Properties

Label 1148.2.d.a.1065.3
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 1065.3
Root \(-1.02366i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.18

$q$-expansion

\(f(q)\) \(=\) \(q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +O(q^{10})\) \(q-2.57093i q^{3} -2.38500 q^{5} -1.00000i q^{7} -3.60969 q^{9} +2.88295i q^{11} +4.97483i q^{13} +6.13166i q^{15} +7.25525i q^{17} -4.84438i q^{19} -2.57093 q^{21} -1.50062 q^{23} +0.688204 q^{25} +1.56748i q^{27} +4.90169i q^{29} -5.66399 q^{31} +7.41187 q^{33} +2.38500i q^{35} +9.73403 q^{37} +12.7900 q^{39} +(4.46933 + 4.58531i) q^{41} -4.00048 q^{43} +8.60910 q^{45} -8.40651i q^{47} -1.00000 q^{49} +18.6528 q^{51} +5.39570i q^{53} -6.87582i q^{55} -12.4546 q^{57} +9.99346 q^{59} -10.3215 q^{61} +3.60969i q^{63} -11.8649i q^{65} -1.61869i q^{67} +3.85799i q^{69} +15.4780i q^{71} -1.66466 q^{73} -1.76933i q^{75} +2.88295 q^{77} +13.1796i q^{79} -6.79919 q^{81} -7.34592 q^{83} -17.3037i q^{85} +12.6019 q^{87} -6.68089i q^{89} +4.97483 q^{91} +14.5617i q^{93} +11.5538i q^{95} +3.42621i q^{97} -10.4066i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

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
\(3\) 2.57093i 1.48433i −0.670218 0.742164i \(-0.733799\pi\)
0.670218 0.742164i \(-0.266201\pi\)
\(4\) 0 0
\(5\) −2.38500 −1.06660 −0.533301 0.845925i \(-0.679049\pi\)
−0.533301 + 0.845925i \(0.679049\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.60969 −1.20323
\(10\) 0 0
\(11\) 2.88295i 0.869242i 0.900614 + 0.434621i \(0.143118\pi\)
−0.900614 + 0.434621i \(0.856882\pi\)
\(12\) 0 0
\(13\) 4.97483i 1.37977i 0.723919 + 0.689885i \(0.242339\pi\)
−0.723919 + 0.689885i \(0.757661\pi\)
\(14\) 0 0
\(15\) 6.13166i 1.58319i
\(16\) 0 0
\(17\) 7.25525i 1.75966i 0.475291 + 0.879829i \(0.342343\pi\)
−0.475291 + 0.879829i \(0.657657\pi\)
\(18\) 0 0
\(19\) 4.84438i 1.11138i −0.831390 0.555689i \(-0.812455\pi\)
0.831390 0.555689i \(-0.187545\pi\)
\(20\) 0 0
\(21\) −2.57093 −0.561023
\(22\) 0 0
\(23\) −1.50062 −0.312901 −0.156450 0.987686i \(-0.550005\pi\)
−0.156450 + 0.987686i \(0.550005\pi\)
\(24\) 0 0
\(25\) 0.688204 0.137641
\(26\) 0 0
\(27\) 1.56748i 0.301662i
\(28\) 0 0
\(29\) 4.90169i 0.910222i 0.890435 + 0.455111i \(0.150400\pi\)
−0.890435 + 0.455111i \(0.849600\pi\)
\(30\) 0 0
\(31\) −5.66399 −1.01728 −0.508641 0.860978i \(-0.669852\pi\)
−0.508641 + 0.860978i \(0.669852\pi\)
\(32\) 0 0
\(33\) 7.41187 1.29024
\(34\) 0 0
\(35\) 2.38500i 0.403138i
\(36\) 0 0
\(37\) 9.73403 1.60026 0.800132 0.599823i \(-0.204763\pi\)
0.800132 + 0.599823i \(0.204763\pi\)
\(38\) 0 0
\(39\) 12.7900 2.04803
\(40\) 0 0
\(41\) 4.46933 + 4.58531i 0.697992 + 0.716105i
\(42\) 0 0
\(43\) −4.00048 −0.610068 −0.305034 0.952341i \(-0.598668\pi\)
−0.305034 + 0.952341i \(0.598668\pi\)
\(44\) 0 0
\(45\) 8.60910 1.28337
\(46\) 0 0
\(47\) 8.40651i 1.22622i −0.789999 0.613108i \(-0.789919\pi\)
0.789999 0.613108i \(-0.210081\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 18.6528 2.61191
\(52\) 0 0
\(53\) 5.39570i 0.741157i 0.928801 + 0.370578i \(0.120840\pi\)
−0.928801 + 0.370578i \(0.879160\pi\)
\(54\) 0 0
\(55\) 6.87582i 0.927135i
\(56\) 0 0
\(57\) −12.4546 −1.64965
\(58\) 0 0
\(59\) 9.99346 1.30104 0.650519 0.759490i \(-0.274552\pi\)
0.650519 + 0.759490i \(0.274552\pi\)
\(60\) 0 0
\(61\) −10.3215 −1.32153 −0.660765 0.750593i \(-0.729768\pi\)
−0.660765 + 0.750593i \(0.729768\pi\)
\(62\) 0 0
\(63\) 3.60969i 0.454779i
\(64\) 0 0
\(65\) 11.8649i 1.47167i
\(66\) 0 0
\(67\) 1.61869i 0.197754i −0.995100 0.0988771i \(-0.968475\pi\)
0.995100 0.0988771i \(-0.0315251\pi\)
\(68\) 0 0
\(69\) 3.85799i 0.464448i
\(70\) 0 0
\(71\) 15.4780i 1.83690i 0.395533 + 0.918452i \(0.370560\pi\)
−0.395533 + 0.918452i \(0.629440\pi\)
\(72\) 0 0
\(73\) −1.66466 −0.194834 −0.0974169 0.995244i \(-0.531058\pi\)
−0.0974169 + 0.995244i \(0.531058\pi\)
\(74\) 0 0
\(75\) 1.76933i 0.204304i
\(76\) 0 0
\(77\) 2.88295 0.328542
\(78\) 0 0
\(79\) 13.1796i 1.48282i 0.671050 + 0.741412i \(0.265844\pi\)
−0.671050 + 0.741412i \(0.734156\pi\)
\(80\) 0 0
\(81\) −6.79919 −0.755466
\(82\) 0 0
\(83\) −7.34592 −0.806319 −0.403160 0.915130i \(-0.632088\pi\)
−0.403160 + 0.915130i \(0.632088\pi\)
\(84\) 0 0
\(85\) 17.3037i 1.87685i
\(86\) 0 0
\(87\) 12.6019 1.35107
\(88\) 0 0
\(89\) 6.68089i 0.708173i −0.935213 0.354086i \(-0.884792\pi\)
0.935213 0.354086i \(-0.115208\pi\)
\(90\) 0 0
\(91\) 4.97483 0.521504
\(92\) 0 0
\(93\) 14.5617i 1.50998i
\(94\) 0 0
\(95\) 11.5538i 1.18540i
\(96\) 0 0
\(97\) 3.42621i 0.347879i 0.984756 + 0.173940i \(0.0556497\pi\)
−0.984756 + 0.173940i \(0.944350\pi\)
\(98\) 0 0
\(99\) 10.4066i 1.04590i
\(100\) 0 0
\(101\) 9.43120i 0.938439i −0.883081 0.469220i \(-0.844535\pi\)
0.883081 0.469220i \(-0.155465\pi\)
\(102\) 0 0
\(103\) −6.58092 −0.648438 −0.324219 0.945982i \(-0.605101\pi\)
−0.324219 + 0.945982i \(0.605101\pi\)
\(104\) 0 0
\(105\) 6.13166 0.598389
\(106\) 0 0
\(107\) 14.2759 1.38010 0.690050 0.723761i \(-0.257588\pi\)
0.690050 + 0.723761i \(0.257588\pi\)
\(108\) 0 0
\(109\) 12.0066i 1.15002i 0.818146 + 0.575011i \(0.195002\pi\)
−0.818146 + 0.575011i \(0.804998\pi\)
\(110\) 0 0
\(111\) 25.0255i 2.37532i
\(112\) 0 0
\(113\) −12.2164 −1.14922 −0.574610 0.818428i \(-0.694846\pi\)
−0.574610 + 0.818428i \(0.694846\pi\)
\(114\) 0 0
\(115\) 3.57897 0.333741
\(116\) 0 0
\(117\) 17.9576i 1.66018i
\(118\) 0 0
\(119\) 7.25525 0.665088
\(120\) 0 0
\(121\) 2.68861 0.244419
\(122\) 0 0
\(123\) 11.7885 11.4904i 1.06294 1.03605i
\(124\) 0 0
\(125\) 10.2836 0.919794
\(126\) 0 0
\(127\) −10.9331 −0.970158 −0.485079 0.874470i \(-0.661209\pi\)
−0.485079 + 0.874470i \(0.661209\pi\)
\(128\) 0 0
\(129\) 10.2850i 0.905541i
\(130\) 0 0
\(131\) 11.6438 1.01732 0.508661 0.860967i \(-0.330141\pi\)
0.508661 + 0.860967i \(0.330141\pi\)
\(132\) 0 0
\(133\) −4.84438 −0.420061
\(134\) 0 0
\(135\) 3.73844i 0.321753i
\(136\) 0 0
\(137\) 9.83027i 0.839856i 0.907557 + 0.419928i \(0.137945\pi\)
−0.907557 + 0.419928i \(0.862055\pi\)
\(138\) 0 0
\(139\) −6.62459 −0.561890 −0.280945 0.959724i \(-0.590648\pi\)
−0.280945 + 0.959724i \(0.590648\pi\)
\(140\) 0 0
\(141\) −21.6126 −1.82011
\(142\) 0 0
\(143\) −14.3422 −1.19935
\(144\) 0 0
\(145\) 11.6905i 0.970844i
\(146\) 0 0
\(147\) 2.57093i 0.212047i
\(148\) 0 0
\(149\) 10.3049i 0.844211i 0.906547 + 0.422105i \(0.138709\pi\)
−0.906547 + 0.422105i \(0.861291\pi\)
\(150\) 0 0
\(151\) 7.00670i 0.570198i 0.958498 + 0.285099i \(0.0920264\pi\)
−0.958498 + 0.285099i \(0.907974\pi\)
\(152\) 0 0
\(153\) 26.1892i 2.11727i
\(154\) 0 0
\(155\) 13.5086 1.08504
\(156\) 0 0
\(157\) 16.0449i 1.28052i 0.768157 + 0.640262i \(0.221174\pi\)
−0.768157 + 0.640262i \(0.778826\pi\)
\(158\) 0 0
\(159\) 13.8720 1.10012
\(160\) 0 0
\(161\) 1.50062i 0.118265i
\(162\) 0 0
\(163\) −11.6231 −0.910395 −0.455198 0.890390i \(-0.650431\pi\)
−0.455198 + 0.890390i \(0.650431\pi\)
\(164\) 0 0
\(165\) −17.6773 −1.37617
\(166\) 0 0
\(167\) 14.5888i 1.12892i 0.825462 + 0.564458i \(0.190915\pi\)
−0.825462 + 0.564458i \(0.809085\pi\)
\(168\) 0 0
\(169\) −11.7489 −0.903764
\(170\) 0 0
\(171\) 17.4867i 1.33724i
\(172\) 0 0
\(173\) −15.2827 −1.16192 −0.580961 0.813931i \(-0.697323\pi\)
−0.580961 + 0.813931i \(0.697323\pi\)
\(174\) 0 0
\(175\) 0.688204i 0.0520233i
\(176\) 0 0
\(177\) 25.6925i 1.93117i
\(178\) 0 0
\(179\) 9.23350i 0.690144i 0.938576 + 0.345072i \(0.112146\pi\)
−0.938576 + 0.345072i \(0.887854\pi\)
\(180\) 0 0
\(181\) 14.0782i 1.04643i −0.852202 0.523213i \(-0.824733\pi\)
0.852202 0.523213i \(-0.175267\pi\)
\(182\) 0 0
\(183\) 26.5358i 1.96159i
\(184\) 0 0
\(185\) −23.2156 −1.70685
\(186\) 0 0
\(187\) −20.9165 −1.52957
\(188\) 0 0
\(189\) 1.56748 0.114018
\(190\) 0 0
\(191\) 5.26930i 0.381273i 0.981661 + 0.190636i \(0.0610552\pi\)
−0.981661 + 0.190636i \(0.938945\pi\)
\(192\) 0 0
\(193\) 12.3281i 0.887399i −0.896176 0.443700i \(-0.853666\pi\)
0.896176 0.443700i \(-0.146334\pi\)
\(194\) 0 0
\(195\) −30.5040 −2.18444
\(196\) 0 0
\(197\) 6.90635 0.492057 0.246029 0.969263i \(-0.420874\pi\)
0.246029 + 0.969263i \(0.420874\pi\)
\(198\) 0 0
\(199\) 14.7055i 1.04244i −0.853422 0.521221i \(-0.825477\pi\)
0.853422 0.521221i \(-0.174523\pi\)
\(200\) 0 0
\(201\) −4.16154 −0.293532
\(202\) 0 0
\(203\) 4.90169 0.344031
\(204\) 0 0
\(205\) −10.6593 10.9359i −0.744480 0.763799i
\(206\) 0 0
\(207\) 5.41678 0.376492
\(208\) 0 0
\(209\) 13.9661 0.966056
\(210\) 0 0
\(211\) 10.3928i 0.715469i 0.933823 + 0.357735i \(0.116451\pi\)
−0.933823 + 0.357735i \(0.883549\pi\)
\(212\) 0 0
\(213\) 39.7930 2.72657
\(214\) 0 0
\(215\) 9.54114 0.650700
\(216\) 0 0
\(217\) 5.66399i 0.384497i
\(218\) 0 0
\(219\) 4.27973i 0.289197i
\(220\) 0 0
\(221\) −36.0937 −2.42792
\(222\) 0 0
\(223\) 3.33749 0.223495 0.111747 0.993737i \(-0.464355\pi\)
0.111747 + 0.993737i \(0.464355\pi\)
\(224\) 0 0
\(225\) −2.48420 −0.165614
\(226\) 0 0
\(227\) 18.3494i 1.21789i −0.793213 0.608945i \(-0.791593\pi\)
0.793213 0.608945i \(-0.208407\pi\)
\(228\) 0 0
\(229\) 5.76073i 0.380680i −0.981718 0.190340i \(-0.939041\pi\)
0.981718 0.190340i \(-0.0609590\pi\)
\(230\) 0 0
\(231\) 7.41187i 0.487665i
\(232\) 0 0
\(233\) 3.97693i 0.260537i 0.991479 + 0.130269i \(0.0415840\pi\)
−0.991479 + 0.130269i \(0.958416\pi\)
\(234\) 0 0
\(235\) 20.0495i 1.30788i
\(236\) 0 0
\(237\) 33.8839 2.20100
\(238\) 0 0
\(239\) 15.3470i 0.992714i 0.868118 + 0.496357i \(0.165329\pi\)
−0.868118 + 0.496357i \(0.834671\pi\)
\(240\) 0 0
\(241\) −20.3337 −1.30981 −0.654904 0.755712i \(-0.727291\pi\)
−0.654904 + 0.755712i \(0.727291\pi\)
\(242\) 0 0
\(243\) 22.1827i 1.42302i
\(244\) 0 0
\(245\) 2.38500 0.152372
\(246\) 0 0
\(247\) 24.1000 1.53345
\(248\) 0 0
\(249\) 18.8859i 1.19684i
\(250\) 0 0
\(251\) −15.8791 −1.00228 −0.501141 0.865366i \(-0.667086\pi\)
−0.501141 + 0.865366i \(0.667086\pi\)
\(252\) 0 0
\(253\) 4.32621i 0.271986i
\(254\) 0 0
\(255\) −44.4868 −2.78587
\(256\) 0 0
\(257\) 3.03747i 0.189472i 0.995502 + 0.0947362i \(0.0302008\pi\)
−0.995502 + 0.0947362i \(0.969799\pi\)
\(258\) 0 0
\(259\) 9.73403i 0.604843i
\(260\) 0 0
\(261\) 17.6936i 1.09521i
\(262\) 0 0
\(263\) 25.2239i 1.55537i −0.628654 0.777685i \(-0.716394\pi\)
0.628654 0.777685i \(-0.283606\pi\)
\(264\) 0 0
\(265\) 12.8687i 0.790519i
\(266\) 0 0
\(267\) −17.1761 −1.05116
\(268\) 0 0
\(269\) −26.3116 −1.60425 −0.802125 0.597156i \(-0.796297\pi\)
−0.802125 + 0.597156i \(0.796297\pi\)
\(270\) 0 0
\(271\) −31.2970 −1.90116 −0.950578 0.310487i \(-0.899508\pi\)
−0.950578 + 0.310487i \(0.899508\pi\)
\(272\) 0 0
\(273\) 12.7900i 0.774083i
\(274\) 0 0
\(275\) 1.98406i 0.119643i
\(276\) 0 0
\(277\) 24.6439 1.48071 0.740355 0.672217i \(-0.234658\pi\)
0.740355 + 0.672217i \(0.234658\pi\)
\(278\) 0 0
\(279\) 20.4453 1.22403
\(280\) 0 0
\(281\) 14.3107i 0.853706i −0.904321 0.426853i \(-0.859622\pi\)
0.904321 0.426853i \(-0.140378\pi\)
\(282\) 0 0
\(283\) 33.2933 1.97908 0.989541 0.144252i \(-0.0460775\pi\)
0.989541 + 0.144252i \(0.0460775\pi\)
\(284\) 0 0
\(285\) 29.7041 1.75952
\(286\) 0 0
\(287\) 4.58531 4.46933i 0.270662 0.263816i
\(288\) 0 0
\(289\) −35.6387 −2.09639
\(290\) 0 0
\(291\) 8.80856 0.516367
\(292\) 0 0
\(293\) 22.9000i 1.33783i 0.743339 + 0.668915i \(0.233241\pi\)
−0.743339 + 0.668915i \(0.766759\pi\)
\(294\) 0 0
\(295\) −23.8344 −1.38769
\(296\) 0 0
\(297\) −4.51897 −0.262217
\(298\) 0 0
\(299\) 7.46533i 0.431731i
\(300\) 0 0
\(301\) 4.00048i 0.230584i
\(302\) 0 0
\(303\) −24.2470 −1.39295
\(304\) 0 0
\(305\) 24.6167 1.40955
\(306\) 0 0
\(307\) 9.86379 0.562956 0.281478 0.959568i \(-0.409175\pi\)
0.281478 + 0.959568i \(0.409175\pi\)
\(308\) 0 0
\(309\) 16.9191i 0.962494i
\(310\) 0 0
\(311\) 16.4799i 0.934490i 0.884128 + 0.467245i \(0.154753\pi\)
−0.884128 + 0.467245i \(0.845247\pi\)
\(312\) 0 0
\(313\) 26.4767i 1.49655i −0.663388 0.748276i \(-0.730882\pi\)
0.663388 0.748276i \(-0.269118\pi\)
\(314\) 0 0
\(315\) 8.60910i 0.485068i
\(316\) 0 0
\(317\) 8.10018i 0.454951i −0.973784 0.227476i \(-0.926953\pi\)
0.973784 0.227476i \(-0.0730472\pi\)
\(318\) 0 0
\(319\) −14.1313 −0.791202
\(320\) 0 0
\(321\) 36.7023i 2.04852i
\(322\) 0 0
\(323\) 35.1472 1.95564
\(324\) 0 0
\(325\) 3.42370i 0.189912i
\(326\) 0 0
\(327\) 30.8681 1.70701
\(328\) 0 0
\(329\) −8.40651 −0.463466
\(330\) 0 0
\(331\) 8.90873i 0.489668i −0.969565 0.244834i \(-0.921266\pi\)
0.969565 0.244834i \(-0.0787335\pi\)
\(332\) 0 0
\(333\) −35.1369 −1.92549
\(334\) 0 0
\(335\) 3.86057i 0.210925i
\(336\) 0 0
\(337\) 13.1099 0.714141 0.357071 0.934077i \(-0.383776\pi\)
0.357071 + 0.934077i \(0.383776\pi\)
\(338\) 0 0
\(339\) 31.4075i 1.70582i
\(340\) 0 0
\(341\) 16.3290i 0.884265i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 9.20129i 0.495381i
\(346\) 0 0
\(347\) 28.4940i 1.52964i −0.644246 0.764818i \(-0.722829\pi\)
0.644246 0.764818i \(-0.277171\pi\)
\(348\) 0 0
\(349\) 21.2620 1.13813 0.569065 0.822292i \(-0.307305\pi\)
0.569065 + 0.822292i \(0.307305\pi\)
\(350\) 0 0
\(351\) −7.79796 −0.416224
\(352\) 0 0
\(353\) 7.83587 0.417061 0.208531 0.978016i \(-0.433132\pi\)
0.208531 + 0.978016i \(0.433132\pi\)
\(354\) 0 0
\(355\) 36.9150i 1.95925i
\(356\) 0 0
\(357\) 18.6528i 0.987209i
\(358\) 0 0
\(359\) −19.8342 −1.04681 −0.523406 0.852084i \(-0.675339\pi\)
−0.523406 + 0.852084i \(0.675339\pi\)
\(360\) 0 0
\(361\) −4.46806 −0.235161
\(362\) 0 0
\(363\) 6.91223i 0.362798i
\(364\) 0 0
\(365\) 3.97021 0.207810
\(366\) 0 0
\(367\) −21.6274 −1.12894 −0.564470 0.825453i \(-0.690920\pi\)
−0.564470 + 0.825453i \(0.690920\pi\)
\(368\) 0 0
\(369\) −16.1329 16.5516i −0.839846 0.861640i
\(370\) 0 0
\(371\) 5.39570 0.280131
\(372\) 0 0
\(373\) 36.3537 1.88232 0.941162 0.337955i \(-0.109735\pi\)
0.941162 + 0.337955i \(0.109735\pi\)
\(374\) 0 0
\(375\) 26.4385i 1.36528i
\(376\) 0 0
\(377\) −24.3851 −1.25590
\(378\) 0 0
\(379\) −26.2551 −1.34864 −0.674318 0.738441i \(-0.735562\pi\)
−0.674318 + 0.738441i \(0.735562\pi\)
\(380\) 0 0
\(381\) 28.1083i 1.44003i
\(382\) 0 0
\(383\) 16.9081i 0.863964i 0.901882 + 0.431982i \(0.142186\pi\)
−0.901882 + 0.431982i \(0.857814\pi\)
\(384\) 0 0
\(385\) −6.87582 −0.350424
\(386\) 0 0
\(387\) 14.4405 0.734053
\(388\) 0 0
\(389\) 24.7825 1.25652 0.628261 0.778003i \(-0.283767\pi\)
0.628261 + 0.778003i \(0.283767\pi\)
\(390\) 0 0
\(391\) 10.8874i 0.550598i
\(392\) 0 0
\(393\) 29.9354i 1.51004i
\(394\) 0 0
\(395\) 31.4334i 1.58158i
\(396\) 0 0
\(397\) 6.05367i 0.303825i 0.988394 + 0.151913i \(0.0485432\pi\)
−0.988394 + 0.151913i \(0.951457\pi\)
\(398\) 0 0
\(399\) 12.4546i 0.623509i
\(400\) 0 0
\(401\) 6.92393 0.345764 0.172882 0.984943i \(-0.444692\pi\)
0.172882 + 0.984943i \(0.444692\pi\)
\(402\) 0 0
\(403\) 28.1774i 1.40362i
\(404\) 0 0
\(405\) 16.2160 0.805782
\(406\) 0 0
\(407\) 28.0627i 1.39102i
\(408\) 0 0
\(409\) −18.5030 −0.914912 −0.457456 0.889232i \(-0.651239\pi\)
−0.457456 + 0.889232i \(0.651239\pi\)
\(410\) 0 0
\(411\) 25.2730 1.24662
\(412\) 0 0
\(413\) 9.99346i 0.491746i
\(414\) 0 0
\(415\) 17.5200 0.860022
\(416\) 0 0
\(417\) 17.0314i 0.834029i
\(418\) 0 0
\(419\) 25.2270 1.23242 0.616210 0.787582i \(-0.288667\pi\)
0.616210 + 0.787582i \(0.288667\pi\)
\(420\) 0 0
\(421\) 2.83208i 0.138027i −0.997616 0.0690135i \(-0.978015\pi\)
0.997616 0.0690135i \(-0.0219852\pi\)
\(422\) 0 0
\(423\) 30.3449i 1.47542i
\(424\) 0 0
\(425\) 4.99309i 0.242201i
\(426\) 0 0
\(427\) 10.3215i 0.499491i
\(428\) 0 0
\(429\) 36.8728i 1.78023i
\(430\) 0 0
\(431\) −15.1590 −0.730183 −0.365091 0.930972i \(-0.618962\pi\)
−0.365091 + 0.930972i \(0.618962\pi\)
\(432\) 0 0
\(433\) −10.7792 −0.518014 −0.259007 0.965875i \(-0.583395\pi\)
−0.259007 + 0.965875i \(0.583395\pi\)
\(434\) 0 0
\(435\) −30.0555 −1.44105
\(436\) 0 0
\(437\) 7.26958i 0.347751i
\(438\) 0 0
\(439\) 1.11594i 0.0532608i −0.999645 0.0266304i \(-0.991522\pi\)
0.999645 0.0266304i \(-0.00847772\pi\)
\(440\) 0 0
\(441\) 3.60969 0.171890
\(442\) 0 0
\(443\) −13.8984 −0.660334 −0.330167 0.943923i \(-0.607105\pi\)
−0.330167 + 0.943923i \(0.607105\pi\)
\(444\) 0 0
\(445\) 15.9339i 0.755339i
\(446\) 0 0
\(447\) 26.4932 1.25309
\(448\) 0 0
\(449\) 0.294012 0.0138753 0.00693764 0.999976i \(-0.497792\pi\)
0.00693764 + 0.999976i \(0.497792\pi\)
\(450\) 0 0
\(451\) −13.2192 + 12.8849i −0.622468 + 0.606724i
\(452\) 0 0
\(453\) 18.0138 0.846360
\(454\) 0 0
\(455\) −11.8649 −0.556237
\(456\) 0 0
\(457\) 11.7879i 0.551416i 0.961241 + 0.275708i \(0.0889122\pi\)
−0.961241 + 0.275708i \(0.911088\pi\)
\(458\) 0 0
\(459\) −11.3725 −0.530822
\(460\) 0 0
\(461\) −11.2768 −0.525214 −0.262607 0.964903i \(-0.584582\pi\)
−0.262607 + 0.964903i \(0.584582\pi\)
\(462\) 0 0
\(463\) 28.6116i 1.32970i 0.746979 + 0.664848i \(0.231504\pi\)
−0.746979 + 0.664848i \(0.768496\pi\)
\(464\) 0 0
\(465\) 34.7297i 1.61055i
\(466\) 0 0
\(467\) 19.3804 0.896818 0.448409 0.893828i \(-0.351991\pi\)
0.448409 + 0.893828i \(0.351991\pi\)
\(468\) 0 0
\(469\) −1.61869 −0.0747441
\(470\) 0 0
\(471\) 41.2504 1.90072
\(472\) 0 0
\(473\) 11.5332i 0.530297i
\(474\) 0 0
\(475\) 3.33392i 0.152971i
\(476\) 0 0
\(477\) 19.4768i 0.891783i
\(478\) 0 0
\(479\) 14.3718i 0.656663i 0.944563 + 0.328332i \(0.106486\pi\)
−0.944563 + 0.328332i \(0.893514\pi\)
\(480\) 0 0
\(481\) 48.4251i 2.20800i
\(482\) 0 0
\(483\) 3.85799 0.175545
\(484\) 0 0
\(485\) 8.17150i 0.371049i
\(486\) 0 0
\(487\) 28.6259 1.29716 0.648581 0.761146i \(-0.275363\pi\)
0.648581 + 0.761146i \(0.275363\pi\)
\(488\) 0 0
\(489\) 29.8823i 1.35133i
\(490\) 0 0
\(491\) 15.6508 0.706312 0.353156 0.935564i \(-0.385108\pi\)
0.353156 + 0.935564i \(0.385108\pi\)
\(492\) 0 0
\(493\) −35.5630 −1.60168
\(494\) 0 0
\(495\) 24.8196i 1.11556i
\(496\) 0 0
\(497\) 15.4780 0.694284
\(498\) 0 0
\(499\) 24.6188i 1.10209i −0.834477 0.551043i \(-0.814230\pi\)
0.834477 0.551043i \(-0.185770\pi\)
\(500\) 0 0
\(501\) 37.5069 1.67568
\(502\) 0 0
\(503\) 35.4258i 1.57956i 0.613392 + 0.789779i \(0.289805\pi\)
−0.613392 + 0.789779i \(0.710195\pi\)
\(504\) 0 0
\(505\) 22.4934i 1.00094i
\(506\) 0 0
\(507\) 30.2057i 1.34148i
\(508\) 0 0
\(509\) 16.0385i 0.710894i 0.934697 + 0.355447i \(0.115671\pi\)
−0.934697 + 0.355447i \(0.884329\pi\)
\(510\) 0 0
\(511\) 1.66466i 0.0736403i
\(512\) 0 0
\(513\) 7.59348 0.335261
\(514\) 0 0
\(515\) 15.6955 0.691625
\(516\) 0 0
\(517\) 24.2355 1.06588
\(518\) 0 0
\(519\) 39.2908i 1.72467i
\(520\) 0 0
\(521\) 29.4260i 1.28918i −0.764529 0.644589i \(-0.777028\pi\)
0.764529 0.644589i \(-0.222972\pi\)
\(522\) 0 0
\(523\) 21.5468 0.942178 0.471089 0.882086i \(-0.343861\pi\)
0.471089 + 0.882086i \(0.343861\pi\)
\(524\) 0 0
\(525\) −1.76933 −0.0772197
\(526\) 0 0
\(527\) 41.0937i 1.79007i
\(528\) 0 0
\(529\) −20.7481 −0.902093
\(530\) 0 0
\(531\) −36.0733 −1.56545
\(532\) 0 0
\(533\) −22.8111 + 22.2342i −0.988060 + 0.963069i
\(534\) 0 0
\(535\) −34.0479 −1.47202
\(536\) 0 0
\(537\) 23.7387 1.02440
\(538\) 0 0
\(539\) 2.88295i 0.124177i
\(540\) 0 0
\(541\) 15.6475 0.672740 0.336370 0.941730i \(-0.390801\pi\)
0.336370 + 0.941730i \(0.390801\pi\)
\(542\) 0 0
\(543\) −36.1942 −1.55324
\(544\) 0 0
\(545\) 28.6356i 1.22662i
\(546\) 0 0
\(547\) 33.4259i 1.42919i −0.699538 0.714595i \(-0.746611\pi\)
0.699538 0.714595i \(-0.253389\pi\)
\(548\) 0 0
\(549\) 37.2574 1.59011
\(550\) 0 0
\(551\) 23.7457 1.01160
\(552\) 0 0
\(553\) 13.1796 0.560455
\(554\) 0 0
\(555\) 59.6858i 2.53352i
\(556\) 0 0
\(557\) 1.78645i 0.0756942i −0.999284 0.0378471i \(-0.987950\pi\)
0.999284 0.0378471i \(-0.0120500\pi\)
\(558\) 0 0
\(559\) 19.9017i 0.841753i
\(560\) 0 0
\(561\) 53.7750i 2.27038i
\(562\) 0 0
\(563\) 35.2787i 1.48682i 0.668836 + 0.743410i \(0.266793\pi\)
−0.668836 + 0.743410i \(0.733207\pi\)
\(564\) 0 0
\(565\) 29.1360 1.22576
\(566\) 0 0
\(567\) 6.79919i 0.285539i
\(568\) 0 0
\(569\) −30.0684 −1.26053 −0.630267 0.776379i \(-0.717054\pi\)
−0.630267 + 0.776379i \(0.717054\pi\)
\(570\) 0 0
\(571\) 22.0288i 0.921876i −0.887432 0.460938i \(-0.847513\pi\)
0.887432 0.460938i \(-0.152487\pi\)
\(572\) 0 0
\(573\) 13.5470 0.565934
\(574\) 0 0
\(575\) −1.03273 −0.0430679
\(576\) 0 0
\(577\) 32.9259i 1.37072i −0.728202 0.685362i \(-0.759644\pi\)
0.728202 0.685362i \(-0.240356\pi\)
\(578\) 0 0
\(579\) −31.6948 −1.31719
\(580\) 0 0
\(581\) 7.34592i 0.304760i
\(582\) 0 0
\(583\) −15.5555 −0.644244
\(584\) 0 0
\(585\) 42.8288i 1.77075i
\(586\) 0 0
\(587\) 4.53262i 0.187081i 0.995615 + 0.0935407i \(0.0298185\pi\)
−0.995615 + 0.0935407i \(0.970181\pi\)
\(588\) 0 0
\(589\) 27.4385i 1.13059i
\(590\) 0 0
\(591\) 17.7558i 0.730375i
\(592\) 0 0
\(593\) 18.3442i 0.753305i −0.926355 0.376653i \(-0.877075\pi\)
0.926355 0.376653i \(-0.122925\pi\)
\(594\) 0 0
\(595\) −17.3037 −0.709384
\(596\) 0 0
\(597\) −37.8068 −1.54733
\(598\) 0 0
\(599\) −35.8389 −1.46434 −0.732168 0.681124i \(-0.761491\pi\)
−0.732168 + 0.681124i \(0.761491\pi\)
\(600\) 0 0
\(601\) 36.9317i 1.50648i 0.657748 + 0.753238i \(0.271509\pi\)
−0.657748 + 0.753238i \(0.728491\pi\)
\(602\) 0 0
\(603\) 5.84297i 0.237944i
\(604\) 0 0
\(605\) −6.41232 −0.260698
\(606\) 0 0
\(607\) 30.6081 1.24235 0.621173 0.783674i \(-0.286656\pi\)
0.621173 + 0.783674i \(0.286656\pi\)
\(608\) 0 0
\(609\) 12.6019i 0.510656i
\(610\) 0 0
\(611\) 41.8210 1.69189
\(612\) 0 0
\(613\) 6.21870 0.251171 0.125586 0.992083i \(-0.459919\pi\)
0.125586 + 0.992083i \(0.459919\pi\)
\(614\) 0 0
\(615\) −28.1156 + 27.4044i −1.13373 + 1.10505i
\(616\) 0 0
\(617\) 11.4734 0.461900 0.230950 0.972966i \(-0.425817\pi\)
0.230950 + 0.972966i \(0.425817\pi\)
\(618\) 0 0
\(619\) 4.62250 0.185794 0.0928970 0.995676i \(-0.470387\pi\)
0.0928970 + 0.995676i \(0.470387\pi\)
\(620\) 0 0
\(621\) 2.35219i 0.0943903i
\(622\) 0 0
\(623\) −6.68089 −0.267664
\(624\) 0 0
\(625\) −27.9674 −1.11870
\(626\) 0 0
\(627\) 35.9059i 1.43394i
\(628\) 0 0
\(629\) 70.6229i 2.81592i
\(630\) 0 0
\(631\) −20.3436 −0.809866 −0.404933 0.914346i \(-0.632705\pi\)
−0.404933 + 0.914346i \(0.632705\pi\)
\(632\) 0 0
\(633\) 26.7192 1.06199
\(634\) 0 0
\(635\) 26.0754 1.03477
\(636\) 0 0
\(637\) 4.97483i 0.197110i
\(638\) 0 0
\(639\) 55.8709i 2.21022i
\(640\) 0 0
\(641\) 16.3439i 0.645545i −0.946477 0.322772i \(-0.895385\pi\)
0.946477 0.322772i \(-0.104615\pi\)
\(642\) 0 0
\(643\) 25.1933i 0.993525i −0.867887 0.496762i \(-0.834522\pi\)
0.867887 0.496762i \(-0.165478\pi\)
\(644\) 0 0
\(645\) 24.5296i 0.965853i
\(646\) 0 0
\(647\) 4.09131 0.160846 0.0804231 0.996761i \(-0.474373\pi\)
0.0804231 + 0.996761i \(0.474373\pi\)
\(648\) 0 0
\(649\) 28.8106i 1.13092i
\(650\) 0 0
\(651\) 14.5617 0.570720
\(652\) 0 0
\(653\) 8.55620i 0.334830i −0.985887 0.167415i \(-0.946458\pi\)
0.985887 0.167415i \(-0.0535419\pi\)
\(654\) 0 0
\(655\) −27.7704 −1.08508
\(656\) 0 0
\(657\) 6.00892 0.234430
\(658\) 0 0
\(659\) 12.6234i 0.491739i 0.969303 + 0.245869i \(0.0790734\pi\)
−0.969303 + 0.245869i \(0.920927\pi\)
\(660\) 0 0
\(661\) −24.1049 −0.937572 −0.468786 0.883312i \(-0.655308\pi\)
−0.468786 + 0.883312i \(0.655308\pi\)
\(662\) 0 0
\(663\) 92.7943i 3.60383i
\(664\) 0 0
\(665\) 11.5538 0.448038
\(666\) 0 0
\(667\) 7.35558i 0.284809i
\(668\) 0 0
\(669\) 8.58045i 0.331739i
\(670\) 0 0
\(671\) 29.7563i 1.14873i
\(672\) 0 0
\(673\) 21.7532i 0.838524i 0.907865 + 0.419262i \(0.137711\pi\)
−0.907865 + 0.419262i \(0.862289\pi\)
\(674\) 0 0
\(675\) 1.07875i 0.0415210i
\(676\) 0 0
\(677\) −3.31433 −0.127380 −0.0636900 0.997970i \(-0.520287\pi\)
−0.0636900 + 0.997970i \(0.520287\pi\)
\(678\) 0 0
\(679\) 3.42621 0.131486
\(680\) 0 0
\(681\) −47.1749 −1.80775
\(682\) 0 0
\(683\) 9.68245i 0.370489i 0.982692 + 0.185244i \(0.0593077\pi\)
−0.982692 + 0.185244i \(0.940692\pi\)
\(684\) 0 0
\(685\) 23.4451i 0.895793i
\(686\) 0 0
\(687\) −14.8105 −0.565054
\(688\) 0 0
\(689\) −26.8427 −1.02263
\(690\) 0 0
\(691\) 11.4847i 0.436900i 0.975848 + 0.218450i \(0.0701000\pi\)
−0.975848 + 0.218450i \(0.929900\pi\)
\(692\) 0 0
\(693\) −10.4066 −0.395313
\(694\) 0 0
\(695\) 15.7996 0.599313
\(696\) 0 0
\(697\) −33.2676 + 32.4261i −1.26010 + 1.22823i
\(698\) 0 0
\(699\) 10.2244 0.386723
\(700\) 0 0
\(701\) 9.74836 0.368191 0.184095 0.982908i \(-0.441064\pi\)
0.184095 + 0.982908i \(0.441064\pi\)
\(702\) 0 0
\(703\) 47.1554i 1.77850i
\(704\) 0 0
\(705\) 51.5459 1.94133
\(706\) 0 0
\(707\) −9.43120 −0.354697
\(708\) 0 0
\(709\) 32.8261i 1.23281i −0.787430 0.616404i \(-0.788589\pi\)
0.787430 0.616404i \(-0.211411\pi\)
\(710\) 0 0
\(711\) 47.5744i 1.78418i
\(712\) 0 0
\(713\) 8.49950 0.318309
\(714\) 0 0
\(715\) 34.2060 1.27923
\(716\) 0 0
\(717\) 39.4561 1.47351
\(718\) 0 0
\(719\) 15.4578i 0.576480i −0.957558 0.288240i \(-0.906930\pi\)
0.957558 0.288240i \(-0.0930701\pi\)
\(720\) 0 0
\(721\) 6.58092i 0.245086i
\(722\) 0 0
\(723\) 52.2766i 1.94419i
\(724\) 0 0
\(725\) 3.37336i 0.125284i
\(726\) 0 0
\(727\) 37.8376i 1.40332i −0.712512 0.701660i \(-0.752443\pi\)
0.712512 0.701660i \(-0.247557\pi\)
\(728\) 0 0
\(729\) 36.6327 1.35677
\(730\) 0 0
\(731\) 29.0245i 1.07351i
\(732\) 0 0
\(733\) 30.5423 1.12811 0.564053 0.825738i \(-0.309241\pi\)
0.564053 + 0.825738i \(0.309241\pi\)
\(734\) 0 0
\(735\) 6.13166i 0.226170i
\(736\) 0 0
\(737\) 4.66660 0.171896
\(738\) 0 0
\(739\) 20.7293 0.762539 0.381269 0.924464i \(-0.375487\pi\)
0.381269 + 0.924464i \(0.375487\pi\)
\(740\) 0 0
\(741\) 61.9594i 2.27614i
\(742\) 0 0
\(743\) 18.0275 0.661364 0.330682 0.943742i \(-0.392721\pi\)
0.330682 + 0.943742i \(0.392721\pi\)
\(744\) 0 0
\(745\) 24.5772i 0.900437i
\(746\) 0 0
\(747\) 26.5165 0.970189
\(748\) 0 0
\(749\) 14.2759i 0.521629i
\(750\) 0 0
\(751\) 51.8386i 1.89162i −0.324726 0.945808i \(-0.605272\pi\)
0.324726 0.945808i \(-0.394728\pi\)
\(752\) 0 0
\(753\) 40.8242i 1.48771i
\(754\) 0 0
\(755\) 16.7110i 0.608174i
\(756\) 0 0
\(757\) 23.7894i 0.864641i 0.901720 + 0.432321i \(0.142305\pi\)
−0.901720 + 0.432321i \(0.857695\pi\)
\(758\) 0 0
\(759\) −11.1224 −0.403717
\(760\) 0 0
\(761\) −17.1518 −0.621751 −0.310875 0.950451i \(-0.600622\pi\)
−0.310875 + 0.950451i \(0.600622\pi\)
\(762\) 0 0
\(763\) 12.0066 0.434667
\(764\) 0 0
\(765\) 62.4612i 2.25829i
\(766\) 0 0
\(767\) 49.7158i 1.79513i
\(768\) 0 0
\(769\) 16.7004 0.602233 0.301117 0.953587i \(-0.402641\pi\)
0.301117 + 0.953587i \(0.402641\pi\)
\(770\) 0 0
\(771\) 7.80914 0.281239
\(772\) 0 0
\(773\) 35.9480i 1.29296i 0.762931 + 0.646480i \(0.223760\pi\)
−0.762931 + 0.646480i \(0.776240\pi\)
\(774\) 0 0
\(775\) −3.89798 −0.140020
\(776\) 0 0
\(777\) −25.0255 −0.897786
\(778\) 0 0
\(779\) 22.2130 21.6512i 0.795863 0.775733i
\(780\) 0 0
\(781\) −44.6223 −1.59671
\(782\) 0 0
\(783\) −7.68332 −0.274579
\(784\) 0 0
\(785\) 38.2670i 1.36581i
\(786\) 0 0
\(787\) 9.36484 0.333821 0.166910 0.985972i \(-0.446621\pi\)
0.166910 + 0.985972i \(0.446621\pi\)
\(788\) 0 0
\(789\) −64.8488 −2.30868
\(790\) 0 0
\(791\) 12.2164i 0.434364i
\(792\) 0 0
\(793\) 51.3476i 1.82341i
\(794\) 0 0
\(795\) −33.0846 −1.17339
\(796\) 0 0
\(797\) 28.4440 1.00754 0.503769 0.863839i \(-0.331946\pi\)
0.503769 + 0.863839i \(0.331946\pi\)
\(798\) 0 0
\(799\) 60.9914 2.15772
\(800\) 0 0
\(801\) 24.1160i 0.852096i
\(802\) 0 0
\(803\) 4.79913i 0.169358i
\(804\) 0 0
\(805\) 3.57897i 0.126142i
\(806\) 0 0