Properties

Label 1089.3.b.i.485.5
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 485.5
Root \(-1.96467i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.i.485.12

$q$-expansion

\(f(q)\) \(=\) \(q-1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} -5.23056 q^{7} -8.13389i q^{8} +O(q^{10})\) \(q-1.96467i q^{2} +0.140085 q^{4} +9.62261i q^{5} -5.23056 q^{7} -8.13389i q^{8} +18.9052 q^{10} +17.2247 q^{13} +10.2763i q^{14} -15.4200 q^{16} +7.71825i q^{17} +14.0290 q^{19} +1.34798i q^{20} +5.68512i q^{23} -67.5947 q^{25} -33.8407i q^{26} -0.732721 q^{28} +24.0845i q^{29} +15.0159 q^{31} -2.24031i q^{32} +15.1638 q^{34} -50.3317i q^{35} -7.65533 q^{37} -27.5624i q^{38} +78.2692 q^{40} +18.1953i q^{41} -53.6955 q^{43} +11.1694 q^{46} +38.8113i q^{47} -21.6412 q^{49} +132.801i q^{50} +2.41291 q^{52} +81.6140i q^{53} +42.5448i q^{56} +47.3181 q^{58} +0.000434171i q^{59} +4.77983 q^{61} -29.5012i q^{62} -66.0816 q^{64} +165.746i q^{65} +111.796 q^{67} +1.08121i q^{68} -98.8850 q^{70} +99.6336i q^{71} -34.2957 q^{73} +15.0402i q^{74} +1.96525 q^{76} -61.7290 q^{79} -148.381i q^{80} +35.7478 q^{82} +65.2314i q^{83} -74.2697 q^{85} +105.494i q^{86} +92.5976i q^{89} -90.0947 q^{91} +0.796398i q^{92} +76.2513 q^{94} +134.996i q^{95} -52.4236 q^{97} +42.5178i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{4} - 8q^{7} + O(q^{10}) \) \( 16q - 32q^{4} - 8q^{7} + 24q^{10} + 4q^{13} + 28q^{16} - 20q^{19} - 44q^{25} + 16q^{28} + 28q^{31} + 148q^{34} - 148q^{37} + 224q^{40} + 272q^{43} + 208q^{46} + 348q^{49} + 520q^{52} - 44q^{58} + 224q^{61} + 436q^{64} + 24q^{67} - 664q^{70} - 4q^{73} + 1052q^{76} + 216q^{79} + 348q^{82} + 416q^{85} - 168q^{91} + 1140q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.96467i − 0.982333i −0.871066 0.491167i \(-0.836571\pi\)
0.871066 0.491167i \(-0.163429\pi\)
\(3\) 0 0
\(4\) 0.140085 0.0350211
\(5\) 9.62261i 1.92452i 0.272127 + 0.962261i \(0.412273\pi\)
−0.272127 + 0.962261i \(0.587727\pi\)
\(6\) 0 0
\(7\) −5.23056 −0.747223 −0.373612 0.927585i \(-0.621881\pi\)
−0.373612 + 0.927585i \(0.621881\pi\)
\(8\) − 8.13389i − 1.01674i
\(9\) 0 0
\(10\) 18.9052 1.89052
\(11\) 0 0
\(12\) 0 0
\(13\) 17.2247 1.32497 0.662487 0.749073i \(-0.269501\pi\)
0.662487 + 0.749073i \(0.269501\pi\)
\(14\) 10.2763i 0.734022i
\(15\) 0 0
\(16\) −15.4200 −0.963752
\(17\) 7.71825i 0.454015i 0.973893 + 0.227007i \(0.0728941\pi\)
−0.973893 + 0.227007i \(0.927106\pi\)
\(18\) 0 0
\(19\) 14.0290 0.738370 0.369185 0.929356i \(-0.379637\pi\)
0.369185 + 0.929356i \(0.379637\pi\)
\(20\) 1.34798i 0.0673989i
\(21\) 0 0
\(22\) 0 0
\(23\) 5.68512i 0.247179i 0.992333 + 0.123590i \(0.0394407\pi\)
−0.992333 + 0.123590i \(0.960559\pi\)
\(24\) 0 0
\(25\) −67.5947 −2.70379
\(26\) − 33.8407i − 1.30157i
\(27\) 0 0
\(28\) −0.732721 −0.0261686
\(29\) 24.0845i 0.830501i 0.909707 + 0.415250i \(0.136306\pi\)
−0.909707 + 0.415250i \(0.863694\pi\)
\(30\) 0 0
\(31\) 15.0159 0.484383 0.242192 0.970228i \(-0.422134\pi\)
0.242192 + 0.970228i \(0.422134\pi\)
\(32\) − 2.24031i − 0.0700097i
\(33\) 0 0
\(34\) 15.1638 0.445994
\(35\) − 50.3317i − 1.43805i
\(36\) 0 0
\(37\) −7.65533 −0.206901 −0.103450 0.994635i \(-0.532988\pi\)
−0.103450 + 0.994635i \(0.532988\pi\)
\(38\) − 27.5624i − 0.725325i
\(39\) 0 0
\(40\) 78.2692 1.95673
\(41\) 18.1953i 0.443789i 0.975071 + 0.221894i \(0.0712240\pi\)
−0.975071 + 0.221894i \(0.928776\pi\)
\(42\) 0 0
\(43\) −53.6955 −1.24873 −0.624366 0.781132i \(-0.714642\pi\)
−0.624366 + 0.781132i \(0.714642\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 11.1694 0.242812
\(47\) 38.8113i 0.825772i 0.910783 + 0.412886i \(0.135479\pi\)
−0.910783 + 0.412886i \(0.864521\pi\)
\(48\) 0 0
\(49\) −21.6412 −0.441658
\(50\) 132.801i 2.65602i
\(51\) 0 0
\(52\) 2.41291 0.0464021
\(53\) 81.6140i 1.53989i 0.638112 + 0.769944i \(0.279716\pi\)
−0.638112 + 0.769944i \(0.720284\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 42.5448i 0.759728i
\(57\) 0 0
\(58\) 47.3181 0.815829
\(59\) 0 0.000434171i 0 7.35883e-6i 1.00000 3.67942e-6i \(1.17119e-6\pi\)
−1.00000 3.67942e-6i \(0.999999\pi\)
\(60\) 0 0
\(61\) 4.77983 0.0783579 0.0391789 0.999232i \(-0.487526\pi\)
0.0391789 + 0.999232i \(0.487526\pi\)
\(62\) − 29.5012i − 0.475826i
\(63\) 0 0
\(64\) −66.0816 −1.03253
\(65\) 165.746i 2.54994i
\(66\) 0 0
\(67\) 111.796 1.66860 0.834298 0.551313i \(-0.185873\pi\)
0.834298 + 0.551313i \(0.185873\pi\)
\(68\) 1.08121i 0.0159001i
\(69\) 0 0
\(70\) −98.8850 −1.41264
\(71\) 99.6336i 1.40329i 0.712526 + 0.701645i \(0.247551\pi\)
−0.712526 + 0.701645i \(0.752449\pi\)
\(72\) 0 0
\(73\) −34.2957 −0.469804 −0.234902 0.972019i \(-0.575477\pi\)
−0.234902 + 0.972019i \(0.575477\pi\)
\(74\) 15.0402i 0.203246i
\(75\) 0 0
\(76\) 1.96525 0.0258585
\(77\) 0 0
\(78\) 0 0
\(79\) −61.7290 −0.781380 −0.390690 0.920522i \(-0.627764\pi\)
−0.390690 + 0.920522i \(0.627764\pi\)
\(80\) − 148.381i − 1.85476i
\(81\) 0 0
\(82\) 35.7478 0.435948
\(83\) 65.2314i 0.785921i 0.919555 + 0.392961i \(0.128549\pi\)
−0.919555 + 0.392961i \(0.871451\pi\)
\(84\) 0 0
\(85\) −74.2697 −0.873761
\(86\) 105.494i 1.22667i
\(87\) 0 0
\(88\) 0 0
\(89\) 92.5976i 1.04042i 0.854038 + 0.520211i \(0.174147\pi\)
−0.854038 + 0.520211i \(0.825853\pi\)
\(90\) 0 0
\(91\) −90.0947 −0.990051
\(92\) 0.796398i 0.00865650i
\(93\) 0 0
\(94\) 76.2513 0.811184
\(95\) 134.996i 1.42101i
\(96\) 0 0
\(97\) −52.4236 −0.540450 −0.270225 0.962797i \(-0.587098\pi\)
−0.270225 + 0.962797i \(0.587098\pi\)
\(98\) 42.5178i 0.433855i
\(99\) 0 0
\(100\) −9.46897 −0.0946897
\(101\) 58.9222i 0.583389i 0.956512 + 0.291694i \(0.0942190\pi\)
−0.956512 + 0.291694i \(0.905781\pi\)
\(102\) 0 0
\(103\) −96.7152 −0.938982 −0.469491 0.882937i \(-0.655563\pi\)
−0.469491 + 0.882937i \(0.655563\pi\)
\(104\) − 140.103i − 1.34715i
\(105\) 0 0
\(106\) 160.344 1.51268
\(107\) 18.3916i 0.171884i 0.996300 + 0.0859422i \(0.0273900\pi\)
−0.996300 + 0.0859422i \(0.972610\pi\)
\(108\) 0 0
\(109\) 50.6495 0.464674 0.232337 0.972635i \(-0.425363\pi\)
0.232337 + 0.972635i \(0.425363\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 80.6555 0.720138
\(113\) 9.83566i 0.0870412i 0.999053 + 0.0435206i \(0.0138574\pi\)
−0.999053 + 0.0435206i \(0.986143\pi\)
\(114\) 0 0
\(115\) −54.7058 −0.475702
\(116\) 3.37387i 0.0290851i
\(117\) 0 0
\(118\) 0.000853002 0 7.22883e−6 0
\(119\) − 40.3708i − 0.339250i
\(120\) 0 0
\(121\) 0 0
\(122\) − 9.39077i − 0.0769736i
\(123\) 0 0
\(124\) 2.10349 0.0169637
\(125\) − 409.872i − 3.27898i
\(126\) 0 0
\(127\) 215.481 1.69670 0.848349 0.529437i \(-0.177597\pi\)
0.848349 + 0.529437i \(0.177597\pi\)
\(128\) 120.867i 0.944274i
\(129\) 0 0
\(130\) 325.636 2.50489
\(131\) − 166.588i − 1.27166i −0.771827 0.635832i \(-0.780657\pi\)
0.771827 0.635832i \(-0.219343\pi\)
\(132\) 0 0
\(133\) −73.3797 −0.551727
\(134\) − 219.642i − 1.63912i
\(135\) 0 0
\(136\) 62.7794 0.461613
\(137\) − 43.7479i − 0.319328i −0.987171 0.159664i \(-0.948959\pi\)
0.987171 0.159664i \(-0.0510411\pi\)
\(138\) 0 0
\(139\) −103.074 −0.741539 −0.370769 0.928725i \(-0.620906\pi\)
−0.370769 + 0.928725i \(0.620906\pi\)
\(140\) − 7.05069i − 0.0503620i
\(141\) 0 0
\(142\) 195.747 1.37850
\(143\) 0 0
\(144\) 0 0
\(145\) −231.756 −1.59832
\(146\) 67.3796i 0.461504i
\(147\) 0 0
\(148\) −1.07239 −0.00724590
\(149\) 90.4963i 0.607358i 0.952774 + 0.303679i \(0.0982150\pi\)
−0.952774 + 0.303679i \(0.901785\pi\)
\(150\) 0 0
\(151\) 240.354 1.59175 0.795873 0.605464i \(-0.207012\pi\)
0.795873 + 0.605464i \(0.207012\pi\)
\(152\) − 114.110i − 0.750727i
\(153\) 0 0
\(154\) 0 0
\(155\) 144.492i 0.932207i
\(156\) 0 0
\(157\) −29.6036 −0.188558 −0.0942789 0.995546i \(-0.530055\pi\)
−0.0942789 + 0.995546i \(0.530055\pi\)
\(158\) 121.277i 0.767576i
\(159\) 0 0
\(160\) 21.5576 0.134735
\(161\) − 29.7364i − 0.184698i
\(162\) 0 0
\(163\) 210.084 1.28886 0.644430 0.764663i \(-0.277095\pi\)
0.644430 + 0.764663i \(0.277095\pi\)
\(164\) 2.54888i 0.0155420i
\(165\) 0 0
\(166\) 128.158 0.772036
\(167\) 15.2464i 0.0912955i 0.998958 + 0.0456478i \(0.0145352\pi\)
−0.998958 + 0.0456478i \(0.985465\pi\)
\(168\) 0 0
\(169\) 127.689 0.755557
\(170\) 145.915i 0.858325i
\(171\) 0 0
\(172\) −7.52190 −0.0437320
\(173\) − 214.396i − 1.23928i −0.784885 0.619642i \(-0.787278\pi\)
0.784885 0.619642i \(-0.212722\pi\)
\(174\) 0 0
\(175\) 353.558 2.02033
\(176\) 0 0
\(177\) 0 0
\(178\) 181.923 1.02204
\(179\) − 204.595i − 1.14299i −0.820605 0.571495i \(-0.806364\pi\)
0.820605 0.571495i \(-0.193636\pi\)
\(180\) 0 0
\(181\) 136.759 0.755572 0.377786 0.925893i \(-0.376685\pi\)
0.377786 + 0.925893i \(0.376685\pi\)
\(182\) 177.006i 0.972561i
\(183\) 0 0
\(184\) 46.2422 0.251316
\(185\) − 73.6643i − 0.398185i
\(186\) 0 0
\(187\) 0 0
\(188\) 5.43686i 0.0289195i
\(189\) 0 0
\(190\) 265.222 1.39590
\(191\) 244.731i 1.28132i 0.767827 + 0.640658i \(0.221338\pi\)
−0.767827 + 0.640658i \(0.778662\pi\)
\(192\) 0 0
\(193\) −15.7364 −0.0815358 −0.0407679 0.999169i \(-0.512980\pi\)
−0.0407679 + 0.999169i \(0.512980\pi\)
\(194\) 102.995i 0.530902i
\(195\) 0 0
\(196\) −3.03160 −0.0154674
\(197\) − 95.4172i − 0.484351i −0.970232 0.242176i \(-0.922139\pi\)
0.970232 0.242176i \(-0.0778610\pi\)
\(198\) 0 0
\(199\) −65.2214 −0.327746 −0.163873 0.986481i \(-0.552399\pi\)
−0.163873 + 0.986481i \(0.552399\pi\)
\(200\) 549.807i 2.74904i
\(201\) 0 0
\(202\) 115.763 0.573082
\(203\) − 125.976i − 0.620569i
\(204\) 0 0
\(205\) −175.087 −0.854081
\(206\) 190.013i 0.922394i
\(207\) 0 0
\(208\) −265.605 −1.27695
\(209\) 0 0
\(210\) 0 0
\(211\) −275.123 −1.30390 −0.651951 0.758261i \(-0.726049\pi\)
−0.651951 + 0.758261i \(0.726049\pi\)
\(212\) 11.4329i 0.0539286i
\(213\) 0 0
\(214\) 36.1334 0.168848
\(215\) − 516.691i − 2.40321i
\(216\) 0 0
\(217\) −78.5415 −0.361942
\(218\) − 99.5094i − 0.456465i
\(219\) 0 0
\(220\) 0 0
\(221\) 132.944i 0.601558i
\(222\) 0 0
\(223\) 182.909 0.820218 0.410109 0.912037i \(-0.365491\pi\)
0.410109 + 0.912037i \(0.365491\pi\)
\(224\) 11.7181i 0.0523128i
\(225\) 0 0
\(226\) 19.3238 0.0855035
\(227\) − 250.411i − 1.10313i −0.834132 0.551565i \(-0.814031\pi\)
0.834132 0.551565i \(-0.185969\pi\)
\(228\) 0 0
\(229\) −350.485 −1.53050 −0.765252 0.643731i \(-0.777386\pi\)
−0.765252 + 0.643731i \(0.777386\pi\)
\(230\) 107.479i 0.467298i
\(231\) 0 0
\(232\) 195.901 0.844400
\(233\) − 364.712i − 1.56529i −0.622471 0.782643i \(-0.713871\pi\)
0.622471 0.782643i \(-0.286129\pi\)
\(234\) 0 0
\(235\) −373.466 −1.58922
\(236\) 0 6.08206e-5i 0 2.57715e-7i
\(237\) 0 0
\(238\) −79.3151 −0.333257
\(239\) 242.816i 1.01597i 0.861367 + 0.507983i \(0.169609\pi\)
−0.861367 + 0.507983i \(0.830391\pi\)
\(240\) 0 0
\(241\) −418.345 −1.73587 −0.867935 0.496677i \(-0.834553\pi\)
−0.867935 + 0.496677i \(0.834553\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.669580 0.00274418
\(245\) − 208.245i − 0.849980i
\(246\) 0 0
\(247\) 241.645 0.978321
\(248\) − 122.138i − 0.492490i
\(249\) 0 0
\(250\) −805.262 −3.22105
\(251\) 276.186i 1.10034i 0.835051 + 0.550172i \(0.185438\pi\)
−0.835051 + 0.550172i \(0.814562\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 423.348i − 1.66672i
\(255\) 0 0
\(256\) −26.8629 −0.104933
\(257\) − 161.198i − 0.627231i −0.949550 0.313615i \(-0.898460\pi\)
0.949550 0.313615i \(-0.101540\pi\)
\(258\) 0 0
\(259\) 40.0417 0.154601
\(260\) 23.2185i 0.0893019i
\(261\) 0 0
\(262\) −327.290 −1.24920
\(263\) 55.4784i 0.210944i 0.994422 + 0.105472i \(0.0336354\pi\)
−0.994422 + 0.105472i \(0.966365\pi\)
\(264\) 0 0
\(265\) −785.340 −2.96355
\(266\) 144.167i 0.541980i
\(267\) 0 0
\(268\) 15.6609 0.0584361
\(269\) − 58.2016i − 0.216363i −0.994131 0.108181i \(-0.965497\pi\)
0.994131 0.108181i \(-0.0345027\pi\)
\(270\) 0 0
\(271\) 425.905 1.57161 0.785803 0.618477i \(-0.212250\pi\)
0.785803 + 0.618477i \(0.212250\pi\)
\(272\) − 119.016i − 0.437558i
\(273\) 0 0
\(274\) −85.9501 −0.313686
\(275\) 0 0
\(276\) 0 0
\(277\) −127.884 −0.461677 −0.230838 0.972992i \(-0.574147\pi\)
−0.230838 + 0.972992i \(0.574147\pi\)
\(278\) 202.506i 0.728438i
\(279\) 0 0
\(280\) −409.392 −1.46211
\(281\) − 74.3983i − 0.264763i −0.991199 0.132381i \(-0.957738\pi\)
0.991199 0.132381i \(-0.0422624\pi\)
\(282\) 0 0
\(283\) 494.336 1.74677 0.873386 0.487029i \(-0.161919\pi\)
0.873386 + 0.487029i \(0.161919\pi\)
\(284\) 13.9571i 0.0491448i
\(285\) 0 0
\(286\) 0 0
\(287\) − 95.1718i − 0.331609i
\(288\) 0 0
\(289\) 229.429 0.793871
\(290\) 455.323i 1.57008i
\(291\) 0 0
\(292\) −4.80429 −0.0164531
\(293\) − 13.6119i − 0.0464568i −0.999730 0.0232284i \(-0.992606\pi\)
0.999730 0.0232284i \(-0.00739450\pi\)
\(294\) 0 0
\(295\) −0.00417786 −1.41622e−5 0
\(296\) 62.2676i 0.210363i
\(297\) 0 0
\(298\) 177.795 0.596628
\(299\) 97.9244i 0.327506i
\(300\) 0 0
\(301\) 280.857 0.933081
\(302\) − 472.215i − 1.56362i
\(303\) 0 0
\(304\) −216.328 −0.711605
\(305\) 45.9945i 0.150801i
\(306\) 0 0
\(307\) −215.268 −0.701198 −0.350599 0.936526i \(-0.614022\pi\)
−0.350599 + 0.936526i \(0.614022\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 283.879 0.915738
\(311\) 90.0470i 0.289540i 0.989465 + 0.144770i \(0.0462442\pi\)
−0.989465 + 0.144770i \(0.953756\pi\)
\(312\) 0 0
\(313\) 258.266 0.825130 0.412565 0.910928i \(-0.364633\pi\)
0.412565 + 0.910928i \(0.364633\pi\)
\(314\) 58.1611i 0.185227i
\(315\) 0 0
\(316\) −8.64728 −0.0273648
\(317\) 105.042i 0.331363i 0.986179 + 0.165681i \(0.0529824\pi\)
−0.986179 + 0.165681i \(0.947018\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 635.878i − 1.98712i
\(321\) 0 0
\(322\) −58.4221 −0.181435
\(323\) 108.279i 0.335231i
\(324\) 0 0
\(325\) −1164.30 −3.58245
\(326\) − 412.746i − 1.26609i
\(327\) 0 0
\(328\) 147.999 0.451216
\(329\) − 203.005i − 0.617036i
\(330\) 0 0
\(331\) 258.731 0.781664 0.390832 0.920462i \(-0.372187\pi\)
0.390832 + 0.920462i \(0.372187\pi\)
\(332\) 9.13791i 0.0275238i
\(333\) 0 0
\(334\) 29.9540 0.0896826
\(335\) 1075.77i 3.21125i
\(336\) 0 0
\(337\) −367.989 −1.09196 −0.545978 0.837799i \(-0.683842\pi\)
−0.545978 + 0.837799i \(0.683842\pi\)
\(338\) − 250.867i − 0.742209i
\(339\) 0 0
\(340\) −10.4040 −0.0306001
\(341\) 0 0
\(342\) 0 0
\(343\) 369.493 1.07724
\(344\) 436.753i 1.26963i
\(345\) 0 0
\(346\) −421.217 −1.21739
\(347\) − 417.073i − 1.20194i −0.799272 0.600969i \(-0.794781\pi\)
0.799272 0.600969i \(-0.205219\pi\)
\(348\) 0 0
\(349\) 279.831 0.801807 0.400903 0.916120i \(-0.368696\pi\)
0.400903 + 0.916120i \(0.368696\pi\)
\(350\) − 694.624i − 1.98464i
\(351\) 0 0
\(352\) 0 0
\(353\) 333.271i 0.944111i 0.881569 + 0.472056i \(0.156488\pi\)
−0.881569 + 0.472056i \(0.843512\pi\)
\(354\) 0 0
\(355\) −958.736 −2.70066
\(356\) 12.9715i 0.0364368i
\(357\) 0 0
\(358\) −401.962 −1.12280
\(359\) − 496.763i − 1.38374i −0.722022 0.691871i \(-0.756787\pi\)
0.722022 0.691871i \(-0.243213\pi\)
\(360\) 0 0
\(361\) −164.187 −0.454810
\(362\) − 268.685i − 0.742224i
\(363\) 0 0
\(364\) −12.6209 −0.0346727
\(365\) − 330.014i − 0.904148i
\(366\) 0 0
\(367\) 165.189 0.450106 0.225053 0.974346i \(-0.427744\pi\)
0.225053 + 0.974346i \(0.427744\pi\)
\(368\) − 87.6648i − 0.238220i
\(369\) 0 0
\(370\) −144.726 −0.391151
\(371\) − 426.887i − 1.15064i
\(372\) 0 0
\(373\) 393.501 1.05496 0.527481 0.849567i \(-0.323136\pi\)
0.527481 + 0.849567i \(0.323136\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 315.687 0.839592
\(377\) 414.848i 1.10039i
\(378\) 0 0
\(379\) 189.140 0.499049 0.249525 0.968368i \(-0.419726\pi\)
0.249525 + 0.968368i \(0.419726\pi\)
\(380\) 18.9108i 0.0497653i
\(381\) 0 0
\(382\) 480.815 1.25868
\(383\) − 468.923i − 1.22434i −0.790725 0.612171i \(-0.790296\pi\)
0.790725 0.612171i \(-0.209704\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 30.9168i 0.0800954i
\(387\) 0 0
\(388\) −7.34374 −0.0189272
\(389\) 454.980i 1.16961i 0.811173 + 0.584807i \(0.198830\pi\)
−0.811173 + 0.584807i \(0.801170\pi\)
\(390\) 0 0
\(391\) −43.8792 −0.112223
\(392\) 176.027i 0.449049i
\(393\) 0 0
\(394\) −187.463 −0.475794
\(395\) − 593.994i − 1.50378i
\(396\) 0 0
\(397\) 168.638 0.424782 0.212391 0.977185i \(-0.431875\pi\)
0.212391 + 0.977185i \(0.431875\pi\)
\(398\) 128.138i 0.321956i
\(399\) 0 0
\(400\) 1042.31 2.60578
\(401\) 69.7963i 0.174056i 0.996206 + 0.0870278i \(0.0277369\pi\)
−0.996206 + 0.0870278i \(0.972263\pi\)
\(402\) 0 0
\(403\) 258.644 0.641796
\(404\) 8.25409i 0.0204309i
\(405\) 0 0
\(406\) −247.500 −0.609606
\(407\) 0 0
\(408\) 0 0
\(409\) 553.915 1.35431 0.677157 0.735838i \(-0.263212\pi\)
0.677157 + 0.735838i \(0.263212\pi\)
\(410\) 343.987i 0.838992i
\(411\) 0 0
\(412\) −13.5483 −0.0328842
\(413\) − 0.00227096i 0 5.49869e-6i
\(414\) 0 0
\(415\) −627.697 −1.51252
\(416\) − 38.5886i − 0.0927610i
\(417\) 0 0
\(418\) 0 0
\(419\) − 307.485i − 0.733855i −0.930250 0.366927i \(-0.880410\pi\)
0.930250 0.366927i \(-0.119590\pi\)
\(420\) 0 0
\(421\) −3.54659 −0.00842420 −0.00421210 0.999991i \(-0.501341\pi\)
−0.00421210 + 0.999991i \(0.501341\pi\)
\(422\) 540.526i 1.28087i
\(423\) 0 0
\(424\) 663.839 1.56566
\(425\) − 521.713i − 1.22756i
\(426\) 0 0
\(427\) −25.0012 −0.0585508
\(428\) 2.57638i 0.00601959i
\(429\) 0 0
\(430\) −1015.12 −2.36076
\(431\) − 118.926i − 0.275930i −0.990437 0.137965i \(-0.955944\pi\)
0.990437 0.137965i \(-0.0440562\pi\)
\(432\) 0 0
\(433\) 199.452 0.460628 0.230314 0.973116i \(-0.426025\pi\)
0.230314 + 0.973116i \(0.426025\pi\)
\(434\) 154.308i 0.355548i
\(435\) 0 0
\(436\) 7.09521 0.0162734
\(437\) 79.7567i 0.182510i
\(438\) 0 0
\(439\) −7.94474 −0.0180973 −0.00904867 0.999959i \(-0.502880\pi\)
−0.00904867 + 0.999959i \(0.502880\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 261.191 0.590930
\(443\) 469.310i 1.05939i 0.848188 + 0.529696i \(0.177694\pi\)
−0.848188 + 0.529696i \(0.822306\pi\)
\(444\) 0 0
\(445\) −891.031 −2.00232
\(446\) − 359.354i − 0.805728i
\(447\) 0 0
\(448\) 345.644 0.771527
\(449\) 738.513i 1.64479i 0.568913 + 0.822397i \(0.307364\pi\)
−0.568913 + 0.822397i \(0.692636\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.37782i 0.00304828i
\(453\) 0 0
\(454\) −491.974 −1.08364
\(455\) − 866.946i − 1.90538i
\(456\) 0 0
\(457\) −181.188 −0.396473 −0.198236 0.980154i \(-0.563521\pi\)
−0.198236 + 0.980154i \(0.563521\pi\)
\(458\) 688.587i 1.50347i
\(459\) 0 0
\(460\) −7.66343 −0.0166596
\(461\) 561.409i 1.21781i 0.793244 + 0.608903i \(0.208390\pi\)
−0.793244 + 0.608903i \(0.791610\pi\)
\(462\) 0 0
\(463\) 379.155 0.818909 0.409454 0.912331i \(-0.365719\pi\)
0.409454 + 0.912331i \(0.365719\pi\)
\(464\) − 371.384i − 0.800397i
\(465\) 0 0
\(466\) −716.537 −1.53763
\(467\) − 456.305i − 0.977098i −0.872536 0.488549i \(-0.837526\pi\)
0.872536 0.488549i \(-0.162474\pi\)
\(468\) 0 0
\(469\) −584.756 −1.24681
\(470\) 733.737i 1.56114i
\(471\) 0 0
\(472\) 0.00353150 7.48199e−6 0
\(473\) 0 0
\(474\) 0 0
\(475\) −948.287 −1.99639
\(476\) − 5.65532i − 0.0118809i
\(477\) 0 0
\(478\) 477.052 0.998017
\(479\) − 538.316i − 1.12383i −0.827194 0.561917i \(-0.810064\pi\)
0.827194 0.561917i \(-0.189936\pi\)
\(480\) 0 0
\(481\) −131.861 −0.274138
\(482\) 821.908i 1.70520i
\(483\) 0 0
\(484\) 0 0
\(485\) − 504.452i − 1.04011i
\(486\) 0 0
\(487\) 194.690 0.399775 0.199887 0.979819i \(-0.435942\pi\)
0.199887 + 0.979819i \(0.435942\pi\)
\(488\) − 38.8786i − 0.0796693i
\(489\) 0 0
\(490\) −409.132 −0.834964
\(491\) − 203.856i − 0.415185i −0.978215 0.207593i \(-0.933437\pi\)
0.978215 0.207593i \(-0.0665628\pi\)
\(492\) 0 0
\(493\) −185.890 −0.377060
\(494\) − 474.752i − 0.961037i
\(495\) 0 0
\(496\) −231.546 −0.466826
\(497\) − 521.140i − 1.04857i
\(498\) 0 0
\(499\) −855.119 −1.71367 −0.856833 0.515595i \(-0.827571\pi\)
−0.856833 + 0.515595i \(0.827571\pi\)
\(500\) − 57.4167i − 0.114833i
\(501\) 0 0
\(502\) 542.614 1.08090
\(503\) 272.283i 0.541319i 0.962675 + 0.270659i \(0.0872417\pi\)
−0.962675 + 0.270659i \(0.912758\pi\)
\(504\) 0 0
\(505\) −566.986 −1.12274
\(506\) 0 0
\(507\) 0 0
\(508\) 30.1855 0.0594203
\(509\) − 943.531i − 1.85370i −0.375437 0.926848i \(-0.622507\pi\)
0.375437 0.926848i \(-0.377493\pi\)
\(510\) 0 0
\(511\) 179.386 0.351048
\(512\) 536.245i 1.04735i
\(513\) 0 0
\(514\) −316.701 −0.616150
\(515\) − 930.653i − 1.80709i
\(516\) 0 0
\(517\) 0 0
\(518\) − 78.6685i − 0.151870i
\(519\) 0 0
\(520\) 1348.16 2.59262
\(521\) 324.370i 0.622591i 0.950313 + 0.311295i \(0.100763\pi\)
−0.950313 + 0.311295i \(0.899237\pi\)
\(522\) 0 0
\(523\) 121.137 0.231620 0.115810 0.993271i \(-0.463054\pi\)
0.115810 + 0.993271i \(0.463054\pi\)
\(524\) − 23.3364i − 0.0445351i
\(525\) 0 0
\(526\) 108.997 0.207218
\(527\) 115.896i 0.219917i
\(528\) 0 0
\(529\) 496.679 0.938902
\(530\) 1542.93i 2.91119i
\(531\) 0 0
\(532\) −10.2794 −0.0193221
\(533\) 313.408i 0.588008i
\(534\) 0 0
\(535\) −176.976 −0.330795
\(536\) − 909.336i − 1.69652i
\(537\) 0 0
\(538\) −114.347 −0.212540
\(539\) 0 0
\(540\) 0 0
\(541\) −596.978 −1.10347 −0.551736 0.834019i \(-0.686034\pi\)
−0.551736 + 0.834019i \(0.686034\pi\)
\(542\) − 836.762i − 1.54384i
\(543\) 0 0
\(544\) 17.2913 0.0317854
\(545\) 487.381i 0.894277i
\(546\) 0 0
\(547\) −338.385 −0.618620 −0.309310 0.950961i \(-0.600098\pi\)
−0.309310 + 0.950961i \(0.600098\pi\)
\(548\) − 6.12841i − 0.0111832i
\(549\) 0 0
\(550\) 0 0
\(551\) 337.882i 0.613217i
\(552\) 0 0
\(553\) 322.877 0.583865
\(554\) 251.250i 0.453520i
\(555\) 0 0
\(556\) −14.4391 −0.0259695
\(557\) 477.662i 0.857562i 0.903408 + 0.428781i \(0.141057\pi\)
−0.903408 + 0.428781i \(0.858943\pi\)
\(558\) 0 0
\(559\) −924.886 −1.65454
\(560\) 776.116i 1.38592i
\(561\) 0 0
\(562\) −146.168 −0.260085
\(563\) 471.956i 0.838288i 0.907920 + 0.419144i \(0.137670\pi\)
−0.907920 + 0.419144i \(0.862330\pi\)
\(564\) 0 0
\(565\) −94.6447 −0.167513
\(566\) − 971.206i − 1.71591i
\(567\) 0 0
\(568\) 810.409 1.42678
\(569\) − 128.554i − 0.225929i −0.993599 0.112965i \(-0.963965\pi\)
0.993599 0.112965i \(-0.0360346\pi\)
\(570\) 0 0
\(571\) 266.234 0.466259 0.233130 0.972446i \(-0.425103\pi\)
0.233130 + 0.972446i \(0.425103\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −186.981 −0.325751
\(575\) − 384.284i − 0.668320i
\(576\) 0 0
\(577\) 720.993 1.24955 0.624777 0.780803i \(-0.285190\pi\)
0.624777 + 0.780803i \(0.285190\pi\)
\(578\) − 450.751i − 0.779846i
\(579\) 0 0
\(580\) −32.4654 −0.0559749
\(581\) − 341.197i − 0.587258i
\(582\) 0 0
\(583\) 0 0
\(584\) 278.957i 0.477667i
\(585\) 0 0
\(586\) −26.7428 −0.0456361
\(587\) − 411.867i − 0.701648i −0.936441 0.350824i \(-0.885902\pi\)
0.936441 0.350824i \(-0.114098\pi\)
\(588\) 0 0
\(589\) 210.658 0.357654
\(590\) 0.00820810i 0 1.39120e-5i
\(591\) 0 0
\(592\) 118.045 0.199401
\(593\) 750.336i 1.26532i 0.774429 + 0.632661i \(0.218037\pi\)
−0.774429 + 0.632661i \(0.781963\pi\)
\(594\) 0 0
\(595\) 388.472 0.652895
\(596\) 12.6771i 0.0212704i
\(597\) 0 0
\(598\) 192.389 0.321720
\(599\) 543.614i 0.907535i 0.891120 + 0.453768i \(0.149920\pi\)
−0.891120 + 0.453768i \(0.850080\pi\)
\(600\) 0 0
\(601\) −736.195 −1.22495 −0.612475 0.790490i \(-0.709826\pi\)
−0.612475 + 0.790490i \(0.709826\pi\)
\(602\) − 551.791i − 0.916596i
\(603\) 0 0
\(604\) 33.6698 0.0557447
\(605\) 0 0
\(606\) 0 0
\(607\) 559.641 0.921979 0.460989 0.887406i \(-0.347495\pi\)
0.460989 + 0.887406i \(0.347495\pi\)
\(608\) − 31.4293i − 0.0516930i
\(609\) 0 0
\(610\) 90.3638 0.148137
\(611\) 668.512i 1.09413i
\(612\) 0 0
\(613\) −109.915 −0.179306 −0.0896530 0.995973i \(-0.528576\pi\)
−0.0896530 + 0.995973i \(0.528576\pi\)
\(614\) 422.929i 0.688810i
\(615\) 0 0
\(616\) 0 0
\(617\) 61.1594i 0.0991238i 0.998771 + 0.0495619i \(0.0157825\pi\)
−0.998771 + 0.0495619i \(0.984217\pi\)
\(618\) 0 0
\(619\) 81.8184 0.132178 0.0660892 0.997814i \(-0.478948\pi\)
0.0660892 + 0.997814i \(0.478948\pi\)
\(620\) 20.2411i 0.0326469i
\(621\) 0 0
\(622\) 176.912 0.284425
\(623\) − 484.337i − 0.777427i
\(624\) 0 0
\(625\) 2254.17 3.60668
\(626\) − 507.406i − 0.810553i
\(627\) 0 0
\(628\) −4.14700 −0.00660351
\(629\) − 59.0857i − 0.0939360i
\(630\) 0 0
\(631\) 9.29210 0.0147260 0.00736300 0.999973i \(-0.497656\pi\)
0.00736300 + 0.999973i \(0.497656\pi\)
\(632\) 502.097i 0.794457i
\(633\) 0 0
\(634\) 206.373 0.325509
\(635\) 2073.49i 3.26533i
\(636\) 0 0
\(637\) −372.763 −0.585185
\(638\) 0 0
\(639\) 0 0
\(640\) −1163.06 −1.81728
\(641\) − 349.041i − 0.544525i −0.962223 0.272263i \(-0.912228\pi\)
0.962223 0.272263i \(-0.0877719\pi\)
\(642\) 0 0
\(643\) 906.166 1.40928 0.704639 0.709566i \(-0.251109\pi\)
0.704639 + 0.709566i \(0.251109\pi\)
\(644\) − 4.16561i − 0.00646833i
\(645\) 0 0
\(646\) 212.733 0.329308
\(647\) − 56.5419i − 0.0873909i −0.999045 0.0436954i \(-0.986087\pi\)
0.999045 0.0436954i \(-0.0139131\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2287.45i 3.51916i
\(651\) 0 0
\(652\) 29.4296 0.0451373
\(653\) − 568.547i − 0.870669i −0.900269 0.435334i \(-0.856630\pi\)
0.900269 0.435334i \(-0.143370\pi\)
\(654\) 0 0
\(655\) 1603.01 2.44735
\(656\) − 280.573i − 0.427702i
\(657\) 0 0
\(658\) −398.837 −0.606135
\(659\) 380.801i 0.577847i 0.957352 + 0.288923i \(0.0932973\pi\)
−0.957352 + 0.288923i \(0.906703\pi\)
\(660\) 0 0
\(661\) 350.773 0.530671 0.265335 0.964156i \(-0.414517\pi\)
0.265335 + 0.964156i \(0.414517\pi\)
\(662\) − 508.319i − 0.767854i
\(663\) 0 0
\(664\) 530.585 0.799074
\(665\) − 706.104i − 1.06181i
\(666\) 0 0
\(667\) −136.924 −0.205283
\(668\) 2.13578i 0.00319727i
\(669\) 0 0
\(670\) 2113.53 3.15452
\(671\) 0 0
\(672\) 0 0
\(673\) −482.896 −0.717527 −0.358763 0.933429i \(-0.616802\pi\)
−0.358763 + 0.933429i \(0.616802\pi\)
\(674\) 722.976i 1.07267i
\(675\) 0 0
\(676\) 17.8873 0.0264605
\(677\) − 849.679i − 1.25507i −0.778590 0.627533i \(-0.784065\pi\)
0.778590 0.627533i \(-0.215935\pi\)
\(678\) 0 0
\(679\) 274.205 0.403836
\(680\) 604.101i 0.888384i
\(681\) 0 0
\(682\) 0 0
\(683\) − 831.602i − 1.21757i −0.793334 0.608786i \(-0.791657\pi\)
0.793334 0.608786i \(-0.208343\pi\)
\(684\) 0 0
\(685\) 420.969 0.614554
\(686\) − 725.931i − 1.05821i
\(687\) 0 0
\(688\) 827.986 1.20347
\(689\) 1405.77i 2.04031i
\(690\) 0 0
\(691\) 910.760 1.31803 0.659016 0.752129i \(-0.270973\pi\)
0.659016 + 0.752129i \(0.270973\pi\)
\(692\) − 30.0336i − 0.0434011i
\(693\) 0 0
\(694\) −819.409 −1.18070
\(695\) − 991.840i − 1.42711i
\(696\) 0 0
\(697\) −140.436 −0.201486
\(698\) − 549.774i − 0.787642i
\(699\) 0 0
\(700\) 49.5280 0.0707543
\(701\) − 900.411i − 1.28447i −0.766509 0.642234i \(-0.778008\pi\)
0.766509 0.642234i \(-0.221992\pi\)
\(702\) 0 0
\(703\) −107.397 −0.152769
\(704\) 0 0
\(705\) 0 0
\(706\) 654.767 0.927432
\(707\) − 308.196i − 0.435921i
\(708\) 0 0
\(709\) −922.297 −1.30084 −0.650421 0.759574i \(-0.725408\pi\)
−0.650421 + 0.759574i \(0.725408\pi\)
\(710\) 1883.60i 2.65295i
\(711\) 0 0
\(712\) 753.178 1.05783
\(713\) 85.3672i 0.119730i
\(714\) 0 0
\(715\) 0 0
\(716\) − 28.6606i − 0.0400288i
\(717\) 0 0
\(718\) −975.974 −1.35930
\(719\) − 10.3358i − 0.0143753i −0.999974 0.00718763i \(-0.997712\pi\)
0.999974 0.00718763i \(-0.00228791\pi\)
\(720\) 0 0
\(721\) 505.875 0.701629
\(722\) 322.572i 0.446775i
\(723\) 0 0
\(724\) 19.1578 0.0264610
\(725\) − 1627.99i − 2.24550i
\(726\) 0 0
\(727\) 119.809 0.164799 0.0823995 0.996599i \(-0.473742\pi\)
0.0823995 + 0.996599i \(0.473742\pi\)
\(728\) 732.820i 1.00662i
\(729\) 0 0
\(730\) −648.368 −0.888175
\(731\) − 414.435i − 0.566942i
\(732\) 0 0
\(733\) 1272.26 1.73570 0.867848 0.496830i \(-0.165503\pi\)
0.867848 + 0.496830i \(0.165503\pi\)
\(734\) − 324.541i − 0.442154i
\(735\) 0 0
\(736\) 12.7364 0.0173049
\(737\) 0 0
\(738\) 0 0
\(739\) −168.978 −0.228658 −0.114329 0.993443i \(-0.536472\pi\)
−0.114329 + 0.993443i \(0.536472\pi\)
\(740\) − 10.3192i − 0.0139449i
\(741\) 0 0
\(742\) −838.691 −1.13031
\(743\) 963.291i 1.29649i 0.761432 + 0.648245i \(0.224497\pi\)
−0.761432 + 0.648245i \(0.775503\pi\)
\(744\) 0 0
\(745\) −870.811 −1.16887
\(746\) − 773.098i − 1.03632i
\(747\) 0 0
\(748\) 0 0
\(749\) − 96.1986i − 0.128436i
\(750\) 0 0
\(751\) −412.835 −0.549713 −0.274857 0.961485i \(-0.588630\pi\)
−0.274857 + 0.961485i \(0.588630\pi\)
\(752\) − 598.472i − 0.795840i
\(753\) 0 0
\(754\) 815.038 1.08095
\(755\) 2312.83i 3.06335i
\(756\) 0 0
\(757\) 902.073 1.19164 0.595821 0.803117i \(-0.296827\pi\)
0.595821 + 0.803117i \(0.296827\pi\)
\(758\) − 371.596i − 0.490233i
\(759\) 0 0
\(760\) 1098.04 1.44479
\(761\) − 972.116i − 1.27742i −0.769448 0.638709i \(-0.779469\pi\)
0.769448 0.638709i \(-0.220531\pi\)
\(762\) 0 0
\(763\) −264.925 −0.347215
\(764\) 34.2830i 0.0448731i
\(765\) 0 0
\(766\) −921.278 −1.20271
\(767\) 0.00747845i 0 9.75027e-6i
\(768\) 0 0
\(769\) 1404.29 1.82613 0.913064 0.407817i \(-0.133710\pi\)
0.913064 + 0.407817i \(0.133710\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.20443 −0.00285548
\(773\) 145.730i 0.188525i 0.995547 + 0.0942627i \(0.0300494\pi\)
−0.995547 + 0.0942627i \(0.969951\pi\)
\(774\) 0 0
\(775\) −1014.99 −1.30967
\(776\) 426.408i 0.549494i
\(777\) 0 0
\(778\) 893.883 1.14895
\(779\) 255.263i 0.327680i
\(780\) 0 0
\(781\) 0 0
\(782\) 86.2080i 0.110240i
\(783\) 0 0
\(784\) 333.709 0.425649
\(785\) − 284.864i − 0.362884i
\(786\) 0 0
\(787\) −914.791 −1.16238 −0.581189 0.813769i \(-0.697412\pi\)
−0.581189 + 0.813769i \(0.697412\pi\)
\(788\) − 13.3665i − 0.0169625i
\(789\) 0 0
\(790\) −1167.00 −1.47722
\(791\) − 51.4460i − 0.0650392i
\(792\) 0 0
\(793\) 82.3310 0.103822
\(794\) − 331.318i − 0.417277i
\(795\) 0 0
\(796\) −9.13651 −0.0114780
\(797\) 373.062i 0.468082i 0.972227 + 0.234041i \(0.0751951\pi\)
−0.972227 + 0.234041i \(0.924805\pi\)
\(798\) 0 0
\(799\) −299.555 −0.374913
\(800\) 151.433i 0.189291i
\(801\) 0 0
\(802\) 137.126 0.170981
\(803\) 0 0
\(804\) 0 0
\(805\) 286.142 0.355456
\(806\) − 508.149i − 0.630457i
\(807\) 0 0
\(808\) 479.267 0.593152
\(809\) − 1578.26i − 1.95088i −0.220263 0.975440i \(-0.570692\pi\)
0.220263 0.975440i \(-0.429308\pi\)
\(810\) 0 0
\(811\) 720.185 0.888021 0.444010 0.896022i \(-0.353555\pi\)
0.444010 + 0.896022i \(0.353555\pi\)
\(812\) − 17.6472i − 0.0217330i
\(813\) 0 0
\(814\) 0 0
\(815\) 2021.56i 2.48044i
\(816\) 0 0
\(817\) −753.295 −0.922025
\(818\) − 1088.26i − 1.33039i
\(819\) 0 0
\(820\) −24.5269 −0.0299109
\(821\) − 932.683i − 1.13603i −0.823017 0.568017i \(-0.807711\pi\)
0.823017 0.568017i \(-0.192289\pi\)
\(822\) 0 0
\(823\) 499.235 0.606604 0.303302 0.952895i \(-0.401911\pi\)
0.303302 + 0.952895i \(0.401911\pi\)
\(824\) 786.670i 0.954697i
\(825\) 0 0
\(826\) −0.00446168 −5.40155e−6 0
\(827\) 729.473i 0.882072i 0.897490 + 0.441036i \(0.145389\pi\)
−0.897490 + 0.441036i \(0.854611\pi\)
\(828\) 0 0
\(829\) −1480.29 −1.78564 −0.892818 0.450418i \(-0.851275\pi\)
−0.892818 + 0.450418i \(0.851275\pi\)
\(830\) 1233.22i 1.48580i
\(831\) 0 0
\(832\) −1138.23 −1.36807
\(833\) − 167.032i − 0.200519i
\(834\) 0 0
\(835\) −146.710 −0.175700
\(836\) 0 0
\(837\) 0 0
\(838\) −604.106 −0.720890
\(839\) − 980.006i − 1.16806i −0.811730 0.584032i \(-0.801474\pi\)
0.811730 0.584032i \(-0.198526\pi\)
\(840\) 0 0
\(841\) 260.936 0.310268
\(842\) 6.96787i 0.00827538i
\(843\) 0 0
\(844\) −38.5405 −0.0456641
\(845\) 1228.70i 1.45409i
\(846\) 0 0
\(847\) 0 0
\(848\) − 1258.49i − 1.48407i
\(849\) 0 0
\(850\) −1024.99 −1.20587
\(851\) − 43.5215i − 0.0511416i
\(852\) 0 0
\(853\) −1390.45 −1.63007 −0.815037 0.579408i \(-0.803284\pi\)
−0.815037 + 0.579408i \(0.803284\pi\)
\(854\) 49.1190i 0.0575164i
\(855\) 0 0
\(856\) 149.595 0.174761
\(857\) 1044.64i 1.21895i 0.792804 + 0.609477i \(0.208621\pi\)
−0.792804 + 0.609477i \(0.791379\pi\)
\(858\) 0 0
\(859\) 480.571 0.559454 0.279727 0.960080i \(-0.409756\pi\)
0.279727 + 0.960080i \(0.409756\pi\)
\(860\) − 72.3803i − 0.0841632i
\(861\) 0 0
\(862\) −233.650 −0.271056
\(863\) 652.084i 0.755602i 0.925887 + 0.377801i \(0.123320\pi\)
−0.925887 + 0.377801i \(0.876680\pi\)
\(864\) 0 0
\(865\) 2063.05 2.38503
\(866\) − 391.856i − 0.452490i
\(867\) 0 0
\(868\) −11.0024 −0.0126756
\(869\) 0 0
\(870\) 0 0
\(871\) 1925.65 2.21085
\(872\) − 411.977i − 0.472451i
\(873\) 0 0
\(874\) 156.695 0.179285
\(875\) 2143.86i 2.45013i
\(876\) 0 0
\(877\) −348.132 −0.396958 −0.198479 0.980105i \(-0.563600\pi\)
−0.198479 + 0.980105i \(0.563600\pi\)
\(878\) 15.6088i 0.0177776i
\(879\) 0 0
\(880\) 0 0
\(881\) − 81.0204i − 0.0919641i −0.998942 0.0459821i \(-0.985358\pi\)
0.998942 0.0459821i \(-0.0146417\pi\)
\(882\) 0 0
\(883\) 975.621 1.10489 0.552447 0.833548i \(-0.313694\pi\)
0.552447 + 0.833548i \(0.313694\pi\)
\(884\) 18.6234i 0.0210672i
\(885\) 0 0
\(886\) 922.039 1.04068
\(887\) − 1534.16i − 1.72960i −0.502113 0.864802i \(-0.667444\pi\)
0.502113 0.864802i \(-0.332556\pi\)
\(888\) 0 0
\(889\) −1127.08 −1.26781
\(890\) 1750.58i 1.96694i
\(891\) 0 0
\(892\) 25.6227 0.0287250
\(893\) 544.485i 0.609725i
\(894\) 0 0
\(895\) 1968.74 2.19971
\(896\) − 632.203i − 0.705584i
\(897\) 0 0
\(898\) 1450.93 1.61574
\(899\) 361.650i 0.402281i
\(900\) 0 0
\(901\) −629.917 −0.699131
\(902\) 0 0
\(903\) 0 0
\(904\) 80.0021 0.0884979
\(905\) 1315.97i 1.45412i
\(906\) 0 0
\(907\) 1040.31 1.14698 0.573490 0.819213i \(-0.305589\pi\)
0.573490 + 0.819213i \(0.305589\pi\)
\(908\) − 35.0787i − 0.0386329i
\(909\) 0 0
\(910\) −1703.26 −1.87171
\(911\) 373.647i 0.410150i 0.978746 + 0.205075i \(0.0657439\pi\)
−0.978746 + 0.205075i \(0.934256\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 355.974i 0.389468i
\(915\) 0 0
\(916\) −49.0976 −0.0536000
\(917\) 871.349i 0.950217i
\(918\) 0 0
\(919\) 4.74930 0.00516790 0.00258395 0.999997i \(-0.499178\pi\)
0.00258395 + 0.999997i \(0.499178\pi\)
\(920\) 444.970i 0.483663i
\(921\) 0 0
\(922\) 1102.98 1.19629
\(923\) 1716.16i 1.85932i
\(924\) 0 0
\(925\) 517.460 0.559416
\(926\) − 744.913i − 0.804442i
\(927\) 0 0
\(928\) 53.9568 0.0581431
\(929\) − 493.733i − 0.531467i −0.964047 0.265733i \(-0.914386\pi\)
0.964047 0.265733i \(-0.0856141\pi\)
\(930\) 0 0
\(931\) −303.605 −0.326107
\(932\) − 51.0904i − 0.0548181i
\(933\) 0 0
\(934\) −896.487 −0.959836
\(935\) 0 0
\(936\) 0 0
\(937\) −587.140 −0.626617 −0.313308 0.949651i \(-0.601437\pi\)
−0.313308 + 0.949651i \(0.601437\pi\)
\(938\) 1148.85i 1.22479i
\(939\) 0 0
\(940\) −52.3168 −0.0556562
\(941\) 1066.36i 1.13322i 0.823986 + 0.566610i \(0.191745\pi\)
−0.823986 + 0.566610i \(0.808255\pi\)
\(942\) 0 0
\(943\) −103.443 −0.109695
\(944\) − 0.00669494i 0 7.09209e-6i
\(945\) 0 0
\(946\) 0 0
\(947\) 198.618i 0.209734i 0.994486 + 0.104867i \(0.0334416\pi\)
−0.994486 + 0.104867i \(0.966558\pi\)
\(948\) 0 0
\(949\) −590.732 −0.622478
\(950\) 1863.07i 1.96112i
\(951\) 0 0
\(952\) −328.371 −0.344928
\(953\) − 1114.25i − 1.16920i −0.811321 0.584601i \(-0.801251\pi\)
0.811321 0.584601i \(-0.198749\pi\)
\(954\) 0 0
\(955\) −2354.95 −2.46592
\(956\) 34.0147i 0.0355802i
\(957\) 0 0
\(958\) −1057.61 −1.10398
\(959\) 228.826i 0.238609i
\(960\) 0 0
\(961\) −735.523 −0.765373
\(962\) 259.062i 0.269295i
\(963\) 0 0
\(964\) −58.6036 −0.0607921
\(965\) − 151.425i − 0.156918i
\(966\) 0 0
\(967\) −983.820 −1.01739 −0.508697 0.860945i \(-0.669873\pi\)
−0.508697 + 0.860945i \(0.669873\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −991.080 −1.02173
\(971\) − 115.916i − 0.119378i −0.998217 0.0596890i \(-0.980989\pi\)
0.998217 0.0596890i \(-0.0190109\pi\)
\(972\) 0 0
\(973\) 539.134 0.554095
\(974\) − 382.502i − 0.392712i
\(975\) 0 0
\(976\) −73.7052 −0.0755176
\(977\) 114.282i 0.116973i 0.998288 + 0.0584863i \(0.0186274\pi\)
−0.998288 + 0.0584863i \(0.981373\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 29.1719i − 0.0297673i
\(981\) 0 0
\(982\) −400.509 −0.407850
\(983\) − 1020.67i − 1.03832i −0.854677 0.519160i \(-0.826245\pi\)
0.854677 0.519160i \(-0.173755\pi\)
\(984\) 0 0
\(985\) 918.163 0.932145
\(986\) 365.213i 0.370398i
\(987\) 0 0
\(988\) 33.8508 0.0342619
\(989\) − 305.265i − 0.308661i
\(990\) 0 0
\(991\) 544.638 0.549584 0.274792 0.961504i \(-0.411391\pi\)
0.274792 + 0.961504i \(0.411391\pi\)
\(992\) − 33.6402i − 0.0339115i
\(993\) 0 0
\(994\) −1023.87 −1.03005
\(995\) − 627.600i − 0.630754i
\(996\) 0 0
\(997\) 1249.59 1.25335 0.626677 0.779279i \(-0.284415\pi\)
0.626677 + 0.779279i \(0.284415\pi\)
\(998\) 1680.02i 1.68339i
\(999\) 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.485.5 16
3.2 odd 2 inner 1089.3.b.i.485.12 16
11.5 even 5 99.3.l.a.80.6 yes 32
11.9 even 5 99.3.l.a.26.3 32
11.10 odd 2 1089.3.b.j.485.12 16
33.5 odd 10 99.3.l.a.80.3 yes 32
33.20 odd 10 99.3.l.a.26.6 yes 32
33.32 even 2 1089.3.b.j.485.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.26.3 32 11.9 even 5
99.3.l.a.26.6 yes 32 33.20 odd 10
99.3.l.a.80.3 yes 32 33.5 odd 10
99.3.l.a.80.6 yes 32 11.5 even 5
1089.3.b.i.485.5 16 1.1 even 1 trivial
1089.3.b.i.485.12 16 3.2 odd 2 inner
1089.3.b.j.485.5 16 33.32 even 2
1089.3.b.j.485.12 16 11.10 odd 2