Properties

Label 1089.3.b.i.485.3
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.3
Root \(-2.99491i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.i.485.14

$q$-expansion

\(f(q)\) \(=\) \(q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +O(q^{10})\) \(q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +4.82698 q^{10} +7.56350 q^{13} +9.97250i q^{14} -11.1821 q^{16} -28.3526i q^{17} -26.0667 q^{19} -8.00948i q^{20} +19.5656i q^{23} +22.4023 q^{25} -22.6520i q^{26} +16.5475 q^{28} +2.15288i q^{29} -9.22840 q^{31} +45.1036i q^{32} -84.9135 q^{34} -5.36675i q^{35} -67.5342 q^{37} +78.0674i q^{38} -4.67977 q^{40} +29.0597i q^{41} +0.719802 q^{43} +58.5973 q^{46} +24.5249i q^{47} -37.9123 q^{49} -67.0930i q^{50} -37.5868 q^{52} +5.79176i q^{53} -9.66837i q^{56} +6.44768 q^{58} +73.9896i q^{59} +72.1664 q^{61} +27.6383i q^{62} +90.3531 q^{64} +12.1903i q^{65} -79.8310 q^{67} +140.898i q^{68} -16.0730 q^{70} +107.780i q^{71} -90.3799 q^{73} +202.259i q^{74} +129.538 q^{76} +114.802 q^{79} -18.0224i q^{80} +87.0314 q^{82} +125.752i q^{83} +45.6966 q^{85} -2.15574i q^{86} -83.3041i q^{89} -25.1850 q^{91} -97.2313i q^{92} +73.4499 q^{94} -42.0123i q^{95} -78.0275 q^{97} +113.544i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 8 q^{7} + O(q^{10}) \) \( 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.99491i − 1.49746i −0.662877 0.748728i \(-0.730665\pi\)
0.662877 0.748728i \(-0.269335\pi\)
\(3\) 0 0
\(4\) −4.96950 −1.24238
\(5\) 1.61173i 0.322345i 0.986926 + 0.161173i \(0.0515276\pi\)
−0.986926 + 0.161173i \(0.948472\pi\)
\(6\) 0 0
\(7\) −3.32981 −0.475688 −0.237844 0.971303i \(-0.576441\pi\)
−0.237844 + 0.971303i \(0.576441\pi\)
\(8\) 2.90358i 0.362947i
\(9\) 0 0
\(10\) 4.82698 0.482698
\(11\) 0 0
\(12\) 0 0
\(13\) 7.56350 0.581807 0.290904 0.956752i \(-0.406044\pi\)
0.290904 + 0.956752i \(0.406044\pi\)
\(14\) 9.97250i 0.712322i
\(15\) 0 0
\(16\) −11.1821 −0.698878
\(17\) − 28.3526i − 1.66780i −0.551917 0.833899i \(-0.686104\pi\)
0.551917 0.833899i \(-0.313896\pi\)
\(18\) 0 0
\(19\) −26.0667 −1.37193 −0.685965 0.727635i \(-0.740620\pi\)
−0.685965 + 0.727635i \(0.740620\pi\)
\(20\) − 8.00948i − 0.400474i
\(21\) 0 0
\(22\) 0 0
\(23\) 19.5656i 0.850679i 0.905034 + 0.425339i \(0.139845\pi\)
−0.905034 + 0.425339i \(0.860155\pi\)
\(24\) 0 0
\(25\) 22.4023 0.896093
\(26\) − 22.6520i − 0.871231i
\(27\) 0 0
\(28\) 16.5475 0.590983
\(29\) 2.15288i 0.0742372i 0.999311 + 0.0371186i \(0.0118179\pi\)
−0.999311 + 0.0371186i \(0.988182\pi\)
\(30\) 0 0
\(31\) −9.22840 −0.297690 −0.148845 0.988861i \(-0.547556\pi\)
−0.148845 + 0.988861i \(0.547556\pi\)
\(32\) 45.1036i 1.40949i
\(33\) 0 0
\(34\) −84.9135 −2.49745
\(35\) − 5.36675i − 0.153336i
\(36\) 0 0
\(37\) −67.5342 −1.82525 −0.912624 0.408800i \(-0.865948\pi\)
−0.912624 + 0.408800i \(0.865948\pi\)
\(38\) 78.0674i 2.05440i
\(39\) 0 0
\(40\) −4.67977 −0.116994
\(41\) 29.0597i 0.708774i 0.935099 + 0.354387i \(0.115310\pi\)
−0.935099 + 0.354387i \(0.884690\pi\)
\(42\) 0 0
\(43\) 0.719802 0.0167396 0.00836979 0.999965i \(-0.497336\pi\)
0.00836979 + 0.999965i \(0.497336\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 58.5973 1.27385
\(47\) 24.5249i 0.521806i 0.965365 + 0.260903i \(0.0840203\pi\)
−0.965365 + 0.260903i \(0.915980\pi\)
\(48\) 0 0
\(49\) −37.9123 −0.773721
\(50\) − 67.0930i − 1.34186i
\(51\) 0 0
\(52\) −37.5868 −0.722823
\(53\) 5.79176i 0.109278i 0.998506 + 0.0546392i \(0.0174009\pi\)
−0.998506 + 0.0546392i \(0.982599\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 9.66837i − 0.172650i
\(57\) 0 0
\(58\) 6.44768 0.111167
\(59\) 73.9896i 1.25406i 0.778994 + 0.627031i \(0.215730\pi\)
−0.778994 + 0.627031i \(0.784270\pi\)
\(60\) 0 0
\(61\) 72.1664 1.18306 0.591528 0.806284i \(-0.298525\pi\)
0.591528 + 0.806284i \(0.298525\pi\)
\(62\) 27.6383i 0.445778i
\(63\) 0 0
\(64\) 90.3531 1.41177
\(65\) 12.1903i 0.187543i
\(66\) 0 0
\(67\) −79.8310 −1.19151 −0.595754 0.803167i \(-0.703146\pi\)
−0.595754 + 0.803167i \(0.703146\pi\)
\(68\) 140.898i 2.07203i
\(69\) 0 0
\(70\) −16.0730 −0.229614
\(71\) 107.780i 1.51803i 0.651072 + 0.759016i \(0.274319\pi\)
−0.651072 + 0.759016i \(0.725681\pi\)
\(72\) 0 0
\(73\) −90.3799 −1.23808 −0.619041 0.785359i \(-0.712478\pi\)
−0.619041 + 0.785359i \(0.712478\pi\)
\(74\) 202.259i 2.73323i
\(75\) 0 0
\(76\) 129.538 1.70445
\(77\) 0 0
\(78\) 0 0
\(79\) 114.802 1.45319 0.726595 0.687066i \(-0.241102\pi\)
0.726595 + 0.687066i \(0.241102\pi\)
\(80\) − 18.0224i − 0.225280i
\(81\) 0 0
\(82\) 87.0314 1.06136
\(83\) 125.752i 1.51509i 0.652784 + 0.757544i \(0.273601\pi\)
−0.652784 + 0.757544i \(0.726399\pi\)
\(84\) 0 0
\(85\) 45.6966 0.537607
\(86\) − 2.15574i − 0.0250668i
\(87\) 0 0
\(88\) 0 0
\(89\) − 83.3041i − 0.936001i −0.883728 0.468000i \(-0.844975\pi\)
0.883728 0.468000i \(-0.155025\pi\)
\(90\) 0 0
\(91\) −25.1850 −0.276759
\(92\) − 97.2313i − 1.05686i
\(93\) 0 0
\(94\) 73.4499 0.781382
\(95\) − 42.0123i − 0.442235i
\(96\) 0 0
\(97\) −78.0275 −0.804407 −0.402203 0.915550i \(-0.631756\pi\)
−0.402203 + 0.915550i \(0.631756\pi\)
\(98\) 113.544i 1.15861i
\(99\) 0 0
\(100\) −111.328 −1.11328
\(101\) 38.9858i 0.385998i 0.981199 + 0.192999i \(0.0618215\pi\)
−0.981199 + 0.192999i \(0.938179\pi\)
\(102\) 0 0
\(103\) 30.1474 0.292694 0.146347 0.989233i \(-0.453248\pi\)
0.146347 + 0.989233i \(0.453248\pi\)
\(104\) 21.9612i 0.211165i
\(105\) 0 0
\(106\) 17.3458 0.163640
\(107\) − 49.7544i − 0.464994i −0.972597 0.232497i \(-0.925310\pi\)
0.972597 0.232497i \(-0.0746897\pi\)
\(108\) 0 0
\(109\) −149.823 −1.37453 −0.687263 0.726409i \(-0.741188\pi\)
−0.687263 + 0.726409i \(0.741188\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 37.2342 0.332448
\(113\) − 127.469i − 1.12805i −0.825759 0.564024i \(-0.809253\pi\)
0.825759 0.564024i \(-0.190747\pi\)
\(114\) 0 0
\(115\) −31.5344 −0.274212
\(116\) − 10.6987i − 0.0922305i
\(117\) 0 0
\(118\) 221.593 1.87790
\(119\) 94.4088i 0.793351i
\(120\) 0 0
\(121\) 0 0
\(122\) − 216.132i − 1.77157i
\(123\) 0 0
\(124\) 45.8606 0.369843
\(125\) 76.3996i 0.611197i
\(126\) 0 0
\(127\) −237.975 −1.87382 −0.936910 0.349572i \(-0.886327\pi\)
−0.936910 + 0.349572i \(0.886327\pi\)
\(128\) − 90.1853i − 0.704573i
\(129\) 0 0
\(130\) 36.5089 0.280837
\(131\) 148.535i 1.13386i 0.823767 + 0.566929i \(0.191869\pi\)
−0.823767 + 0.566929i \(0.808131\pi\)
\(132\) 0 0
\(133\) 86.7972 0.652610
\(134\) 239.087i 1.78423i
\(135\) 0 0
\(136\) 82.3239 0.605322
\(137\) 20.1586i 0.147143i 0.997290 + 0.0735716i \(0.0234397\pi\)
−0.997290 + 0.0735716i \(0.976560\pi\)
\(138\) 0 0
\(139\) 148.878 1.07106 0.535532 0.844515i \(-0.320111\pi\)
0.535532 + 0.844515i \(0.320111\pi\)
\(140\) 26.6701i 0.190501i
\(141\) 0 0
\(142\) 322.792 2.27319
\(143\) 0 0
\(144\) 0 0
\(145\) −3.46985 −0.0239300
\(146\) 270.680i 1.85397i
\(147\) 0 0
\(148\) 335.611 2.26764
\(149\) − 44.8984i − 0.301332i −0.988585 0.150666i \(-0.951858\pi\)
0.988585 0.150666i \(-0.0481418\pi\)
\(150\) 0 0
\(151\) 46.5427 0.308230 0.154115 0.988053i \(-0.450747\pi\)
0.154115 + 0.988053i \(0.450747\pi\)
\(152\) − 75.6866i − 0.497938i
\(153\) 0 0
\(154\) 0 0
\(155\) − 14.8737i − 0.0959591i
\(156\) 0 0
\(157\) 18.1734 0.115754 0.0578769 0.998324i \(-0.481567\pi\)
0.0578769 + 0.998324i \(0.481567\pi\)
\(158\) − 343.822i − 2.17609i
\(159\) 0 0
\(160\) −72.6947 −0.454342
\(161\) − 65.1498i − 0.404657i
\(162\) 0 0
\(163\) −139.162 −0.853753 −0.426876 0.904310i \(-0.640386\pi\)
−0.426876 + 0.904310i \(0.640386\pi\)
\(164\) − 144.412i − 0.880564i
\(165\) 0 0
\(166\) 376.617 2.26878
\(167\) − 145.214i − 0.869544i −0.900541 0.434772i \(-0.856829\pi\)
0.900541 0.434772i \(-0.143171\pi\)
\(168\) 0 0
\(169\) −111.794 −0.661500
\(170\) − 136.857i − 0.805043i
\(171\) 0 0
\(172\) −3.57706 −0.0207968
\(173\) − 290.416i − 1.67871i −0.543586 0.839353i \(-0.682934\pi\)
0.543586 0.839353i \(-0.317066\pi\)
\(174\) 0 0
\(175\) −74.5956 −0.426261
\(176\) 0 0
\(177\) 0 0
\(178\) −249.488 −1.40162
\(179\) 146.411i 0.817940i 0.912548 + 0.408970i \(0.134112\pi\)
−0.912548 + 0.408970i \(0.865888\pi\)
\(180\) 0 0
\(181\) 223.461 1.23459 0.617295 0.786732i \(-0.288228\pi\)
0.617295 + 0.786732i \(0.288228\pi\)
\(182\) 75.4270i 0.414434i
\(183\) 0 0
\(184\) −56.8102 −0.308751
\(185\) − 108.847i − 0.588361i
\(186\) 0 0
\(187\) 0 0
\(188\) − 121.876i − 0.648279i
\(189\) 0 0
\(190\) −125.823 −0.662228
\(191\) − 23.2178i − 0.121559i −0.998151 0.0607795i \(-0.980641\pi\)
0.998151 0.0607795i \(-0.0193586\pi\)
\(192\) 0 0
\(193\) −125.437 −0.649934 −0.324967 0.945725i \(-0.605353\pi\)
−0.324967 + 0.945725i \(0.605353\pi\)
\(194\) 233.685i 1.20456i
\(195\) 0 0
\(196\) 188.405 0.961252
\(197\) − 12.3847i − 0.0628665i −0.999506 0.0314333i \(-0.989993\pi\)
0.999506 0.0314333i \(-0.0100072\pi\)
\(198\) 0 0
\(199\) 112.611 0.565885 0.282943 0.959137i \(-0.408689\pi\)
0.282943 + 0.959137i \(0.408689\pi\)
\(200\) 65.0469i 0.325235i
\(201\) 0 0
\(202\) 116.759 0.578016
\(203\) − 7.16868i − 0.0353137i
\(204\) 0 0
\(205\) −46.8364 −0.228470
\(206\) − 90.2890i − 0.438296i
\(207\) 0 0
\(208\) −84.5754 −0.406613
\(209\) 0 0
\(210\) 0 0
\(211\) 153.910 0.729433 0.364716 0.931119i \(-0.381166\pi\)
0.364716 + 0.931119i \(0.381166\pi\)
\(212\) − 28.7822i − 0.135765i
\(213\) 0 0
\(214\) −149.010 −0.696309
\(215\) 1.16012i 0.00539593i
\(216\) 0 0
\(217\) 30.7289 0.141608
\(218\) 448.708i 2.05829i
\(219\) 0 0
\(220\) 0 0
\(221\) − 214.445i − 0.970337i
\(222\) 0 0
\(223\) 282.384 1.26630 0.633148 0.774031i \(-0.281763\pi\)
0.633148 + 0.774031i \(0.281763\pi\)
\(224\) − 150.187i − 0.670476i
\(225\) 0 0
\(226\) −381.760 −1.68920
\(227\) − 202.865i − 0.893677i −0.894615 0.446839i \(-0.852550\pi\)
0.894615 0.446839i \(-0.147450\pi\)
\(228\) 0 0
\(229\) −419.120 −1.83022 −0.915108 0.403208i \(-0.867895\pi\)
−0.915108 + 0.403208i \(0.867895\pi\)
\(230\) 94.4428i 0.410621i
\(231\) 0 0
\(232\) −6.25105 −0.0269442
\(233\) − 111.121i − 0.476916i −0.971153 0.238458i \(-0.923358\pi\)
0.971153 0.238458i \(-0.0766420\pi\)
\(234\) 0 0
\(235\) −39.5274 −0.168202
\(236\) − 367.692i − 1.55802i
\(237\) 0 0
\(238\) 282.746 1.18801
\(239\) − 314.056i − 1.31404i −0.753872 0.657021i \(-0.771816\pi\)
0.753872 0.657021i \(-0.228184\pi\)
\(240\) 0 0
\(241\) −104.293 −0.432753 −0.216376 0.976310i \(-0.569424\pi\)
−0.216376 + 0.976310i \(0.569424\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −358.631 −1.46980
\(245\) − 61.1043i − 0.249405i
\(246\) 0 0
\(247\) −197.155 −0.798199
\(248\) − 26.7954i − 0.108046i
\(249\) 0 0
\(250\) 228.810 0.915241
\(251\) − 333.537i − 1.32883i −0.747363 0.664416i \(-0.768680\pi\)
0.747363 0.664416i \(-0.231320\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 712.714i 2.80596i
\(255\) 0 0
\(256\) 91.3152 0.356700
\(257\) 306.989i 1.19451i 0.802052 + 0.597254i \(0.203742\pi\)
−0.802052 + 0.597254i \(0.796258\pi\)
\(258\) 0 0
\(259\) 224.876 0.868248
\(260\) − 60.5797i − 0.232999i
\(261\) 0 0
\(262\) 444.851 1.69790
\(263\) − 249.060i − 0.946997i −0.880794 0.473499i \(-0.842991\pi\)
0.880794 0.473499i \(-0.157009\pi\)
\(264\) 0 0
\(265\) −9.33473 −0.0352254
\(266\) − 259.950i − 0.977255i
\(267\) 0 0
\(268\) 396.720 1.48030
\(269\) 196.927i 0.732071i 0.930601 + 0.366035i \(0.119285\pi\)
−0.930601 + 0.366035i \(0.880715\pi\)
\(270\) 0 0
\(271\) −463.464 −1.71020 −0.855099 0.518465i \(-0.826504\pi\)
−0.855099 + 0.518465i \(0.826504\pi\)
\(272\) 317.040i 1.16559i
\(273\) 0 0
\(274\) 60.3733 0.220340
\(275\) 0 0
\(276\) 0 0
\(277\) −23.4550 −0.0846751 −0.0423376 0.999103i \(-0.513480\pi\)
−0.0423376 + 0.999103i \(0.513480\pi\)
\(278\) − 445.876i − 1.60387i
\(279\) 0 0
\(280\) 15.5828 0.0556528
\(281\) 263.447i 0.937534i 0.883322 + 0.468767i \(0.155302\pi\)
−0.883322 + 0.468767i \(0.844698\pi\)
\(282\) 0 0
\(283\) −35.8237 −0.126585 −0.0632927 0.997995i \(-0.520160\pi\)
−0.0632927 + 0.997995i \(0.520160\pi\)
\(284\) − 535.614i − 1.88597i
\(285\) 0 0
\(286\) 0 0
\(287\) − 96.7636i − 0.337155i
\(288\) 0 0
\(289\) −514.868 −1.78155
\(290\) 10.3919i 0.0358342i
\(291\) 0 0
\(292\) 449.143 1.53816
\(293\) 114.195i 0.389745i 0.980829 + 0.194873i \(0.0624294\pi\)
−0.980829 + 0.194873i \(0.937571\pi\)
\(294\) 0 0
\(295\) −119.251 −0.404241
\(296\) − 196.091i − 0.662469i
\(297\) 0 0
\(298\) −134.467 −0.451231
\(299\) 147.984i 0.494931i
\(300\) 0 0
\(301\) −2.39681 −0.00796281
\(302\) − 139.391i − 0.461560i
\(303\) 0 0
\(304\) 291.479 0.958812
\(305\) 116.313i 0.381353i
\(306\) 0 0
\(307\) −224.438 −0.731070 −0.365535 0.930798i \(-0.619114\pi\)
−0.365535 + 0.930798i \(0.619114\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −44.5453 −0.143695
\(311\) − 171.175i − 0.550401i −0.961387 0.275201i \(-0.911256\pi\)
0.961387 0.275201i \(-0.0887442\pi\)
\(312\) 0 0
\(313\) −274.110 −0.875749 −0.437875 0.899036i \(-0.644269\pi\)
−0.437875 + 0.899036i \(0.644269\pi\)
\(314\) − 54.4276i − 0.173336i
\(315\) 0 0
\(316\) −570.509 −1.80541
\(317\) 618.549i 1.95126i 0.219427 + 0.975629i \(0.429581\pi\)
−0.219427 + 0.975629i \(0.570419\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 145.625i 0.455077i
\(321\) 0 0
\(322\) −195.118 −0.605957
\(323\) 739.057i 2.28810i
\(324\) 0 0
\(325\) 169.440 0.521354
\(326\) 416.777i 1.27846i
\(327\) 0 0
\(328\) −84.3772 −0.257248
\(329\) − 81.6633i − 0.248217i
\(330\) 0 0
\(331\) −160.328 −0.484376 −0.242188 0.970229i \(-0.577865\pi\)
−0.242188 + 0.970229i \(0.577865\pi\)
\(332\) − 624.926i − 1.88231i
\(333\) 0 0
\(334\) −434.903 −1.30210
\(335\) − 128.666i − 0.384077i
\(336\) 0 0
\(337\) −110.004 −0.326422 −0.163211 0.986591i \(-0.552185\pi\)
−0.163211 + 0.986591i \(0.552185\pi\)
\(338\) 334.812i 0.990568i
\(339\) 0 0
\(340\) −227.089 −0.667910
\(341\) 0 0
\(342\) 0 0
\(343\) 289.402 0.843738
\(344\) 2.09000i 0.00607558i
\(345\) 0 0
\(346\) −869.771 −2.51379
\(347\) − 221.736i − 0.639008i −0.947585 0.319504i \(-0.896484\pi\)
0.947585 0.319504i \(-0.103516\pi\)
\(348\) 0 0
\(349\) −466.352 −1.33625 −0.668126 0.744049i \(-0.732903\pi\)
−0.668126 + 0.744049i \(0.732903\pi\)
\(350\) 223.407i 0.638307i
\(351\) 0 0
\(352\) 0 0
\(353\) 494.404i 1.40058i 0.713860 + 0.700289i \(0.246945\pi\)
−0.713860 + 0.700289i \(0.753055\pi\)
\(354\) 0 0
\(355\) −173.712 −0.489330
\(356\) 413.980i 1.16286i
\(357\) 0 0
\(358\) 438.489 1.22483
\(359\) − 28.1698i − 0.0784673i −0.999230 0.0392336i \(-0.987508\pi\)
0.999230 0.0392336i \(-0.0124917\pi\)
\(360\) 0 0
\(361\) 318.471 0.882191
\(362\) − 669.246i − 1.84875i
\(363\) 0 0
\(364\) 125.157 0.343838
\(365\) − 145.668i − 0.399090i
\(366\) 0 0
\(367\) 29.7976 0.0811925 0.0405962 0.999176i \(-0.487074\pi\)
0.0405962 + 0.999176i \(0.487074\pi\)
\(368\) − 218.784i − 0.594521i
\(369\) 0 0
\(370\) −325.986 −0.881044
\(371\) − 19.2855i − 0.0519824i
\(372\) 0 0
\(373\) −313.887 −0.841520 −0.420760 0.907172i \(-0.638237\pi\)
−0.420760 + 0.907172i \(0.638237\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −71.2099 −0.189388
\(377\) 16.2833i 0.0431917i
\(378\) 0 0
\(379\) −31.4155 −0.0828904 −0.0414452 0.999141i \(-0.513196\pi\)
−0.0414452 + 0.999141i \(0.513196\pi\)
\(380\) 208.780i 0.549422i
\(381\) 0 0
\(382\) −69.5352 −0.182029
\(383\) − 474.897i − 1.23994i −0.784625 0.619970i \(-0.787145\pi\)
0.784625 0.619970i \(-0.212855\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 375.674i 0.973249i
\(387\) 0 0
\(388\) 387.758 0.999376
\(389\) 42.3470i 0.108861i 0.998518 + 0.0544306i \(0.0173344\pi\)
−0.998518 + 0.0544306i \(0.982666\pi\)
\(390\) 0 0
\(391\) 554.735 1.41876
\(392\) − 110.081i − 0.280820i
\(393\) 0 0
\(394\) −37.0911 −0.0941399
\(395\) 185.029i 0.468429i
\(396\) 0 0
\(397\) −560.270 −1.41126 −0.705630 0.708580i \(-0.749336\pi\)
−0.705630 + 0.708580i \(0.749336\pi\)
\(398\) − 337.261i − 0.847389i
\(399\) 0 0
\(400\) −250.504 −0.626260
\(401\) 565.289i 1.40970i 0.709357 + 0.704850i \(0.248986\pi\)
−0.709357 + 0.704850i \(0.751014\pi\)
\(402\) 0 0
\(403\) −69.7990 −0.173198
\(404\) − 193.740i − 0.479555i
\(405\) 0 0
\(406\) −21.4696 −0.0528807
\(407\) 0 0
\(408\) 0 0
\(409\) 174.254 0.426048 0.213024 0.977047i \(-0.431669\pi\)
0.213024 + 0.977047i \(0.431669\pi\)
\(410\) 140.271i 0.342124i
\(411\) 0 0
\(412\) −149.818 −0.363635
\(413\) − 246.372i − 0.596542i
\(414\) 0 0
\(415\) −202.678 −0.488382
\(416\) 341.141i 0.820050i
\(417\) 0 0
\(418\) 0 0
\(419\) 279.267i 0.666508i 0.942837 + 0.333254i \(0.108147\pi\)
−0.942837 + 0.333254i \(0.891853\pi\)
\(420\) 0 0
\(421\) 387.008 0.919259 0.459630 0.888111i \(-0.347982\pi\)
0.459630 + 0.888111i \(0.347982\pi\)
\(422\) − 460.948i − 1.09229i
\(423\) 0 0
\(424\) −16.8168 −0.0396623
\(425\) − 635.164i − 1.49450i
\(426\) 0 0
\(427\) −240.301 −0.562765
\(428\) 247.255i 0.577698i
\(429\) 0 0
\(430\) 3.47447 0.00808017
\(431\) − 319.975i − 0.742402i −0.928552 0.371201i \(-0.878946\pi\)
0.928552 0.371201i \(-0.121054\pi\)
\(432\) 0 0
\(433\) 526.145 1.21512 0.607558 0.794276i \(-0.292149\pi\)
0.607558 + 0.794276i \(0.292149\pi\)
\(434\) − 92.0303i − 0.212051i
\(435\) 0 0
\(436\) 744.547 1.70768
\(437\) − 510.010i − 1.16707i
\(438\) 0 0
\(439\) 22.7817 0.0518946 0.0259473 0.999663i \(-0.491740\pi\)
0.0259473 + 0.999663i \(0.491740\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −642.243 −1.45304
\(443\) 60.0022i 0.135445i 0.997704 + 0.0677226i \(0.0215733\pi\)
−0.997704 + 0.0677226i \(0.978427\pi\)
\(444\) 0 0
\(445\) 134.263 0.301716
\(446\) − 845.715i − 1.89622i
\(447\) 0 0
\(448\) −300.859 −0.671560
\(449\) − 761.224i − 1.69538i −0.530495 0.847688i \(-0.677994\pi\)
0.530495 0.847688i \(-0.322006\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 633.459i 1.40146i
\(453\) 0 0
\(454\) −607.562 −1.33824
\(455\) − 40.5914i − 0.0892119i
\(456\) 0 0
\(457\) −788.938 −1.72634 −0.863171 0.504912i \(-0.831525\pi\)
−0.863171 + 0.504912i \(0.831525\pi\)
\(458\) 1255.23i 2.74067i
\(459\) 0 0
\(460\) 156.710 0.340675
\(461\) − 726.842i − 1.57666i −0.615250 0.788332i \(-0.710945\pi\)
0.615250 0.788332i \(-0.289055\pi\)
\(462\) 0 0
\(463\) 301.548 0.651291 0.325646 0.945492i \(-0.394418\pi\)
0.325646 + 0.945492i \(0.394418\pi\)
\(464\) − 24.0736i − 0.0518827i
\(465\) 0 0
\(466\) −332.799 −0.714161
\(467\) − 292.485i − 0.626306i −0.949703 0.313153i \(-0.898615\pi\)
0.949703 0.313153i \(-0.101385\pi\)
\(468\) 0 0
\(469\) 265.822 0.566785
\(470\) 118.381i 0.251875i
\(471\) 0 0
\(472\) −214.835 −0.455158
\(473\) 0 0
\(474\) 0 0
\(475\) −583.954 −1.22938
\(476\) − 469.165i − 0.985640i
\(477\) 0 0
\(478\) −940.571 −1.96772
\(479\) 496.880i 1.03733i 0.854978 + 0.518664i \(0.173570\pi\)
−0.854978 + 0.518664i \(0.826430\pi\)
\(480\) 0 0
\(481\) −510.795 −1.06194
\(482\) 312.350i 0.648029i
\(483\) 0 0
\(484\) 0 0
\(485\) − 125.759i − 0.259297i
\(486\) 0 0
\(487\) 691.718 1.42037 0.710183 0.704017i \(-0.248612\pi\)
0.710183 + 0.704017i \(0.248612\pi\)
\(488\) 209.541i 0.429387i
\(489\) 0 0
\(490\) −183.002 −0.373474
\(491\) 296.243i 0.603346i 0.953412 + 0.301673i \(0.0975450\pi\)
−0.953412 + 0.301673i \(0.902455\pi\)
\(492\) 0 0
\(493\) 61.0396 0.123813
\(494\) 590.462i 1.19527i
\(495\) 0 0
\(496\) 103.192 0.208049
\(497\) − 358.888i − 0.722109i
\(498\) 0 0
\(499\) −134.004 −0.268544 −0.134272 0.990944i \(-0.542870\pi\)
−0.134272 + 0.990944i \(0.542870\pi\)
\(500\) − 379.668i − 0.759336i
\(501\) 0 0
\(502\) −998.914 −1.98987
\(503\) − 190.003i − 0.377740i −0.982002 0.188870i \(-0.939518\pi\)
0.982002 0.188870i \(-0.0604824\pi\)
\(504\) 0 0
\(505\) −62.8345 −0.124425
\(506\) 0 0
\(507\) 0 0
\(508\) 1182.62 2.32799
\(509\) − 119.100i − 0.233989i −0.993133 0.116995i \(-0.962674\pi\)
0.993133 0.116995i \(-0.0373260\pi\)
\(510\) 0 0
\(511\) 300.948 0.588940
\(512\) − 634.222i − 1.23872i
\(513\) 0 0
\(514\) 919.404 1.78872
\(515\) 48.5895i 0.0943485i
\(516\) 0 0
\(517\) 0 0
\(518\) − 673.485i − 1.30016i
\(519\) 0 0
\(520\) −35.3955 −0.0680682
\(521\) − 637.297i − 1.22322i −0.791160 0.611609i \(-0.790522\pi\)
0.791160 0.611609i \(-0.209478\pi\)
\(522\) 0 0
\(523\) 129.698 0.247988 0.123994 0.992283i \(-0.460430\pi\)
0.123994 + 0.992283i \(0.460430\pi\)
\(524\) − 738.147i − 1.40868i
\(525\) 0 0
\(526\) −745.914 −1.41809
\(527\) 261.649i 0.496487i
\(528\) 0 0
\(529\) 146.187 0.276346
\(530\) 27.9567i 0.0527485i
\(531\) 0 0
\(532\) −431.339 −0.810787
\(533\) 219.793i 0.412370i
\(534\) 0 0
\(535\) 80.1905 0.149889
\(536\) − 231.795i − 0.432454i
\(537\) 0 0
\(538\) 589.779 1.09624
\(539\) 0 0
\(540\) 0 0
\(541\) −493.991 −0.913108 −0.456554 0.889696i \(-0.650916\pi\)
−0.456554 + 0.889696i \(0.650916\pi\)
\(542\) 1388.03i 2.56095i
\(543\) 0 0
\(544\) 1278.80 2.35074
\(545\) − 241.474i − 0.443072i
\(546\) 0 0
\(547\) −106.128 −0.194018 −0.0970089 0.995284i \(-0.530928\pi\)
−0.0970089 + 0.995284i \(0.530928\pi\)
\(548\) − 100.178i − 0.182807i
\(549\) 0 0
\(550\) 0 0
\(551\) − 56.1183i − 0.101848i
\(552\) 0 0
\(553\) −382.269 −0.691264
\(554\) 70.2457i 0.126797i
\(555\) 0 0
\(556\) −739.849 −1.33066
\(557\) − 827.637i − 1.48588i −0.669356 0.742942i \(-0.733430\pi\)
0.669356 0.742942i \(-0.266570\pi\)
\(558\) 0 0
\(559\) 5.44422 0.00973921
\(560\) 60.0113i 0.107163i
\(561\) 0 0
\(562\) 789.001 1.40392
\(563\) 470.917i 0.836442i 0.908345 + 0.418221i \(0.137346\pi\)
−0.908345 + 0.418221i \(0.862654\pi\)
\(564\) 0 0
\(565\) 205.446 0.363621
\(566\) 107.289i 0.189556i
\(567\) 0 0
\(568\) −312.948 −0.550965
\(569\) 200.074i 0.351624i 0.984424 + 0.175812i \(0.0562551\pi\)
−0.984424 + 0.175812i \(0.943745\pi\)
\(570\) 0 0
\(571\) 658.692 1.15358 0.576788 0.816894i \(-0.304306\pi\)
0.576788 + 0.816894i \(0.304306\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −289.798 −0.504875
\(575\) 438.315i 0.762287i
\(576\) 0 0
\(577\) −100.977 −0.175004 −0.0875020 0.996164i \(-0.527888\pi\)
−0.0875020 + 0.996164i \(0.527888\pi\)
\(578\) 1541.98i 2.66779i
\(579\) 0 0
\(580\) 17.2434 0.0297301
\(581\) − 418.732i − 0.720709i
\(582\) 0 0
\(583\) 0 0
\(584\) − 262.425i − 0.449358i
\(585\) 0 0
\(586\) 342.005 0.583627
\(587\) − 349.390i − 0.595213i −0.954689 0.297606i \(-0.903812\pi\)
0.954689 0.297606i \(-0.0961883\pi\)
\(588\) 0 0
\(589\) 240.554 0.408410
\(590\) 357.147i 0.605333i
\(591\) 0 0
\(592\) 755.171 1.27563
\(593\) − 670.238i − 1.13025i −0.825005 0.565125i \(-0.808828\pi\)
0.825005 0.565125i \(-0.191172\pi\)
\(594\) 0 0
\(595\) −152.161 −0.255733
\(596\) 223.123i 0.374367i
\(597\) 0 0
\(598\) 443.200 0.741138
\(599\) − 771.328i − 1.28769i −0.765155 0.643847i \(-0.777338\pi\)
0.765155 0.643847i \(-0.222662\pi\)
\(600\) 0 0
\(601\) −185.722 −0.309021 −0.154511 0.987991i \(-0.549380\pi\)
−0.154511 + 0.987991i \(0.549380\pi\)
\(602\) 7.17823i 0.0119240i
\(603\) 0 0
\(604\) −231.294 −0.382937
\(605\) 0 0
\(606\) 0 0
\(607\) −488.185 −0.804259 −0.402129 0.915583i \(-0.631730\pi\)
−0.402129 + 0.915583i \(0.631730\pi\)
\(608\) − 1175.70i − 1.93372i
\(609\) 0 0
\(610\) 348.346 0.571059
\(611\) 185.494i 0.303591i
\(612\) 0 0
\(613\) 510.461 0.832726 0.416363 0.909199i \(-0.363305\pi\)
0.416363 + 0.909199i \(0.363305\pi\)
\(614\) 672.174i 1.09475i
\(615\) 0 0
\(616\) 0 0
\(617\) − 1142.14i − 1.85111i −0.378608 0.925557i \(-0.623597\pi\)
0.378608 0.925557i \(-0.376403\pi\)
\(618\) 0 0
\(619\) −908.213 −1.46723 −0.733613 0.679567i \(-0.762168\pi\)
−0.733613 + 0.679567i \(0.762168\pi\)
\(620\) 73.9147i 0.119217i
\(621\) 0 0
\(622\) −512.653 −0.824202
\(623\) 277.387i 0.445244i
\(624\) 0 0
\(625\) 436.923 0.699077
\(626\) 820.934i 1.31140i
\(627\) 0 0
\(628\) −90.3126 −0.143810
\(629\) 1914.77i 3.04415i
\(630\) 0 0
\(631\) 944.402 1.49668 0.748338 0.663318i \(-0.230852\pi\)
0.748338 + 0.663318i \(0.230852\pi\)
\(632\) 333.336i 0.527431i
\(633\) 0 0
\(634\) 1852.50 2.92192
\(635\) − 383.551i − 0.604017i
\(636\) 0 0
\(637\) −286.750 −0.450157
\(638\) 0 0
\(639\) 0 0
\(640\) 145.354 0.227116
\(641\) − 359.081i − 0.560188i −0.959973 0.280094i \(-0.909634\pi\)
0.959973 0.280094i \(-0.0903657\pi\)
\(642\) 0 0
\(643\) 928.146 1.44346 0.721731 0.692174i \(-0.243347\pi\)
0.721731 + 0.692174i \(0.243347\pi\)
\(644\) 323.762i 0.502737i
\(645\) 0 0
\(646\) 2213.41 3.42633
\(647\) 436.289i 0.674326i 0.941446 + 0.337163i \(0.109467\pi\)
−0.941446 + 0.337163i \(0.890533\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 507.458i − 0.780705i
\(651\) 0 0
\(652\) 691.565 1.06068
\(653\) − 243.417i − 0.372767i −0.982477 0.186383i \(-0.940323\pi\)
0.982477 0.186383i \(-0.0596767\pi\)
\(654\) 0 0
\(655\) −239.399 −0.365494
\(656\) − 324.948i − 0.495347i
\(657\) 0 0
\(658\) −244.574 −0.371694
\(659\) − 348.053i − 0.528153i −0.964502 0.264076i \(-0.914933\pi\)
0.964502 0.264076i \(-0.0850671\pi\)
\(660\) 0 0
\(661\) 601.901 0.910592 0.455296 0.890340i \(-0.349533\pi\)
0.455296 + 0.890340i \(0.349533\pi\)
\(662\) 480.170i 0.725332i
\(663\) 0 0
\(664\) −365.131 −0.549897
\(665\) 139.893i 0.210366i
\(666\) 0 0
\(667\) −42.1224 −0.0631520
\(668\) 721.640i 1.08030i
\(669\) 0 0
\(670\) −385.343 −0.575138
\(671\) 0 0
\(672\) 0 0
\(673\) 738.885 1.09790 0.548949 0.835856i \(-0.315028\pi\)
0.548949 + 0.835856i \(0.315028\pi\)
\(674\) 329.453i 0.488803i
\(675\) 0 0
\(676\) 555.558 0.821832
\(677\) 42.1168i 0.0622109i 0.999516 + 0.0311055i \(0.00990277\pi\)
−0.999516 + 0.0311055i \(0.990097\pi\)
\(678\) 0 0
\(679\) 259.817 0.382647
\(680\) 132.684i 0.195123i
\(681\) 0 0
\(682\) 0 0
\(683\) − 543.110i − 0.795183i −0.917562 0.397592i \(-0.869846\pi\)
0.917562 0.397592i \(-0.130154\pi\)
\(684\) 0 0
\(685\) −32.4902 −0.0474309
\(686\) − 866.734i − 1.26346i
\(687\) 0 0
\(688\) −8.04886 −0.0116989
\(689\) 43.8059i 0.0635790i
\(690\) 0 0
\(691\) −428.173 −0.619643 −0.309821 0.950795i \(-0.600269\pi\)
−0.309821 + 0.950795i \(0.600269\pi\)
\(692\) 1443.22i 2.08558i
\(693\) 0 0
\(694\) −664.080 −0.956887
\(695\) 239.951i 0.345253i
\(696\) 0 0
\(697\) 823.918 1.18209
\(698\) 1396.68i 2.00098i
\(699\) 0 0
\(700\) 370.703 0.529576
\(701\) 2.78056i 0.00396656i 0.999998 + 0.00198328i \(0.000631299\pi\)
−0.999998 + 0.00198328i \(0.999369\pi\)
\(702\) 0 0
\(703\) 1760.39 2.50411
\(704\) 0 0
\(705\) 0 0
\(706\) 1480.70 2.09730
\(707\) − 129.816i − 0.183615i
\(708\) 0 0
\(709\) 1077.51 1.51976 0.759878 0.650065i \(-0.225258\pi\)
0.759878 + 0.650065i \(0.225258\pi\)
\(710\) 520.253i 0.732751i
\(711\) 0 0
\(712\) 241.880 0.339719
\(713\) − 180.559i − 0.253239i
\(714\) 0 0
\(715\) 0 0
\(716\) − 727.591i − 1.01619i
\(717\) 0 0
\(718\) −84.3660 −0.117501
\(719\) 1073.42i 1.49293i 0.665424 + 0.746465i \(0.268251\pi\)
−0.665424 + 0.746465i \(0.731749\pi\)
\(720\) 0 0
\(721\) −100.385 −0.139231
\(722\) − 953.792i − 1.32104i
\(723\) 0 0
\(724\) −1110.49 −1.53383
\(725\) 48.2295i 0.0665234i
\(726\) 0 0
\(727\) −212.371 −0.292119 −0.146060 0.989276i \(-0.546659\pi\)
−0.146060 + 0.989276i \(0.546659\pi\)
\(728\) − 73.1267i − 0.100449i
\(729\) 0 0
\(730\) −436.262 −0.597620
\(731\) − 20.4082i − 0.0279182i
\(732\) 0 0
\(733\) −942.223 −1.28543 −0.642717 0.766104i \(-0.722193\pi\)
−0.642717 + 0.766104i \(0.722193\pi\)
\(734\) − 89.2414i − 0.121582i
\(735\) 0 0
\(736\) −882.479 −1.19902
\(737\) 0 0
\(738\) 0 0
\(739\) −720.862 −0.975456 −0.487728 0.872996i \(-0.662174\pi\)
−0.487728 + 0.872996i \(0.662174\pi\)
\(740\) 540.914i 0.730965i
\(741\) 0 0
\(742\) −57.7583 −0.0778414
\(743\) 836.564i 1.12593i 0.826482 + 0.562963i \(0.190339\pi\)
−0.826482 + 0.562963i \(0.809661\pi\)
\(744\) 0 0
\(745\) 72.3640 0.0971329
\(746\) 940.064i 1.26014i
\(747\) 0 0
\(748\) 0 0
\(749\) 165.673i 0.221192i
\(750\) 0 0
\(751\) −35.2668 −0.0469598 −0.0234799 0.999724i \(-0.507475\pi\)
−0.0234799 + 0.999724i \(0.507475\pi\)
\(752\) − 274.238i − 0.364679i
\(753\) 0 0
\(754\) 48.7670 0.0646777
\(755\) 75.0141i 0.0993564i
\(756\) 0 0
\(757\) 454.700 0.600660 0.300330 0.953835i \(-0.402903\pi\)
0.300330 + 0.953835i \(0.402903\pi\)
\(758\) 94.0866i 0.124125i
\(759\) 0 0
\(760\) 121.986 0.160508
\(761\) − 104.689i − 0.137568i −0.997632 0.0687840i \(-0.978088\pi\)
0.997632 0.0687840i \(-0.0219119\pi\)
\(762\) 0 0
\(763\) 498.884 0.653845
\(764\) 115.381i 0.151022i
\(765\) 0 0
\(766\) −1422.28 −1.85676
\(767\) 559.620i 0.729622i
\(768\) 0 0
\(769\) −93.7806 −0.121951 −0.0609757 0.998139i \(-0.519421\pi\)
−0.0609757 + 0.998139i \(0.519421\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 623.361 0.807463
\(773\) − 358.726i − 0.464070i −0.972707 0.232035i \(-0.925462\pi\)
0.972707 0.232035i \(-0.0745384\pi\)
\(774\) 0 0
\(775\) −206.738 −0.266758
\(776\) − 226.559i − 0.291957i
\(777\) 0 0
\(778\) 126.826 0.163015
\(779\) − 757.491i − 0.972388i
\(780\) 0 0
\(781\) 0 0
\(782\) − 1661.38i − 2.12453i
\(783\) 0 0
\(784\) 423.938 0.540737
\(785\) 29.2905i 0.0373127i
\(786\) 0 0
\(787\) 276.916 0.351863 0.175931 0.984402i \(-0.443706\pi\)
0.175931 + 0.984402i \(0.443706\pi\)
\(788\) 61.5458i 0.0781038i
\(789\) 0 0
\(790\) 554.147 0.701452
\(791\) 424.449i 0.536598i
\(792\) 0 0
\(793\) 545.830 0.688311
\(794\) 1677.96i 2.11330i
\(795\) 0 0
\(796\) −559.622 −0.703042
\(797\) − 528.612i − 0.663252i −0.943411 0.331626i \(-0.892403\pi\)
0.943411 0.331626i \(-0.107597\pi\)
\(798\) 0 0
\(799\) 695.343 0.870267
\(800\) 1010.43i 1.26303i
\(801\) 0 0
\(802\) 1692.99 2.11096
\(803\) 0 0
\(804\) 0 0
\(805\) 105.004 0.130439
\(806\) 209.042i 0.259357i
\(807\) 0 0
\(808\) −113.198 −0.140097
\(809\) − 1257.86i − 1.55483i −0.628985 0.777417i \(-0.716529\pi\)
0.628985 0.777417i \(-0.283471\pi\)
\(810\) 0 0
\(811\) −1041.19 −1.28383 −0.641916 0.766775i \(-0.721860\pi\)
−0.641916 + 0.766775i \(0.721860\pi\)
\(812\) 35.6248i 0.0438729i
\(813\) 0 0
\(814\) 0 0
\(815\) − 224.291i − 0.275203i
\(816\) 0 0
\(817\) −18.7628 −0.0229655
\(818\) − 521.875i − 0.637989i
\(819\) 0 0
\(820\) 232.754 0.283846
\(821\) 435.664i 0.530650i 0.964159 + 0.265325i \(0.0854793\pi\)
−0.964159 + 0.265325i \(0.914521\pi\)
\(822\) 0 0
\(823\) 587.220 0.713512 0.356756 0.934198i \(-0.383883\pi\)
0.356756 + 0.934198i \(0.383883\pi\)
\(824\) 87.5354i 0.106232i
\(825\) 0 0
\(826\) −737.862 −0.893296
\(827\) − 274.955i − 0.332473i −0.986086 0.166236i \(-0.946839\pi\)
0.986086 0.166236i \(-0.0531615\pi\)
\(828\) 0 0
\(829\) −916.540 −1.10560 −0.552799 0.833315i \(-0.686440\pi\)
−0.552799 + 0.833315i \(0.686440\pi\)
\(830\) 607.004i 0.731330i
\(831\) 0 0
\(832\) 683.385 0.821376
\(833\) 1074.91i 1.29041i
\(834\) 0 0
\(835\) 234.045 0.280293
\(836\) 0 0
\(837\) 0 0
\(838\) 836.380 0.998067
\(839\) 1058.83i 1.26202i 0.775777 + 0.631008i \(0.217358\pi\)
−0.775777 + 0.631008i \(0.782642\pi\)
\(840\) 0 0
\(841\) 836.365 0.994489
\(842\) − 1159.06i − 1.37655i
\(843\) 0 0
\(844\) −764.858 −0.906229
\(845\) − 180.181i − 0.213232i
\(846\) 0 0
\(847\) 0 0
\(848\) − 64.7637i − 0.0763723i
\(849\) 0 0
\(850\) −1902.26 −2.23795
\(851\) − 1321.35i − 1.55270i
\(852\) 0 0
\(853\) −262.534 −0.307777 −0.153888 0.988088i \(-0.549180\pi\)
−0.153888 + 0.988088i \(0.549180\pi\)
\(854\) 719.680i 0.842716i
\(855\) 0 0
\(856\) 144.466 0.168768
\(857\) − 139.937i − 0.163287i −0.996662 0.0816437i \(-0.973983\pi\)
0.996662 0.0816437i \(-0.0260169\pi\)
\(858\) 0 0
\(859\) −466.124 −0.542635 −0.271318 0.962490i \(-0.587459\pi\)
−0.271318 + 0.962490i \(0.587459\pi\)
\(860\) − 5.76524i − 0.00670377i
\(861\) 0 0
\(862\) −958.299 −1.11172
\(863\) 223.415i 0.258882i 0.991587 + 0.129441i \(0.0413183\pi\)
−0.991587 + 0.129441i \(0.958682\pi\)
\(864\) 0 0
\(865\) 468.072 0.541123
\(866\) − 1575.76i − 1.81958i
\(867\) 0 0
\(868\) −152.707 −0.175930
\(869\) 0 0
\(870\) 0 0
\(871\) −603.801 −0.693228
\(872\) − 435.023i − 0.498880i
\(873\) 0 0
\(874\) −1527.44 −1.74764
\(875\) − 254.397i − 0.290739i
\(876\) 0 0
\(877\) 446.074 0.508637 0.254318 0.967121i \(-0.418149\pi\)
0.254318 + 0.967121i \(0.418149\pi\)
\(878\) − 68.2292i − 0.0777099i
\(879\) 0 0
\(880\) 0 0
\(881\) 1631.61i 1.85200i 0.377522 + 0.926001i \(0.376776\pi\)
−0.377522 + 0.926001i \(0.623224\pi\)
\(882\) 0 0
\(883\) 1491.22 1.68881 0.844404 0.535707i \(-0.179955\pi\)
0.844404 + 0.535707i \(0.179955\pi\)
\(884\) 1065.68i 1.20552i
\(885\) 0 0
\(886\) 179.701 0.202823
\(887\) − 623.153i − 0.702540i −0.936274 0.351270i \(-0.885750\pi\)
0.936274 0.351270i \(-0.114250\pi\)
\(888\) 0 0
\(889\) 792.413 0.891353
\(890\) − 402.107i − 0.451806i
\(891\) 0 0
\(892\) −1403.31 −1.57321
\(893\) − 639.282i − 0.715881i
\(894\) 0 0
\(895\) −235.975 −0.263659
\(896\) 300.300i 0.335157i
\(897\) 0 0
\(898\) −2279.80 −2.53875
\(899\) − 19.8676i − 0.0220997i
\(900\) 0 0
\(901\) 164.211 0.182254
\(902\) 0 0
\(903\) 0 0
\(904\) 370.117 0.409422
\(905\) 360.158i 0.397965i
\(906\) 0 0
\(907\) −966.557 −1.06566 −0.532832 0.846221i \(-0.678872\pi\)
−0.532832 + 0.846221i \(0.678872\pi\)
\(908\) 1008.14i 1.11028i
\(909\) 0 0
\(910\) −121.568 −0.133591
\(911\) − 617.042i − 0.677324i −0.940908 0.338662i \(-0.890026\pi\)
0.940908 0.338662i \(-0.109974\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2362.80i 2.58512i
\(915\) 0 0
\(916\) 2082.82 2.27382
\(917\) − 494.595i − 0.539362i
\(918\) 0 0
\(919\) 1104.16 1.20148 0.600742 0.799443i \(-0.294872\pi\)
0.600742 + 0.799443i \(0.294872\pi\)
\(920\) − 91.5626i − 0.0995246i
\(921\) 0 0
\(922\) −2176.83 −2.36099
\(923\) 815.195i 0.883202i
\(924\) 0 0
\(925\) −1512.92 −1.63559
\(926\) − 903.109i − 0.975280i
\(927\) 0 0
\(928\) −97.1025 −0.104636
\(929\) 1078.61i 1.16104i 0.814245 + 0.580521i \(0.197151\pi\)
−0.814245 + 0.580521i \(0.802849\pi\)
\(930\) 0 0
\(931\) 988.248 1.06149
\(932\) 552.218i 0.592509i
\(933\) 0 0
\(934\) −875.967 −0.937866
\(935\) 0 0
\(936\) 0 0
\(937\) −587.691 −0.627205 −0.313602 0.949554i \(-0.601536\pi\)
−0.313602 + 0.949554i \(0.601536\pi\)
\(938\) − 796.115i − 0.848736i
\(939\) 0 0
\(940\) 196.432 0.208970
\(941\) − 1069.27i − 1.13631i −0.822921 0.568155i \(-0.807657\pi\)
0.822921 0.568155i \(-0.192343\pi\)
\(942\) 0 0
\(943\) −568.572 −0.602939
\(944\) − 827.356i − 0.876437i
\(945\) 0 0
\(946\) 0 0
\(947\) 435.924i 0.460321i 0.973153 + 0.230160i \(0.0739251\pi\)
−0.973153 + 0.230160i \(0.926075\pi\)
\(948\) 0 0
\(949\) −683.588 −0.720325
\(950\) 1748.89i 1.84094i
\(951\) 0 0
\(952\) −274.123 −0.287945
\(953\) − 670.977i − 0.704068i −0.935987 0.352034i \(-0.885490\pi\)
0.935987 0.352034i \(-0.114510\pi\)
\(954\) 0 0
\(955\) 37.4207 0.0391840
\(956\) 1560.70i 1.63253i
\(957\) 0 0
\(958\) 1488.11 1.55335
\(959\) − 67.1244i − 0.0699942i
\(960\) 0 0
\(961\) −875.837 −0.911380
\(962\) 1529.79i 1.59021i
\(963\) 0 0
\(964\) 518.287 0.537642
\(965\) − 202.171i − 0.209503i
\(966\) 0 0
\(967\) −940.682 −0.972784 −0.486392 0.873741i \(-0.661687\pi\)
−0.486392 + 0.873741i \(0.661687\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −376.637 −0.388286
\(971\) 1077.62i 1.10980i 0.831916 + 0.554901i \(0.187244\pi\)
−0.831916 + 0.554901i \(0.812756\pi\)
\(972\) 0 0
\(973\) −495.736 −0.509492
\(974\) − 2071.64i − 2.12694i
\(975\) 0 0
\(976\) −806.969 −0.826812
\(977\) 1006.05i 1.02973i 0.857270 + 0.514866i \(0.172158\pi\)
−0.857270 + 0.514866i \(0.827842\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 303.658i 0.309855i
\(981\) 0 0
\(982\) 887.221 0.903484
\(983\) − 1173.82i − 1.19412i −0.802197 0.597060i \(-0.796335\pi\)
0.802197 0.597060i \(-0.203665\pi\)
\(984\) 0 0
\(985\) 19.9608 0.0202647
\(986\) − 182.808i − 0.185404i
\(987\) 0 0
\(988\) 979.763 0.991663
\(989\) 14.0834i 0.0142400i
\(990\) 0 0
\(991\) −989.386 −0.998372 −0.499186 0.866495i \(-0.666368\pi\)
−0.499186 + 0.866495i \(0.666368\pi\)
\(992\) − 416.234i − 0.419591i
\(993\) 0 0
\(994\) −1074.84 −1.08133
\(995\) 181.499i 0.182411i
\(996\) 0 0
\(997\) −1465.59 −1.47000 −0.735000 0.678067i \(-0.762818\pi\)
−0.735000 + 0.678067i \(0.762818\pi\)
\(998\) 401.329i 0.402133i
\(999\) 0 0
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 1089.3.b.i.485.3 16
3.2 odd 2 inner 1089.3.b.i.485.14 16
11.3 even 5 99.3.l.a.53.2 32
11.4 even 5 99.3.l.a.71.7 yes 32
11.10 odd 2 1089.3.b.j.485.14 16
33.14 odd 10 99.3.l.a.53.7 yes 32
33.26 odd 10 99.3.l.a.71.2 yes 32
33.32 even 2 1089.3.b.j.485.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.53.2 32 11.3 even 5
99.3.l.a.53.7 yes 32 33.14 odd 10
99.3.l.a.71.2 yes 32 33.26 odd 10
99.3.l.a.71.7 yes 32 11.4 even 5
1089.3.b.i.485.3 16 1.1 even 1 trivial
1089.3.b.i.485.14 16 3.2 odd 2 inner
1089.3.b.j.485.3 16 33.32 even 2
1089.3.b.j.485.14 16 11.10 odd 2