Properties

Label 1089.3.b.d.485.3
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{15})\)
Defining polynomial: \(x^{4} + 16 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.3
Root \(2.03151i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.d.485.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} +1.12702 q^{7} +7.86799i q^{8} +O(q^{10})\) \(q+2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} +1.12702 q^{7} +7.86799i q^{8} +1.61895 q^{10} +9.25403 q^{13} +2.28954i q^{14} -16.4919 q^{16} +11.7514i q^{17} +17.2379 q^{19} +0.101222i q^{20} +5.11797i q^{23} +24.3649 q^{25} +18.7996i q^{26} -0.143150 q^{28} -29.0812i q^{29} -52.7298 q^{31} -2.03151i q^{32} -23.8730 q^{34} -0.898143i q^{35} +37.3649 q^{37} +35.0189i q^{38} +6.27017 q^{40} +75.0089i q^{41} +68.8407 q^{43} -10.3972 q^{46} +9.36061i q^{47} -47.7298 q^{49} +49.4975i q^{50} -1.17542 q^{52} -34.0195i q^{53} +8.86735i q^{56} +59.0786 q^{58} +69.2965i q^{59} -24.9839 q^{61} -107.121i q^{62} -61.8407 q^{64} -7.37473i q^{65} -16.8730 q^{67} -1.49262i q^{68} +1.82458 q^{70} +77.0632i q^{71} +55.2379 q^{73} +75.9071i q^{74} -2.18950 q^{76} +64.9839 q^{79} +13.1428i q^{80} -152.381 q^{82} +120.342i q^{83} +9.36492 q^{85} +139.850i q^{86} +110.926i q^{89} +10.4294 q^{91} -0.650067i q^{92} -19.0161 q^{94} -13.7372i q^{95} -14.0948 q^{97} -96.9634i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 20 q^{7} + O(q^{10}) \) \( 4 q - 16 q^{4} + 20 q^{7} - 40 q^{10} + 68 q^{13} - 4 q^{16} - 24 q^{19} + 20 q^{25} - 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} + 180 q^{40} + 12 q^{43} - 212 q^{46} - 36 q^{49} - 392 q^{52} - 120 q^{58} + 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} + 128 q^{73} + 456 q^{76} + 136 q^{79} - 656 q^{82} - 40 q^{85} + 460 q^{91} - 200 q^{94} + 176 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.03151i 1.01575i 0.861430 + 0.507877i \(0.169569\pi\)
−0.861430 + 0.507877i \(0.830431\pi\)
\(3\) 0 0
\(4\) −0.127017 −0.0317542
\(5\) − 0.796921i − 0.159384i −0.996820 0.0796921i \(-0.974606\pi\)
0.996820 0.0796921i \(-0.0253937\pi\)
\(6\) 0 0
\(7\) 1.12702 0.161002 0.0805012 0.996755i \(-0.474348\pi\)
0.0805012 + 0.996755i \(0.474348\pi\)
\(8\) 7.86799i 0.983499i
\(9\) 0 0
\(10\) 1.61895 0.161895
\(11\) 0 0
\(12\) 0 0
\(13\) 9.25403 0.711849 0.355924 0.934515i \(-0.384166\pi\)
0.355924 + 0.934515i \(0.384166\pi\)
\(14\) 2.28954i 0.163539i
\(15\) 0 0
\(16\) −16.4919 −1.03075
\(17\) 11.7514i 0.691257i 0.938371 + 0.345629i \(0.112334\pi\)
−0.938371 + 0.345629i \(0.887666\pi\)
\(18\) 0 0
\(19\) 17.2379 0.907258 0.453629 0.891191i \(-0.350129\pi\)
0.453629 + 0.891191i \(0.350129\pi\)
\(20\) 0.101222i 0.00506111i
\(21\) 0 0
\(22\) 0 0
\(23\) 5.11797i 0.222520i 0.993791 + 0.111260i \(0.0354887\pi\)
−0.993791 + 0.111260i \(0.964511\pi\)
\(24\) 0 0
\(25\) 24.3649 0.974597
\(26\) 18.7996i 0.723062i
\(27\) 0 0
\(28\) −0.143150 −0.00511250
\(29\) − 29.0812i − 1.00280i −0.865216 0.501400i \(-0.832819\pi\)
0.865216 0.501400i \(-0.167181\pi\)
\(30\) 0 0
\(31\) −52.7298 −1.70096 −0.850481 0.526005i \(-0.823689\pi\)
−0.850481 + 0.526005i \(0.823689\pi\)
\(32\) − 2.03151i − 0.0634846i
\(33\) 0 0
\(34\) −23.8730 −0.702147
\(35\) − 0.898143i − 0.0256612i
\(36\) 0 0
\(37\) 37.3649 1.00986 0.504931 0.863160i \(-0.331518\pi\)
0.504931 + 0.863160i \(0.331518\pi\)
\(38\) 35.0189i 0.921550i
\(39\) 0 0
\(40\) 6.27017 0.156754
\(41\) 75.0089i 1.82949i 0.404037 + 0.914743i \(0.367607\pi\)
−0.404037 + 0.914743i \(0.632393\pi\)
\(42\) 0 0
\(43\) 68.8407 1.60095 0.800473 0.599368i \(-0.204581\pi\)
0.800473 + 0.599368i \(0.204581\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.3972 −0.226026
\(47\) 9.36061i 0.199162i 0.995029 + 0.0995809i \(0.0317502\pi\)
−0.995029 + 0.0995809i \(0.968250\pi\)
\(48\) 0 0
\(49\) −47.7298 −0.974078
\(50\) 49.4975i 0.989949i
\(51\) 0 0
\(52\) −1.17542 −0.0226042
\(53\) − 34.0195i − 0.641878i −0.947100 0.320939i \(-0.896002\pi\)
0.947100 0.320939i \(-0.103998\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.86735i 0.158346i
\(57\) 0 0
\(58\) 59.0786 1.01860
\(59\) 69.2965i 1.17452i 0.809400 + 0.587258i \(0.199793\pi\)
−0.809400 + 0.587258i \(0.800207\pi\)
\(60\) 0 0
\(61\) −24.9839 −0.409572 −0.204786 0.978807i \(-0.565650\pi\)
−0.204786 + 0.978807i \(0.565650\pi\)
\(62\) − 107.121i − 1.72776i
\(63\) 0 0
\(64\) −61.8407 −0.966261
\(65\) − 7.37473i − 0.113457i
\(66\) 0 0
\(67\) −16.8730 −0.251836 −0.125918 0.992041i \(-0.540188\pi\)
−0.125918 + 0.992041i \(0.540188\pi\)
\(68\) − 1.49262i − 0.0219503i
\(69\) 0 0
\(70\) 1.82458 0.0260655
\(71\) 77.0632i 1.08540i 0.839927 + 0.542699i \(0.182597\pi\)
−0.839927 + 0.542699i \(0.817403\pi\)
\(72\) 0 0
\(73\) 55.2379 0.756684 0.378342 0.925666i \(-0.376494\pi\)
0.378342 + 0.925666i \(0.376494\pi\)
\(74\) 75.9071i 1.02577i
\(75\) 0 0
\(76\) −2.18950 −0.0288092
\(77\) 0 0
\(78\) 0 0
\(79\) 64.9839 0.822581 0.411290 0.911504i \(-0.365078\pi\)
0.411290 + 0.911504i \(0.365078\pi\)
\(80\) 13.1428i 0.164285i
\(81\) 0 0
\(82\) −152.381 −1.85831
\(83\) 120.342i 1.44991i 0.688799 + 0.724953i \(0.258138\pi\)
−0.688799 + 0.724953i \(0.741862\pi\)
\(84\) 0 0
\(85\) 9.36492 0.110175
\(86\) 139.850i 1.62617i
\(87\) 0 0
\(88\) 0 0
\(89\) 110.926i 1.24636i 0.782079 + 0.623179i \(0.214159\pi\)
−0.782079 + 0.623179i \(0.785841\pi\)
\(90\) 0 0
\(91\) 10.4294 0.114609
\(92\) − 0.650067i − 0.00706595i
\(93\) 0 0
\(94\) −19.0161 −0.202299
\(95\) − 13.7372i − 0.144603i
\(96\) 0 0
\(97\) −14.0948 −0.145307 −0.0726534 0.997357i \(-0.523147\pi\)
−0.0726534 + 0.997357i \(0.523147\pi\)
\(98\) − 96.9634i − 0.989423i
\(99\) 0 0
\(100\) −3.09475 −0.0309475
\(101\) 25.1194i 0.248707i 0.992238 + 0.124353i \(0.0396857\pi\)
−0.992238 + 0.124353i \(0.960314\pi\)
\(102\) 0 0
\(103\) −56.3649 −0.547232 −0.273616 0.961839i \(-0.588220\pi\)
−0.273616 + 0.961839i \(0.588220\pi\)
\(104\) 72.8106i 0.700102i
\(105\) 0 0
\(106\) 69.1109 0.651989
\(107\) 80.7670i 0.754832i 0.926044 + 0.377416i \(0.123187\pi\)
−0.926044 + 0.377416i \(0.876813\pi\)
\(108\) 0 0
\(109\) 105.698 0.969702 0.484851 0.874597i \(-0.338874\pi\)
0.484851 + 0.874597i \(0.338874\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −18.5867 −0.165953
\(113\) − 182.444i − 1.61454i −0.590179 0.807272i \(-0.700943\pi\)
0.590179 0.807272i \(-0.299057\pi\)
\(114\) 0 0
\(115\) 4.07862 0.0354662
\(116\) 3.69380i 0.0318431i
\(117\) 0 0
\(118\) −140.776 −1.19302
\(119\) 13.2440i 0.111294i
\(120\) 0 0
\(121\) 0 0
\(122\) − 50.7549i − 0.416024i
\(123\) 0 0
\(124\) 6.69757 0.0540126
\(125\) − 39.3399i − 0.314720i
\(126\) 0 0
\(127\) −143.492 −1.12986 −0.564929 0.825140i \(-0.691096\pi\)
−0.564929 + 0.825140i \(0.691096\pi\)
\(128\) − 133.756i − 1.04497i
\(129\) 0 0
\(130\) 14.9818 0.115245
\(131\) − 69.8125i − 0.532920i −0.963846 0.266460i \(-0.914146\pi\)
0.963846 0.266460i \(-0.0858540\pi\)
\(132\) 0 0
\(133\) 19.4274 0.146071
\(134\) − 34.2776i − 0.255803i
\(135\) 0 0
\(136\) −92.4597 −0.679850
\(137\) 227.754i 1.66244i 0.555946 + 0.831219i \(0.312356\pi\)
−0.555946 + 0.831219i \(0.687644\pi\)
\(138\) 0 0
\(139\) 102.808 0.739629 0.369815 0.929106i \(-0.379421\pi\)
0.369815 + 0.929106i \(0.379421\pi\)
\(140\) 0.114079i 0 0.000814851i
\(141\) 0 0
\(142\) −156.554 −1.10250
\(143\) 0 0
\(144\) 0 0
\(145\) −23.1754 −0.159830
\(146\) 112.216i 0.768604i
\(147\) 0 0
\(148\) −4.74597 −0.0320673
\(149\) 2.27958i 0.0152992i 0.999971 + 0.00764960i \(0.00243497\pi\)
−0.999971 + 0.00764960i \(0.997565\pi\)
\(150\) 0 0
\(151\) −233.714 −1.54777 −0.773886 0.633324i \(-0.781690\pi\)
−0.773886 + 0.633324i \(0.781690\pi\)
\(152\) 135.628i 0.892287i
\(153\) 0 0
\(154\) 0 0
\(155\) 42.0215i 0.271107i
\(156\) 0 0
\(157\) −170.157 −1.08380 −0.541902 0.840442i \(-0.682296\pi\)
−0.541902 + 0.840442i \(0.682296\pi\)
\(158\) 132.015i 0.835539i
\(159\) 0 0
\(160\) −1.61895 −0.0101184
\(161\) 5.76804i 0.0358263i
\(162\) 0 0
\(163\) 118.333 0.725967 0.362984 0.931796i \(-0.381758\pi\)
0.362984 + 0.931796i \(0.381758\pi\)
\(164\) − 9.52738i − 0.0580938i
\(165\) 0 0
\(166\) −244.476 −1.47275
\(167\) − 195.857i − 1.17280i −0.810023 0.586399i \(-0.800545\pi\)
0.810023 0.586399i \(-0.199455\pi\)
\(168\) 0 0
\(169\) −83.3629 −0.493271
\(170\) 19.0249i 0.111911i
\(171\) 0 0
\(172\) −8.74392 −0.0508367
\(173\) − 171.185i − 0.989511i −0.869032 0.494755i \(-0.835258\pi\)
0.869032 0.494755i \(-0.164742\pi\)
\(174\) 0 0
\(175\) 27.4597 0.156912
\(176\) 0 0
\(177\) 0 0
\(178\) −225.347 −1.26599
\(179\) − 215.228i − 1.20239i −0.799101 0.601197i \(-0.794691\pi\)
0.799101 0.601197i \(-0.205309\pi\)
\(180\) 0 0
\(181\) 237.730 1.31342 0.656712 0.754141i \(-0.271947\pi\)
0.656712 + 0.754141i \(0.271947\pi\)
\(182\) 21.1875i 0.116415i
\(183\) 0 0
\(184\) −40.2681 −0.218848
\(185\) − 29.7769i − 0.160956i
\(186\) 0 0
\(187\) 0 0
\(188\) − 1.18895i − 0.00632422i
\(189\) 0 0
\(190\) 27.9073 0.146881
\(191\) − 337.929i − 1.76926i −0.466294 0.884630i \(-0.654411\pi\)
0.466294 0.884630i \(-0.345589\pi\)
\(192\) 0 0
\(193\) −252.903 −1.31038 −0.655190 0.755464i \(-0.727411\pi\)
−0.655190 + 0.755464i \(0.727411\pi\)
\(194\) − 28.6336i − 0.147596i
\(195\) 0 0
\(196\) 6.06248 0.0309310
\(197\) − 94.1578i − 0.477959i −0.971025 0.238979i \(-0.923187\pi\)
0.971025 0.238979i \(-0.0768128\pi\)
\(198\) 0 0
\(199\) 179.571 0.902365 0.451182 0.892432i \(-0.351002\pi\)
0.451182 + 0.892432i \(0.351002\pi\)
\(200\) 191.703i 0.958514i
\(201\) 0 0
\(202\) −51.0302 −0.252625
\(203\) − 32.7750i − 0.161453i
\(204\) 0 0
\(205\) 59.7762 0.291591
\(206\) − 114.506i − 0.555853i
\(207\) 0 0
\(208\) −152.617 −0.733735
\(209\) 0 0
\(210\) 0 0
\(211\) 200.175 0.948699 0.474349 0.880337i \(-0.342683\pi\)
0.474349 + 0.880337i \(0.342683\pi\)
\(212\) 4.32105i 0.0203823i
\(213\) 0 0
\(214\) −164.079 −0.766723
\(215\) − 54.8606i − 0.255166i
\(216\) 0 0
\(217\) −59.4274 −0.273859
\(218\) 214.725i 0.984978i
\(219\) 0 0
\(220\) 0 0
\(221\) 108.748i 0.492071i
\(222\) 0 0
\(223\) −182.984 −0.820555 −0.410278 0.911961i \(-0.634568\pi\)
−0.410278 + 0.911961i \(0.634568\pi\)
\(224\) − 2.28954i − 0.0102212i
\(225\) 0 0
\(226\) 370.635 1.63998
\(227\) 321.608i 1.41678i 0.705823 + 0.708388i \(0.250577\pi\)
−0.705823 + 0.708388i \(0.749423\pi\)
\(228\) 0 0
\(229\) 201.000 0.877729 0.438865 0.898553i \(-0.355381\pi\)
0.438865 + 0.898553i \(0.355381\pi\)
\(230\) 8.28573i 0.0360249i
\(231\) 0 0
\(232\) 228.810 0.986252
\(233\) − 39.3399i − 0.168841i −0.996430 0.0844205i \(-0.973096\pi\)
0.996430 0.0844205i \(-0.0269039\pi\)
\(234\) 0 0
\(235\) 7.45967 0.0317433
\(236\) − 8.80181i − 0.0372958i
\(237\) 0 0
\(238\) −26.9052 −0.113047
\(239\) 23.7707i 0.0994592i 0.998763 + 0.0497296i \(0.0158359\pi\)
−0.998763 + 0.0497296i \(0.984164\pi\)
\(240\) 0 0
\(241\) 166.952 0.692745 0.346373 0.938097i \(-0.387413\pi\)
0.346373 + 0.938097i \(0.387413\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 3.17337 0.0130056
\(245\) 38.0369i 0.155253i
\(246\) 0 0
\(247\) 159.520 0.645830
\(248\) − 414.878i − 1.67289i
\(249\) 0 0
\(250\) 79.9193 0.319677
\(251\) − 247.487i − 0.986005i −0.870028 0.493003i \(-0.835899\pi\)
0.870028 0.493003i \(-0.164101\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 291.505i − 1.14766i
\(255\) 0 0
\(256\) 24.3629 0.0951675
\(257\) 69.1040i 0.268887i 0.990921 + 0.134444i \(0.0429247\pi\)
−0.990921 + 0.134444i \(0.957075\pi\)
\(258\) 0 0
\(259\) 42.1109 0.162590
\(260\) 0.936714i 0.00360275i
\(261\) 0 0
\(262\) 141.825 0.541315
\(263\) 162.520i 0.617949i 0.951070 + 0.308974i \(0.0999857\pi\)
−0.951070 + 0.308974i \(0.900014\pi\)
\(264\) 0 0
\(265\) −27.1109 −0.102305
\(266\) 39.4669i 0.148372i
\(267\) 0 0
\(268\) 2.14315 0.00799683
\(269\) − 445.060i − 1.65450i −0.561837 0.827248i \(-0.689905\pi\)
0.561837 0.827248i \(-0.310095\pi\)
\(270\) 0 0
\(271\) −177.317 −0.654304 −0.327152 0.944972i \(-0.606089\pi\)
−0.327152 + 0.944972i \(0.606089\pi\)
\(272\) − 193.803i − 0.712510i
\(273\) 0 0
\(274\) −462.683 −1.68863
\(275\) 0 0
\(276\) 0 0
\(277\) 51.9193 0.187434 0.0937172 0.995599i \(-0.470125\pi\)
0.0937172 + 0.995599i \(0.470125\pi\)
\(278\) 208.856i 0.751281i
\(279\) 0 0
\(280\) 7.06658 0.0252378
\(281\) 440.088i 1.56615i 0.621927 + 0.783075i \(0.286350\pi\)
−0.621927 + 0.783075i \(0.713650\pi\)
\(282\) 0 0
\(283\) 402.490 1.42223 0.711113 0.703078i \(-0.248191\pi\)
0.711113 + 0.703078i \(0.248191\pi\)
\(284\) − 9.78831i − 0.0344659i
\(285\) 0 0
\(286\) 0 0
\(287\) 84.5363i 0.294552i
\(288\) 0 0
\(289\) 150.905 0.522163
\(290\) − 47.0810i − 0.162348i
\(291\) 0 0
\(292\) −7.01613 −0.0240279
\(293\) 137.303i 0.468610i 0.972163 + 0.234305i \(0.0752814\pi\)
−0.972163 + 0.234305i \(0.924719\pi\)
\(294\) 0 0
\(295\) 55.2238 0.187199
\(296\) 293.987i 0.993198i
\(297\) 0 0
\(298\) −4.63098 −0.0155402
\(299\) 47.3618i 0.158401i
\(300\) 0 0
\(301\) 77.5846 0.257756
\(302\) − 474.791i − 1.57215i
\(303\) 0 0
\(304\) −284.286 −0.935152
\(305\) 19.9102i 0.0652792i
\(306\) 0 0
\(307\) 359.413 1.17073 0.585364 0.810771i \(-0.300952\pi\)
0.585364 + 0.810771i \(0.300952\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −85.3670 −0.275377
\(311\) 29.2965i 0.0942009i 0.998890 + 0.0471005i \(0.0149981\pi\)
−0.998890 + 0.0471005i \(0.985002\pi\)
\(312\) 0 0
\(313\) 274.681 0.877576 0.438788 0.898591i \(-0.355408\pi\)
0.438788 + 0.898591i \(0.355408\pi\)
\(314\) − 345.675i − 1.10088i
\(315\) 0 0
\(316\) −8.25403 −0.0261204
\(317\) − 585.684i − 1.84758i −0.382895 0.923792i \(-0.625073\pi\)
0.382895 0.923792i \(-0.374927\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 49.2822i 0.154007i
\(321\) 0 0
\(322\) −11.7178 −0.0363907
\(323\) 202.569i 0.627149i
\(324\) 0 0
\(325\) 225.474 0.693765
\(326\) 240.393i 0.737403i
\(327\) 0 0
\(328\) −590.169 −1.79930
\(329\) 10.5496i 0.0320655i
\(330\) 0 0
\(331\) −88.3831 −0.267018 −0.133509 0.991048i \(-0.542625\pi\)
−0.133509 + 0.991048i \(0.542625\pi\)
\(332\) − 15.2855i − 0.0460405i
\(333\) 0 0
\(334\) 397.885 1.19127
\(335\) 13.4464i 0.0401386i
\(336\) 0 0
\(337\) −335.429 −0.995340 −0.497670 0.867367i \(-0.665811\pi\)
−0.497670 + 0.867367i \(0.665811\pi\)
\(338\) − 169.352i − 0.501042i
\(339\) 0 0
\(340\) −1.18950 −0.00349853
\(341\) 0 0
\(342\) 0 0
\(343\) −109.016 −0.317831
\(344\) 541.638i 1.57453i
\(345\) 0 0
\(346\) 347.764 1.00510
\(347\) − 440.046i − 1.26814i −0.773274 0.634072i \(-0.781382\pi\)
0.773274 0.634072i \(-0.218618\pi\)
\(348\) 0 0
\(349\) 470.252 1.34743 0.673713 0.738993i \(-0.264698\pi\)
0.673713 + 0.738993i \(0.264698\pi\)
\(350\) 55.7845i 0.159384i
\(351\) 0 0
\(352\) 0 0
\(353\) − 459.088i − 1.30053i −0.759707 0.650266i \(-0.774658\pi\)
0.759707 0.650266i \(-0.225342\pi\)
\(354\) 0 0
\(355\) 61.4133 0.172995
\(356\) − 14.0894i − 0.0395771i
\(357\) 0 0
\(358\) 437.238 1.22133
\(359\) 403.162i 1.12301i 0.827472 + 0.561507i \(0.189778\pi\)
−0.827472 + 0.561507i \(0.810222\pi\)
\(360\) 0 0
\(361\) −63.8548 −0.176883
\(362\) 482.950i 1.33411i
\(363\) 0 0
\(364\) −1.32471 −0.00363932
\(365\) − 44.0202i − 0.120603i
\(366\) 0 0
\(367\) 634.458 1.72877 0.864384 0.502833i \(-0.167709\pi\)
0.864384 + 0.502833i \(0.167709\pi\)
\(368\) − 84.4052i − 0.229362i
\(369\) 0 0
\(370\) 60.4919 0.163492
\(371\) − 38.3406i − 0.103344i
\(372\) 0 0
\(373\) 43.3629 0.116254 0.0581272 0.998309i \(-0.481487\pi\)
0.0581272 + 0.998309i \(0.481487\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −73.6492 −0.195875
\(377\) − 269.118i − 0.713842i
\(378\) 0 0
\(379\) −230.458 −0.608068 −0.304034 0.952661i \(-0.598334\pi\)
−0.304034 + 0.952661i \(0.598334\pi\)
\(380\) 1.74486i 0.00459173i
\(381\) 0 0
\(382\) 686.504 1.79713
\(383\) − 715.367i − 1.86780i −0.357536 0.933899i \(-0.616383\pi\)
0.357536 0.933899i \(-0.383617\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 513.774i − 1.33102i
\(387\) 0 0
\(388\) 1.79027 0.00461409
\(389\) − 373.539i − 0.960255i −0.877199 0.480127i \(-0.840590\pi\)
0.877199 0.480127i \(-0.159410\pi\)
\(390\) 0 0
\(391\) −60.1431 −0.153819
\(392\) − 375.538i − 0.958005i
\(393\) 0 0
\(394\) 191.282 0.485488
\(395\) − 51.7870i − 0.131106i
\(396\) 0 0
\(397\) 265.429 0.668588 0.334294 0.942469i \(-0.391502\pi\)
0.334294 + 0.942469i \(0.391502\pi\)
\(398\) 364.799i 0.916580i
\(399\) 0 0
\(400\) −401.825 −1.00456
\(401\) − 634.698i − 1.58279i −0.611306 0.791394i \(-0.709356\pi\)
0.611306 0.791394i \(-0.290644\pi\)
\(402\) 0 0
\(403\) −487.964 −1.21083
\(404\) − 3.19058i − 0.00789748i
\(405\) 0 0
\(406\) 66.5826 0.163997
\(407\) 0 0
\(408\) 0 0
\(409\) 483.014 1.18096 0.590482 0.807051i \(-0.298938\pi\)
0.590482 + 0.807051i \(0.298938\pi\)
\(410\) 121.436i 0.296185i
\(411\) 0 0
\(412\) 7.15928 0.0173769
\(413\) 78.0983i 0.189100i
\(414\) 0 0
\(415\) 95.9032 0.231092
\(416\) − 18.7996i − 0.0451914i
\(417\) 0 0
\(418\) 0 0
\(419\) 565.911i 1.35062i 0.737533 + 0.675311i \(0.235991\pi\)
−0.737533 + 0.675311i \(0.764009\pi\)
\(420\) 0 0
\(421\) 632.871 1.50326 0.751628 0.659587i \(-0.229269\pi\)
0.751628 + 0.659587i \(0.229269\pi\)
\(422\) 406.658i 0.963644i
\(423\) 0 0
\(424\) 267.665 0.631286
\(425\) 286.321i 0.673697i
\(426\) 0 0
\(427\) −28.1572 −0.0659420
\(428\) − 10.2588i − 0.0239690i
\(429\) 0 0
\(430\) 111.450 0.259185
\(431\) − 325.762i − 0.755829i −0.925840 0.377915i \(-0.876641\pi\)
0.925840 0.377915i \(-0.123359\pi\)
\(432\) 0 0
\(433\) 242.427 0.559879 0.279939 0.960018i \(-0.409686\pi\)
0.279939 + 0.960018i \(0.409686\pi\)
\(434\) − 120.727i − 0.278173i
\(435\) 0 0
\(436\) −13.4254 −0.0307921
\(437\) 88.2230i 0.201883i
\(438\) 0 0
\(439\) −43.2702 −0.0985653 −0.0492826 0.998785i \(-0.515694\pi\)
−0.0492826 + 0.998785i \(0.515694\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −220.921 −0.499822
\(443\) 650.758i 1.46898i 0.678620 + 0.734490i \(0.262578\pi\)
−0.678620 + 0.734490i \(0.737422\pi\)
\(444\) 0 0
\(445\) 88.3992 0.198650
\(446\) − 371.733i − 0.833482i
\(447\) 0 0
\(448\) −69.6955 −0.155570
\(449\) − 18.7996i − 0.0418700i −0.999781 0.0209350i \(-0.993336\pi\)
0.999781 0.0209350i \(-0.00666430\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 23.1734i 0.0512685i
\(453\) 0 0
\(454\) −653.349 −1.43909
\(455\) − 8.31145i − 0.0182669i
\(456\) 0 0
\(457\) 20.4456 0.0447387 0.0223693 0.999750i \(-0.492879\pi\)
0.0223693 + 0.999750i \(0.492879\pi\)
\(458\) 408.333i 0.891556i
\(459\) 0 0
\(460\) −0.518052 −0.00112620
\(461\) 231.268i 0.501666i 0.968030 + 0.250833i \(0.0807045\pi\)
−0.968030 + 0.250833i \(0.919295\pi\)
\(462\) 0 0
\(463\) −741.663 −1.60186 −0.800932 0.598755i \(-0.795662\pi\)
−0.800932 + 0.598755i \(0.795662\pi\)
\(464\) 479.605i 1.03363i
\(465\) 0 0
\(466\) 79.9193 0.171501
\(467\) − 551.632i − 1.18122i −0.806956 0.590612i \(-0.798886\pi\)
0.806956 0.590612i \(-0.201114\pi\)
\(468\) 0 0
\(469\) −19.0161 −0.0405461
\(470\) 15.1544i 0.0322433i
\(471\) 0 0
\(472\) −545.224 −1.15514
\(473\) 0 0
\(474\) 0 0
\(475\) 420.000 0.884211
\(476\) − 1.68221i − 0.00353405i
\(477\) 0 0
\(478\) −48.2904 −0.101026
\(479\) 659.380i 1.37658i 0.725438 + 0.688288i \(0.241637\pi\)
−0.725438 + 0.688288i \(0.758363\pi\)
\(480\) 0 0
\(481\) 345.776 0.718869
\(482\) 339.163i 0.703658i
\(483\) 0 0
\(484\) 0 0
\(485\) 11.2324i 0.0231596i
\(486\) 0 0
\(487\) −10.7480 −0.0220698 −0.0110349 0.999939i \(-0.503513\pi\)
−0.0110349 + 0.999939i \(0.503513\pi\)
\(488\) − 196.573i − 0.402813i
\(489\) 0 0
\(490\) −77.2722 −0.157698
\(491\) − 186.899i − 0.380649i −0.981721 0.190324i \(-0.939046\pi\)
0.981721 0.190324i \(-0.0609540\pi\)
\(492\) 0 0
\(493\) 341.744 0.693193
\(494\) 324.066i 0.656004i
\(495\) 0 0
\(496\) 869.617 1.75326
\(497\) 86.8515i 0.174752i
\(498\) 0 0
\(499\) −433.631 −0.869000 −0.434500 0.900672i \(-0.643075\pi\)
−0.434500 + 0.900672i \(0.643075\pi\)
\(500\) 4.99683i 0.00999366i
\(501\) 0 0
\(502\) 502.772 1.00154
\(503\) 72.8434i 0.144818i 0.997375 + 0.0724090i \(0.0230687\pi\)
−0.997375 + 0.0724090i \(0.976931\pi\)
\(504\) 0 0
\(505\) 20.0182 0.0396400
\(506\) 0 0
\(507\) 0 0
\(508\) 18.2259 0.0358777
\(509\) − 665.644i − 1.30775i −0.756603 0.653874i \(-0.773142\pi\)
0.756603 0.653874i \(-0.226858\pi\)
\(510\) 0 0
\(511\) 62.2540 0.121828
\(512\) − 485.530i − 0.948301i
\(513\) 0 0
\(514\) −140.385 −0.273123
\(515\) 44.9184i 0.0872202i
\(516\) 0 0
\(517\) 0 0
\(518\) 85.5485i 0.165152i
\(519\) 0 0
\(520\) 58.0243 0.111585
\(521\) 30.7763i 0.0590715i 0.999564 + 0.0295358i \(0.00940289\pi\)
−0.999564 + 0.0295358i \(0.990597\pi\)
\(522\) 0 0
\(523\) 77.4274 0.148045 0.0740224 0.997257i \(-0.476416\pi\)
0.0740224 + 0.997257i \(0.476416\pi\)
\(524\) 8.86735i 0.0169224i
\(525\) 0 0
\(526\) −330.161 −0.627683
\(527\) − 619.648i − 1.17580i
\(528\) 0 0
\(529\) 502.806 0.950485
\(530\) − 55.0759i − 0.103917i
\(531\) 0 0
\(532\) −2.46760 −0.00463835
\(533\) 694.135i 1.30232i
\(534\) 0 0
\(535\) 64.3649 0.120308
\(536\) − 132.756i − 0.247680i
\(537\) 0 0
\(538\) 904.141 1.68056
\(539\) 0 0
\(540\) 0 0
\(541\) −574.375 −1.06169 −0.530846 0.847469i \(-0.678126\pi\)
−0.530846 + 0.847469i \(0.678126\pi\)
\(542\) − 360.220i − 0.664612i
\(543\) 0 0
\(544\) 23.8730 0.0438842
\(545\) − 84.2326i − 0.154555i
\(546\) 0 0
\(547\) −1018.85 −1.86262 −0.931312 0.364224i \(-0.881334\pi\)
−0.931312 + 0.364224i \(0.881334\pi\)
\(548\) − 28.9285i − 0.0527893i
\(549\) 0 0
\(550\) 0 0
\(551\) − 501.299i − 0.909798i
\(552\) 0 0
\(553\) 73.2379 0.132437
\(554\) 105.474i 0.190387i
\(555\) 0 0
\(556\) −13.0584 −0.0234863
\(557\) − 373.330i − 0.670251i −0.942174 0.335125i \(-0.891221\pi\)
0.942174 0.335125i \(-0.108779\pi\)
\(558\) 0 0
\(559\) 637.054 1.13963
\(560\) 14.8121i 0.0264502i
\(561\) 0 0
\(562\) −894.042 −1.59082
\(563\) − 1014.70i − 1.80232i −0.433491 0.901158i \(-0.642718\pi\)
0.433491 0.901158i \(-0.357282\pi\)
\(564\) 0 0
\(565\) −145.393 −0.257333
\(566\) 817.661i 1.44463i
\(567\) 0 0
\(568\) −606.333 −1.06749
\(569\) − 580.458i − 1.02014i −0.860134 0.510068i \(-0.829620\pi\)
0.860134 0.510068i \(-0.170380\pi\)
\(570\) 0 0
\(571\) −604.454 −1.05859 −0.529294 0.848439i \(-0.677543\pi\)
−0.529294 + 0.848439i \(0.677543\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −171.736 −0.299192
\(575\) 124.699i 0.216868i
\(576\) 0 0
\(577\) −8.90320 −0.0154302 −0.00771508 0.999970i \(-0.502456\pi\)
−0.00771508 + 0.999970i \(0.502456\pi\)
\(578\) 306.565i 0.530389i
\(579\) 0 0
\(580\) 2.94366 0.00507528
\(581\) 135.628i 0.233438i
\(582\) 0 0
\(583\) 0 0
\(584\) 434.611i 0.744197i
\(585\) 0 0
\(586\) −278.931 −0.475992
\(587\) − 680.972i − 1.16009i −0.814585 0.580045i \(-0.803035\pi\)
0.814585 0.580045i \(-0.196965\pi\)
\(588\) 0 0
\(589\) −908.952 −1.54321
\(590\) 112.188i 0.190148i
\(591\) 0 0
\(592\) −616.220 −1.04091
\(593\) − 855.518i − 1.44269i −0.692574 0.721347i \(-0.743523\pi\)
0.692574 0.721347i \(-0.256477\pi\)
\(594\) 0 0
\(595\) 10.5544 0.0177385
\(596\) − 0.289545i 0 0.000485814i
\(597\) 0 0
\(598\) −96.2159 −0.160896
\(599\) 186.251i 0.310937i 0.987841 + 0.155469i \(0.0496887\pi\)
−0.987841 + 0.155469i \(0.950311\pi\)
\(600\) 0 0
\(601\) 362.843 0.603732 0.301866 0.953350i \(-0.402391\pi\)
0.301866 + 0.953350i \(0.402391\pi\)
\(602\) 157.614i 0.261817i
\(603\) 0 0
\(604\) 29.6855 0.0491482
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000 0.0263591 0.0131796 0.999913i \(-0.495805\pi\)
0.0131796 + 0.999913i \(0.495805\pi\)
\(608\) − 35.0189i − 0.0575969i
\(609\) 0 0
\(610\) −40.4476 −0.0663076
\(611\) 86.6234i 0.141773i
\(612\) 0 0
\(613\) −1161.74 −1.89517 −0.947585 0.319502i \(-0.896484\pi\)
−0.947585 + 0.319502i \(0.896484\pi\)
\(614\) 730.150i 1.18917i
\(615\) 0 0
\(616\) 0 0
\(617\) − 163.905i − 0.265648i −0.991140 0.132824i \(-0.957595\pi\)
0.991140 0.132824i \(-0.0424045\pi\)
\(618\) 0 0
\(619\) −305.982 −0.494316 −0.247158 0.968975i \(-0.579497\pi\)
−0.247158 + 0.968975i \(0.579497\pi\)
\(620\) − 5.33743i − 0.00860876i
\(621\) 0 0
\(622\) −59.5160 −0.0956849
\(623\) 125.015i 0.200667i
\(624\) 0 0
\(625\) 577.772 0.924435
\(626\) 558.017i 0.891401i
\(627\) 0 0
\(628\) 21.6128 0.0344153
\(629\) 439.089i 0.698075i
\(630\) 0 0
\(631\) 123.764 0.196140 0.0980698 0.995180i \(-0.468733\pi\)
0.0980698 + 0.995180i \(0.468733\pi\)
\(632\) 511.292i 0.809007i
\(633\) 0 0
\(634\) 1189.82 1.87669
\(635\) 114.352i 0.180081i
\(636\) 0 0
\(637\) −441.693 −0.693396
\(638\) 0 0
\(639\) 0 0
\(640\) −106.593 −0.166551
\(641\) − 261.417i − 0.407827i −0.978989 0.203913i \(-0.934634\pi\)
0.978989 0.203913i \(-0.0653661\pi\)
\(642\) 0 0
\(643\) 218.000 0.339036 0.169518 0.985527i \(-0.445779\pi\)
0.169518 + 0.985527i \(0.445779\pi\)
\(644\) − 0.732637i − 0.00113763i
\(645\) 0 0
\(646\) −411.520 −0.637028
\(647\) 858.262i 1.32653i 0.748387 + 0.663263i \(0.230829\pi\)
−0.748387 + 0.663263i \(0.769171\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 458.051i 0.704694i
\(651\) 0 0
\(652\) −15.0302 −0.0230525
\(653\) 968.691i 1.48345i 0.670706 + 0.741723i \(0.265991\pi\)
−0.670706 + 0.741723i \(0.734009\pi\)
\(654\) 0 0
\(655\) −55.6351 −0.0849391
\(656\) − 1237.04i − 1.88573i
\(657\) 0 0
\(658\) −21.4315 −0.0325707
\(659\) − 690.156i − 1.04728i −0.851940 0.523639i \(-0.824574\pi\)
0.851940 0.523639i \(-0.175426\pi\)
\(660\) 0 0
\(661\) −1183.04 −1.78977 −0.894885 0.446297i \(-0.852743\pi\)
−0.894885 + 0.446297i \(0.852743\pi\)
\(662\) − 179.551i − 0.271225i
\(663\) 0 0
\(664\) −946.851 −1.42598
\(665\) − 15.4821i − 0.0232814i
\(666\) 0 0
\(667\) 148.837 0.223143
\(668\) 24.8771i 0.0372412i
\(669\) 0 0
\(670\) −27.3165 −0.0407709
\(671\) 0 0
\(672\) 0 0
\(673\) −1006.88 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(674\) − 681.427i − 1.01102i
\(675\) 0 0
\(676\) 10.5885 0.0156634
\(677\) 907.102i 1.33989i 0.742413 + 0.669943i \(0.233681\pi\)
−0.742413 + 0.669943i \(0.766319\pi\)
\(678\) 0 0
\(679\) −15.8850 −0.0233947
\(680\) 73.6831i 0.108357i
\(681\) 0 0
\(682\) 0 0
\(683\) 687.644i 1.00680i 0.864054 + 0.503400i \(0.167918\pi\)
−0.864054 + 0.503400i \(0.832082\pi\)
\(684\) 0 0
\(685\) 181.502 0.264966
\(686\) − 221.467i − 0.322838i
\(687\) 0 0
\(688\) −1135.32 −1.65017
\(689\) − 314.818i − 0.456920i
\(690\) 0 0
\(691\) 199.099 0.288131 0.144066 0.989568i \(-0.453982\pi\)
0.144066 + 0.989568i \(0.453982\pi\)
\(692\) 21.7434i 0.0314211i
\(693\) 0 0
\(694\) 893.955 1.28812
\(695\) − 81.9302i − 0.117885i
\(696\) 0 0
\(697\) −881.458 −1.26465
\(698\) 955.320i 1.36865i
\(699\) 0 0
\(700\) −3.48784 −0.00498262
\(701\) − 81.6124i − 0.116423i −0.998304 0.0582114i \(-0.981460\pi\)
0.998304 0.0582114i \(-0.0185398\pi\)
\(702\) 0 0
\(703\) 644.093 0.916206
\(704\) 0 0
\(705\) 0 0
\(706\) 932.639 1.32102
\(707\) 28.3100i 0.0400424i
\(708\) 0 0
\(709\) −1014.66 −1.43112 −0.715558 0.698553i \(-0.753827\pi\)
−0.715558 + 0.698553i \(0.753827\pi\)
\(710\) 124.762i 0.175720i
\(711\) 0 0
\(712\) −872.764 −1.22579
\(713\) − 269.870i − 0.378499i
\(714\) 0 0
\(715\) 0 0
\(716\) 27.3376i 0.0381810i
\(717\) 0 0
\(718\) −819.026 −1.14070
\(719\) − 613.361i − 0.853075i −0.904470 0.426537i \(-0.859733\pi\)
0.904470 0.426537i \(-0.140267\pi\)
\(720\) 0 0
\(721\) −63.5242 −0.0881057
\(722\) − 129.721i − 0.179670i
\(723\) 0 0
\(724\) −30.1956 −0.0417067
\(725\) − 708.561i − 0.977325i
\(726\) 0 0
\(727\) 169.163 0.232687 0.116343 0.993209i \(-0.462883\pi\)
0.116343 + 0.993209i \(0.462883\pi\)
\(728\) 82.0588i 0.112718i
\(729\) 0 0
\(730\) 89.4274 0.122503
\(731\) 808.973i 1.10667i
\(732\) 0 0
\(733\) 165.734 0.226104 0.113052 0.993589i \(-0.463937\pi\)
0.113052 + 0.993589i \(0.463937\pi\)
\(734\) 1288.90i 1.75600i
\(735\) 0 0
\(736\) 10.3972 0.0141266
\(737\) 0 0
\(738\) 0 0
\(739\) 1001.21 1.35481 0.677406 0.735610i \(-0.263104\pi\)
0.677406 + 0.735610i \(0.263104\pi\)
\(740\) 3.78216i 0.00511103i
\(741\) 0 0
\(742\) 77.8891 0.104972
\(743\) − 110.639i − 0.148909i −0.997224 0.0744544i \(-0.976278\pi\)
0.997224 0.0744544i \(-0.0237215\pi\)
\(744\) 0 0
\(745\) 1.81665 0.00243845
\(746\) 88.0919i 0.118086i
\(747\) 0 0
\(748\) 0 0
\(749\) 91.0257i 0.121530i
\(750\) 0 0
\(751\) −80.0827 −0.106635 −0.0533174 0.998578i \(-0.516980\pi\)
−0.0533174 + 0.998578i \(0.516980\pi\)
\(752\) − 154.375i − 0.205285i
\(753\) 0 0
\(754\) 546.715 0.725087
\(755\) 186.251i 0.246691i
\(756\) 0 0
\(757\) 254.464 0.336148 0.168074 0.985774i \(-0.446245\pi\)
0.168074 + 0.985774i \(0.446245\pi\)
\(758\) − 468.176i − 0.617646i
\(759\) 0 0
\(760\) 108.085 0.142216
\(761\) − 280.475i − 0.368561i −0.982874 0.184280i \(-0.941005\pi\)
0.982874 0.184280i \(-0.0589954\pi\)
\(762\) 0 0
\(763\) 119.123 0.156124
\(764\) 42.9226i 0.0561814i
\(765\) 0 0
\(766\) 1453.27 1.89722
\(767\) 641.272i 0.836078i
\(768\) 0 0
\(769\) −788.423 −1.02526 −0.512629 0.858610i \(-0.671328\pi\)
−0.512629 + 0.858610i \(0.671328\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.1229 0.0416100
\(773\) − 420.198i − 0.543594i −0.962355 0.271797i \(-0.912382\pi\)
0.962355 0.271797i \(-0.0876179\pi\)
\(774\) 0 0
\(775\) −1284.76 −1.65775
\(776\) − 110.897i − 0.142909i
\(777\) 0 0
\(778\) 758.847 0.975382
\(779\) 1293.00i 1.65982i
\(780\) 0 0
\(781\) 0 0
\(782\) − 122.181i − 0.156242i
\(783\) 0 0
\(784\) 787.157 1.00403
\(785\) 135.602i 0.172741i
\(786\) 0 0
\(787\) −246.730 −0.313507 −0.156753 0.987638i \(-0.550103\pi\)
−0.156753 + 0.987638i \(0.550103\pi\)
\(788\) 11.9596i 0.0151772i
\(789\) 0 0
\(790\) 105.206 0.133172
\(791\) − 205.617i − 0.259945i
\(792\) 0 0
\(793\) −231.202 −0.291553
\(794\) 539.222i 0.679120i
\(795\) 0 0
\(796\) −22.8085 −0.0286538
\(797\) 166.005i 0.208287i 0.994562 + 0.104144i \(0.0332101\pi\)
−0.994562 + 0.104144i \(0.966790\pi\)
\(798\) 0 0
\(799\) −110.000 −0.137672
\(800\) − 49.4975i − 0.0618718i
\(801\) 0 0
\(802\) 1289.39 1.60772
\(803\) 0 0
\(804\) 0 0
\(805\) 4.59667 0.00571015
\(806\) − 991.301i − 1.22990i
\(807\) 0 0
\(808\) −197.639 −0.244603
\(809\) 447.222i 0.552809i 0.961041 + 0.276404i \(0.0891429\pi\)
−0.961041 + 0.276404i \(0.910857\pi\)
\(810\) 0 0
\(811\) −885.903 −1.09236 −0.546180 0.837668i \(-0.683918\pi\)
−0.546180 + 0.837668i \(0.683918\pi\)
\(812\) 4.16297i 0.00512681i
\(813\) 0 0
\(814\) 0 0
\(815\) − 94.3018i − 0.115708i
\(816\) 0 0
\(817\) 1186.67 1.45247
\(818\) 981.246i 1.19957i
\(819\) 0 0
\(820\) −7.59257 −0.00925923
\(821\) − 1575.63i − 1.91916i −0.281430 0.959582i \(-0.590809\pi\)
0.281430 0.959582i \(-0.409191\pi\)
\(822\) 0 0
\(823\) −1433.92 −1.74231 −0.871153 0.491012i \(-0.836627\pi\)
−0.871153 + 0.491012i \(0.836627\pi\)
\(824\) − 443.479i − 0.538202i
\(825\) 0 0
\(826\) −158.657 −0.192079
\(827\) − 9.09551i − 0.0109982i −0.999985 0.00549910i \(-0.998250\pi\)
0.999985 0.00549910i \(-0.00175043\pi\)
\(828\) 0 0
\(829\) 206.996 0.249693 0.124847 0.992176i \(-0.460156\pi\)
0.124847 + 0.992176i \(0.460156\pi\)
\(830\) 194.828i 0.234732i
\(831\) 0 0
\(832\) −572.276 −0.687832
\(833\) − 560.891i − 0.673339i
\(834\) 0 0
\(835\) −156.083 −0.186925
\(836\) 0 0
\(837\) 0 0
\(838\) −1149.65 −1.37190
\(839\) 924.237i 1.10159i 0.834639 + 0.550797i \(0.185676\pi\)
−0.834639 + 0.550797i \(0.814324\pi\)
\(840\) 0 0
\(841\) −4.71575 −0.00560731
\(842\) 1285.68i 1.52694i
\(843\) 0 0
\(844\) −25.4256 −0.0301251
\(845\) 66.4336i 0.0786197i
\(846\) 0 0
\(847\) 0 0
\(848\) 561.048i 0.661613i
\(849\) 0 0
\(850\) −581.663 −0.684310
\(851\) 191.232i 0.224715i
\(852\) 0 0
\(853\) −215.111 −0.252182 −0.126091 0.992019i \(-0.540243\pi\)
−0.126091 + 0.992019i \(0.540243\pi\)
\(854\) − 57.2016i − 0.0669808i
\(855\) 0 0
\(856\) −635.474 −0.742376
\(857\) 775.528i 0.904933i 0.891781 + 0.452467i \(0.149456\pi\)
−0.891781 + 0.452467i \(0.850544\pi\)
\(858\) 0 0
\(859\) −1070.94 −1.24673 −0.623363 0.781933i \(-0.714234\pi\)
−0.623363 + 0.781933i \(0.714234\pi\)
\(860\) 6.96821i 0.00810257i
\(861\) 0 0
\(862\) 661.788 0.767736
\(863\) 698.664i 0.809576i 0.914411 + 0.404788i \(0.132655\pi\)
−0.914411 + 0.404788i \(0.867345\pi\)
\(864\) 0 0
\(865\) −136.421 −0.157712
\(866\) 492.493i 0.568698i
\(867\) 0 0
\(868\) 7.54827 0.00869616
\(869\) 0 0
\(870\) 0 0
\(871\) −156.143 −0.179269
\(872\) 831.627i 0.953701i
\(873\) 0 0
\(874\) −179.226 −0.205064
\(875\) − 44.3368i − 0.0506706i
\(876\) 0 0
\(877\) 1205.04 1.37405 0.687023 0.726636i \(-0.258917\pi\)
0.687023 + 0.726636i \(0.258917\pi\)
\(878\) − 87.9036i − 0.100118i
\(879\) 0 0
\(880\) 0 0
\(881\) 37.9228i 0.0430452i 0.999768 + 0.0215226i \(0.00685139\pi\)
−0.999768 + 0.0215226i \(0.993149\pi\)
\(882\) 0 0
\(883\) 444.786 0.503722 0.251861 0.967763i \(-0.418957\pi\)
0.251861 + 0.967763i \(0.418957\pi\)
\(884\) − 13.8128i − 0.0156253i
\(885\) 0 0
\(886\) −1322.02 −1.49212
\(887\) 238.917i 0.269354i 0.990890 + 0.134677i \(0.0429996\pi\)
−0.990890 + 0.134677i \(0.957000\pi\)
\(888\) 0 0
\(889\) −161.718 −0.181910
\(890\) 179.584i 0.201779i
\(891\) 0 0
\(892\) 23.2420 0.0260561
\(893\) 161.357i 0.180691i
\(894\) 0 0
\(895\) −171.520 −0.191643
\(896\) − 150.745i − 0.168242i
\(897\) 0 0
\(898\) 38.1915 0.0425296
\(899\) 1533.45i 1.70572i
\(900\) 0 0
\(901\) 399.776 0.443703
\(902\) 0 0
\(903\) 0 0
\(904\) 1435.46 1.58790
\(905\) − 189.452i − 0.209339i
\(906\) 0 0
\(907\) 83.7360 0.0923219 0.0461610 0.998934i \(-0.485301\pi\)
0.0461610 + 0.998934i \(0.485301\pi\)
\(908\) − 40.8496i − 0.0449885i
\(909\) 0 0
\(910\) 16.8848 0.0185547
\(911\) − 682.361i − 0.749024i −0.927222 0.374512i \(-0.877810\pi\)
0.927222 0.374512i \(-0.122190\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 41.5353i 0.0454435i
\(915\) 0 0
\(916\) −25.5303 −0.0278716
\(917\) − 78.6799i − 0.0858014i
\(918\) 0 0
\(919\) 1516.74 1.65043 0.825214 0.564820i \(-0.191054\pi\)
0.825214 + 0.564820i \(0.191054\pi\)
\(920\) 32.0905i 0.0348810i
\(921\) 0 0
\(922\) −469.823 −0.509569
\(923\) 713.146i 0.772639i
\(924\) 0 0
\(925\) 910.393 0.984209
\(926\) − 1506.69i − 1.62710i
\(927\) 0 0
\(928\) −59.0786 −0.0636623
\(929\) − 901.919i − 0.970849i −0.874279 0.485425i \(-0.838665\pi\)
0.874279 0.485425i \(-0.161335\pi\)
\(930\) 0 0
\(931\) −822.762 −0.883740
\(932\) 4.99683i 0.00536140i
\(933\) 0 0
\(934\) 1120.64 1.19983
\(935\) 0 0
\(936\) 0 0
\(937\) −389.472 −0.415658 −0.207829 0.978165i \(-0.566640\pi\)
−0.207829 + 0.978165i \(0.566640\pi\)
\(938\) − 38.6314i − 0.0411848i
\(939\) 0 0
\(940\) −0.947502 −0.00100798
\(941\) 372.011i 0.395336i 0.980269 + 0.197668i \(0.0633367\pi\)
−0.980269 + 0.197668i \(0.936663\pi\)
\(942\) 0 0
\(943\) −383.893 −0.407098
\(944\) − 1142.83i − 1.21063i
\(945\) 0 0
\(946\) 0 0
\(947\) − 301.071i − 0.317920i −0.987285 0.158960i \(-0.949186\pi\)
0.987285 0.158960i \(-0.0508142\pi\)
\(948\) 0 0
\(949\) 511.173 0.538644
\(950\) 853.233i 0.898139i
\(951\) 0 0
\(952\) −104.204 −0.109458
\(953\) − 1174.88i − 1.23283i −0.787422 0.616414i \(-0.788585\pi\)
0.787422 0.616414i \(-0.211415\pi\)
\(954\) 0 0
\(955\) −269.302 −0.281992
\(956\) − 3.01928i − 0.00315824i
\(957\) 0 0
\(958\) −1339.53 −1.39826
\(959\) 256.682i 0.267656i
\(960\) 0 0
\(961\) 1819.44 1.89327
\(962\) 702.446i 0.730194i
\(963\) 0 0
\(964\) −21.2056 −0.0219975
\(965\) 201.544i 0.208854i
\(966\) 0 0
\(967\) −293.663 −0.303685 −0.151842 0.988405i \(-0.548521\pi\)
−0.151842 + 0.988405i \(0.548521\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −22.8187 −0.0235244
\(971\) 787.038i 0.810544i 0.914196 + 0.405272i \(0.132823\pi\)
−0.914196 + 0.405272i \(0.867177\pi\)
\(972\) 0 0
\(973\) 115.867 0.119082
\(974\) − 21.8347i − 0.0224175i
\(975\) 0 0
\(976\) 412.032 0.422164
\(977\) − 692.629i − 0.708935i −0.935068 0.354467i \(-0.884662\pi\)
0.935068 0.354467i \(-0.115338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 4.83132i − 0.00492992i
\(981\) 0 0
\(982\) 379.686 0.386645
\(983\) 1798.69i 1.82979i 0.403690 + 0.914896i \(0.367728\pi\)
−0.403690 + 0.914896i \(0.632272\pi\)
\(984\) 0 0
\(985\) −75.0364 −0.0761791
\(986\) 694.255i 0.704112i
\(987\) 0 0
\(988\) −20.2617 −0.0205078
\(989\) 352.325i 0.356243i
\(990\) 0 0
\(991\) 736.790 0.743482 0.371741 0.928337i \(-0.378761\pi\)
0.371741 + 0.928337i \(0.378761\pi\)
\(992\) 107.121i 0.107985i
\(993\) 0 0
\(994\) −176.439 −0.177504
\(995\) − 143.104i − 0.143823i
\(996\) 0 0
\(997\) 1277.49 1.28133 0.640667 0.767819i \(-0.278658\pi\)
0.640667 + 0.767819i \(0.278658\pi\)
\(998\) − 880.924i − 0.882689i
\(999\) 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.d.485.3 yes 4
3.2 odd 2 inner 1089.3.b.d.485.2 yes 4
11.10 odd 2 1089.3.b.c.485.2 4
33.32 even 2 1089.3.b.c.485.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.c.485.2 4 11.10 odd 2
1089.3.b.c.485.3 yes 4 33.32 even 2
1089.3.b.d.485.2 yes 4 3.2 odd 2 inner
1089.3.b.d.485.3 yes 4 1.1 even 1 trivial