50292 −0.247839
\(494\) 0 0
\(495\) 1.60657i 0.0722099i
\(496\) 0 0
\(497\) 13.1629 0.590438
\(498\) 0 0
\(499\) 25.9159i 1.16015i 0.814562 + 0.580077i \(0.196978\pi\)
−0.814562 + 0.580077i \(0.803022\pi\)
\(500\) 0 0
\(501\) −32.2021 −1.43868
\(502\) 0 0
\(503\) 18.1880i 0.810961i 0.914104 + 0.405480i \(0.132896\pi\)
−0.914104 + 0.405480i \(0.867104\pi\)
\(504\) 0 0
\(505\) 4.28139i 0.190520i
\(506\) 0 0
\(507\) 14.3663i 0.638031i
\(508\) 0 0
\(509\) 4.58230i 0.203107i −0.994830 0.101553i \(-0.967619\pi\)
0.994830 0.101553i \(-0.0323813\pi\)
\(510\) 0 0
\(511\) 8.45968i 0.374234i
\(512\) 0 0
\(513\) −28.5012 −1.25836
\(514\) 0 0
\(515\) −8.72160 −0.384320
\(516\) 0 0
\(517\) 15.4719 0.680454
\(518\) 0 0
\(519\) 26.3302i 1.15577i
\(520\) 0 0
\(521\) 4.35877i 0.190961i −0.995431 0.0954806i \(-0.969561\pi\)
0.995431 0.0954806i \(-0.0304388\pi\)
\(522\) 0 0
\(523\) 16.3327 0.714179 0.357090 0.934070i \(-0.383769\pi\)
0.357090 + 0.934070i \(0.383769\pi\)
\(524\) 0 0
\(525\) −4.43176 −0.193418
\(526\) 0 0
\(527\) 12.6673i 0.551797i
\(528\) 0 0
\(529\) −6.94455 −0.301937
\(530\) 0 0
\(531\) −7.93627 −0.344405
\(532\) 0 0
\(533\) 2.55781 + 10.3499i 0.110791 + 0.448304i
\(534\) 0 0
\(535\) −3.65263 −0.157917
\(536\) 0 0
\(537\) −18.7636 −0.809710
\(538\) 0 0
\(539\) 1.15173i 0.0496085i
\(540\) 0 0
\(541\) 28.9309 1.24384 0.621919 0.783082i \(-0.286353\pi\)
0.621919 + 0.783082i \(0.286353\pi\)
\(542\) 0 0
\(543\) 18.4789 0.793006
\(544\) 0 0
\(545\) 1.14597i 0.0490880i
\(546\) 0 0
\(547\) 23.9217i 1.02282i −0.859337 0.511409i \(-0.829124\pi\)
0.859337 0.511409i \(-0.170876\pi\)
\(548\) 0 0
\(549\) 3.86741 0.165057
\(550\) 0 0
\(551\) −12.1159 −0.516154
\(552\) 0 0
\(553\) 1.01755 0.0432707
\(554\) 0 0
\(555\) 1.59925i 0.0678843i
\(556\) 0 0
\(557\) 2.18023i 0.0923795i 0.998933 + 0.0461897i \(0.0147079\pi\)
−0.998933 + 0.0461897i \(0.985292\pi\)
\(558\) 0 0
\(559\) 17.5427i 0.741979i
\(560\) 0 0
\(561\) 3.70231i 0.156312i
\(562\) 0 0
\(563\) 38.7732i 1.63409i 0.576571 + 0.817047i \(0.304391\pi\)
−0.576571 + 0.817047i \(0.695609\pi\)
\(564\) 0 0
\(565\) −3.34557 −0.140749
\(566\) 0 0
\(567\) 4.86441i 0.204286i
\(568\) 0 0
\(569\) 11.0270 0.462278 0.231139 0.972921i \(-0.425755\pi\)
0.231139 + 0.972921i \(0.425755\pi\)
\(570\) 0 0
\(571\) 20.3493i 0.851592i 0.904819 + 0.425796i \(0.140006\pi\)
−0.904819 + 0.425796i \(0.859994\pi\)
\(572\) 0 0
\(573\) −5.55748 −0.232167
\(574\) 0 0
\(575\) −12.6421 −0.527214
\(576\) 0 0
\(577\) 20.9769i 0.873279i 0.899637 + 0.436640i \(0.143831\pi\)
−0.899637 + 0.436640i \(0.856169\pi\)
\(578\) 0 0
\(579\) 35.5344 1.47676
\(580\) 0 0
\(581\) 17.7169i 0.735020i
\(582\) 0 0
\(583\) 7.83181 0.324360
\(584\) 0 0
\(585\) 2.32255i 0.0960258i
\(586\) 0 0
\(587\) 37.6543i 1.55416i −0.629402 0.777080i \(-0.716700\pi\)
0.629402 0.777080i \(-0.283300\pi\)
\(588\) 0 0
\(589\) 27.8899i 1.14918i
\(590\) 0 0
\(591\) 7.37574i 0.303398i
\(592\) 0 0
\(593\) 11.0466i 0.453629i 0.973938 + 0.226814i \(0.0728311\pi\)
−0.973938 + 0.226814i \(0.927169\pi\)
\(594\) 0 0
\(595\) −3.10846 −0.127434
\(596\) 0 0
\(597\) 15.0695 0.616753
\(598\) 0 0
\(599\) −10.6972 −0.437075 −0.218537 0.975829i \(-0.570129\pi\)
−0.218537 + 0.975829i \(0.570129\pi\)
\(600\) 0 0
\(601\) 14.4952i 0.591272i −0.955301 0.295636i \(-0.904468\pi\)
0.955301 0.295636i \(-0.0955316\pi\)
\(602\) 0 0
\(603\) 6.21936i 0.253272i
\(604\) 0 0
\(605\) −13.1394 −0.534191
\(606\) 0 0
\(607\) 2.74552 0.111437 0.0557186 0.998447i \(-0.482255\pi\)
0.0557186 + 0.998447i \(0.482255\pi\)
\(608\) 0 0
\(609\) 3.37756i 0.136866i
\(610\) 0 0
\(611\) −22.3671 −0.904878
\(612\) 0 0
\(613\) 1.29092 0.0521396 0.0260698 0.999660i \(-0.491701\pi\)
0.0260698 + 0.999660i \(0.491701\pi\)
\(614\) 0 0
\(615\) −2.93094 11.8597i −0.118187 0.478231i
\(616\) 0 0
\(617\) 26.6193 1.07165 0.535826 0.844328i \(-0.320000\pi\)
0.535826 + 0.844328i \(0.320000\pi\)
\(618\) 0 0
\(619\) −6.50185 −0.261331 −0.130666 0.991426i \(-0.541711\pi\)
−0.130666 + 0.991426i \(0.541711\pi\)
\(620\) 0 0
\(621\) 22.6650i 0.909517i
\(622\) 0 0
\(623\) 7.88443 0.315883
\(624\) 0 0
\(625\) 0.729856 0.0291942
\(626\) 0 0
\(627\) 8.15146i 0.325538i
\(628\) 0 0
\(629\) 1.91830i 0.0764875i
\(630\) 0 0
\(631\) 14.4938 0.576988 0.288494 0.957482i \(-0.406845\pi\)
0.288494 + 0.957482i \(0.406845\pi\)
\(632\) 0 0
\(633\) 19.9818 0.794206
\(634\) 0 0
\(635\) 25.3884 1.00751
\(636\) 0 0
\(637\) 1.66501i 0.0659701i
\(638\) 0 0
\(639\) 13.5180i 0.534763i
\(640\) 0 0
\(641\) 42.0599i 1.66127i −0.556820 0.830633i \(-0.687979\pi\)
0.556820 0.830633i \(-0.312021\pi\)
\(642\) 0 0
\(643\) 36.1067i 1.42391i 0.702225 + 0.711955i \(0.252190\pi\)
−0.702225 + 0.711955i \(0.747810\pi\)
\(644\) 0 0
\(645\) 20.1019i 0.791510i
\(646\) 0 0
\(647\) −5.49156 −0.215895 −0.107948 0.994157i \(-0.534428\pi\)
−0.107948 + 0.994157i \(0.534428\pi\)
\(648\) 0 0
\(649\) 8.90037i 0.349370i
\(650\) 0 0
\(651\) 7.77491 0.304723
\(652\) 0 0
\(653\) 41.9683i 1.64235i 0.570680 + 0.821173i \(0.306680\pi\)
−0.570680 + 0.821173i \(0.693320\pi\)
\(654\) 0 0
\(655\) 4.98927 0.194947
\(656\) 0 0
\(657\) 8.68786 0.338946
\(658\) 0 0
\(659\) 14.4600i 0.563283i −0.959520 0.281641i \(-0.909121\pi\)
0.959520 0.281641i \(-0.0908788\pi\)
\(660\) 0 0
\(661\) 11.5392 0.448824 0.224412 0.974494i \(-0.427954\pi\)
0.224412 + 0.974494i \(0.427954\pi\)
\(662\) 0 0
\(663\) 5.35229i 0.207866i
\(664\) 0 0
\(665\) −6.84396 −0.265398
\(666\) 0 0
\(667\) 9.63493i 0.373066i
\(668\) 0 0
\(669\) 26.4515i 1.02267i
\(670\) 0 0
\(671\) 4.33722i 0.167437i
\(672\) 0 0
\(673\) 32.0633i 1.23595i −0.786198 0.617975i \(-0.787953\pi\)
0.786198 0.617975i \(-0.212047\pi\)
\(674\) 0 0
\(675\) 17.8466i 0.686915i
\(676\) 0 0
\(677\) 9.06923 0.348559 0.174279 0.984696i \(-0.444240\pi\)
0.174279 + 0.984696i \(0.444240\pi\)
\(678\) 0 0
\(679\) −9.62923 −0.369536
\(680\) 0 0
\(681\) 1.97593 0.0757178
\(682\) 0 0
\(683\) 2.83942i 0.108647i −0.998523 0.0543237i \(-0.982700\pi\)
0.998523 0.0543237i \(-0.0173003\pi\)
\(684\) 0 0
\(685\) 18.6053i 0.710872i
\(686\) 0 0
\(687\) −22.2000 −0.846982
\(688\) 0 0
\(689\) −11.3221 −0.431339
\(690\) 0 0
\(691\) 10.8854i 0.414098i −0.978331 0.207049i \(-0.933614\pi\)
0.978331 0.207049i \(-0.0663860\pi\)
\(692\) 0 0
\(693\) −1.18280 −0.0449307
\(694\) 0 0
\(695\) −24.4408 −0.927093
\(696\) 0 0
\(697\) 3.51566 + 14.2257i 0.133165 + 0.538838i
\(698\) 0 0
\(699\) 1.55696 0.0588895
\(700\) 0 0
\(701\) −15.5887 −0.588775 −0.294388 0.955686i \(-0.595116\pi\)
−0.294388 + 0.955686i \(0.595116\pi\)
\(702\) 0 0
\(703\) 4.22355i 0.159294i
\(704\) 0 0
\(705\) 25.6300 0.965283
\(706\) 0 0
\(707\) −3.15207 −0.118546
\(708\) 0 0
\(709\) 21.5808i 0.810485i 0.914209 + 0.405242i \(0.132813\pi\)
−0.914209 + 0.405242i \(0.867187\pi\)
\(710\) 0 0
\(711\) 1.04500i 0.0391905i
\(712\) 0 0
\(713\) 22.1789 0.830606
\(714\) 0 0
\(715\) −2.60470 −0.0974102
\(716\) 0 0
\(717\) 24.8338 0.927437
\(718\) 0 0
\(719\) 6.91985i 0.258067i 0.991640 + 0.129033i \(0.0411874\pi\)
−0.991640 + 0.129033i \(0.958813\pi\)
\(720\) 0 0
\(721\) 6.42106i 0.239133i
\(722\) 0 0
\(723\) 12.5613i 0.467158i
\(724\) 0 0
\(725\) 7.58659i 0.281759i
\(726\) 0 0
\(727\) 25.5733i 0.948462i 0.880401 + 0.474231i \(0.157274\pi\)
−0.880401 + 0.474231i \(0.842726\pi\)
\(728\) 0 0
\(729\) −28.8316 −1.06784
\(730\) 0 0
\(731\) 24.1122i 0.891820i
\(732\) 0 0
\(733\) −31.6366 −1.16852 −0.584262 0.811565i \(-0.698616\pi\)
−0.584262 + 0.811565i \(0.698616\pi\)
\(734\) 0 0
\(735\) 1.90790i 0.0703740i
\(736\) 0 0
\(737\) −6.97488 −0.256923
\(738\) 0 0
\(739\) 15.6332 0.575077 0.287539 0.957769i \(-0.407163\pi\)
0.287539 + 0.957769i \(0.407163\pi\)
\(740\) 0 0
\(741\) 11.7842i 0.432905i
\(742\) 0 0
\(743\) −33.9097 −1.24403 −0.622013 0.783007i \(-0.713685\pi\)
−0.622013 + 0.783007i \(0.713685\pi\)
\(744\) 0 0
\(745\) 10.8374i 0.397053i
\(746\) 0 0
\(747\) −18.1948 −0.665711
\(748\) 0 0
\(749\) 2.68916i 0.0982596i
\(750\) 0 0
\(751\) 27.7263i 1.01175i −0.862607 0.505874i \(-0.831170\pi\)
0.862607 0.505874i \(-0.168830\pi\)
\(752\) 0 0
\(753\) 10.1233i 0.368915i
\(754\) 0 0
\(755\) 6.15336i 0.223944i
\(756\) 0 0
\(757\) 17.9832i 0.653611i −0.945092 0.326806i \(-0.894028\pi\)
0.945092 0.326806i \(-0.105972\pi\)
\(758\) 0 0
\(759\) 6.48229 0.235292
\(760\) 0 0
\(761\) 34.5712 1.25320 0.626602 0.779340i \(-0.284445\pi\)
0.626602 + 0.779340i \(0.284445\pi\)
\(762\) 0 0
\(763\) −0.843693 −0.0305437
\(764\) 0 0
\(765\) 3.19231i 0.115418i
\(766\) 0 0
\(767\) 12.8669i 0.464597i
\(768\) 0 0
\(769\) −17.4330 −0.628649 −0.314324 0.949316i \(-0.601778\pi\)
−0.314324 + 0.949316i \(0.601778\pi\)
\(770\) 0 0
\(771\) −7.86422 −0.283223
\(772\) 0 0
\(773\) 1.09666i 0.0394441i −0.999806 0.0197220i \(-0.993722\pi\)
0.999806 0.0197220i \(-0.00627813\pi\)
\(774\) 0 0
\(775\) −17.4638 −0.627317
\(776\) 0 0
\(777\) −1.17741 −0.0422392
\(778\) 0 0
\(779\) 7.74051 + 31.3211i 0.277333 + 1.12219i
\(780\) 0 0
\(781\) 15.1601 0.542472
\(782\) 0 0
\(783\) −13.6014 −0.486073
\(784\) 0 0
\(785\) 3.45700i 0.123386i
\(786\) 0 0
\(787\) −32.3899 −1.15458 −0.577288 0.816541i \(-0.695889\pi\)
−0.577288 + 0.816541i \(0.695889\pi\)
\(788\) 0 0
\(789\) −20.3360 −0.723979
\(790\) 0 0
\(791\) 2.46309i 0.0875775i
\(792\) 0 0
\(793\) 6.27015i 0.222660i
\(794\) 0 0
\(795\) 12.9738 0.460133
\(796\) 0 0
\(797\) −35.3364 −1.25168 −0.625840 0.779951i \(-0.715244\pi\)
−0.625840 + 0.779951i \(0.715244\pi\)
\(798\) 0 0
\(799\) −30.7432 −1.08762
\(800\) 0 0
\(801\) 8.09710i 0.286097i
\(802\) 0 0
\(803\) 9.74326i 0.343832i
\(804\) 0 0
\(805\) 5.44253i 0.191824i
\(806\)